{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_70620/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_70620","id":"gutenberg_70620","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/70620","summary":null,"tags":["Category: Science - Physics","Physics -- Philosophy"],"text":"THE LOGIC OF\n                       MODERN PHYSICS\n\n\n\n\n                             BY\n\n                      P. W. BRIDGMAN\n\n       HOLLIS PROFESSOR OF MATHEMATICS AND NATURAL HISTORY\n                       IN HARVARD UNIVERSITY\n\n\n\n\n\n                          NEW YORK\n                   THE MACMILLAN COMPANY\n\n\n\n\n                       COPYRIGHT, 1927,\n                  By THE MACMILLAN COMPANY.\n\n\n\n\nPREFACE\n\n\nThis excursion into the field of fundamental criticism by one whose\nactivities have hitherto been confined almost entirely to experiment is\nnot evidence of senile decay, as might be cynically assumed. I have\nalways, throughout all my experimental work, felt an imperative need of\na better understanding of the foundations of our physical thought and\nhave for a long time made more or less unsystematic attempts to reach\nsuch an understanding. Only now, however, has a half sabbatical year\ngiven me leisure to attempt a more or less orderly exposition.\n\nIn spite of previous writings on the broad fundamentals by Clifford,\nStallo, Mach, and Poincar\u00e9, to mention only a few, I believe a new\nessay of this critical character needs no apology. For entirely apart\nfrom the question of whether many of the points of view of these essays\ncan be maintained, the discovery of new facts in the domain of\nrelativity and quantum theory has shifted the center of interest and\nemphasis. All the quite recent activity with the new quantum mechanics\nseems to call for a new examination of fundamental matters which shall\nrecognize, at least by implication, the existence of the special\nphenomena of the quantum domain. However, the necessity for\nre-examination does not mean at all that many of the results of previous\ncriticism may not still be accepted; some of these results have become\nso thoroughly incorporated into physical thinking that we can assume\nthem without mention. Thus the fundamental attitude of this essay is\nempiricism, which is now justified as the attitude of the physicist in\nlarge part by the inquiry into the physiological origin of our concepts\nof space, time, and mechanics with which the previous essays were\nlargely concerned.\n\nNone of the previous essays have consciously or immediately affected the\ndetails of this; in fact I have not read any of them within several\nyears. If passages here recall passages already written, it is because\nthe ideas have been assimilated and the precise origin forgotten; it is\nprobably worth while to let such passages stand without revision,\nbecause such ideas gain in plausibility through having been found\nacceptable to independent thought.\n\nI am much indebted to Professor R. F. Alfred Hoernl\u00e9 of the Department\nof Philosophy of Johannesburg University, South Africa, for suggesting\nseveral modifications to make the text more acceptable to a philosopher,\nand slight amplifications for the benefit of readers not familiar with\nall the details of recent technical developments in physics.\n\n\n\n\nINTRODUCTION\n\n\nOne of the most noteworthy movements in recent physics is a change of\nattitude toward what may be called the interpretative aspect of physics.\nIt is being increasingly recognized, both in the writings and the\nconversation of physicists, that the world of experiment is not\nunderstandable without some examination of the purpose of physics and of\nthe nature of its fundamental concepts. It is no new thing to attempt a\nmore critical understanding of the nature of physics, but until recently\nall such attempts have been regarded with a certain suspicion or even\nsometimes contempt. The average physicist is likely to deprecate his own\nconcern with such questions, and is inclined to dismiss the speculations\nof fellow physicists with the epithet \"metaphysical.\" This attitude has\nno doubt had a certain justification in the utter unintelligibility to\nthe physicist of many metaphysical speculations and the sterility of\nsuch speculations in yielding physical results. However, the growing\nreaction favoring a better understanding of the interpretative\nfundamentals of physics is not a pendulum swing of the fashion of\nthought toward metaphysics, originating in the upheaval of moral values\nproduced by the great war, or anything of the sort, but is a reaction\nabsolutely forced upon us by a rapidly increasing array of cold\nexperimental facts.\n\nThis reaction, or rather new movement, was without doubt initiated by\nthe restricted theory of relativity of Einstein. Before Einstein, an\never increasing number of experimental facts concerning bodies in rapid\nmotion required increasingly complicated modifications in our na\u00efve\nnotions in order to preserve self-consistency, until Einstein showed\nthat everything could be restored again to a wonderful simplicity by a\nslight change in some of our fundamental concepts. The concepts which\nwere most obviously touched by Einstein were those of space and time,\nand much of the writing consciously inspired by Einstein has been\nconcerned with these concepts. But that experiment compels a critique of\nmuch more than the concepts of space and time is made increasingly\nevident by all the new facts being discovered in the quantum realm.\n\nThe situation presented to us by these new quantum facts is twofold. In\nthe first place, all these experiments are concerned with things so\nsmall as to be forever beyond the possibility of direct experience, so\nthat we have the problem of translating the evidence of experiment into\nother language. Thus we observe an emission line in a spectroscope and\nmay infer an electron jumping from one energy level to another in an\natom. In the second place, we have the problem of understanding the\ntranslated experimental evidence. Now of course every one knows that\nthis problem is making us the greatest difficulty. The experimental\nfacts are so utterly different from those of our ordinary experience\nthat not only do we apparently have to give up generalizations from past\nexperience as broad as the field equations of electrodynamics, for\ninstance, but it is even being questioned whether our ordinary forms of\nthought are applicable in the new domain; it is often suggested, for\nexample, that the concepts of space and time break down.\n\nThe situation is rapidly becoming acute. Since I began writing this\nessay, there has been a striking increase in critical activity inspired\nby the new quantum mechanics of 1925-26, and it is common to hear\nexpositions of the new ideas prefaced by analysis of what experiment\nreally gives to us or what our fundamental concepts really mean. The\nchange in ideas is now so rapid that a number of the statements of this\nessay are already antiquated as expressions of the best current opinion;\nhowever I have allowed these statements to stand, since the fundamental\narguments are in nowise affected and we have no reason to think that\npresent best opinions are in any way final. We have the impression of\nbeing in an important formative period; if we are, the complexion of\nphysics for a long time in the future will be determined by our present\nattitude toward fundamental questions of interpretation. To meet this\nsituation it seems to me that something more is needed than the\nhand-to-mouth philosophy that is now growing up to meet special\nemergencies, something approaching more closely to a systematic\nphilosophy of all physics which shall cover the experimental domains\nalready consolidated as well as those which are now making us so much\ntrouble. It is the attempt of this essay to give a more or less\ninclusive critique of all physics. Our problem is the double one of\nunderstanding what we are trying to do and what our ideals should be in\nphysics, and of understanding the nature of the structure of physics as\nit now exists. These two ends are together furthered by an analysis of\nthe fundamental concepts of physics; an understanding of the concepts we\nhow have discloses the present structure of physics and a realization of\nwhat the concepts should be involves the ideals of physics. This essay\nwill be largely concerned with the fundamental concepts; it will appear\nthat almost all the concepts can profit from re-examination.\n\nThe material of this essay is largely obtained by observation of the\nactual currents of opinion in physics; much of what I have to say is\nmore or less common property and doubtless every reader will find\npassages that he will feel have been taken out of his own mouth. On\ncertain broad tendencies in present day physics, however, I have put my\nown interpretation, and it is more than likely that this interpretation\nwill be unacceptable to many. But even if not acceptable, I hope that\nthe stimulus of combatting the ideas offered here may be of value.\n\nCertain limitations will have to be set to our inquiry in order to keep\nit within manageable compass. It is of course the merest truism that all\nour experimental knowledge and our understanding of nature is impossible\nand non-existent apart from our own mental processes, so that strictly\nspeaking no aspect of psychology or epistemology is without pertinence.\nFortunately we shall be able to get along with a more or less naive\nattitude toward many of these matters. We shall accept as significant\nour common sense judgment that there is a world external to us, and\nshall limit as far as possible our inquiry to the behavior and\ninterpretation of this \"external\" world. We shall rule out inquiries\ninto our states of consciousness as such. In spite, however, of the best\nintentions, we shall not be able to eliminate completely considerations\nsavoring of the metaphysical, because it is evident that the nature of\nour thinking mechanism essentially colors any picture that we can form\nof nature, and we shall have to recognize that unavoidable\ncharacteristics of any outlook of ours are imposed in this way.\n\n\n\n\nCONTENTS\n\nPreface\n\nIntroduction\n\nCHAPTER\n\nI. BROAD POINTS OF VIEW\n\n    New Kinds of Experience Always Possible\n    The Operational Character of Concepts\n      Einstein's Contribution in Changing Our Attitude\n          toward Concepts\n      Detailed Discussion of the Concept of Length\n      The Relative Character of Knowledge\n      Meaningless Questions\n    General Comments on the Operational Point of\n          View\n\nII. OTHER GENERAL CONSIDERATIONS\n    The Approximate Character of Empirical Knowledge\n    Explanations and Mechanisms\n    Models and Constructs\n    The R\u00f4le of Mathematics in Physics\n\nIII. DETAILED CONSIDERATION OF VARIOUS CONCEPTS\n        OF PHYSICS\n\n    The Concept of Space\n    The Concept of Time\n    The Causality Concept\n    The Concept of Identity\n    The Concept of Velocity\n    The Concepts of Force and Mass\n    The Concept of Energy\n    The Concepts of Thermodynamics\n    Electrical Concepts\n    The Nature of Light and the Concepts of Relativity\n    Other Relativity Concepts\n    Rotational Motion and Relativity\n    Quantum Concepts\n\nIV. SPECIAL VIEWS OF NATURE\n\n    The Simplicity of Nature\n    Determinism\n    On the Possibility of Describing Nature Completely\n          in Terms of Analysis\n    A Glimpse Ahead\n\n   INDEX\n\n\n\n\nTHE LOGIC OF MODERN PHYSICS\n\n\n\n\nCHAPTER I\n\nBROAD POINTS OF VIEW\n\n\nWHATEVER may be one's opinion as to our permanent acceptance of the\nanalytical details of Einstein's restricted and general theories of\nrelativity, there can be no doubt that through these theories physics is\npermanently changed. It was a great shock to discover that classical\nconcepts, accepted unquestioningly, were inadequate to meet the actual\nsituation, and the shock of this discovery has resulted in a critical\nattitude toward our whole conceptual structure which must at least in\npart be permanent. Reflection on the situation after the event shows\nthat it should not have needed the new experimental facts which led to\nrelativity to convince us of the inadequacy of our previous concepts,\nbut that a sufficiently shrewd analysis should have prepared us for at\nleast the possibility of what Einstein did.\n\nLooking now to the future, our ideas of what external nature is will\nalways be subject to change as we gain new experimental knowledge, but\nthere is a part of our attitude to nature which should not be subject to\nfuture change, namely that part which rests on the permanent basis of\nthe character of our minds. It is precisely here, in an improved\nunderstanding of our mental relations to nature, that the permanent\ncontribution of relativity is to be found. We should now make it our\nbusiness to understand so thoroughly the character of our permanent\nmental relations to nature that another change in our attitude, such as\nthat due to Einstein, shall be forever impossible. It was perhaps\nexcusable that a revolution in mental attitude should occur once,\nbecause after all physics is a young science, and physicists have been\nvery busy, but it would certainly be a reproach if such a revolution\nshould ever prove necessary again.\n\n\n\n\nNEW KINDS OF EXPERIENCE ALWAYS POSSIBLE\n\n\nThe first lesson of our recent experience with relativity is merely an\nintensification and emphasis of the lesson which all past experience has\nalso taught, namely, that when experiment is pushed into new domains, we\nmust be prepared for new facts, of an entirely different character from\nthose of our former experience. This is taught not only by the discovery\nof those unsuspected properties of matter moving with high velocities,\nwhich inspired the theory of relativity, but also even more emphatically\nby the new facts in the quantum domain. To a certain extent, of course,\nthe recognition of all this does not involve a change of former\nattitude; the _fact_ has always been for the physicist the one ultimate\nthing from which there is no appeal, and in the face of which the only\npossible attitude is a humility almost religious. The new feature in the\npresent situation is an intensified conviction that in reality new\norders of experience do exist, and that we may expect to meet them\ncontinually. We have already encountered new phenomena in going to high\nvelocities, and in going to small scales of magnitude: we may similarly\nexpect to find them, for example, in dealing with relations of cosmic\nmagnitudes, or in dealing with the properties of matter of enormous\ndensities, such as is supposed to exist in the stars.\n\nImplied in this recognition of the possibility of new experience beyond\nour present range, is the recognition that no element of a physical\nsituation, no matter how apparently irrelevant or trivial, may be\ndismissed as without effect on the final result until proved to be\nwithout effect by actual experiment.\n\nThe attitude of the physicist must therefore be one of pure empiricism.\nHe recognizes no _a priori_ principles which determine or limit the\npossibilities of new experience. Experience is determined only by\nexperience. This practically means that we must give up the demand that\nall nature be embraced in any formula, either simple or complicated. It\nmay perhaps turn out eventually that as a matter of fact nature can be\nembraced in a formula, but we must so organize our thinking as not to\ndemand it as a necessity.\n\n\n\n\nTHE OPERATIONAL CHARACTER OF CONCEPTS\n\n_Einstein's Contribution in Changing Our Attitude\nToward Concepts_\n\n\nRecognizing the essential unpredictability of experiment beyond our\npresent range, the physicist, if he is to escape continually revising\nhis attitude, must use in describing and correlating nature concepts of\nsuch a character that our present experience does not exact hostages of\nthe future. Now here it seems to me is the greatest contribution of\nEinstein. Although he himself does not explicitly state or emphasize it,\nI believe that a study of what he has done will show that he has\nessentially modified our view of what the concepts useful in physics are\nand should be. Hitherto many of the concepts of physics have been\ndefined in terms of their properties. An excellent example is afforded\nby Newton's concept of absolute time. The following quotation from the\nScholium in Book I of the _Principia_ is illuminating:\n\n\nI do not define Time, Space, Place or Motion, as being well known to\nall. Only I must observe that the vulgar conceive those quantities under\nno other notions but from the relation they bear to sensible objects.\nAnd thence arise certain prejudices, for the removing of which, it will\nbe convenient to distinguish them into Absolute and Relative, True and\nApparent, Mathematical and Common.\n\n(I) Absolute, True, and Mathematical Time, of itself, and from its own\nnature flows equably without regard to anything external, and by another\nname is called Duration.\n\n\nNow there is no assurance whatever that there exists in nature anything\nwith properties like those assumed in the definition, and physics, when\nreduced to concepts of this character, becomes as purely an abstract\nscience and as far removed from reality as the abstract geometry of the\nmathematicians, built on postulates. It is a task for experiment to\ndiscover whether concepts so defined correspond to anything in nature,\nand we must always be prepared to find that the concepts correspond to\nnothing or only partially correspond. In particular, if we examine the\ndefinition of absolute time in the light of experiment, we find nothing\nin nature with such properties.\n\nThe new attitude toward a concept is entirely different. We may\nillustrate by considering the concept of length: what do we mean by the\nlength of an object? We evidently know what we mean by length if we can\ntell what the length of any and every object is, and for the physicist\nnothing more is required. To find the length of an object, we have to\nperform certain physical operations. The concept of length is therefore\nfixed when the operations by which length is measured are fixed: that\nis, the concept of length involves as much as and nothing more than the\nset of operations by which length is determined. In general, we mean by\nany concept nothing more than a set of operations; _the concept is\nsynonymous with the corresponding set of operations_. If the concept is\nphysical, as of length, the operations are actual physical operations,\nnamely, those by which length is measured; or if the concept is mental,\nas of mathematical continuity, the operations are mental operations,\nnamely those by which we determine whether a given aggregate of\nmagnitudes is continuous. It is not intended to imply that there is a\nhard and fast division between physical and mental concepts, or that one\nkind of concept does not always contain an element of the other; this\nclassification of concept is not important for our future\nconsiderations.\n\nWe must demand that the set of operations equivalent to any concept be a\nunique set, for otherwise there are possibilities of ambiguity in\npractical applications which we cannot admit.\n\nApplying this idea of \"concept\" to absolute time, we do not understand\nthe meaning of absolute time unless we can tell how to determine the\nabsolute time of any concrete event, _i.e._, unless we can measure\nabsolute time. Now we merely have to examine any of the possible\noperations by which we measure time to see that all such operations are\nrelative operations. Therefore the previous statement that absolute time\ndoes not exist is replaced by the statement that absolute time is\nmeaningless. And in making this statement we are not saying something\nnew about nature, but are merely bringing to light implications already\ncontained in the physical operations used in measuring time.\n\nIt is evident that if we adopt this point of view toward concepts,\nnamely that the proper definition of a concept is not in terms of its\nproperties but in terms of actual operations, we need run no danger of\nhaving to revise our attitude toward nature. For if experience is always\ndescribed in terms of experience, there must always be correspondence\nbetween experience and our description of it, and we need never be\nembarrassed, as we were in attempting to find in nature the prototype of\nNewton's absolute time. Furthermore, if we remember that the operations\nto which a physical concept are equivalent are actual physical\noperations, the concepts can be defined only in the range of actual\nexperiment, and are undefined and meaningless in regions as yet\nuntouched by experiment. It follows that strictly speaking we cannot\nmake statements at all about regions as yet untouched, and that when we\ndo make such statements, as we inevitably shall, we are making a\nconventionalized extrapolation, of the looseness of which we must be\nfully conscious, and the justification of which is in the experiment of\nthe future.\n\nThere probably is no statement either in Einstein or other writers that\nthe change described above in the use of \"concept\" has been\nself-consciously made, but that such is the case is proved, I believe,\nby an examination of the way concepts are now handled by Einstein and\nothers. For of course the true meaning of a term is to be found by\nobserving what a man does with it, not by what he says about it. We may\nshow that this is the actual sense in which concept is coming to be used\nby examining in particular Einstein's treatment of simultaneity.\n\nBefore Einstein, the concept of simultaneity was defined in terms of\nproperties. It was a property of two events, when described with respect\nto their relation in time, that one event was either before the other,\nor after it, or simultaneous with it. Simultaneity was a property of the\ntwo events alone and nothing else; either two events were simultaneous\nor they were not. The justification for using this term in this way was\nthat it seemed to describe the behavior of actual things. But of course\nexperience then was restricted to a narrow range. When the range of\nexperience was broadened, as by going to high velocities, it was found\nthat the concepts no longer applied, because there was no counterpart in\nexperience for this absolute relation between two events. Einstein now\nsubjected the concept of simultaneity to a critique, which consisted\nessentially in showing that the operations which enable two events to be\ndescribed as simultaneous involve measurements on the two events made by\nan observer, so that \"simultaneity\" is, therefore, not an absolute\nproperty of the two events and nothing else, but must also involve the\nrelation of the events to the observer. Until therefore we have\nexperimental proof to the contrary, we must be prepared to find that the\nsimultaneity of two events depends on their relation to the observer,\nand in particular on their velocity. Einstein, in thus analyzing what is\ninvolved in making a judgment of simultaneity, and in seizing on the act\nof the observer as the essence of the situation, is actually adopting a\nnew point of view as to what the concepts of physics should be, namely,\nthe operational view.\n\nOf course Einstein actually went much further than this, and found\nprecisely how the operations for judging simultaneity change when the\nobserver moves, and obtained quantitative expressions for the effect of\nthe motion of the observer on the relative time of two events. We may\nnotice, parenthetically, that there is much freedom of choice in\nselecting the exact operations; those which Einstein chose were\ndetermined by convenience and simplicity with relation to light beams.\nEntirely apart from the precise quantitative relations of Einstein's\ntheory, however, the important point for us is that if we had adopted\nthe operational point of view, we would, before the discovery of the\nactual physical facts, have seen that simultaneity is essentially a\nrelative concept, and would have left room in our thinking for the\ndiscovery of such effects as were later found.\n\n\n\n\n_Detailed Discussion of the Concept of Length_\n\n\nWe may now gain further familiarity with the operational attitude toward\na concept and some of its implications by examining from this point of\nview the concept of length. Our task is to find the operations by which\nwe measure the length of any concrete physical object. We begin with\nobjects of our commonest experience, such as a house or a house lot.\nWhat we do is sufficiently indicated by the following rough description.\nWe start with a measuring rod, lay it on the object so that one of its\nends coincides with one end of the object, mark on the object the\nposition of the other end of the rod, then move the rod along in a\nstraight line extension of its previous position until the first end\ncoincides with the previous position of the second end, repeat this\nprocess as often as we can, and call the length the total number of\ntimes the rod was applied. This procedure, apparently so simple, is in\npractice exceedingly complicated, and doubtless a full description of\nall the precautions that must be taken would fill a large treatise. We\nmust, for example, be sure that the temperature of the rod is the\nstandard temperature at which its length is defined, or else we must\nmake a correction for it; or we must correct for the gravitational\ndistortion of the rod if we measure a vertical length; or we must be\nsure that the rod is not a magnet or is not subject to electrical\nforces. All these precautions would occur to every physicist. But we\nmust also go further and specify all the details by which the rod is\nmoved from one position to the next on the object--its precise path\nthrough space and its velocity and acceleration in getting from one\nposition to another. Practically of course, precautions such as these\nare not mentioned, but the justification is in our experience that\nvariations of procedure of this kind are without effect on the final\nresult. But we always have to recognize that all our experience is\nsubject to error, and that at some time in the future we may have to\nspecify more carefully the acceleration, for example, of the rod in\nmoving from one position to another, if experimental accuracy should be\nso increased as to show a measurable effect. In _principle_ the\noperations by which length is measured should be _uniquely_ specified.\nIf we have more than one set of operations, we have more than one\nconcept, and strictly there should be a separate name to correspond to\neach different set of operations.\n\nSo much for the length of a stationary object, which is complicated\nenough. Now suppose we have to measure a moving street car. The\nsimplest, and what we may call the \"na\u00efve\" procedure, is to board the\ncar with our meter stick and repeat the operations we would apply to a\nstationary body. Notice that this procedure reduces to that already\nadopted in the limiting case when the velocity of the street car\nvanishes. But here there may be new questions of detail. How shall we\njump on to the car with our stick in hand? Shall we run and jump on from\nbehind, or shall we let it pick us up from in front? Or perhaps does now\nthe material of which the stick is composed make a difference, although\npreviously it did not? All these questions must be answered by\nexperiment. We believe from present evidence that it makes no difference\nhow we jump on to the car, or of what material the rod is made, and that\nthe length of the car found in this way will be the same as if it were\nat rest. But the experiments are more difficult, and we are not so sure\nof our conclusions as before. Now there are very obvious limitations to\nthe procedure just given. If the street car is going too fast, we can\nnot board it directly, but must use devices, such as getting on from a\nmoving automobile; and, more important still, there are limitations to\nthe velocity that can be given to street cars or to meter sticks by any\npractical means in our control, so that the moving bodies which can be\nmeasured in this way are restricted to a low range of velocity. If we\nwant to be able to measure the length of bodies moving with higher\nvelocities such as we find existing in nature (stars or cathode\nparticles), we must adopt another definition and other operations for\nmeasuring length, which also reduce to the operations already adopted in\nthe static case. This is precisely what Einstein did. Since Einstein's\noperations were different from our operations above, _his \"length\" does\nnot mean the same as our \"length.\"_ We must accordingly be prepared to\nfind that the length of a moving body measured by the procedure of\nEinstein is not the same as that above; this of course is the fact, and\nthe transformation formulas of relativity give the precise connection\nbetween the two lengths.\n\nEinstein's procedure for measuring the length of bodies in motion was\ndictated not only by the consideration that it must be applicable to\nbodies with high velocities, but also by mathematical convenience, in\nthat Einstein describes the world mathematically by a system of\nco\u00f6rdinate geometry, and the \"length\" of an object is connected simply\nwith quantities in the analytic equations.\n\nIt is of interest to describe briefly Einstein's actual operations for\nmeasuring the length of a body in motion; it will show how operations\nwhich may be simple from a mathematical point of view may appear\ncomplicated from a physical viewpoint. The observer who is to measure\nthe length of a moving object must first extend over his entire plane of\nreference (for simplicity the problem is considered two-dimensional) a\nsystem of time coordinates, _i.e._, at each point of his plane of\nreference there must be a clock, and all these clocks must be\nsynchronized. At each clock an observer must be situated. Now to find\nthe length of the moving object at a specified instant of time (it is a\nsubject for later investigation to find whether its length is a function\nof time), the two observers who happen to coincide in position with the\ntwo ends of the object at the specified time on their clocks are\nrequired to find the distance between their two positions by the\nprocedure for measuring the length of a stationary object, and this\ndistance is by definition the length of the moving object in the given\nreference system. This procedure for measuring the length of a body in\nmotion hence involves the idea of simultaneity, through the simultaneous\nposition of the two ends of the rod, and we have seen that the\noperations by which simultaneity are determined are relative, changing\nwhen the motion of the system changes. We hence are prepared to find a\nchange in the length of a body when the velocity of the measuring system\nchanges, and this in fact is what happens. The precise numerical\ndependence is worked out by Einstein, and involves other considerations,\nin which we are not interested at present.\n\nThe two sorts of length, the naive one and that of Einstein, have\ncertain features in common. In either case in the limit, as the velocity\nof the measuring system approaches zero, the operations approach those\nfor measuring the length of a stationary object. This, of course, is a\nrequirement in any good definition, imposed by considerations of\nconvenience, and it is too obvious a matter to need elaboration. Another\nfeature is that the operations equivalent to either concept both involve\nthe motion of the system, so that we must recognize the possibility that\nthe length of a moving object may be a function of its velocity. It is a\nmatter of experiment, unpredictable until tried, that within the limits\nof present experimental error the naive length is not affected by\nmotion, and Einstein's length is.\n\nSo far, we have extended the concept of length in only one way beyond\nthe range of ordinary experience, namely to high velocities. The\nextension may obviously be made in other directions. Let us inquire what\nare the operations by which we measure the length of a very large\nobject. In practice we probably first meet the desirability of a change\nof procedure in measuring large pieces of land. Here our procedure\ndepends on measurements with a surveyor's theodolite. This involves\nextending over the surface of the land a system of coordinates, starting\nfrom a base line measured with a tape in the conventional way, sighting\non distant points from the extremities of the line, and measuring the\nangles. Now in this extension we have made one very essential change:\nthe angles between the lines connecting distant points are now angles\nbetween beams of light. We assume that a beam of light travels in a\nstraight line. Furthermore, we assume in extending our system of\ntriangulation over the surface of the earth that the geometry of light\nbeams is Euclidean. We do the best we can to check the assumptions, but\nat most can never get more than a partial check.\n\nThus Gauss[1] checked whether the angles of a large terrestrial triangle\nadd to two right angles and found agreement within experimental error.\nWe now know from the experiments of Michelson[2] that if his\nmeasurements had been accurate enough he would not have got a check, but\nwould have had an excess or defect according to the direction in which\nthe beam of light travelled around the triangle with respect to the\nrotation of the earth. But if the geometry of light beams is Euclidean,\nthen not only must the angles of a triangle add to two right angles, but\nthere are definite relations between the lengths of the sides and the\nangles, and to check these relations the sides should be measured by the\nold procedure with a meter stick. Such a check on a large scale has\nnever been attempted, and is not feasible. It seems, then, that our\nchecks on the Euclidean character of optical space are all of restricted\ncharacter. We have apparently proved that up to a certain scale of\nmagnitude optical space is Euclidean with respect to measures of angle,\nbut this may not necessarily involve that space is also Euclidean with\nrespect to measures of length, so that space need not be completely\nEuclidean. There is a further most important restriction in that our\nstudies of non-Euclidean geometry have shown that the _percentage_\nexcess of the angles of a non-Euclidean triangle over 180\u00b0 may depend\non the magnitude of the triangle, so that it may well be that we have\nnot detected the non-Euclidean character of space simply because our\nmeasurements have not been on a large enough scale.\n\n[Footnote 1: C. F. Gauss, Gesammelte Werke, especially vol. IV.]\n\n[Footnote 2: See a discussion of the theory of this experiment by L.\nSilberstein, Jour. Opt. Soc. Amer. 5, 291-307. 1921.]\n\nWe thus see that the concept of length has undergone a very essential\nchange of character even within the range of terrestrial measurements,\nin that we have substituted for what I may call the tactual concept an\noptical concept, complicated by an assumption about the nature of our\ngeometry. From a very direct concept we have come to a very indirect\nconcept with a most complicated set of operations. Strictly speaking,\nlength when measured in this way by light beams should be called by\nanother name, since the operations are different. The practical\njustification for retaining the same name is that within our present\nexperimental limits a numerical difference between the results of the\ntwo sorts of operations has not been detected.\n\nWe are still worse off when we make the extension to solar and stellar\ndistances. Here space is entirely optical in character, and we never\nhave an opportunity of even partially comparing tactual with optical\nspace. No direct measures of length have ever been made, nor can we even\nmeasure the three angles of a triangle and so check our assumption that\nthe use of Euclidean geometry in extending the concept of space is\njustified. We never have under observation more than two angles of a\ntriangle, as when we measure the distance of the moon by observation\nfrom the two ends of the earth's diameter. To extend to still greater\ndistance our measures of length, we have to make still further\nassumptions, such as that inferences from the Newtonian laws of\nmechanics are valid. The accuracy of our inferences about lengths from\nsuch measurements is not high. Astronomy is usually regarded as a\nscience of extraordinarily high accuracy, but its accuracy is very\nrestricted in character, namely to the measurement of angles. It is\nprobably safe to say that no astronomical distance, except perhaps that\nof the moon, is known with an accuracy greater than 0.1%. When we push\nour estimates to distances beyond the confines of the solar system in\nwhich we are assisted by the laws of mechanics, we are reduced in the\nfirst place to measurements of parallax, which at best have a quite\ninferior accuracy, and which furthermore fail entirely outside a rather\nrestricted range. For greater stellar distances we are driven to other\nand much rougher estimates, resting for instance on the extension to\ngreat distances of connections found within the range of parallax\nbetween brightness and spectral type of a star, or on such assumptions\nas that, because a group of stars looks as if it were all together in\nspace and had a common origin, it actually is so. Thus at greater and\ngreater distances not only does experimental accuracy become less, but\nthe very nature of the operations by which length is to be determined\nbecomes indefinite, so that the distances of the most remote stellar\nobjects as estimated by different observers or by different methods may\nbe very divergent. A particular consequence of the inaccuracy of the\nastronomical measures of great distances is that the question of whether\nlarge scale space is Euclidean or not is merely academic.\n\nWe thus see that in the extension from terrestrial to great stellar\ndistances the concept of length has changed completely in character. To\nsay that a certain star is 10^5 light years distant is actually and\nconceptually an entire different _kind_ of thing from saying that a\ncertain goal post is 100 meters distant. Because of our conviction that\nthe character of our experience may change when the range of phenomena\nchanges, we feel the importance of such a question as whether the space\nof distances of 10^5 light years is Euclidean or not, and are\ncorrespondingly dissatisfied that at present there seems no way of\ngiving meaning to it.\n\nWe encounter difficulties similar to those above, and are also compelled\nto modify our procedures, when we go to small distances. Down to the\nscale of microscopic dimensions a fairly straightforward extension of\nthe ordinary measuring procedure is sufficient, as when we measure a\nlength in a micrometer eyepiece of a microscope. This is of course a\ncombination of tactual and optical measurements, and certain\nassumptions, justified as far as possible by experience, have to be made\nabout the behavior of light beams. These assumptions are of a quite\ndifferent character from those which give us concern on the astronomical\nscale, because here we meet difficulty from interference effects due to\nthe finite scale of the structure of light, and are not concerned with a\npossible curvature of light beams in the long reaches of space. Apart\nfrom the matter of convenience, we might also measure small distances by\nthe tactual method.\n\nAs the dimensions become smaller, certain difficulties become\nincreasingly important that were negligible on a larger scale. In\ncarrying out physically the operations equivalent to our concepts, there\nare a host of practical precautions to be taken which could be\nexplicitly enumerated with difficulty, but of which nevertheless any\npractical physicist is conscious. Suppose, for example, we measure\nlength tactually by a combination of Johanssen gauges. In piling these\ntogether, we must be sure that they are clean, and are thus in actual\ncontact. Particles of mechanical dirt first engage our attention. Then\nas we go to smaller dimensions we perhaps have to pay attention to\nadsorbed films of moisture, then at still smaller dimensions to adsorbed\nfilms of gas, until finally we have to work in a vacuum, which must be\nthe more nearly complete the smaller the dimensions. About the time that\nwe discover the necessity for a complete vacuum, we discover that the\ngauges themselves are atomic in structure, that they have no definite\nboundaries, and therefore no definite length, but that the length is a\nhazy thing, varying rapidly in time between certain limits. We treat\nthis situation as best we can by taking a time average of the apparent\npositions of the boundaries, assuming that along with the decrease of\ndimensions we have acquired a corresponding extravagant increase in\nnimbleness. But as the dimensions get smaller continually, the\ndifficulties due to this haziness increase indefinitely in percentage\neffect, and we are eventually driven to give up altogether. We have made\nthe discovery that there are _essential_ physical limitations to the\noperations which defined the concept of length. [We perhaps do not\nregard the substitution of optical for tactual space on the astronomical\nscale as compelled by the same sort of physical necessity, because I\nsuppose the possible eventual landing of men in the moon will always be\none of the dreams of humanity.] At the same time that we have come to\nthe end of our rope with our Johanssen gauge procedure, our companion\nwith the microscope has been encountering difficulties due to the finite\nwave length of light; this difficulty he has been able to minimize by\nusing light of progressively shorter wave lengths, but he has eventually\nhad to stop on reaching X-rays. Of course this optical procedure with\nthe microscope is more convenient, and is therefore adopted in practice.\n\nLet us now see what is implied in our concept of length extended to\nultramicroscopic dimensions. What, for instance, is the meaning of the\nstatement that the distance between the planes of atoms in a certain\ncrystal is 3 x 10^-8 cm.? What we would like to mean is that \u2153 x 10^8\nof these planes piled on top of each other give a thickness of 1 cm.;\nbut of course such a meaning is not the actual one. The actual meaning\nmay be found by examining the operations by which we arrived at the\nnumber 3 x 10^-8. As a matter of fact, 3 x 10^-8 was the number obtained\nby solving a general equation derived from the wave theory of light,\ninto which certain numerical data obtained by experiments with X-rays\nhad been substituted. Thus not only has the character of the concept of\nlength changed from tactual to optical, but we have gone much further in\ncommitting ourselves to a definite optical theory. If this were the\nwhole story, we would be most uncomfortable with respect to this branch\nof physics, because we are so uncertain of the correctness of our\noptical theories, but actually a number of checks can be applied which\ngreatly restore our confidence. For instance, from the density of the\ncrystal and the grating space, the weight of the individual atoms may be\ncomputed, and these weights may then be combined with measurements of\nthe dimensions of other sorts of crystal into which the same atoms enter\nto give values of the densities of these crystals, which may be checked\nagainst experiment. All such checks have succeeded within limits of\naccuracy which are fairly high. It is important to notice that, in spite\nof the checks, the character of the concept is changing, and begins to\ninvolve such things as the equations of optics and the assumption of the\nconservation of mass.\n\nWe are not content, however, to stop with dimensions of atomic order,\nbut have to push on to the electron with a diameter of the order of\n10^-18 cm. What is the possible meaning of the statement that the\ndiameter of an electron is 10^-18 cm.? Again the only answer is found by\nexamining the operations by which the number 10^-18 was obtained. This\nnumber came by solving certain equations derived from the field\nequations of electrodynamics, into which certain numerical data obtained\nby experiment had been substituted. The concept of length has therefore\nnow been so modified as to include that theory of electricity embodied\nin the field equations, and, most important, assumes the correctness of\nextending these equations from the dimensions in which they may be\nverified experimentally into a region in which their correctness is one\nof the most important and problematical of present-day questions in\nphysics. To find whether the field equations are correct on a small\nscale, we must verify the relations demanded by the equations between\nthe electric and magnetic forces and the space coordinates, to determine\nwhich involves measurement of lengths. But if these space coordinates\ncannot be given an independent meaning apart from the equations, not\nonly is the attempted verification of the equations impossible, but the\nquestion itself is meaningless. If we stick to the concept of length by\nitself, we are landed in a vicious circle. As a matter of fact, the\nconcept of length disappears as an independent thing, and fuses in a\ncomplicated way with other concepts, all of which are themselves altered\nthereby, with the result that the total number of concepts used in\ndescribing nature at this level is reduced in number. A precise analysis\nof the situation is difficult, and I suppose has never been attempted,\nbut the general character of the situation is evident. Until at least a\npartial analysis is attempted, I do not see how any meaning can be\nattached to such questions as whether space is Euclidean in the small\nscale.\n\nIt is interesting to observe that any increased accuracy in knowledge of\nlarge scale phenomena must, as far as we now can see, arise from an\nincrease in the accuracy of measurement of small things, that is, in the\nmeasurement of small angles or the analysis of minute differences of\nwave lengths in the spectra. To know the very large takes us into the\nsame field of experiment as to know the very small, so that\noperationally the large and the small have features in common.\n\nThis somewhat detailed analysis of the concept of length brings out\nfeatures common to all our concepts. If we deal with phenomena outside\nthe domain in which we originally defined our concepts, we may find\nphysical hindrances to performing the operations of the original\ndefinition, so that the original operations have to be replaced by\nothers. These new operations are, of course, to be so chosen that they\ngive, within experimental error, the same numerical results in the\ndomain in which the two sets of operations may be both applied; but we\nmust recognize in principle that in changing the operations we have\nreally changed the concept, and that to use the same name for these\ndifferent concepts over the entire range is dictated only by\nconsiderations of convenience, which may sometimes prove to have been\npurchased at too high a price in terms of unambiguity. We must always be\nprepared some day to find that an increase in experimental accuracy may\nshow that the two different sets of operations which give the same\nresults in the more ordinary part of the domain of experience, lead to\nmeasurably different results in the more unfamiliar parts of the domain.\nWe must remain aware of these joints in our conceptual structure if we\nhope to render unnecessary the services of the unborn Einsteins.\n\nThe second feature common to all concepts brought out by the detailed\ndiscussion of length is that, as we approach the experimentally\nattainable limit, concepts lose their individuality, fuse together, and\nbecome fewer in number, as we have seen that at dimensions of the order\nof the diameter of an electron the concepts of length and the electric\nfield vectors fuse into an amorphous whole. Not only does nature as\nexperienced by us become different in character on its horizons, but it\nbecomes simpler, and therefore our concepts, which are the building\nstones of our descriptions, become fewer in number. This seems to be an\nentirely natural state of affairs. How the number of concepts is often\nkept formally the same as we approach the horizon will be discussed\nlater in special cases.\n\nA precise analysis of our conceptual structure has never been attempted,\nexcept perhaps in very restricted domains, and it seems to me that there\nis room here for much important future work. Such an analysis is not to\nbe attempted in this essay, but only some of the more important\nqualitative aspects are to be pointed out. It will never be possible to\ngive a clean-cut logical analysis of the conceptual situation, for the\nnature of our concepts, according to our operational point of view, is\nthe same as the nature of experimental knowledge, which is often hazy.\nThus in the transition regions where nature is getting simpler and the\nnumber of operationally independent concepts changes, a certain haziness\nis inevitable, for the actual change in our conceptual structure in\nthese transition regions is continuous, corresponding to the continuity\nof our experimental knowledge, whereas formally the number of concepts\nshould be an integer.\n\n\n\n\n_The Relative Character of Knowledge_\n\n\nTwo other consequences of the operational point of view must now be\nexamined. First is the consequence that all our knowledge is relative.\nThis may be understood in a general or a more particular sense. The\ngeneral sense is illustrated in Haldane's book on the _Reign of\nRelativity_. Relativity in the general sense is the merest truism if the\noperational definition of concept is accepted, for experience is\ndescribed in terms of concepts, and since our concepts are constructed\nof operations, all our knowledge must unescapably be relative to the\noperations selected. But knowledge is also relative in a narrower sense,\nas when we say there is no such thing as absolute rest (or motion) or\nabsolute size, but rest and size are relative terms. Conclusions of this\nkind are involved in the specific character of the operations in terms\nof which rest or size are defined. An examination of the operations by\nwhich we determine whether a body is at rest or in motion shows that the\noperations are relative operations: rest or motion is determined with\nrespect to some other body selected as the standard. In saying that\nthere is no such thing as absolute rest or motion we are not making a\nstatement about nature in the sense that might be supposed, but we are\nmerely making a statement about the character of our descriptive\nprocesses. Similarly with regard to size: examination of the operations\nof the measuring process shows that size is measured relative to the\nfundamental measuring rod.\n\nThe \"absolute\" therefore disappears in the original meaning of the word.\nBut the \"absolute\" may usefully return with an altered meaning, and we\nmay say that a thing has absolute properties if the numerical magnitude\nis the same when measured with the same formal procedure by all\nobservers. Whether a given property is absolute or not can be determined\nonly by experiment, landing us in the paradoxical position that the\nabsolute is absolute only relative to experiment. In some cases, the\nmost superficial observation shows that a property is not absolute, as,\nfor example, it is at once obvious that measured velocity changes with\nthe motion of the observer. But in other cases the decision is more\ndifficult. Thus Michelson thought he had an absolute procedure for\nmeasuring length, by referring to the wave length of the red cadmium\nline as standard;[3] it required difficult and accurate experiment to\nshow that this length varies with the motion of the observer. Even then,\nby changing the definition of the length of a moving object, we believe\nthat length might be made to re-assume its desired absolute character.\n\n[Footnote 3: A. A. Michelson, Light Waves and Their Uses, University of\nChicago Press, 1903, Chap. V.]\n\nTo stop the discussion at this point might leave the impression that\nthis observation of the relative character of knowledge is of only a\nvery tenuous and academic interest, since it appears to be concerned\nmostly with the character of our descriptive processes, and to say\nlittle about external nature. [What this means we leave to the\nmetaphysician to decide.] But I believe there is a deeper significance\nto all this. It must be remembered that all our argument starts with the\nconcepts as given. Now these concepts involve physical operations; in\nthe discovery of what operations may be usefully employed in describing\nnature is buried almost all physical experience. In erecting our\nstructure of physical science, we are building on the work of all the\nages. There is then this purely physical significance in the statement\nthat all motion is relative, namely that no operations of measuring\nmotion have been found to be useful in describing simply the behavior of\nnature which are not operations relative to a single observer; in making\nthis statement we are stating something about nature. It takes an\nenormous amount of real physical experience to discover relations of\nthis sort. The discovery that the number obtained by counting the number\nof times a stick may be applied to an object can be simply used in\ndescribing natural phenomena was one of the most important and\nfundamental discoveries ever made by man.\n\n\n\n\n_Meaningless Questions_\n\n\nAnother consequence of the operational character of our concepts, almost\na corollary of that considered above, is that it is quite possible, nay\neven disquietingly easy, to invent expressions or to ask questions that\nare meaningless. It constitutes a great advance in our critical attitude\ntoward nature to realize that a great many of the questions that we\nuncritically ask are without meaning. If a specific question has\nmeaning, it must be possible to find operations by which an answer may\nbe given to it. It will be found in many cases that the operations\ncannot exist, and the question therefore has no meaning. For instance,\nit means nothing to ask whether a star is at rest or not. Another\nexample is a question proposed by Clifford, namely, whether it is not\npossible that as the solar system moves from one part of space to\nanother the absolute scale of magnitude may be changing, but in such a\nway as to affect all things equally, so that the change of scale can\nnever be detected. An examination of the operations by which length is\nmeasured in terms of measuring rods shows that the operations do not\nexist (because of the nature of our definition of length) for answering\nthe question. The question can be given meaning only from the point of\nview of some imaginary superior being watching from an external point of\nvantage. But the operations by which such a being measures length are\ndifferent from the operations of our definition of length, so that the\nquestion acquires meaning only by changing the significance of our\nterms--in the original sense the question means nothing.\n\nTo state that a certain question about nature is meaningless is to make\na significant statement about nature itself, because the fundamental\noperations are determined by nature, and to state that nature cannot be\ndescribed in terms of certain operations is a significant statement.\n\nIt must be recognized, however, that there is a sense in which no\nserious question is entirely without meaning, because doubtless the\nquestioner had in mind some intention in asking the question. But to\ngive meaning in this sense to a question, one must inquire into the\nmeaning of the concepts as used by the questioner, and it will often be\nfound that these concepts can be defined only in terms of fictitious\nproperties, as Newton's absolute time was defined by its properties, so\nthat the meaning to be ascribed to the question in this way has no\nconnection with reality. I believe that it will enable us to make more\nsignificant and interesting statements, and therefore will be more\nuseful, to adopt exclusively the operational view, and so admit the\npossibility of questions entirely without meaning.\n\nThis matter of meaningless questions is a very subtle thing which may\npoison much more of our thought than that dealing with purely physical\nphenomena. I believe that many of the questions asked about social and\nphilosophical subjects will be found to be meaningless when examined\nfrom the point of view of operations. It would doubtless conduce greatly\nto clarity of thought if the operational mode of thinking were adopted\nin all fields of inquiry as well as in the physical. Just as in the\nphysical domain, so in other domains, one is making a significant\nstatement about his subject in stating that a certain question is\nmeaningless.\n\nIn order to emphasize this matter of meaningless questions, I give here\na list of questions, with which the reader may amuse himself by finding\nwhether they have meaning or not.\n\n(1) Was there ever a time when matter did not exist?\n\n(2) May time have a beginning or an end?\n\n(3) Why does time flow?\n\n(4) May space be bounded?\n\n(5) May space or time be discontinuous?\n\n(6) May space have a fourth dimension, not directly detectible, but\ngiven indirectly by inference?\n\n(7) Are there parts of nature forever beyond our detection?\n\n(8) Is the sensation which I call blue really the _same_ as that which\nmy neighbor calls blue? Is it possible that a blue object may arouse in\nhim the same sensation that a red object does in me and _vice versa_?\n\n(9) May there be missing integers in the series of natural numbers as we\nknow them?\n\n(10) Is a universe possible in which 2 + 2 \u2260 4?\n\n(11) Why does negative electricity attract positive?\n\n(12) Why does nature obey laws?\n\n(13) Is a universe possible in which the laws are different?\n\n(14) If one part of our universe could be completely isolated from the\nrest, would it continue to obey the same laws?\n\n(15) Can we be sure that our logical processes are valid?\n\n\n\n\nGENERAL COMMENTS ON THE OPERATIONAL POINT\nOF VIEW\n\n\nTo adopt the operational point of view involves much more than a mere\nrestriction of the sense in which we understand \"concept,\" but means a\nfar-reaching change in all our habits of thought, in that we shall no\nlonger permit ourselves to use as tools in our thinking concepts of\nwhich we cannot give an adequate account in terms of operations. In some\nrespects thinking becomes simpler, because certain old generalizations\nand idealizations become incapable of use; for instance, many of the\nspeculations of the early natural philosophers become simply unreadable.\nIn other respects, however, thinking becomes much more difficult,\nbecause the operational implications of a concept are often very\ninvolved. For example, it is most difficult to grasp adequately all that\nis contained in the apparently simple concept of \"time,\" and requires\nthe continual correction of mental tendencies which we have long\nunquestioningly accepted.\n\nOperational thinking will at first prove to be an unsocial virtue; one\nwill find oneself perpetually unable to understand the simplest\nconversation of one's friends, and will make oneself universally\nunpopular by demanding the meaning of apparently the simplest terms of\nevery argument. Possibly after every one has schooled himself to this\nbetter way, there will remain a permanent unsocial tendency, because\ndoubtless much of our present conversation will then become unnecessary.\nThe socially optimistic may venture to hope, however, that the ultimate\neffect will be to release one's energies for more stimulating and\ninteresting interchange of ideas.\n\nNot only will operational thinking reform the social art of\nconversation, but all our social relations will be liable to reform. Let\nany one examine in operational terms any popular present-day discussion\nof religious or moral questions to realize the magnitude of the\nreformation awaiting us. Wherever we temporize or compromise in applying\nour theories of conduct to practical life we may suspect a failure of\noperational thinking.\n\n\n\n\nCHAPTER II\n\nOTHER GENERAL CONSIDERATIONS\n\n\nTHE APPROXIMATE CHARACTER OF EMPIRICAL\nKNOWLEDGE\n\n\nALTHOUGH many aspects of the processes by which we obtain knowledge of\nthe external physical world are much beyond the scope of our present\ninquiry, one matter must be mentioned in detail because it tacitly\nunderlies all our discussion, the fact, namely, that all results of\nmeasurement are only approximate. That such is true is evident after the\nmost superficial examination of any measuring process; any statement\nabout numerical relations between measured quantities must always be\nsubject to the qualification that the relation is valid only within\nlimits. Furthermore, all experience seems to be of this character; we\nnever have perfectly clean-cut knowledge of anything, but all our\nexperience is surrounded by a twilight zone, a penumbra of uncertainty,\ninto which we have not yet penetrated. This penumbra is as truly an\nunexplored region as any other region beyond experiment, such as the\nregion of high velocities, for example, and we must hold no preconceived\nnotions as to what will be found within the region. The penumbra is to\nbe penetrated by improving the accuracy of measurement. Within what was\nat one time penumbra has been found the displacement of angular position\nof the stars near the edge of the solar disc, and within the penumbra as\nyet unpenetrated we look for such effects as the equivalence of mass and\nenergy. Many of the great discoveries of the future will probably be\nmade within the penumbra: we have already mentioned that increased\nknowledge of phenomena of a cosmic scale is to be obtained by increasing\nthe accuracy of measurement of the very small.\n\nIt is a general consequence of the approximate character of all\nmeasurement that no empirical science can ever make exact statements.\nThis was fairly obvious in the case of mechanics, but it required a\nGauss[4] to convince us that the geometry in which we are interested as\nphysicists is an empirical subject, and that one cannot say that actual\nspace is Euclidean, but only that actual space approaches to ideal\nEuclidean space within a certain degree of approximation. I believe that\nwe are compelled to go still further, and recognize that arithmetic, so\nfar as it purports to deal with actual physical objects, is also\naffected with the same penumbra of uncertainty as all other empirical\nscience. A typical statement of empirical arithmetic is that 2 objects\nplus 2 objects makes 4 objects. This statement acquires physical meaning\nonly in terms of certain physical operations, and these operations must\nbe performed in time.\n\n[Footnote 4: C. F. Gauss, Gesammelte Werke, especially vols. IV and\nVIII.]\n\nNow the penumbra gets into this situation through the concept of object.\nIf the statement of arithmetic is to be an exact statement in the\nmathematical sense the \"object\" must be a definite clean-cut thing,\nwhich preserves its identity in time with no penumbra. But this sort of\nthing is never experienced, and as far as we know does not correspond\nexactly to anything in experience. It is of course true that in most\nexperience the penumbra is so very thin and snug-fitting that it\nrequires special effort to recognize its presence at all; but scrutiny,\nI believe, shows that it is always there. If our experience had been\nrestricted to phenomena in a vacuum, and the objects we were trying to\ncount had been spheres of a gas which expand and interpenetrate, it is\nobvious that the concept of \"object\" as a thing with identity would have\nbeen much more difficult to form. Or, if our objects are tumblers of\nwater, we discover when our observation reaches a certain stage of\nrefinement that the amount of water is continually changing by\nevaporation and condensation, and we are bothered by the question\nwhether the object is still the same after it has waxed and waned.\nComing to solids, we eventually discover that even solids evaporate, or\ncondense gases on them, and we see that an object with identity is an\nabstraction corresponding exactly to nothing in nature. Of course the\npenumbra of uncertainty which surrounds our arithmetical statements\nbecause of this property of physical objects is so exceedingly tenuous\nthat practically we are not aware of its existence, and expect never to\nfind undiscovered phenomena within the penumbra. But in principle we\nmust recognize its presence, and must further recognize that _all_\nempirical science must be of this character.\n\nIn most empirical sciences, the penumbra is at first prominent, and\nbecomes less important and thinner as the accuracy of physical\nmeasurement is increased. In mechanics, for example, the penumbra is at\nfirst like a thick obscuring veil at the stage where we measure forces\nonly by our muscular sensations, and gradually is attenuated, as the\nprecision of measurements increases. But with the arithmetical concept\nof an individual identifiable object it is the exact reverse; a crude\npoint of view does not suspect the existence of the penumbra at all, and\nwe discover it only by highly refining our methods. Doubtless arithmetic\nowes its early development to this property.\n\nWe may now go still further. Operations themselves are, of course,\nderived from experience, and would be expected also to have a nebulous\nedge of uncertainty. We have to ask such questions as whether the\n_operations_ of arithmetic are clean-cut things. Is the operation of\nmultiplying 2 objects by 2 a definite operation, with no enveloping\nhaze? All our physical experience convinces us that if there is a\npenumbra about the concept of operations of this sort it is so tenuous\nas to be negligible, at least for the present; but the question affords\nan interesting topic for speculation. We also have to ask whether mental\noperations may similarly be enveloped in a haze.\n\n\n\n\nEXPLANATIONS AND MECHANISMS\n\n\nPerhaps the climax of our task of interpreting and correlating nature is\nreached when we are able to find an explanation of phenomena; with the\nfinding of the explanation we are inclined to feel that our\nunderstanding of the situation is complete. We now have to ask what is\nthe nature of the explanation which we set as the goal of our efforts.\nThe answer is not easy to give, and there may be difference of opinion\nabout it. We shall get the best answer to this, as to so many other\nquestions, by adopting the operational point of view, and examining what\nwe do in giving an explanation. I believe that examination will show\nthat the essence of an explanation consists in reducing a situation to\nelements with which we are so familiar that we accept them as a matter\nof course, so that our curiosity rests.[5] \"Reducing a situation to\nelements\" means, from the operational point of view, discovering\nfamiliar correlations between the phenomena of which the situation is\ncomposed.\n\n[Footnote 5: The ultimate elements of explanation are analogous to the\naxioms of formal mathematics.]\n\nThere is involved here the thesis that it is possible to analyze nature\ninto correlations, without, however, any assumption whatever as to the\ncharacter of these correlations. It seems to me that such a thesis is\nthe most general that can be made if nature is to be intelligible at\nall. This thesis underlies all the considerations of this essay, and we\nshall not try to find anything more general. We shall, however,\nrecognize that any assumption as to the character of the correlations\nconstitutes a special hypothesis which may restrict the future, and that\ntherefore these special hypotheses are to be subjected to special\nexamination. We return to this matter in more detail in discussing the\ncausality concept, which is closely related to the concept of\nexplanation.\n\nIn this view of explanation there is no implication that the \"element\"\nis either a smaller or a larger scale thing than the phenomenon being\nexplained; thus we may explain the properties of a gas in terms of its\nconstituent molecules, or perhaps some day we shall become so familiar\nwith the idea of a non-Euclidean space that we shall _explain_ (instead\nof describe) the gravitational attraction of a stone by the earth in\nterms of a space-time curvature imposed by all the rest of the matter in\nthe universe.\n\nIf this is accepted as the true nature of explanation, we see that an\nexplanation is not an absolute sort of thing, but what is satisfactory\nfor one man will not be for another. The savage is satisfied by\nexplaining the thunderstorm as the capricious act of an angry god. The\nphysicist demands more, and requires that the familiar elements to which\nwe reduce a situation be such that we can intuitively predict their\nbehavior. Thus even if the physicist believed in the existence of the\nangry god, he would not be satisfied with this explanation of the\nthunderstorm because he is not so well acquainted with angry gods as to\nbe able to predict when anger is followed by a storm. He would have to\nknow why the god had become angry, and why making a thunderstorm eased\nhis ire. But even with this additional qualification, scientific\nexplanation is obviously still a relative affair--relative to the\nelements or axioms to which we make reduction and which we accept as\nultimate. These elements depend to a certain extent on the purpose in\nview, and also on the range of our previous physical experience. If we\nare explaining the action of a machine, we are satisfied to reduce the\naction to the push and pull of the various members of the machine, it\nbeing accepted as an ultimate that these members transmit pushes or\npulls. But the physicist who has extended his experimental knowledge\nfurther, may want to explain how the members transmit pushes or pulls in\nterms of the action on each other of the electrons in their orbits in\nthe atoms. The character of our explanatory structure will depend on the\ncharacter of our experimental knowledge, and will change as this\nchanges.\n\nFormally, there is no limit to the process of explanation, because we\ncan always ask what is the explanation of the elements in terms of which\nwe have given the last explanation. But the point of view of operations\nshows that this is mere formalism which ends only in meaningless jargon,\nfor we soon arrive at the limit of our experimental knowledge, and\nbeyond this the operations involved in the concepts of our explanations\nbecome impossible and the concepts become meaningless.\n\nAs we extend experimental knowledge and push our explanations further\nand further, we see that the explanatory sequence may be terminated in\nseveral possible ways. In the first place, we may never push our\nexperiments beyond a stage into which the elements with which we are\nalready familiar do not enter. In this case explanation is very simple:\nit involves nothing essentially new, but merely the disentanglement of\ncomplexities. The kinetic theory of gases, in explaining the thermal\nproperties of a gas in terms of ordinary mechanical properties of the\nmolecules, suggests such a situation. Or, secondly, our experiments may\nbring us into contact with situations novel to us, in which we can\nrecognize no familiar elements, or at least must recognize that there is\nsomething in addition to the familiar elements. Such a situation\nconstitutes an explanatory crisis and explanation has to stop by\ndefinition. Or thirdly, we may try to force our explanations into a\npredetermined mold, by formally erecting or inventing beyond the range\nof present experiment ultimates more or less like elements already\nfamiliar to us, and seek to explain all present experience in terms of\nthese chosen ultimates.\n\nLeaving for the present the third possibility, which is within our\ncontrol to accept or reject, and is a formal matter, it is merely a\nquestion of experimental fact which of the first two possibilities\ncorresponds to the actual state of affairs. The most perfunctory\nexamination of the present state of physics shows that we are now facing\nthe second of these possibilities, and that in the new experimental\nfacts of relativity, and in all quantum phenomena, we are confronted\nwith an explanatory crisis. It has often been emphasized that Einstein's\ntheory of gravitation does not seek at all to give an explanation of\ngravitational phenomena, but merely describes and correlates these\nphenomena in comparatively simple mathematical language. No more attempt\nis made to reduce the gravitational attraction between the earth and the\nsun to simple terms than was made by Newton. In the realm of quantum\nphenomena it is of course the merest commonplace that our old ideas of\nmechanics and electrodynamics have failed, so that it is a matter of the\ngreatest concern to find how many, or indeed whether any, elements of\nthe old situations can be carried over into the new.\n\nAn examination of many of the so-called \"explanations\" of quantum theory\nconstitutes at once a justification of the definition of explanation\ngiven above, and of the statement that in quantum phenomena we are at an\nexplanatory crisis. For the endeavor of all these quantum explanations\nis to find in every new or more complicated situation the same elements\nwhich have already been met in simpler situations, and which are\ntherefore relatively more familiar. For example, many quantum phenomena\nare made to involve the emission of energy when an electron jumps from\none orbit to another. But always the elements to which reduction is made\nare themselves quantum phenomena, and these are still so new and\nunfamiliar that we feel an instinctive need for explanation in other\nterms. We seek to understand why the electron emits when it jumps.\n\nThe explanatory crisis which now confronts us in relativity and quantum\nphenomena is but a repetition of what has occurred many time in the\npast. A similar crisis confronted Prometheus when he discovered fire,\nand the first man who observed a straw sticking to a piece of rubbed\namber, or a suspended lodestone seeking the north star. Every kitten is\nconfronted with such a crisis at the end of nine days. Whenever\nexperience takes us into new and unfamiliar realms, we are to be at\nleast prepared for a new crisis.\n\nNow what are we to do in such a crisis? It seems to me that the only\nsensible course is to do exactly what the kitten does, namely, to wait\nuntil we have amassed so much experience of the new kind that it is\nperfectly familiar to us, and then to resume the process of explanation\nwith elements from our new experience included in our list of axioms.\nNot only will observation show that this is what is now actually being\ndone with respect to quantum and gravitational phenomena, but it is in\nharmony with the entire spirit of our outlook on nature. All our\nknowledge is in terms of experience; we should not expect or desire to\nerect an explanatory structure different in character from that of\nexperience. Our experience is finite; on the confines of the\nexperimentally attainable it becomes hazy, and the concepts in terms of\nwhich we describe it fuse together and lose independent meaning.\nFurthermore, at every extension of our experimental range we must be\nprepared to find, and as a matter of fact we have often found, that we\nencounter phenomena of an entirely novel character for which previous\nexperience has given us no preparation. The explanatory structure\nproposed above has all these properties; it is finite, being terminated\nby the edge of experiment, the final stages of our explanations are hazy\nin that it becomes more and more difficult to distinguish elements of\nfamiliar experience, and every now and then we must admit new elements\ninto our explanations.\n\nThe first step in resuming our explanatory progress, after we have been\nconfronted with such a crisis, is to seek for various sorts of\ncorrelation between the elements of our new experience, in the confident\nexpectation that these elements will eventually become so familiar to us\nthat they may be used as the ultimates of a new explanation. This is\nexactly what is now happening in quantum theory.\n\nDiametrically opposed to the views above, there is another ideal of the\nexplanatory process which is held by many physicists, and which has been\nmentioned above as the third possible way in which the explanatory\nsequence may be terminated, namely, the endeavor to devise beyond the\nlimit of present experiment a structure built of elements like some of\nthose of our present experience, in the action of which we endeavor to\nfind the explanation of phenomena in the present range. Now a program\nsuch as this, as a serious program for the final correlation of nature,\nis entirely opposed to the spirit of the considerations expounded here.\nThere is no warrant whatever in experience for the conviction that as we\npenetrate deeper and deeper we shall find the elements of previous\nexperience repeated, although sometimes we do find such repetitions, as\nin the behavior of gases. Yet this has been the attitude of many eminent\nphysicists, for example, Faraday and Maxwell, in seeking to explain\ndistant electrical action by the propagation through a medium of a\nmechanical push or pull, or by Hertz, who sought in all phenomena the\neffect of concealed masses with ordinary mechanical inertia. Although as\na general principle this program seems to be absolutely without\njustification, nevertheless it may be justified if the specific\ncharacter of the physical facts seems to indicate a repetition at lower\nlevels of elements familiar higher up. Hertz undoubtedly had this\njustification, as did also Maxwell to a certain extent, in the discovery\nthat the general equations of electrodynamics are of the same form as\nthe generalized Lagrangean equations of mechanics. For Faraday, however,\nthere seems no such justification; the urge to this sort of thing in\nFaraday came from an uncritical acceptance of his own temperamental\nreactions.\n\nFrom a less serious point of view it may, however, be quite justified to\nmake such a working hypothesis as that in the action of electrical\nforces may be discovered the same elements with which we are familiar in\nthe everyday experiences of mechanics. For such a hypothesis often\nenables us to make partial correlations which suggest new experimental\ntests, and thus gives the stimulus to an extension of our experimental\nhorizon. Many physicists recognize the tentative character of such\nattempted explanations, but others apparently take them more seriously,\nas for example Lord Kelvin in his continuous life-long attempts to find\na mechanical explanation of all physical phenomena. This quotation from\nKelvin is illuminating. \"I never satisfy myself until I can make a\nmechanical model of a thing. If I can make a mechanical model, I can\nunderstand it. As long as I cannot make a mechanical model all the way\nthrough, I cannot understand it.... But I want to understand light as\nwell as I can without introducing things that we understand even less\nof.\"\n\nSo much for general considerations on the nature of explanation. Coming\nnow to greater detail, many explanations involve what may be described\nas a mechanism. It is difficult to characterize exactly what we mean by\nmechanism, but it seems to be associated with an attitude of mind that\nstrives to realize the third of the possibilities mentioned above. As a\nmatter of fact, the mechanism sought for is usually of a particular\ntype, in that the ultimate elements selected are mechanical elements.\nThis point of view is particularly characteristic of the English school\nof physicists. Although \"mechanism\" usually implies mechanical elements,\nwe may show by specific examples that we do actually use the word in a\nbroader significance. If, for example, we could devise within the core\nof an atom a revolving system of electrical charges, acting on each\nother with the ordinary inverse square forces of electrostatics, such\nthat every now and then the system becomes unstable and breaks up, we\nshould doubtless say that we had found a mechanism for explaining\nradioactive disintegration.\n\nHowever, the formulation of a precise definition of mechanism is of\nsecondary concern to us; we are primarily interested in understanding\nthe attitude of mind that feels a mechanism is necessary. A typical\nexample of such an urge to a mechanism is afforded by the gravitational\naction between distant bodies. To many minds the concept of action at a\ndistance is absolutely abhorrent, not to be tolerated for an instant.\nSuch an intolerable situation is avoided by the invention of a medium\nfilling all space, which transmits a force from one body to the other\nthrough the successive action on each other of its contiguous parts. Or\nthe dilemma of action at a distance may be avoided in other ways, as by\nBoscovitch in the eighteenth century, who, in order to explain\ngravitation, filled space with a triply infinite horde of infinitesimal\nprojectiles. Now of course it is a matter for experiment to decide\nwhether any physical reality can be ascribed to a medium which makes\ngravitation possible by the action of its adjacent parts, but I can see\nno justification whatever for the attitude which refuses on purely _a\npriori_ grounds to accept action at a distance as a possible axiom or\nultimate of explanation. It is difficult to conceive anything more\nscientifically bigoted than to postulate that all possible experience\nconforms to the same type as that with which we are already familiar,\nand therefore to demand that explanation use only elements familiar in\neveryday experience. Such an attitude bespeaks an unimaginativeness, a\nmental obtuseness and obstinacy, which might be expected to have\nexhausted their pragmatic justification at a lower plane of mental\nactivity.\n\nAlthough it will probably be fairly easy to give intellectual assent to\nthe strictures of the last paragraph, I believe many will discover in\nthemselves a longing for mechanical explanation which has all the\ntenacity of original sin. The discovery of such a desire need not\noccasion any particular alarm, because it is easy to see how the demand\nfor this sort of explanation has had its origin in the enormous\npreponderance of the mechanical in our physical experience. But\nnevertheless, just as the old monks struggled to subdue the flesh, so\nmust the physicist struggle to subdue this sometimes nearly\nirresistible, but perfectly unjustifiable desire. One of the large\npurposes of this exposition will be attained if it carries the\nconviction that this longing is unjustifiable, and is worth making the\neffort to subdue.\n\nThe situation with respect to action at a distance is typical of the\ngeneral situation. I believe the essence of the explanatory process is\nsuch that we must be prepared to accept as an ultimate for our\nexplanations the mere statement of a correlation between phenomena or\nsituations with which we are sufficiently familiar. Thus, in quantum\ntheory, there is no reason why we should not be willing to accept as an\nultimate the fundamental fact that when an electron jumps radiation is\nemitted, provided always that we can give independent meaning in terms\nof operations to the jumping of an electron. If there is no experiment\nsuggesting other and intermediate phenomena, we ought to be able to rest\nintellectually satisfied with this. Of course it is quite a different\nmatter, and entirely justified, to imagine what the assumption of finer\ndetails in the process would involve experimentally, and then to seek\nfor these possible new experimental facts.\n\nIt is a consequence of this view that any correlation is adapted to be\nan absolutely final element of explanation, and can never be superseded\nby the discovery of new experimental facts, if the correlation is by\n_definition_ beyond the reach of further experiment. Such a possibility,\nfor example, is contained in a correlation between the numerical\nmagnitude of the gravitational constant and the total mass of the\nuniverse. Something of this sort may be well attempted by those who\ndesire their explanations to take a formally final shape. We shall\nreturn to this subject later.\n\nThe instinctive demand for a mechanism is fortified by observation of\nthe many important cases in which mechanisms have been discovered or\ninvented. However, the significance of such successful attempts must be\nsubject to most careful scrutiny. The matter has been discussed by\nPoincar\u00e9,[6] who showed that not only is it always possible to find a\nmechanistic explanation of any phenomenon (Hertz's program was a\nperfectly possible one), but there are always an infinite number of such\nexplanations. This is very unsatisfactory.\n\n[Footnote 6: Henri Poincar\u00e9. Wissenschaft and Hypothese, Translated\ninto German by F. and L. Lindemann, Teubner, Leipzig, 1906. See\nespecially p. 217.]\n\nWe want to be able to find the _real_ mechanism. Now an examination of\nspecific proposed mechanisms will show that most mechanisms are more\ncomplicated than the simple physical phenomenon which they are invented\nto explain, in that they have more independently variable attributes\nthan the phenomenon has been yet proved to have. An example is afforded\nby the mechanical models invented to facilitate the study of the\nproperties of simple inductive electrical circuits. The great number of\nsuch models which have been proposed is sufficient indication of their\npossible infinite number. But if the mechanism has more independently\nvariable attributes than the original phenomenon, it is obvious that the\nquestion is without meaning whether the mechanism is the real one or\nnot, for in the mechanism there must be simple motions or combinations\nof motions which have no counterpart in features of the original\nphenomenon as yet discovered. Obviously, then, the operations do not\nexist by which we may set up a one to one correspondence between the\nproperties of the mechanism and the natural phenomenon, and the question\nof reality has no meaning. If, then, a mechanism is to be taken\nseriously as actually corresponding to reality, we must demand that it\nhave no more degrees of freedom than the original phenomenon, and we\nmust also be sure that the phenomenon has no undiscovered features.\nPhysical experience shows that such conditions are most difficult to\nmeet, and indeed the probability is that they are impossible.\n\nA mechanism with more independently variable attributes than the\nphenomenon may prove to be a very useful tool of thought, and therefore\nworth inventing and studying, but it is to be regarded no more seriously\nthan is a mnemonic device, or any other artifice by which a man forces\nhis mind to give him better service.\n\nThere is another possible program of explanation, the converse of that\nconsidered above, namely, to explain all familiar facts of ordinary\nexperience in terms of less familiar facts found at a deeper level. The\nmost striking example of this is the recent attempt to give a complete\nelectrical explanation of the universe. The original attempt was to\nexplain electrical effects in mechanical terms; this attempt failed. At\nabout the same time the existence of the electron was experimentally\nestablished, so that it was evident that electricity is a very\nfundamental constituent of matter. The program of explanation was\nreversed, and an electrical explanation sought for all mechanical\nphenomena, including in particular mechanical mass. But this attempt has\nalso failed; we recognize that part of mass may be non-electrical in\ncharacter, we postulate non-electrical forces inside the electron, and\nfurther, we usually postulate for electrons and protons the property of\nimpenetrability, a property derived from experience on a higher scale of\nmagnitude.\n\nA program of this general sort is likely to be regarded with\nconsiderable sympathy, and indeed the chances of success seem much\ngreater than do those of the converse program, for in our experience\nlarge scale phenomena are more often built up from and analyzed into\nsmall scale phenomena than the converse. But as a matter of principle we\nmust again recognize that the only appeal is to experiment, and that we\nhave to ask just one question: \"Is it true, as a matter of fact, that\nall large scale phenomena can be built up of elements of small scale\nphenomena?\" It seems to me that the experimental warrant for this\nconviction has not yet been given. The failure of the attempted\nelectrical explanation of the universe is a case in point. However, the\nfailure to prove a proposition is no guarantee that some time it may not\nbe proved, and many physicists are convinced of the ultimate feasibility\nof this program. Personally I feel that the large may not always be\nanalyzed into the smaller; the subject will be discussed again.\n\nA conviction of the significance of microscopic analysis has many\nfeatures in common with the usual conviction of the ultimate simplicity\nof nature. The thesis of simplicity involves in addition the assumption\nthat the kinds of small scale elements are few in number, but actually\nthis involves no important difference between the two convictions,\nbecause we have seen that the elements of which we build our structure\nbecome fewer in number as we approach the limit of the experimentally\nattainable. We may properly grant to convictions of this sort pragmatic\nvalue in suggesting new correlations and experiments, but a recognition\nof the empirical basis of all physics will not allow us to go further.\n\n\n\n\nMODELS AND CONSTRUCTS\n\n\nIn discussing the concept of length, we could find no meaning in\nquestions such as: \"Is space on a scale of 10^-8 cm. Euclidean?\"\nNevertheless it will seem to many that they do attach a perfectly\ndefinite meaning to a question of this kind. Of course it must be agreed\nthat magnitudes of 10^-8 cm. cannot be thought of in terms of immediate\nsensation. When one thinks of an atom as a thing with any geometrical\nproperties at all, I believe he will find that what he essentially does\nis to imagine a model, multiplying all the hypothetical dimensions by a\nfactor large enough to bring it to a magnitude of ordinary experience.\nThis large scale model is given properties corresponding to those of the\nphysical thing. For example, the model of the atom which was accepted in\nthe fall of 1925 contains electrons rotating in orbits, and every now\nand then an electron jumps from one orbit to another, and simultaneously\nenergy is radiated from the atom. Such a model is satisfactory if it\noffers the counterpart of all the phenomena of the original atom. Now I\nbelieve the only meaning that any one can find in his statement that the\nspace of the atom is Euclidean is that he believes that he can construct\nin Euclidean space a model with all the observed properties of the atom.\nThis possibility may or may not be sufficient to give real physical\nsignificance to the statement that the space of the atom is Euclidean.\nThe situation here is very much the same as it was with respect to\nmechanisms. The model may have many more properties than correspond to\nmeasurable properties of the atom, and in particular, the operations by\nwhich the space of the model is tested for its Euclidean character may\n[and as a matter of fact I believe _do_] not have any counterpart in\noperations which can be carried out on the atom. Further, we cannot\nattach any _real_ significance to the statement that the space of the\natom is Euclidean unless we can show that no model constructed in\nnon-Euclidean space can reproduce the measurable properties of the atom.\n\nIn spite of all this, I believe that the model is a useful and indeed\nunescapable tool of thought, in that it enables us to think about the\nunfamiliar in terms of the familiar. There are, however, dangers in its\nuse: it is the function of criticism to disclose these dangers, so that\nthe tool may be used with confidence.\n\nClosely related to the mental model are mental constructs, of which\nphysics is full. There are many sorts of constructs: those in which we\nare interested are made by us to enable us to deal with physical\nsituations which we cannot directly experience through our senses, but\nwith which we have contact indirectly and by inference. Such constructs\nusually involve the element of invention to a greater or less degree. A\nconstruct containing very little of invention is that of the inside of\nan opaque solid body. We can never experience directly through our\nsenses the inside of such a solid body, because the instant we directly\nexperience it, it ceases by definition to be the inside. We have here a\nconstruct, but so natural a one as to be practically unavoidable. An\nexample of a construct involving a greater amount of invention is the\nstress in an elastic body. A stress is by definition a property of the\ninterior points of a body which is connected mathematically in a simple\nway with the forces acting across the free surface of the body. A stress\nis then, by its very nature, forever beyond the reach of direct\nexperience, and it is therefore a construct. The entire structure of a\nstress corresponds to nothing in direct experience; it is related to\nforce, but is itself a six-fold magnitude, whereas a force is only\nthree-fold.\n\nWe have next to ask whether the stress, which we have invented to meet\nthe situation in a body exposed to forces, is a good construct. In the\nfirst place, a stress has the same number of degrees of freedom as the\nobservable phenomenon, for it is one of the propositions of the\nmathematical theory of elasticity that the boundary conditions, which\nare the experimental variables, uniquely determine the stress in a\n_given_ body [_i.e._ a body of given elastic constants]; and of course\nit is at once obvious, by an inspection of the equations, that\nconversely a possible stress system uniquely determines the boundary\nconditions to the significant amount. There is, therefore, a unique\none-to-one correspondence between a stress and the physical situation it\nwas made to meet, and so far a stress is a good construct. Up to this\npoint a stress, from the point of view of the operations in terms of\nwhich it is defined, is a purely mathematical invention, which is\njustified because it is convenient in describing the behavior of bodies\nsubjected to the action of force. But we wish now to go farther and\nascribe physical reality to a stress, meaning by this that a stress in a\nsolid body shall correspond to some real physical state of the interior\npoints. Let us examine, from the point of view of operations, what the\nmeaning of a statement like this may be. Since we now wish to ascribe an\nadditional physical meaning to a stress beyond that of the mathematical\noperations in terms of which the stress was determined, there must exist\nadditional physical operations corresponding to this meaning, or else\nour statement is meaningless. Now of course it is a matter of the most\nelementary experience that physical phenomena do exist which allow these\nother independent operations. A body under stress is also in a state of\nstrain, which may be determined from the external deformations, or the\nstrain at internal points may be made more vividly real by those optical\neffects of double refraction in transparent bodies which are now so\nextensively used in model experiments, or if the stress is pushed beyond\na certain point, we have such new phenomena as permanent set or finally,\nrupture.\n\nWe have, then, every reason to be satisfied with our construct of\nstress. In the first place, from the formal point of view, it is a good\nconstruct because there is a unique correspondence between it and the\nphysical data in terms of which it is defined; and in the second place\nwe have a right to ascribe physical reality to it because it is uniquely\nconnected with other physical phenomena, _independent of those which\nentered its definition_. This last requirement, in fact, from the\noperational point of view, amounts to nothing more than a definition of\nwhat we mean by the reality of things not given directly by experience.\nSince now in addition to satisfying the formal requirements, experience\nshows that a stress is most useful in correlating phenomena, we are\njustified in giving to this construct of stress a prominent place among\nour concepts.\n\nConsider now another construct, one of the most important of physics,\nthat of the electric field. In the first place, an examination of the\noperations by which we determine the electric field at any point will\nshow that it is a construct, in that it is not a direct datum of\nexperience. To determine the electric field at a point, we place an\nexploring charge at the point, measure the force on it, and then\ncalculate the ratio of the force to the charge. We then allow the\nexploring charge to become smaller and smaller, repeating our\nmeasurement of force on each smaller charge, and define the limit of the\nratio of the force to the charge as the electric field intensity at the\npoint in question, and the limiting direction of the force on a small\ncharge as the direction of the field. We may extend this process to\nevery point of space, and so obtain the concept of a field of force, by\nwhich every point of the space surrounding electric charges is tagged\nwith the appropriate number and direction, the exploring charge having\ncompletely disappeared. The field is, then, clearly a construct. Next,\nfrom the formal point of view of mathematics, it is a good construct,\nbecause there is a one to one correspondence between the electric field\nand the electric charges in terms of which it is defined, the field\nbeing uniquely determined by the charges, and conversely there being\nonly one possible set of charges corresponding to a given field. Now\nnearly every physicist takes the next step, and ascribes physical\nreality to the electric field, in that he thinks that at every point of\nthe field there is some real physical phenomenon taking place which is\nconnected in a way not yet precisely determined with the number and\ndirection which tag the point. At first this view most naturally\ninvolved as a corollary the existence of a medium, but lately it has\nbecome the fashion to say that the medium does not exist, and that only\nthe field is real. The reality of the field is self-consciously\ninculcated in our elementary teaching, often with considerable\ndifficulty for the student. This view is usually credited to Faraday,\nand is considered the most fundamental concept of all modern electrical\ntheory. Yet in spite of this, I believe that a critical examination will\nshow that the ascription of physical reality to the electric field is\nentirely without justification. I cannot find a single physical\nphenomenon or a single physical operation by which evidence of the\nexistence of the field may be obtained independent of the operations\nwhich entered the definition. The only physical evidence we ever have of\nthe existence of a field is obtained by going there with an electric\ncharge and observing the action on the charge [when the charges are\ninside atoms we may have optical phenomena], which is precisely the\noperation of the definition. It is then either meaningless to say that a\nfield has physical reality, or we are guilty of adopting a definition of\nreality which is the crassest tautology.\n\nThere can be no question whatever of the tremendous importance of the\nconcept of the electric field as a tool in thinking about, describing,\ncorrelating, and predicting the properties of electrical systems;\nelectrical science is inconceivable without this or something\nequivalent. But in addition to this aspect of the field concept, the\nfurther tacit implication of physical reality is almost always present,\nand has had the greatest influence on the character of physical thought\nand experiment. Yet I do not believe that the additional implication of\nphysical reality has justified itself by bringing to light a single\npositive result, or can offer more than the pragmatic plea of having\nstimulated a large number of experiments, all with persistently negative\nresults. It is sufficient to mention the fate of the attempt of Faraday\nand Maxwell to ascribe a stress like that of ordinary matter to the\nether, which failed because, among other reasons, nothing can exist in\nthe ether analogous to the _strain_ of ordinary matter, to indicate the\nunfruitfulness of the idea of physical reality. It seems to me that any\npragmatic justification in postulating reality for the electric field\nhas now been exhausted, and that we have reached a stage where we should\nattempt to get closer to the actual facts by ridding the field concept\nof the implications of reality.\n\nAnother indispensable and most interesting construct is that of the\natom. This is evidently a construct, because no one ever directly\nexperienced an atom, and its existence is entirely inferential. The atom\nwas invented to explain constant combining weights in chemistry. For a\nlong time there was no other experimental evidence of its existence, and\nit remained a pure invention, without physical reality, useful in\ndiscussing a certain group of phenomena. It is one of the most\nfascinating things in physics to trace the accumulation of independent\nnew physical information all pointing to the atom, until now we are as\nconvinced of its physical reality as of our hands and feet.\n\nA construct which had to be abandoned because it did not turn out to\nhave physical reality, and which furthermore was not sufficiently useful\nin the light of newly discovered phenomena, was that of a caloric fluid.\n\nThe notion of \"physical reality\" is not of prime importance to this\ndiscussion of the character of our constructs; our definition of the\nmeaning of physical reality may not appeal to everyone. The essential\npoint is that our constructs fall into two classes: those to which no\nphysical operations correspond other than those which enter the\ndefinition of the construct, and those which admit of other operations,\nor which could be defined in several alternative ways in terms of\nphysically distinct operations. This difference in the character of\nconstructs may be expected to correspond to essential physical\ndifferences, and these physical differences are much too likely to be\noverlooked in the thinking of physicists. We must always be on our guard\nnot to forget the physical differences between a thing like a stress in\nan elastic body and an electromagnetic field.\n\nThe moral of all this is that constructs are most useful and even\nunavoidable things, but that they may have great dangers, and that a\ncareful critique may be necessary to avoid reading into them\nimplications for which there is no warrant in experience, and which may\nmost profoundly affect our physical outlook and course of action.\n\n\n\n\nTHE R\u00d4LE OF MATHEMATICS IN PHYSICS\n\n\nPractically all the formulations of theoretical physics are made in\nmathematical terms; in fact to obtain such formulations is generally\nfelt to be the goal of theoretical physics. It is then evidently\npertinent to consider what the nature of the mathematics is to which we\nassign so prominent a r\u00f4le.\n\nWe have in the first place to understand why it is possible to express\nphysical relations in mathematical language at all. I am not sure that\nthere is much meaning in this question. It is the merest truism, evident\nat once to unsophisticated observation, that mathematics is a human\ninvention. Furthermore, the mathematics in which the physicist is\ninterested was developed for the explicit purpose of describing the\nbehavior of the external world, so that it is certainly no accident that\nthere is a correspondence between mathematics and nature. The\ncorrespondence is not by any means perfect, however, but there is always\nin mathematics a precise quality to which none of our information about\nnature ever attains. The theorems of Euclid's geometry illustrate this\nin a preeminent degree. The statement that there is just one straight\nline between two points and that this is the shortest possible path\nbetween the points is entirely different in character from any\ninformation ever given by physical measurement, for all our measurements\nare subject to error. It is possible, nevertheless, to give a certain\nreal physical meaning to the ideally precise statements of geometry,\nbecause it is a result of everyday experience that as we refine the\naccuracy of our physical measurements the quantitative statements of\ngeometry are verified within an ever decreasing margin of error. From\nthis arises that view of the nature of mathematics which apparently is\nmost commonly held; namely that if we could eliminate the imperfections\nof our measurements, the relations of mathematics would be exactly\nverified. Abstract mathematical principles are supposed to be active in\nnature, controlling natural phenomena, as Pythagoras long ago tried to\nexpress with his harmony of the spheres and the mystic relations of\nnumbers.\n\nThis idealized view of the connection of mathematics with nature could\nbe maintained only during that historical period when the accuracy of\nphysical measurement was low, and must now be abandoned. For it is no\nlonger true that the precise relations of Euclid's geometry may be\nindefinitely approximated to by increasing the refinements of the\nmeasuring process, but there are essential physical limitations to the\nvery concepts of length, etc., which enter the geometrical formulations,\nset by the discrete structure of matter and of radiation. This is no\nacademic matter, but touches the essence of the situation. There is no\nlonger any basis for the idealization of mathematics, and for the view\nthat our imperfect knowledge of nature is responsible for failure to\nfind in nature the precise relations of mathematics. It is the\nmathematics made by us which is imperfect and not our knowledge of\nnature. [From the operational point of view it is meaningless to attempt\nto separate \"nature\" from \"knowledge of nature\".] The concepts of\nmathematics are inventions made by us in the attempt to describe nature.\nNow we shall repeatedly see that it is the most difficult thing in the\nworld to invent concepts which exactly correspond to what we know about\nnature, and we apparently never achieve success. Mathematics is no\nexception; we doubtless come closer to the ideal here than anywhere\nelse, but we have seen that even arithmetic does not completely\nreproduce the physical situation.\n\nMathematics appears to fail to correspond exactly to the physical\nsituation in at least two respects. In the first place, there is the\nmatter of errors of measurement in the range of ordinary experience. Now\nmathematics can deal with this situation, although somewhat clumsily,\nand only approximately, by specifically supplementing its equations by\nstatements about the limit of error, or replacing equations by\ninequalities--in short, the sort of thing done in every discussion of\nthe propagation of error of measurement. In the second place, and much\nmore important, mathematics does not recognize that as the physical\nrange increases, the fundamental concepts become hazy, and eventually\ncease entirely to have physical meaning, and therefore must be replaced\nby other concepts which are operationally quite different. For instance,\nthe equations of motion make no distinction between the motion of a star\ninto our galaxy from external space, and the motion of an electron about\nthe nucleus, although physically the meaning in terms of operations of\nthe quantities in the equations is entirely different in the two cases.\nThe structure of our mathematics is such that we are almost forced,\nwhether we want to or not, to talk about the inside of an electron,\nalthough physically we cannot assign any meaning to such statements. As\nat present constructed, mathematics reminds one of the loquacious and\nnot always coherent orator, who was said to be able to set his mouth\ngoing and go off and leave it. What we would like is some development of\nmathematics by which the equations could be made to cease to have\nmeaning outside the range of numerical magnitude in which the physical\nconcepts themselves have meaning. In other words, the problem is to make\nour equations correspond more closely to the physical experience back of\nthem; it evidently needs some sort of new invention to accomplish this.\n\nWe return later, in discussing Lorentz's equations of electrodynamics,\nto the disadvantages arising from the present undiscriminating character\nof mathematics. In the meantime, we must recognize that there are very\nimportant advantages here, as well as disadvantages. All experience\njustifies the expectation that the laws of nature with which we are\nalready familiar hold at least approximately and without violent change\nin the unexplored regions immediately beyond our present reach. By\nassuming an unlimited validity for the laws as we now know them,\nmathematics enables us to penetrate the twilight zone, and make\npredictions which may be later verified. It is only when we are carried\ntoo far afield that we must deprecate this characteristic of our\nmathematics.\n\nThere is another aspect of the use of mathematics in describing nature\nthat is often lost sight of; namely, that any system of equations can\ncontain only a very small part of the actual physical situation; there\nis behind the equations an enormous descriptive background through which\nthe equations make connection with nature. This background includes a\ndescription of all the physical operations by which the data are\nobtained which enter the equations. For instance, when Einstein\nformulates the behavior of the universe in terms of the world lines of\nevents, the events as they enter the equations are entirely colorless\nthings, merely 3 space and 1 time co\u00f6rdinate. To make connection with\nexperience there must be a descriptive background giving the physical\ncontents of the events; for example, there may be the statement that\nsome of the events are light signals. This descriptive background is\nsupposed to remain fixed, unaffected by any operations to which the\nequations themselves are subject. If, for example, the frame of\nreference of the equations is altered by changing its velocity, the\nphysical significance of the descriptive background is supposed to\nremain unaltered, or rather no mention is usually made of this question\nat all. It would seem, however, that this matter needs some discussion.\nThe descriptive background gets its meaning only in terms of certain\nphysical operations. If the descriptive background remains unaltered\nwhen the uniform velocity of the frame of reference is changed, for\ninstance, this means that the motion of the frame of reference does not\nat all affect the possibility of carrying out certain operations. This\nis pretty close to the restricted principle of relativity itself, which\nstates that the form of natural laws is not affected by uniform\nvelocity. Until a more careful analysis of the situation is made it\nwould seem therefore that there is some ground for the suspicion that\nthe principle of relativity is involved in the possibility of giving to\nphysical phenomena a _complete_ mathematical formulation, understanding\n\"complete\" to mean \"including the descriptive background.\"\n\n\n\n\nCHAPTER III\n\nDETAILED CONSIDERATION OF VARIOUS\nCONCEPTS OF PHYSICS\n\n\nWE now begin our detailed consideration of the most important concepts\nof physics. It is entirely beyond the scope of this essay to attempt\nmore than an indication of some of the most important matters. Neither\nis it to be expected that the parts of this discussion will always have\na very close connection with each other; the purpose of the discussion\nis to aid in acquiring the greatest possible self-consciousness of the\nwhole structure of physics.\n\n\n\n\nTHE CONCEPT OF SPACE\n\n\nA logically satisfying definition of what we understand by the concept\nof space is doubtless difficult to give, but we shall not be far from\nthe mark if we think of it as the aggregate of all those concepts which\nhave to do with position. Position means position of something. The\nposition of things is determined by some system of measurement; perhaps\nthe simplest is that implied in a Cartesian co\u00f6rdinate system with its\nthree measurements of length. Hence much of the essential discussion of\nspace has already been given in connection with the concept of length.\nWe have seen that measurements of length are made with physical\nmeasuring rods applied to some physical object. We cannot measure the\ndistance between two points in empty space, because if space were empty\nthere would be nothing to identify the position of the ends of the\nmeasuring rod when we move it from one position to the next. We see,\nthen, from the point of view of operations that the framework of\nCartesian geometry, often imagined in an ideal mathematical sense, is\nreally a physical framework, and that what we mean by spatial properties\nis nothing but the properties of this framework. When we say that space\nis Euclidean, we mean that the physical space of meter sticks is\nEuclidean: it is meaningless to ask whether empty space is Euclidean.\nGeometry, therefore, in so far as its results are expected to apply to\nthe external physical world, and in as far as it is not a logical system\nbuilt up from postulates, is an experimental science. This view is now\nwell understood and accepted, but there was a time when it was not\naccepted, but vigorously attacked; the change of attitude toward this\nquestion is symptomatic of a change of attitude toward many other\nsimilar questions.\n\nWe have already emphasized that the space of astronomy is not a physical\nspace of meter sticks, but is a space of light waves. We may, therefore,\nhave different kinds of space, depending on the fundamental operations.\nThe space of meter sticks we have called \"tactual space\", and the space\nof light beams \"optical space\". If we ask whether astronomical space is\nEuclidean, we mean merely to ask whether those features of optical space\nwhich are within the reach of astronomical measurement are Euclidean.\nThe only possible attitude with respect to this question, or such\nrelated questions as whether the total volume of space is finite, or\nwhether space has curvature, is that it is entirely for experiment to\ndecide, and that we have no right to form any preconceived notion\nwhatever. It is therefore beyond the scope of this discussion.\n\nIt is interesting to notice that the restricted theory of relativity\nvirtually assumes, although often without making the explicit statement,\nthat tactual and optical space are the same. This equivalence results\nfrom the properties assumed for light beams. The distance of a mirror\nmay be found equally well by measuring it with meter sticks, or by\ndetermining the time required by a light signal to travel there and\nback. This situation is, however, logically unsatisfying, because it\nmust be assumed that the operations for measuring time are independently\ndefined, and we shall see that they are not. It is a consequence of the\nassumed equivalence of tactual and optical space that the path of a beam\nof light is a straight line, as a straight line is determined by\noperations with meter sticks. When we come to astronomical phenomena,\nthe physical operations with meter sticks can no longer be carried out,\nand it is meaningless to ascribe to beams of light on an astronomical\nscale the same geometrical properties that we do on a small scale.\n\n\n\n\nTHE CONCEPT OF TIME\n\n\nAccording to our viewpoint, the concept of time is determined by the\noperations by which it is measured. We have to distinguish two sorts of\ntime; the time of events taking place near each other in space, or local\ntime, and the time of events taking place at considerably separated\npoints in space, or extended time. As we now know, the concept of\nextended time is inextricably mixed up with that of space. This is not\nprimarily a statement about nature at all, and might have been made\nsimply by the observation that the operations by which extended time is\nmeasured involve those for measuring space. Of course historically the\ndoctrine of relativity was responsible for the critical attitude which\nled to an examination of the operations of measuring time, but\nrelativity was not necessary for a realization of the spatial\nimplications of time, any more than the discovery of Planck's quantum\nunit _h_ was necessary for the invention by Planck of his absolute units\nof measurement, although historically he was inspired to make this\ninvention by discovering _h_, and in his own mind seems to have thought\nof the connection as a necessary one.[7]\n\n[Footnote 7: Max Planck, The Theory of Heat Radiation, translated by\nMasius, P. Blakiston's Son & Co., 1914 edition, p. 174.]\n\nThe physical operations at the basis of the measurement of time have\nnever been subjected to the critical examination which seems to be\nrequired. One method of measurement, for instance, involves the\nproperties of light.\n\nA meter stick is set up with mirrors at the two ends, and a light beam\ntravels back and forth between the two mirrors without absorption. The\ntime required for a single passage back and forth is defined as the unit\nof time, and time is measured simply by counting these intervals. But\nsuch a procedure is unsatisfactory if we are to permit ourselves all\nthose operations which are demanded by even the simplest postulate of\nrelativity, for we must be able to move our clock from place to place,\ntransfer it from one system to another in relative motion, and with it\ndetermine the properties of light beams in the stationary or moving\nsystem. We recognize in principle that the length of the meter stick may\nbe different when it is in motion, that it may change also during the\nacceleration incident to moving it from one place to another, and that\nuntil proved to the contrary the velocity of light may be a function of\nvelocity or acceleration. The complicated interplay of all these\npossibilities leaves us in much doubt as to the physical significance of\nsuch postulates as, for example, that the velocity of light is the same\nin the moving system and the stationary system. In order to ascribe any\nsimple significance to postulates about the velocity of light, it would\nseem that we must have an instrument for measuring this velocity, and\ntherefore for measuring time, which does not itself involve the\nproperties of light. To do this we might seek to specify the measurement\nof time in purely mechanical terms, as for instance in terms of the\nvibration of a tuning fork, or the rotation of a flywheel. But here\nagain we encounter great difficulties, because we recognize that the\ndimensions of our mechanical clock may change when it is set in motion,\nand that the mass of its parts may also change. We want to use the clock\nas a physical instrument in determining the laws of mechanics, which of\ncourse are not determined until we can measure time, and we find that\nthe laws of mechanics enter into the operation of the clock.\n\nThe dilemma which confronts us here is not an impossible one, and is in\nfact of the same nature as that which confronted the first physicist who\nhad to discover simultaneously the approximate laws of mechanics and\ngeometry with a string which stretched when he pulled it. We must first\nguess at what the laws are approximately, then design an experiment so\nthat, in accordance with this guess, the effect of motion on some\nphenomenon is much greater than the expected effect on the clock, then\nfrom measurements with uncorrected clock time find an approximate\nexpression for the effect of motion on mass or length, with which we\ncorrect the clock, and so on ad infinitum. However, so far as I know,\nthe possibility of such a procedure has not been analyzed, and until the\nanalysis is given, our complacency is troubled by a real disquietude,\nthe intensity of which depends on the natural skepticism of our\ntemperament.\n\nIn practice, the difficulties of such a logical treatment are so great\nthat the matter has been entirely glossed over. It is convenient to\npostulate a clock, of unknown construction, but such that the velocity\nof light, when measured in terms of it, has certain properties. Such,\nfor example, is the point of view in Birkhoff's recent book.[8]\n\n[Footnote 8: G. D. Birkhoff. Relativity and Modern Physics, Harvard\nUniversity Press, 1923.]\n\nThe difficulty with this method is that the resulting edifice is as\ndivorced from physical reality as is the logical geometry of postulates.\nWe cannot be at all sure that the properties of light as measured with\nour physical clocks are the same as the theoretical properties. The\ndifficulty is particularly important and fundamental in the general\ntheory of relativity; the basis of the whole theory is the infinitesimal\ninterval _ds_, which is supposed to be given. Once given, the\nmathematics follows. But in a physical world, _ds_ is _not_ given, but\nmust be found by physical operations, and these operations involve\nmeasurements of length and of time with clocks whose construction is not\nspecified. In any actual physical application the question must be\nanswered whether the physical instrument used in measuring the temporal\npart of _ds_ is really a clock or not. There is at present no criterion\nby which this question can be answered. If the vibrating atom is a\nclock, then the light of the sun is shifted toward the infra-red, but\nhow do we know that the atom is a clock (some say yes, others no)? If we\nfind the displacement physically have we thereby proved that general\nrelativity is physically true, or have we proved that the atom is a\nclock, or have we merely proved that there is a particular kind of\nconnection between the atom and the rest of nature, leaving the\npossibility open that neither is the atom a clock nor general relativity\ntrue? In practice, of course, we shall adopt the solution which is\nsimplest and most satisfying aesthetically, and doubtless shall say that\nthe atom is a clock and relativity true. But if we adopt this simple\nview, we must also cultivate the abiding consciousness that at some time\nin the future troubles may have their origin here.\n\nIt seems to me that the logical position of general relativity theory is\nmerely this: Given any physical system, then it is possible to assign\nvalues to _ds_ such that relations mathematically deduced by the\nprinciple of relativity correspond to relations between measurable\nquantities in the physical system; but that the things that we\nphysically call _ds_ are anything more than approximately connected with\nthe _ds's_ required to give the mathematical relations, is at present no\nmore than a pious faith.\n\nTo return to the concept of time, we have already stated that there are\ntwo main problems, that of measuring time at a single point of space,\nand that of spreading a time system over all space. The second aspect of\nthe problem is that to which attention has been directed by relativity\ntheory; the following detailed examination shows how the operations of\nrelativity for setting and synchronizing clocks at distant places\ninvolve the measurement of space. It is a fundamental postulate that the\nadjustment of the clocks is to be accomplished by light signals. The\nsynchronization of the clocks is now simple enough. We merely demand\nthat light signals sent from the master clock at intervals of one second\narrive at any distant clock at intervals of one second as measured by\nit, and we change the rate of the distant clock until it measures these\nintervals as one second. After its rate has been adjusted, the distant\nclock is to be so _set_ that when a light signal is despatched from the\nmaster clock at its indicated zero of time the time of arrival recorded\nat the distant clock shall be such that the distance of the clock from\nthe master clock divided by the time of arrival shall give the velocity\nof light, assumed already known. This operation involves a measurement\nof the distance of the distant clock, so that in spreading the time\ncoordinates over space the measurement of space is involved by\ndefinition, and the measurement of time is, therefore, not a\nself-contained thing. This is the physical basis for the treatment of\nspace and time as a four-dimensional manifold. Although mathematically\nthe numbers measuring space and time enter the formulas symmetrically,\nnevertheless the physical operations by which these numbers are obtained\nare entirely distinct and never fuse, and I believe it can lead only to\nconfusion to see in the possibility of a four dimensional treatment\nanything more than a purely formal matter.\n\nThe notion of extended time, therefore, involves the measurement of\nspace. It is an interesting question whether the notion of local time\nalso involves the measurement of space. A rigorous answer to this\nquestion involves giving the specifications for the construction of a\nclock, which we have seen has not yet been done. It seems to me\nprobable, however, that the construction of even a single local clock\ninvolves in some way the _measurement_ of space. If, for example, we use\na vibrating tuning fork, we must find how the time of vibration depends\non the amplitude of vibration, and this involves space measurement, or\nif we use a rotating flywheel, we have to correct for the change of\nmoment of inertia due to the change of dimensions when it is set into\nmotion or brought into a gravitational field, and all this involves\nspace measurement. However, these considerations are not certain, and\nperhaps the question is not important.\n\nThere is now the further consideration that actually in practice the\nconcept of local time is not entirely divorced from that of extended\ntime, for two bodies cannot occupy the same space at the same time, and\nthe time of any event is actually measured on an instrument at some\ndistance, communication being maintained by light or elastic signals.\nBut experience convinces us that in the limit, as the phenomenon to be\nmeasured gets closer to the clock, there is no measurable difference,\nwhether communication with the clock is maintained by light, or\nacoustical or tactual signals, so that we have come in physical practice\nto accept measurement of the time of events in the immediate\nneighborhood of the clock (local time) as one of the ultimately simple\nthings behind which we do not attempt to go.\n\nLocal time is, therefore, a concept treated by the physicist even now as\nsimple and unanalyzable. This concept is what most people have in mind\nwhen they think of time. Time, according to this concept, is something\nwith the properties of local time; it was something of this kind that\nNewton must have meant by his absolute time, and it is the tacit\nretention of this sort of concept that is responsible for the difficulty\nso often found in grasping the idea of the relativity of simultaneity,\nwhich is of course entirely foreign to our experience of simultaneity in\nlocal time. An examination of the operations involved in extending time\nhas shown how the concept of extended time is different from that of\nsimple local time; this difference leads to appreciably different\nnumerical relations when we are dealing with high velocities or great\ndistances. Local time is proved by experience not to be a satisfactory\nconcept for dealing with events separated by great distances in space or\nwith phenomena involving high velocities. For instance, we must not talk\nabout the age of a beam of light, although the concept of age is one of\nthe simplest derivatives of the concept of local time. Neither must we\nallow ourselves to think of events taking place in Arcturus _now_ with\nall the connotations attached to events taking place _here_ now. It is\ndifficult to inhibit this habit of thought, but we must learn to do it.\nThe na\u00efve feeling is very strong that it does _mean_ something to talk\nabout the entire present state of the universe independent of the\nprocess by which news of the condition of distant parts is determined by\nus. I believe that an examination of this feeling will show that it is\npsychological in character; what we mean by the totality of the present\nis merely the entire present content of our consciousness. This is\napparently a simple direct thing; we do not appreciate until we make\nfurther analysis that our present consciousness of the existence of the\nmoon or a star is due to light signals, and that therefore the\napparently simple immediate consciousness of events distant in space\ninvolves complicated physical operations.\n\nSimilarly, if we continue to use local time, we get into trouble, when\nwe go to high velocities, with our simple concept of velocity, which may\nbe defined in terms of a combination of space and time concepts. The\nconcept of local time thus loses its value and becomes merely a blunted\ntool when we try to carry it out of its original range. But the concept\nof extended time, with which we have to replace local time, is a\ncomplicated thing, to which we have not yet got ourselves accustomed; it\nmay perhaps prove to be so complicated as never to be a very useful\nintuitive tool of thought.\n\nAll these considerations about time have been concerned only with\nintervals of such an order of magnitude that they are readily\nexperienced by any individual. If we have to deal with intervals either\nvery long or very short, it is obvious that our entire procedure\nchanges, and consequently the concept changes. In extending the time\nconcept to eras remote in the past, for example, we try as always, to\nchoose the new operations so as to piece on continuously with those of\nordinary experience. A precise analysis of the change in the concept of\ntime when applied to the remote past does not seem to be of great\nsignificance for our present physical purpose, and will not be attempted\nhere. It is perhaps worth while to point out, however, that all our\nother concepts, as well as that of time, must be modified when applied\nto the remote past; an example is the concept of truth. It is amusing to\ntry to discover what is the precise meaning in terms of operations of a\nstatement like this: \"It is true that Darius the Mede arose at 6:30 on\nthe morning of his thirtieth birthday.\"\n\nOf more concern for our physical purposes is the modification which the\ntime concept undergoes when applied to very short intervals. What is the\nmeaning, for example, in saying that an electron when colliding with a\ncertain atom is brought to rest in 10^-18 seconds? Here I believe the\nsituation is very similar to that with regard to short lengths. The\nnature of the physical operations changes entirely, and as before, comes\nto contain operations of an electrical and optical character. The\nimmediate significance of 10^-18 is that of a number, which when\nsubstituted into the equations of optics, produces agreement with\nobserved facts. Thus short intervals of time acquire meaning only in\nconnection with the equations of electrodynamics, whose validity is\ndoubtful and which can be tested only in terms of the space and time\ncoordinates which enter them. Here is the same vicious circle that we\nfound before. Once again we find that concepts fuse together on the\nlimit of the experimentally attainable.\n\nThis discussion of the concept of time will doubtless be felt by some to\nbe superficial in that it makes no mention of the _properties_ of the\nphysical time to which the concept is designed to apply. For instance,\nwe do not discuss the one dimensional flow of time, or the\nirrevocability of the past. Such a discussion, however, is beyond our\npresent purpose, and would take us deeper than I feel competent to go,\nand perhaps beyond the verge of meaning itself. Our discussion here is\nfrom the point of view of operations: we assume the operations to be\ngiven, and do not attempt to ask why precisely these operations were\nchosen, or whether others might not be more suitable. Such properties of\ntime as its irrevocability are implicitly contained in the operations\nthemselves, and the physical essence of time is buried in that long\nphysical experience that taught us what operations are adapted to\ndescribing and correlating nature. We may digress, however, to consider\none question. It is quite common to talk about a reversal of the\ndirection of flow of time. Particularly, for example, in discussing the\nequations of mechanics, it is shown that if the direction of flow of\ntime is reversed, the whole history of the system is retraced. The\nstatement is sometimes added that such a reversal is actually\nimpossible, because it is one of the properties of physical time to flow\nalways forward. If this last statement is subjected to an operational\nanalysis, I believe that it will be found not to be a statement about\nnature at all, but merely a statement about operations. It is\n_meaningless_ to talk about time moving backward: by definition,\n_forward_ is the direction in which time flows.\n\n\n\n\nTHE CAUSALITY CONCEPT\n\n\nThe causality concept is unquestionably one of the most fundamental,\nperhaps as fundamental as that of space and time, and therefore at least\nequally entitled to a first place in the discussion. But as ordinarily\nunderstood, there are certain spatial and temporal implications in the\ncausality concept, so that it can best be discussed in this order after\nour examination of space and time.\n\nThere is an aspect of the causality concept that in many respects is\nclosely related to the question of \"explanation\", for to find the causes\nof an event usually involves at the same time finding its explanation.\nBut there are nevertheless sufficient differences to warrant a separate\ndiscussion.\n\nIt seems fairly evident that there was originally in the causality\nconcept an animistic element much like that in the concept of force to\nbe discussed later. The physical essence of the concept as we now have\nit, freed as much as possible from the animistic element, seems to be\nsomewhat as follows. We assume in the first place an isolated system on\nwhich we can perform unlimited identical experiments, that is, the\nsystem may be started over again from a definite initial condition as\noften as desired.[9]\n\n[Footnote 9: We must include in general in the concept of \"initial\ncondition\" the past history of the system. In order not to make this\ncondition so broad as to defeat itself, we have to add the observation\nthat actually identity of past history is necessary over only a\ncomparatively short interval of time. Logical precision seems\nunattainable here--the physical concepts themselves have not the\nnecessary precision.]\n\nWe assume further that when so started, the system always runs through\nexactly the same sequence of events in all its parts. This contains the\nassumption that the course of events runs independent of the absolute\ntime at which they occur--there is no change with time of the properties\nof the universe.[10] It is a result of experience that systems with\nthese properties actually exist. An alternative way of stating our\nfundamental hypothesis is that two or more isolated similar systems\nstarted from the same initial condition run through the same future\ncourse of events. Upon the system given in this way, which by itself\nruns a definite course of events, we assume that we can superpose from\nthe outside certain changes, which have no connection with the previous\nhistory of the system, and are completely arbitrary. Now of course in\nnature, as we observe it, there is no such thing as an arbitrary change,\nwithout connection with past history, so that strictly our assumption is\na pure fiction. It is here that the animistic element still seems to\npersist, although perhaps not necessarily.\n\n[Footnote 10: As so often in physics, we appear to be doing two things\nat once here. It is doubtful whether we can give a meaning to \"definite\ninitial condition\" apart from the future behavior of the system, so that\nwe have no real right to infer from uniform future behavior both a\nconstancy of the laws of nature, independent of time, and a constancy of\ninitial condition. I very much question whether a thoroughgoing\noperational analysis would show that there are really two independent\nconcepts here, and whether the use of two formally quite different\nconcepts is anything more than a convenience in expression. It seems to\nme that it may be just as meaningless to ask whether the laws of nature\nare independent of time as it was to ask with Clifford whether the\nabsolute scale of magnitude may not be changing as the solar system\ntravels through space.]\n\nWe regard our acts as not determined by the external world, so that\nchanges produced in the external world by acts of our wills are, to a\ncertain degree of approximation, arbitrary. The system, then, on which\nwe are experimenting, is one capable of isolation from us in that we may\nregard ourselves as outside the system, and having no connection with\nit. The system, furthermore, is capable of isolation from the rest of\nthe physical universe, in that events taking place outside the system\nhave no connection with those taking place inside.[11] Experience gives\nthe justification for assuming that physical isolation of this sort is\npossible. Actually, of course, isolation is never complete, but only\npartial, up to presumably any desired degree of approximation.\n\n[Footnote 11: Here again, the concept of \"isolation\" or \"connection\" is\ndefined only in terms of the behavior of the system, and it is not clear\nwhether this is really an operationally independent concept or not.]\n\nThe statement that two exactly similar isolated systems, starting from\nthe same initial conditions (including past history in the general idea\nof initial condition) will run through the same future course of events\ninvolves as a corollary that if differences develop in the behavior of\ntwo such apparently similar systems these differences are evidence of\nother previous differences. The thesis that this corresponds to\nexperience may be called the thesis of essential connectivity and is\nperhaps the broadest we have: it is the thesis that differences between\nthe behavior of systems do not occur isolated but are associated with\nother differences. It is essentially the same thesis as that already\nmentioned in connection with \"explanation\", namely that it is possible\nto correlate any of the phenomena of nature with other phenomena.\n\nIf now the connectivity or correlation between phenomena is of a special\nkind, we have a causal connection; namely, if whenever we arbitrarily\nimpress event A on a system we find that event B, always occurs, whereas\nif we had not impressed A, B would not have occurred, then we say that A\nis the cause of B, and B the effect of A. By suitably choosing the event\nA, we may find the effect of any event of which the system is\nsusceptible.\n\nThe relation between A and B is an unsymmetrical one, by the very nature\nof the definition, the cause being the arbitrary variable element, and\nthe effect that which accompanies it. Furthermore, A may obviously be\nthe cause of more than one event B, and may cause a whole train of\nevents.\n\nThe causal concept analyzed in this way is not simple by any means. We\ndo not have a simple event A causally connected with a simple event B,\nbut the whole background of the system in which the events occur is\nincluded in the concept, and is a vital part of it. If the system,\nincluding its past history, were different, the nature of the relation\nbetween A and B might change entirely. The causality concept is\ntherefore a relative one, in that it involves the whole system in which\nthe events take place.\n\nIn practice we now take an exceedingly pregnant step and seek to extend\nthe concept, and rid ourselves as much as possible of its relativity. It\nis a matter of experience that there are often a great number of systems\nin which A is the cause of B. In many cases the causal relation persists\nthrough such a very wide range of systems that we lose sight entirely of\nthe system, and come to assume that we have an _absolute_ causal\nconnection between A and B. For instance, when I strike a bell, and hear\nthe sound, the causal connection persists through such a great number of\ndifferent kinds of system that I might think that here is an absolute\ncausal connection. Such an absolute causal connection would mean that\nalways under all circumstances, the striking of the bell is accompanied\nby a sound. But _all_ conditions means only _all_ those conditions\ncovered by experiment. Thus in the case of the bell, all our experiments\nwere made in the presence of the atmosphere. The causal connection\nbetween the striking of the bell and the sound should have been always\nrecognized in principle as relative to the presence of the atmosphere.\nIndeed, later experiments in the absence of the atmosphere show that the\natmosphere does play an essential part. Now as a matter of fact, the\natmosphere is so comparatively easy to remove that we very readily\ninclude the atmosphere in the chain of causal connection. But if the\natmosphere had been impossible to remove, like the old ether of space,\nour idea of the causal connections between the striking of the bell and\nits sound might have been quite different. In actual physical\napplications of the causality concept, the constant background which is\nmaintained during all the variations by which the causal connection is\nestablished usually has to be inferred from the context.\n\nIt is a matter of perhaps universal experience that the event A is\naccompanied by not only one event, which is the effect of A by\ndefinition, but A entails a whole causal train of events. It seems to be\na generalization from experience that the causally connected train of\nevents started by A is a never ending train, provided the system is\nlarge enough. This is perhaps not necessary in the general case, but if\nthe event A involves imparting external energy to the system, or the\naction of external force (momentum change), there can be no question.\n\nThat there is a causal train started by A is particularly evident if A\nand B are separated in space. Thus in the case of the bell, the impulse\ngiven to the air by the vibration of the bell is propagated through the\nair as an elastic wave, which thus constitutes the causal train of\nevents. The phenomenon of propagation is characteristic of causal\nconnections of a mechanical character, and is the justification for the\nintroduction of the time concept in connection with the causality\nconcept, where it now appears for the first time. It is evident that\nwhen a disturbance is propagated to a distant point, the effect\n_follows_ the cause in time, as time is usually measured.\n\nWe extend this result, and usually think that the effect _necessarily_\nfollows the cause. We now examine whether this is a necessary result of\nthe causality concept. If we are to talk about the time of events at\ndifferent places, we must have some way of setting clocks all over\nspace. If this is done arbitrarily, there is no necessary connection\nbetween the local clock times of a cause and its effect, but\nnevertheless the causality concept involves a certain temporal relation\neven in this most general case. Suppose that event A takes place at\npoint 1 and its effect, event B, at point 2. We station a confederate at\n2 who sends a light signal (or any other sort of signal) to 1, as soon\nas the event B occurs at 2. Then it is a consequence of the nature of\nthe causality concept that the signal cannot arrive at 1 before event A\noccurs. For if it did arrive before A, we should merely omit to perform\nA, which by hypothesis is arbitrary, and entirely in our control, and\nthen our assumption would be violated that the system is such that the\nevent B occurs only when A also occurs. The same argument shows _a\nfortiori_ that if the effect B occurs at the same place as its cause A,\nit cannot precede it in time. I cannot see that the nature of the\ncausality concept imposes any further restriction on the time of B. The\nrestricted principle of relativity, however, in postulating that no\nsignal can be propagated faster than a light signal, virtually makes a\nfurther assumption about the temporal connection of causally connected\nevents, namely, that the event B at 2 cannot occur before the arrival at\n2 of a light signal which started from 1 at the instant that A occurred\nat 1. For if B did occur earlier, we could use events A and B as a\nsignaling code, thus violating our hypothesis.\n\nThere is thus a closest connection in time, when time is extended over\nspace as the theory of relativity directs, between cause and effect,\ndepending on their separation in space; from this arises the relativity\nconcept of the causal cone, which in the four dimensional manifold of\nspace-time divides the aggregate of all those events which may be\ncausally related from the aggregate of those which are separated by such\na small interval in time and such a large interval in space that\ncommunication by light signals and therefore a causal connection is not\npossible. Given now two events A and B which are related as cause to\neffect in one system of reference, then they must be causally related\nalso in every other system of reference. For if they were not, we could\nby definition of causality suppress the event A in one of the systems in\nwhich the causal relation does not hold, and this, because of the nature\nof the concept of event, involves suppressing A in all the systems, thus\nviolating our hypothesis of a causal connection in the original system.\nThe concept of event involved in this argument will be examined later. It\nappears then, that the fundamental postulate of relativity (that the\nform of natural laws is the same in all reference systems) demands that\nthe temporal order of events causally connected be the same in all\nreference systems.\n\nThe whole universe at this present moment is often supposed to be\ncausally connected with all succeeding states. This means that if we\ncould repeat experience, starting from the same initial conditions, the\nfuture course of events would always be found to be the same. The truth\nof this conviction can never be tested by direct experiment, but it is\nsomething at which we arrive by the usual physical process of successive\napproximation. It is difficult to formulate precisely what we mean by\n\"present\" state of the universe, and there is every reason to think that\nsuch a formulation is not unique, but the concept contains the necessary\nimplication that none of the events constituting the \"present\" can be\ncausally connected. The events in distant places which constitute the\npresent must be separated by an interval of time less than time required\nby light to travel between the two places.\n\nThe conviction, arising from experience, that the future is determined\nby the present and correspondingly the present by the past, is often\nphrased differently by saying that the present causally determines the\nfuture. This is in a certain sense a generalization of the causality\nconcept. It is one of the principal jobs of physics to analyze this\ncomplex causal connection into components, representing as far as\npossible the future state of the system as the sum of independent trains\nof events started by each individual event of the present. How far such\nan analysis is possible must be decided by experiment. It is certainly\npossible to a very large extent in most cases, but there seems to be no\nreason to expect that a complete analysis is possible. So far as the\nsystem is describable in terms of linear differential equations, the\ncausal trains started by different events propagate themselves in space\nand time without interference and with simple addition of effects, and\nconversely the present may be analyzed back into the simple sum of\nelementary events in the past, but if the equations governing the motion\nof the system are not linear, effects are not additive, and such a\ncausal analysis into elements is not possible. No emphasis is to be laid\nhere on the _differential_ aspect of the equations: it is quite possible\nthat finite difference equations may have the same property of\nadditivity. Although there can be no question that linear equations\nenormously preponderate, neither can there be any doubt that some\nphenomena cannot be described in terms of linear equations (_e.g._,\nferro-magnetism), so that there seems no reason to think that a causal\nanalysis is always possible. I believe, however, that the assumption\nthat such an analysis into small scale elements is possible is tacitly\nmade in the thought of many physicists. If the analysis is not possible,\nwe may expect to find results following the cooperation of several\nevents which cannot be built up from the results of the events occurring\nindividually.\n\nWhen a causal analysis is possible, finding the simplest events which\nact as the origin of independent causal trains is equivalent to finding\nthe ultimate elements in a scheme of explanation, so that here we merge\nwith the concept of explanation, as already mentioned. As was true of\nthe explanatory sequence, so here there can be no _formal_ end of the\ncausal sequence, because we can always ask for the cause of the last\nmember. But it may be physically meaningless to extend the causal\nsequence beyond a certain point. We have seen from the point of view of\noperations that the causal concept demands the possibility of variation\nin the system. It is therefore meaningless to say that A is the cause of\nB unless we can experience systems in which A does not occur. Now if in\nextending the causal sequence, we eventually arrive at a condition so\nbroad that physically no further variation can be made, our causal\nsequence has to stop.\n\nCorresponding to this property of the causality concept, the causal\nsequence may be terminated either formally, by postulate, or naturally,\nby the intrinsic physical nature of the elements of the sequence. Thus\nif we say that light gets from point to point because it is propagated\nby a medium of unalterable properties, which fills all space, which is\nalways present and can never be eliminated physically, we have by the\npostulated properties of the medium brought the possibility of further\ninquiry to a close, because to take the next step and ask the cause of\nthe properties of the ether, demands that we be able to perform\nexperiments with the ether altered or absent. Such an ending of the\nsequence is evidently pure formalism, without physical significance. But\nother considerations may give physical significance. Thus if there are\nother sorts of experiment that can be explained by assuming a universal\nmedium of the same properties, the concept proves not only to be useful,\nbut to have a certain degree of physical significance. An example of an\ninevitable termination of the causal sequence is afforded by the\npossibility, already mentioned, that the value of the gravitational\nconstant may be determined by the total quantity of matter in the\nuniverse. Without further qualification, this is an entirely sterile\nstatement, but if it can be shown that there is a simple numerical\nconnection, the matter takes on interest, and we may seek further for a\ncorrelation between the numerical relation and other things.\n\nThis analysis of the causality concept does not pretend to be complete\nand leaves many interesting questions untouched. Perhaps one of the most\ninteresting of these questions is whether we can separate into cause and\neffect two phenomena which _always_ accompany each other, and whether\ntherefore the classification of phenomena into causally connected groups\nis an exhaustive classification. But the discussion is broad enough for\nour purpose here; the most important points of view to acquire are that\nthe causality concept is relative to the whole background of the system\nwhich contains the causally connected events, and that we must assume\nthe possibility of an unlimited number of identical experiments, so that\nthe causality concept applies only to sub-groups of events separated out\nfrom the aggregate of all events.\n\n\n\n\nTHE CONCEPT OF IDENTITY\n\n\nOne of the most fundamental of all the concepts with which we describe\nthe external world is that of identity; in fact, thinking would be\nalmost inconceivable without such a concept. By this concept we bridge\nthe passage of time; it enables us to say that a particular object in\nour present experience is the same as an object of our past experience.\nFrom the point of view of operations, the meaning of identity is\ndetermined by the operations by which we make the judgment that this\nobject is the _same_ as that one of my past experience. In practice\nthere are many indirect ways of making this judgment, but I believe the\nessence of the situation lies in the possibility of continuous\nconnection between the object of the present and the past by continuous\nobservation (either direct or indirect) through all intermediate time.\nWe must, for example, be able to look continuously at the object, and\nstate that while we look at it, it remains the same. This involves the\npossession by the object of certain characteristics--it must be a\ndiscrete thing, separated from its surroundings by physical\ndiscontinuities which persist. The concept of identifiability applies,\ntherefore, only to certain classes of physical objects; no one thinks of\ntrying to identify the wind of to-day with the wind of yesterday. It is\nsomewhat easier to identify a liquid such as water in its flow in a\nstream, because we can make the motion of the water visible by solid\nparticles suspended in it, but even here it is not easy to prove to a\ncaptious critic that it is really the water and not the suspended\nparticles of solid that we are identifying. Even solids, when our\nmeasurements are sufficiently refined, seem to lose their discontinuous\nedges, as has been suggested in the discussion of the approximate\ncharacter of experimental arithmetic, and the identity concept becomes\nhazy.\n\nThere can be no question that the concept of identity is a tool\nperfectly well adapted to deal approximately with nature in the region\nof our ordinary experience, but we have to ask a more serious question.\nDoes not the apparent demand of our thinking apparatus to be furnished\nwith discrete and identifiable things to think about impose a very\nessential restriction on any picture of the physical universe which we\nare able to form? We are continually surprising ourselves in the\ninvention of discrete structure further and further down in the scale of\nthings, whole sole _raison d'\u00eatre_ is to be found entirely within\nourselves. Thus Osborne Reynolds[12] has speculated seriously and most\nelaborately about an atomic structure in the ether, and we find\nEddington[13] hinting at the existence of structure of an order of\nmagnitude of 10^-40 cm. On a much larger scale of magnitude we also\nthink in the same terms, and conceive positive and negative elementary\ncharges with hard and impenetrable cores, which involves a complete\nchange in the law of force at points sufficiently close. What physical\nassurance have we that an electron in jumping about in an atom preserves\nits identifiability in anything like the way that we suppose, or that\nthe identity concept applies here at all?\n\n[Footnote 12: Osborne Reynolds, The Sub-Mechanics of the Universe, 254\npp., Cambridge University Press, 1903.]\n\n[Footnote 13: A. S. Eddington, Report on Gravitation, Lon. Phys. Soc.,\n1918, p. 91.]\n\nIn fact, the identity concept seems to lose all meaning in terms of\nactual operations on this level of experience.\n\nThe mind seems essentially incapable of dealing with continuity as a\nproperty of physical things; it is not even able to talk about\ncontinuity except in negative terms. To each attempted description of\nthe properties of a truly continuous substance, it can say \"No, it is\nnot that\", but cannot imagine experience which corresponds to what it\nconceives a really continuous thing ought to feel like. In terms of\noperations, continuity has only a sort of negative meaning. Now certain\nimplications of this inability of the mind can be removed by appropriate\npostulates, as, for example, we can postulate the complete annihilation\nof a negative by a positive charge, as is now being done in certain\nspeculations.[14] There is point in doing this, because the annihilation\nof two charges has physical meaning. But it is a question whether all\nimplications of this habit of thought can be removed, and whether any\npicture that we can form of nature will not be tinged\u2014sickbed o'er\nwith the pale cast of thought.\n\n[Footnote 14: For example, J. H. Jeans, Nat. 114, 828-829, 1924.]\n\nThe operational view suggests that in this last we are coming perilously\nclose to a meaningless question, although there is a certain sense in\nwhich there is meaning here. It may turn out as a matter of fact that we\nshall not be able to carry our delving into small-scale phenomena deeper\nthan a certain point, and that nature will appear to be finite downward,\nso that we shall bring up against a wall of some kind. But to ask in\nsuch a situation whether we have come to the end because nature is\n_really_ finite, or whether we only appear to be at an end because of\nsome property of our minds, such as inability to deal with continuity,\nis, I believe, a meaningless question.\n\nIn actual use the identity concept is extended, and identity is used in\nother senses than the fundamental one examined above. For instance, we\nspeak of two observers seeing the same object, or if the object moves or\ndoes something, we may speak of two observers perceiving the same\nhappening. A happening about which the judgment of sameness is possible\nwhen perceived by different observers (or mathematically expressed when\nobserved in two reference frames) is what we mean by an event, which is\none of the fundamental concepts of relativity theory. What now is\ninvolved in this concept of event, or what do we mean when we say that\ntwo observers experience the same event? A first crude attempt might say\nthat the event is the same if it is described in the same way by the two\nobservers. But this leads us into all the complicated questions of the\nmeaning of language, which we would gladly avoid, and is furthermore not\ntrue, because the whistle of a locomotive, for example, does not have\nthe same pitch for two observers moving with different velocities. A\nsatisfactory analysis of the situation is difficult to give, but I\nbelieve the essence lies in the discrete character of the event, just as\nthe identity concept when applied only to objects involved discreteness.\nThe event is bounded on all sides by discontinuities, both in space and\ntime. Now it seems to be a result of experience that discontinuities\nhave a certain absolute significance, in that there is a one-to-one\ncorrespondence between the discontinuities observed in any one reference\nsystem and those observed in any other. Corresponding discontinuities in\ntwo reference systems are by definition the same. An event is by\ndefinition the aggregate of all phenomena bounded by certain\ndiscontinuities, and two reference systems are by definition describing\nthe same event if the discontinuous boundaries of the event are the\nsame, irrespective of the appearance of the event in the two systems.\nThe emission of a light signal, for example, is an event according to\nthis definition, although it may appear as red light in one reference\nsystem and green in another.\n\nWe now see that the concept of event is only an approximate concept, as\nwas also that of identity, and for the same reason, namely, there are no\nsuch things in experience as sharp discontinuities, but as our\nmeasurements become more refined, the edges of supposed discontinuities\nbecome blurred. As we go to smaller scales of magnitude this blurring\nbecomes more important, until the physical possibility of performing\nthose operations by which the discontinuities are detected entirely\ndisappears, and the concept of event acquires, in terms of operations,\nan entirely different meaning. We continue to think of the event in the\nsame way as before in terms of a mental model, but the true operational\nsignificance now depends on the particular phenomenon under\nconsideration. The concept of event is really not the same sort of thing\nwhen applied to the emission of a quantum of radiation from an atom, or\nthe emission of gamma radiation from a radioactive disintegration, or\nthe flashing of a signal from a dark lantern by opening and closing a\nshutter. Here as always, when our range of experience is extended, we\nmust be prepared at some future time to find that, by extending the\nordinary concept of event to small-scale phenomena by the device of the\nmental model, we have by implication smuggled into our picture phenomena\nwhich do not exist, so that it will be necessary to revise our thinking,\ncasting it into terms corresponding to direct experience.\n\n\n\n\nTHE CONCEPT OF VELOCITY\n\n\nThe concept of velocity, as ordinarily defined, involves the two\nconcepts of space and time. The operations by which we measure the\nvelocity of an object are these: we first observe the time at which the\nobject is at one position, and then later observe the time at which it\nis at a second position, divide the distance between the two positions\nby the time interval, and if necessary, when the velocity is variable,\ntake the limit. As long as we deal with fairly low velocities we do not\nhave to inquire carefully as to the kind of time we use in these\noperations, but when the velocities become high, we do have to take care\nto use the local times at the two positions of the body, which means\nthat we must have a time system spread over space, or, in other words,\nthe \"extended\" time system. This velocity concept, defined in this way,\nmay be used as a tool in describing nature, and it will be found that\nnature has certain properties; for example, the velocity of light is 3 x\n10^10 cm./sec. Further, no material thing can be given a velocity as\nhigh as this, but as its velocity is made to approach this value,\nincrements of energy increasing without limit are required.\n\nBut now it is very much a question for examination whether the velocity\nconcept defined in this particular way has been chosen wisely as a tool\nfor describing natural phenomena. It is quite possible to modify the\nvelocity concept, that is, to set up other operations which correspond\nto our instinctive feeling of what velocity is in terms of immediate\nsensation and such that all numerical measures are unmodified at low\nvelocities.[15] For example, a traveller in an automobile measures his\nvelocity by observing the clock on his instrument board and the mile\nstones which he passes on the road. This operation differs from that of\nthe definition above in that the time is no longer extended time, but is\nthe local time of the moving object. The space coordinates used in this\nalternative operation at first seem a hybrid sort of thing, but they are\nwhat the observer would actually most naturally use: they are what he\nwould measure with a tape measure fixed to a point of the road and\nallowed to unwind as he proceeds, or what is measured by a vessel at sea\nwith a log line let out behind.\n\n[Footnote 15: It is an interesting question for the psychologist whether\nthe velocity concept is not a more primitive thing in order of\napprehension than that of time, and whether the concept of time is not\nderived from observing things in motion, or whether indeed there is any\nnecessary connection at all between velocity and time in terms of\nuntutored experience.]\n\nOr there is still another most interesting way of defining velocity, in\nwhich the analysis into space and time is not made at all, but velocity\nis directly measured by building up the given velocity by physical\naddition of a unit velocity selected arbitrarily. This matter is\ndiscussed at some length in my book \"_Dimensional Analysis_\",[16] but is\nof sufficient pertinence here to describe briefly. We may in the first\nplace construct a concrete standard for velocity, as, for example, by\nstretching a string between two pegs on a board with a fixed weight. If\nwe strike the string, a disturbance travels along the string which we\ncan follow with the eye, and we define unit velocity as the velocity of\nthis disturbance. An object has greater than unit velocity if it outruns\nthe disturbance, and less if it lags behind. We may now duplicate our\nstandard, making another board with pegs and stretched string, and check\nthe equality of the two velocities by observing that the two\ndisturbances run together. We now define two units of velocity as the\nvelocity of anything which runs with the disturbance of the string of\nthe second board, when the second board is made to move bodily with such\na velocity that it runs with the disturbance of the first string. The\nprocess may be extended indefinitely, and any velocity measured.\n\n[Footnote 16: Yale University Press, 1922.]\n\nIf either of these two alternative definitions of velocity were adopted,\nit would be found that the velocity of light is infinite. Further, there\nwould be no limit to the velocity which can be imparted to material\nbodies on giving them unlimited energy, which is what we are prepared to\nregard from ordinary experience is natural and simple. The infinite\nvelocity for light, on the other hand, is most unnatural, particularly\nif we favor a medium point of view. We are here faced with a\ndilemma--all sorts of phenomena cannot at the same time be treated\nsimply. If we attach the most fundamental significance to the behavior\nof material bodies, we shall do well to adopt one of the alternative\ndefinitions of velocity. If, on the other hand, we regard the phenomena\nof light as the most fundamental, we shall endeavor to form our\ndefinition so that the properties of light are simple. This was\nprecisely the point of view of Einstein; it is characteristic of his\nentire scheme of restricted relativity that light is the fundamental\nthing, and this influenced him in adopting the first definition of\nvelocity. Now one can have no quarrel with this desire to make light\nfundamental (the wisdom of doing this is to be justified by the\nresults), and if the properties of light are to be treated\nmathematically, one can easily see the desirability of getting rid of\ninfinite attributes, and so admit the desirability of making the\nvelocity of light finite. But all this involves another very insidious\nassumption which we ourselves have tacitly used in all our preceding\ndiscussion, namely, that the notion of velocity properly pertains to\nlight at all. Einstein has very definitely adopted this point of view,\nand so determined the character of the entire structure of relativity. I\nbelieve, on the contrary, that it is very gravely to be questioned\nwhether the identification of light with a _thing travelling_, which is\ninvolved in applying the velocity concept, should be made. This\ndiscussion must be postponed, however, until we deal with the properties\nof light. The important points for us to notice at present are that the\ndefinition of velocity actually used involves the concept of extended\ntime, and that it would be possible to define velocity in different\nways, which would give quite a different complexion to phenomena at high\nvelocities, but which would leave untouched our ordinary experience.\n\nThe velocities at which the precise form of definition becomes important\nare higher than can be reached in ordinary mechanical experiments. Such\nvelocities can be attained in terrestrial laboratories only with\nelectrified particles, as in experiments in high vacua or with\nradioactive disintegrations. It is interesting to notice that we very\nseldom attempt a direct measurement of velocity in such experiments by\nfollowing a discrete particle in its flight and finding the time\nrequired to pass over a measured distance, but the velocities are\nmeasured indirectly, by calculation from the equations of\nelectrodynamics and in terms of such immediately observed things as\ncurvature of path. It is true that one or two experiments have attempted\na more direct measure of velocity, but it seems there is room for more\nwork here.\n\n\n\n\nTHE CONCEPTS OF FORCE AND MASS\n\n\nAnother concept of great importance is that of force. Since the usual\nanalysis finds a connection between force and acceleration, and\nacceleration involves velocity, this is a natural place for the\ndiscussion of force. This concept has been subjected to much analysis by\nvarious writers. In origin the concept doubtless arises from the\nmuscular sensations of resistance experienced from external bodies. This\ncrude concept may at once be put on a quantitative basis by substituting\na spring balance for our muscles, or instead of the spring balance we\nmay use any elastic body, and measure the force exerted by it in terms\nof its deformation. Of course, the various precautions which must be\ntaken in carrying out this idea physically are complex; the matter of\nprecautions against temperature changes, for example, is one of the most\neasily understood. The concept of force so defined is limited to static\nsystems; it is the task of statics to find the relation between the\nforces in systems at rest. We next extend the force concept to systems\nnot in equilibrium, in which there are accelerations, and we must\nconceive that at first all our experiments are made in an isolated\nlaboratory far out in empty space, where there is no gravitational field.\nWe here encounter a new concept, that of mass, which as it is originally\nmet is entangled with the force concept, but may later be disentangled\nby a process of successive approximations. The details of the various\nsteps in the process of approximation are very instructive as typical of\nall methods in physics, but need not be elaborated here. Suffice it to\nsay that we are eventually able to give to each rigid material body a\nnumerical tag characteristic of the body, such that the product of this\nnumber and the acceleration it receives under the action of any given\nforce applied to it by a spring balance is numerically equal to the\nforce, the force being defined, except for a correction, in terms of the\ndeformation of the balance, exactly as it was in the static case. In\nparticularly, the relation found between mass, force, and acceleration\napplies to the spring balance itself by which the force is applied, so\nthat a correction has to be applied for a diminution of the force\nexerted by the balance arising from its own acceleration.\n\nWe now extend the scope of our measurements by bringing our laboratory\ninto the gravitational field of the earth, and immediately our\nexperience is extended, in that we continually see bodies accelerated\nwith no spring balance (that is, no force) acting on them. We extend the\nconcept of force, and say that any body accelerated is acted on by a\nforce, and the magnitude of this force is defined as that which would\nhave been necessary to produce in the same body the same acceleration\nwith a spring balance in empty space. There is physical justification\nfor this extension in that we find we can remove the acceleration which\na body acquires in a gravitational field by exerting on it with a spring\nbalance a force of exactly the specified amount in the opposite\ndirection. This extended idea of force may also be applied to systems in\nwhich there are electrical actions.\n\nWe thus see that in extending the notion of force from bodies in rest to\nbodies in motion, the character of the concept has changed, because the\noperations by which force is measured change--the force acting on a body\nis now measured in terms of its acceleration. But in determining the\nforce from the acceleration, we have to know the mass. This mass has to\nbe independently measured with the original concept of force; otherwise\nwe have no basis for such simple statements as that the force of gravity\non a body is proportional to its mass. All this applies to the ordinary\nrange of experiments with low velocities. If now we extend the range of\nmeasurements, we find phenomena which we had not expected; for example,\nthere seem to be difficulties in the way of indefinitely increasing the\nvelocity of a material body, as of a charged atom. We begin to ask\nsearching questions: is the force of gravity independent of velocity at\nhigh velocities, or is the mass independent of velocity under the same\nconditions or independent of the gravitational field, etc.?\n\nIn attempting to answer these new questions, we find difficulty with the\nconcepts in terms of which they are formulated. There are no operations\nby which we can find whether force is independent of velocity unless we\nfirst know the mass, or any operations by which a mass can be measured\nunless we know a force. The purely mechanical systems with the highest\nvelocities of which we have any experimental knowledge are the heavenly\nbodies. The motion of these is, with the important exception of Mercury,\nthat predicted by the ordinary laws of mechanics, so that at first it\nmight appear that we have here confirmation of the laws of mechanics for\nbodies with comparatively high velocities. But it must be remembered\nthat all we can observe of the heavenly bodies is their positions, and\nthat we cannot perform on these bodies all the operations by which we\ncan check the laws of mechanics for terrestrial phenomena. If, for\nexample, mass and the force with which gravity acts on mass were both\nequally affected by velocity, the motion of the heavenly bodies would be\nexactly the same as that observed now. Hence as we increase the range of\nvelocity, the concepts of force and mass simultaneously lose their\ndefiniteness, and become partially fused. This is typical of what we\nhave now come always to expect near the limit of the experimentally\nattainable; experience becomes less rich, the choice of physical\noperations more restricted, concepts change and become fewer in number.\nIf we are to retain the same formal number of concepts, we must\nintroduce arbitrary conventions or definitions. These definitions are to\nbe determined largely by convenience. In the case of mechanical systems,\nthis motive of convenience is supplied by considerations from outside\nthe domain of mechanical phenomena. The highest velocities of practice\nare not reached in mechanical, but in electrical systems, in experiments\nwith vacuum tubes, etc. Considerations of convenience are therefore\ndictated from the electrical point of view. These considerations will be\ngone into in much more detail later; the conclusion is all that we need\nhere, which is that it is convenient to assume for the charge of the\nelectron a constant number, independent of the velocity, and this\ninvolves making its mass variable in a definite way with velocity. Now\nif the principle of relativity is accepted, the mass of mechanical\nobjects must vary with velocity in the same way as the mass of\nelectrical charges. Since the variability of this latter is fixed,\nmechanical mass becomes a definite function of velocity, and the force\nis therefore also fixed in any specific physical case.\n\nThe fundamental definition of force given above is highly academic,\ninvolving as it does hypothetical experiments in laboratories situated\nfar out in empty space. Some sort of procedure like this seems to\ncorrespond to more or less explicit statements to be found in the\nliterature of mechanics. The meaning in terms of actual operations to be\ngiven to such definitions involves complicated inferential reasoning. We\nwould make much closer connection with the conditions of actual\nexperiment if in the definition we substituted for the hypothetical\noperations in empty space more or less approximately realizable\noperations on bodies sliding on level table tops without friction. I\nsuppose our instinctive feeling for the laws of mechanics is such that\nwe are convinced that definitions in terms of an interstellar space\nlaboratory or a level table top are actually the same. But in principle\nwe must recognize that when the operations are different, the concepts\nare different, and if we adopt something equivalent to the table top\ndefinition, as it seems we are physically forced to do, we must leave\nopen in our thinking the possibility of finding in the present penumbra,\nwhen our accuracy is sufficiently increased, such phenomena perhaps as\ndirectional attributes of mass in a gravitational field.\n\nWe have just considered the sort of problem that we encounter on\nordinary scales of magnitude on going from low to high velocities; what\nbecomes of the concepts of force and mass when we go to a very small\nscale? Down to the atomic scale we may at least slur over the new\nphysical difficulties, for although we cannot of course experiment with\nactual atoms, we can nevertheless make measurements of the Brownian[17]\nmovement of suspensions in liquids settling in a gravitational field,\nfor example, and the extrapolation to the atom is not a very great one.\nThe mass of each individual atom is obtained by what is equivalent to a\nprocess of counting, assuming the law of conservation of mass on an\natomic scale. This is justified by all chemical experience. The mass of\nthe component parts of the atoms, the electrons, may also perhaps be\ngiven a unique significance after we have decided on the laws of the\nelectrical field, by experiments on acceleration in electrical fields.\nThe question which interests in principle here is what meaning, if any,\nshall be attached to the mass of the elements of the electron.\n\n[Footnote 17: This phenomenon is discussed at length in the book by J.\nPerrin, Brownian Movement and Molecular Reality, translated by F. Soddy,\nTaylor and Francis, London, 1909.]\n\nIt is evident that we here go beyond any possible experience, at least\nfor the present, and that experience has again become poorer and our\nconcepts fewer in number. All that we can now demand is that certain\ncombinations of numbers, some of which represent mechanical mass and\nothers electrical charge, have proper relations to each other when\nintegrated throughout the entire body of the electron. Similar questions\nconfront us when we ask what are the forces which the parts of the\nelectron exert on each other. We return to this question in considering\nthe nature of the electrical concepts. In any event, the concepts of\nboth force and mass are entirely altered in this domain.\n\nIt is interesting to note, in passing, that present electrical theory\ngives no meaning to the mass of the elements of the electron, since the\ntotal electromagnetic mass of the electron is built up from the _mutual_\nterms in the action of the elements--the total mass is not a _linear_\nresultant of the action of the elements.\n\n\n\n\nTHE CONCEPT OF ENERGY\n\n\nIn examining the concept of energy, we start with purely mechanical\nenergy. In isolated mechanical systems, in which there are only\nconservative forces, the sum of kinetic and potential energy is\nconstant. The kinetic energy may be defined as \u2211 \u00bd mv^2, formed for\nall parts of the body. The potential energy is determined by the\nposition of the parts of the system, and has physical significance only\nwith reference to a datum position, that is, only changes of potential\nenergy have meaning in terms of operations. The total energy ascribed to\nthe system has therefore an element of arbitrariness in that the datum\nposition may be chosen at random, and energy acquires meaning only on\ntracing the history back to the epoch of the datum position.\n\nThe concept of energy may be extended from mechanical systems to all\nsystems with which we are acquainted; the operations by which meaning is\ngiven to the extended energy concept involve the generalized\nconservation principle, or the first law of thermodynamics. The\nextension to thermal systems is immediate; the inclusion of optical and\nelectrical systems in the scheme was a most important physical step,\nwhich of course required careful experimental justification. Because of\nits wide range of application, the energy concept has now come to be\nregarded as one of the most important in physics; this idea was held by\nOstwald[18] twenty and more years ago, and is now much to the front\nbecause of the connection between mass and energy indicated by the\ntheory of relativity, and the important r\u00f4le assigned to energy levels\nin spectrum analysis.\n\n[Footnote 18: W. Ostwald, Die Energie, Barth, Leipzig, 1908.]\n\nWhat now is the precise nature and significance of the general energy\nconcept? In the first place the conservation property of energy is one\nof the simplest and most obvious of the properties of matter, so that in\nthis property of energy is seen a reason for ascribing to it certain of\nthe properties of matter, in particular and most important, that of\nlocalization in space. We must recognize, however, that this idea of a\nlocation in space is injected into the situation entirely by ourselves,\nand corresponds to nothing directly given by the operations of\nexperiment. The idea has had a most important effect, however. Witness,\nfor instance, the importance ascribed to the discovery by Kelvin of a\nfunction by which the total energy of an electric field can be\nrepresented as distributed through space;[19] this was one of the most\nimportant props of the medium point of view.\n\n[Footnote 19: This function is \u215b\u03c0 times the scalar product of\nelectric force and displacement. If Maxwell's definition of displacement\nis adopted, the factor \u215b\u03c0 is replaced by \u00bd, and an accurate analogy\nresults between the energy stored in the ether and the elastic energy\nstored in a bent spring.]\n\nA more critical examination is likely to diminish considerably our\nsatisfaction with this naive analogy drawn between matter and energy.\nWith regard to matter, we may still be tolerably satisfied with our\nascription to matter of location in space, but it is quite different\nwith regard to conservation of matter. In just what sense is matter\nconserved? Certainly not in terms of mass, as we at one time thought.\nNevertheless we undeniably have a feeling that there is some sort of\nconservation property here, and are driven to formulate it badly in\nterms of a hypothetically constant number of protons and electrons. I\nhave long thought that Newton was groping after some very similar idea\nwhen he so far forgot himself as to define mass as quantity of matter, a\ndefinition perfectly meaningless to a rigorous and unsympathetic\ninterpretation. On the other hand, whatever meaning may reside in our\nidea of conservation of matter, it certainly is not, in at least one\nimportant respect, like the conservation of energy. For the energy of an\nisolated mechanical system is a function of the frame of reference in\nwhich it is described; merely by giving velocity to the reference frame\nand altering in no way the mechanical system we may change its kinetic,\nand so its total, energy by any direct amount. This does not even\nremotely resemble ordinary matter. I cannot see that the operations\nwhich are equivalent to the energy concept justify us in saying more\nthan that energy is a property of a material system; the operations do\nnot seem to give any unique meaning to a location associated with\nenergy.\n\nWe now ask what significance is to be ascribed to the sort of\nconservation that energy does have. We restrict ourselves first to\nmechanical systems. The motions of a mechanical system satisfy certain\ndifferential equations of the second order, and the actual motion is to\nbe found by an integration of the equations. In the integral of a\ndifferential equation certain constants appear which are determined by\nthe initial conditions, and are therefore the same during all the future\nmotion of the system; obviously these constants of motion correspond to\nconservative properties. This reasoning can of course be at once\nextended. Any system, mechanical or not, whose motion is determined by\ndifferential equations, will have certain conservative properties. For\nthe systems of mechanics energy is one of the conservative functions;\nothers are momentum and moment of momentum. Energy is particularly\nsimple, in that it is connected with measurable properties of the system\nby a simple formula (\u2211 \u00bd mv^2), and is furthermore scalar, which is\nalso a property of quantity of matter. But to go further and ascribe to\nenergy other properties of matter, such as localization in space, is\nentirely overlooking the essential difference in the character of the\noperation by which matter and quantity of energy are measured, that is,\noverlooking the essential difference in their physical character.\n\nThe possible extension of the energy concept from mechanics to\nthermodynamics receives a sufficient physical explanation in terms of\nour views of the essentially mechanical character of thermal phenomena.\nThat the idea can be extended also to simple electrical or magnetic\nsystems, in which the effect of velocity of propagation is neglected, is\na consequence of the fact that in these systems the equations of motion\nremain of the same general mechanical type, it having been shown by\nMaxwell that the equations of such systems may be written in the\ngeneralized Lagrangean form. When, however, we extend our formulas to\nsystems in which the velocity of propagation is important (that is, when\nwe consider the field equations in their general form) we find that the\nLagrangean equations no longer apply to matter taken by itself, and\nenergy is no longer conserved in the original sense. A new function\nappears, however, which behaves mathematically in the same way that the\nenergy did before. The equations of motion of the system remain\nLagrangean in form if the mechanical parts of the system are\nsupplemented by the electric and magnetic fields in space. In this\nextended form we have, therefore, a conservative function as before, and\nthe energy concept may be retained in this enlarged aspect. The physical\noperations by which energy is determined are entirely altered, however,\nand the physical character of the concept is changed. No more than\nbefore is there justification for localizing energy in space, or\nascribing to it other properties of matter. Yet the materialization of\nthe energy concept, and the consequent desire that energy be localized\nin space, is one of the strongest arguments in many minds for the\nexistence of a medium.\n\nAs far as I can see, therefore, the existence of conservative functions\nis involved in the possibility of describing natural phenomena with\ndifferential equations. That further there is a conservative function of\nthe precise form found in mechanics is a consequence of the particular\nform of the equations and the nature of the forces. The question of the\nsignificance of the fact that the forces of nature appear to be\nconservative, with respect to this particular function of mechanics, is\nof much interest, but it is not our immediate concern now. We are\ninterested rather to ask under what general conditions we shall have\nconservative functions. Quantum theory strongly suggests that when we\npass to phenomena on a small enough scale, we may no longer be able to\nemploy differential equations in our descriptions, and hence the\nprevious reason for the existence of constants connected with the motion\ndisappears. Now there is one obvious remark to be made about this more\ngeneral situation. Whenever the future history of a system is so\nconnected with its present condition that we can retrace our way to the\npresent from any future configuration, we shall always have conservative\nfunctions. For any future configuration contains certain fixed (or\nconservative) features, in that we can reconstruct the unique present\nfrom any future state. There is no reason to expect that the operations\nby which we find the fixed features will always be simple, as in the\nmechanical case. Now the determination of the future by the present, and\nconversely the possibility of reconstructing the present from the future\n(or the past from the present), is, we are convinced, a property which\nis at least approximately true of phenomena down to a smaller scale of\nmagnitude than we have yet reached, and so we expect to find these\nconservative functions in systems whose ultimate laws of motion are much\nmore general than any with which we are yet familiar. The particular\nform of the conservative function depends on the character of the\nsystem. That there is a scalar conservative function for ordinary\nsystems depends of course on particular properties of the system, but we\nare at least prepared to find that a scalar conservative function does\nnot necessarily mean a differential equation of the second order.\n\nThe potential energy of a system has a particular significance with\nrespect to this point of view. In an ordinary mechanical system, the\npotential energy simply measures the work done by the applied forces in\nbeing displaced from the initial to the final positions; that is, the\npotential energy is a measure of the deviation from the initial\nposition, and so measures a certain feature of the history of the\nsystem. In the more general system, in which we may not have\ndifferential equations, we may look for something analogous to the\npotential energy which shall measure the displacement of the system from\nits initial configuration. Such a measure is always possible as long as\nthe past can be reconstructed from the present (or the present from the\nfuture). We recall a remark of Poincar\u00e9's[20] to the effect that we of\nnecessity must always have conservation, because if we have a system in\nwhich conservation apparently fails, we merely have to invent a new form\nof potential energy. This remark is obviously not of entire generality,\nbut applies only to such systems as those considered here, in which the\npast may be reconstructed from the present.\n\n[Footnote 20: Henri Poincar\u00e9, Wissenschaft und Hypothese, translated\ninto German by F. and L. Lindemann, Teubner, Leipzig, 1906, Chap. VIII.]\n\nOf late there has been much discussion of the advisability, on the basis\nof certain quantum phenomena, of giving up conservation as a principle\napplied to the details of the emission and absorption of light,\nretaining it only in a statistical sense. It seems to me that the\nquestion here in the minds of physicists was always merely one of\nconvenience, and that few, if any, doubted the ultimate applicability of\nthe principle of Poincar\u00e9, or thought that we were here concerned with a\nsystem of such great generality that the past could not be reconstructed\nfrom the present. The question was merely whether those variables in\nterms of which the potential energy is defined make close enough\nconnection with other things of immediate experimental significance, or\nwhether on the whole the retention of a potential energy is not more\ntrouble than is justified by its convenience, making a treatment from\nthe statistical point of view preferable. However, this is all now a\nmatter of more or less past history, because with the recent extension\nof the experiments of Compton,[21] we seem to have experimental evidence\nfor the validity of the conservation law in detail for elementary\nquantum processes, with a corresponding simple potential energy.\n\n[Footnote 21: W. Bothe and H. Geiger, 2S. f. Phys. 32, 639-663, 1925. A.\nH. Compton, Proc. Nat. Acad. Soc., II, 303-306, 1925.]\n\nGoing still deeper, however, there are quantum phenomena which still may\nhave to be treated by statistical methods, and this may mean giving up\nconservation in detail. We have no experimental evidence, for example,\nof what the electron is doing while jumping from one quantum orbit to\nanother. A situation like this merely means that those details which\ndetermine the future in terms of the past may lie so deep in the\nstructure that at present we have no immediate experimental knowledge of\nthem, and we may for the present be compelled to give a treatment from a\nstatistical point of view based on considerations of probability. But I\nsuppose that no one, except perhaps Norman Campbell,[22] will maintain\nthat such a situation is more than temporary, or will cease to search\nfor consequences of these details of structure which may be open to\nexperimental verification.\n\n[Footnote 22: Norman Campbell, Time and Chance, Phil. Mag. I, 1106-1117,\n1926.]\n\nSimilarly, we cannot permanently be satisfied with a picture of\nradioactive phenomena which represents radioactive disintegration as a\nmatter of chance.\n\nThe general conclusion to which all this discussion leads is that energy\nis probably not entitled to the fundamental position that physical\nthought is inclined to give it, but that it is a more or less incidental\nconsequence of more deep-seated properties, and that the character of\nthese deep-seated properties is subject to only the most general\nrestrictions, so that very little about the nature of the details can be\ninferred from the existence of any energy function.\n\n\n\n\nTHE CONCEPTS OF THERMODYNAMICS\n\n\nWe shall not be concerned here with the many technical questions which\nare the proper subject of treatises on thermodynamics, but shall attempt\nan examination only of some fundamental concepts.\n\nThe most fundamental of these, which sets thermodynamics off apart from\nthe simpler subjects, is probably that of temperature. In origin this\nconcept was without question physiological, in much the same way as the\nmechanical concept of force was physiological. But just as the force\nconcept was made more precise, so the temperature concept may be more or\nless divorced from its crude significance in terms of immediate\nsensation and be given a more precise meaning. This precision may be\nobtained through the notion of equilibrium states. We have in the first\nplace the fundamental experimental fact that when a small body is placed\ninside a large system, which we recognize by crude means as\ncomparatively invariable in temperature as time goes on, the small body\nvery soon acquires a steady condition, that is, it comes to equilibrium\nwith its surroundings. We now have the further experimental fact that if\nthe small body A is in equilibrium with its environment, and body B is\nalso in equilibrium with the same environment, there will be no change\nof condition of A and B when they are brought into contact with each\nother--that is, A and B are each in equilibrium with the other and also\nwith the environment and therefore, by definition, at the same\ntemperature as the environment. The temperature of the environment is\nnow measured in terms of some of the properties of A and B which crude\nexperience has shown change with the physiological temperature of A and\nB. The physiological notion of temperature is thus made more precise by\nbeing connected with the physical phenomenon of equilibrium.\n\nNow it is at once evident that stated in this way without qualification\nwe have said things that are not true. It is not true in general that,\nwhen A is in equilibrium with an environment and B is in equilibrium\nwith the same environment, A will be in equilibrium with B. Suppose, for\nexample, that the environment is a stream of water and A is a tiny water\nwheel moving freely in its bearings, and that B is a similar wheel with\nmuch friction. Then we know that B will become warm, and will not be in\nequilibrium with A when brought into contact with it. Or we may choose\nfor A a mercury thermometer with bulb covered with putty, and for B a\nsimilar thermometer with bulb sheathed in platinum, and we know that the\ntwo thermometers will not register the same temperature in the water\nstream. Or still more simply, we may try to read the temperature of the\nair in our garden on any bright day with a silvered and with a blackened\nbulb thermometer; we know that the two thermometers will read different\ntemperatures. It is evident, therefore, that we shall have to specify\nmuch more carefully the conditions under which equilibrium holds if we\nare to give precise significance to the temperature concept.\n\nIt seems fairly evident in the first place that we shall have to rule\nout systems in which there is large scale mechanical motion; the simple\nnotion of temperature does not apply to a system moving with respect to\nus. Only when the two thermometers A and B move with the same velocity\nas the stream do we have three-fold equilibrium between the stream, A\nand B. We may state this in another way by saying that the temperature\nof a moving body must be measured on a thermometer stationary with\nrespect to the body, but this is only a matter of words, and properly\nspeaking the temperature concept applies only to a certain aspect of the\nrelation between two bodies mutually at rest. We here entirely neglect\nrelativity questions such, for example, as the proper way of correcting\nfor the change of dimensions of moving thermometers.\n\nIf now the body whose temperature we are measuring does not move with\nthe same velocity in all its parts, we may still give a meaning to local\ntemperature by dividing the body into parts so small that the velocity\nof each part is essentially uniform, and measuring the temperature of\neach part with a thermometer stationary with respect to it. We are now\nconfronted with the question of how far to carry the process of\nsubdivision. Suppose we have a fluid whose motion is completely\nturbulent when measured with instruments of the ordinary scale of\nmagnitude. For such a fluid the fundamental equilibrium proportions hold\nbetween two measuring bodies A and B and the fluid, provided that the\nbodies A and B are so large that the motion is completely turbulent on\ntheir scale of magnitude. We may then define the temperature of the\nturbulent fluid from the standpoint of these large scale bodies. But we\nmay also define the temperature from the small scale point of view as\nthe average of the temperatures recorded by sufficiently small\nthermometers, each moving with the velocity of a local bit of the fluid.\nThese two temperatures will in general be different, and we must more or\nless arbitrarily select one which we define as the true temperature. It\nwould seem that the small scale temperature is the better one to choose,\nbecause there is a certain degree of arbitrariness in specifying the\nscale from which the motion shall be judged completely turbulent But on\nthe other hand, there are difficulties in the small scale definition,\nbecause the turbulence may become more and more fine grained, until we\nend with the motion of the molecules themselves, when the operations\ncertainly fail which give meaning to the temperature concept. In this\ncase of molecular turbulence, we are driven back to the large scale\ndefinition, which obviously corresponds to ordinary physical practice.\n\nIt appears then that the temperature concept is not a clean cut thing,\nwhich can be made to apply to all experience, but that it is more or\nless arbitrary, involving the scale of our measuring instruments. In any\nspecial case, the meaning of the temperature concept must be set by\nspecial convention. In practice this does not often make difficulty,\nbecause in the majority of cases there is no large scale motion with\nrespect to the thermometers.\n\nConsider now the other aspect of the equilibrium problem suggested by\nthe thermometers with blackened and silvered bulbs in the sunshine. Our\ncommon experience tells us how to deal with this situation effectively\nenough for ordinary purposes. We recognize that the possibility of\ntemperature equilibrium is disturbed by the radiation, and we protect\nthe bulbs of the thermometers from the sun's radiation by appropriate\nshields. But this only minimizes the difficulty. For the shield is\nwarmed by the sun, and in turn warms to a less degree by its radiation\nthe bulb of the thermometer within. We must recognize that every body,\nno matter what its temperature, is always emitting radiation, so that\nthe bulb of our thermometer is always in a radiation field. At first\nthis puts us in a serious quandary as to the whole question of\nequilibrium and the meaning of temperature. The situation is saved by\nthe experimental observation that there is a particular radiation field\nwhich affects all thermometers equally, namely, the field inside an\ninfinite body all at the same temperature. Logically this looks like the\nvicious circle again, for we have not yet defined what we mean by the\nsame temperature. But actually we avoid the circle here, as in so many\nother physical cases, by a process of asymptotic approximation. The\nprocedure is perhaps something like this: we find that if we experiment\nwith larger and larger bodies, isolated and at great distances from\nother bodies at approximately the same temperature as judged by crude\nphysiological sensations, two thermometers, identical except that the\nbulb of one is blackened and that of the other is silvered, record more\nnearly the same temperature as time goes on and as the thermometers are\nsunk to greater depths in the body. In actual practice, of course, the\nradiational opacity of most bodies is so high that these precautions\nagainst the effects of external radiation can usually be entirely\nignored. At high temperatures, on the other hand, radiation has to be\nexplicitly dealt with.\n\nThe conclusion for us from these considerations is that operationally\nthe concept of temperature is tied up with that of radiation--the\nequilibrium concept of temperature is strictly never exactly applicable;\nit is only a limiting sort of concept applicable when the radiation\nfield is of a special sort, namely, that of a black body.\n\nIn spite of the explicit recognition which we have to give radiation in\ndefining temperature, we usually entirely lose sight of it in thinking\nabout the mechanism of ordinary physical processes, as for instance when\nwe picture the temperature of a gas as determined by the kinetic energy\nof its molecules. Now I have no doubt that negligence of this sort can\nbe justified, but the necessary logical analysis is apparently\ncomplicated, and involves a great many different sorts of experiment by\nmethods of asymptotic approximation, by which we establish the existence\nof various sorts of physical constants, such as constants of emission\nand absorption and reflection and scattering and fluorescence and\nthermal conductivity. We do not need to make the analysis here, but I\nbelieve that some time it would be worth while to attempt it. Such an\nanalysis will justify the principle so often used: that if a body is in\nthermal equilibrium the various processes involved, such as radiation or\nthermal conductivity, must when taken separately also be in equilibrium.\nDoubtless, if our experience had been confined to higher temperatures,\nlike that of the sun, this notion of different mechanisms acting\nindependently would have been more difficult to acquire.\n\nWe next consider another fundamental concept of thermodynamics, that of\nquantity of heat. We are at first perhaps inclined to think of this as a\ncomparatively straightforward concept, given immediately in terms of\nexperience, but an analysis of the operations by which we measure\nquantity of heat will show that the situation is really most\ncomplicated. Consider, for example, Joule's experiment in which the\nmechanical equivalent of heat was measured by determining the rise of\ntemperature of the water in a container when stirred by paddles driven\nby a falling weight. We do not question that the rise of temperature of\nthe water has its origin in the mechanical work done on it by the\npaddles. But what about the rise of temperature of the container? We\nshall doubtless say that part of this rise comes from heat communicated\nto it by the warmer water in contact with it, and part from mechanical\nwork done on it by turbulent impact of the water. But by what operations\nshall we measure what part of the energy communicated to the container\nis heat and what part mechanical work? We try to give an idealized\nanswer to this question in terms of Maxwell demons stationed at all\nparts of the boundary of the containing vessel with small scale\nmeasuring instruments. To measure the heat entering at any point I can\nsee nothing else for the Lilliputian observers to do but to determine\nthe temperature gradient at every point of the boundary from temperature\nobservations at two different levels, and calculate the heat inflow from\nthe gradient and the thermal conductivity of the material of the\nwalls--there seems no way of measuring a flow of heat as such. The\ninflow of mechanical energy must be calculated from a detailed knowledge\nof the elastic waves and other large scale deformations of the walls.\nHere again there is an arbitrary element in our procedure; if our\nmechanical measuring instruments are on too gross a scale, we may miss\nmechanical energy which we would catch with finer instruments.\n\nThis situation which we have just submitted to detailed analysis is, I\nbelieve, typical of the general situation; it is not possible in the\ngeneral case to find anything which we can call heat as such. Without\nfurther explicit examination, we can unambiguously speak of a body\nlosing or gaining heat only when there has been no energy interchange of\nany other sort with other bodies. In such a case the heat is measured in\nterms of the temperature change of the body. The heat concept is in the\ngeneral case a sort of wastebasket concept, defined negatively in terms\nof the energy left over when all other forms of energy have been allowed\nfor.\n\nThe essential fact that a quantity of heat can by itself be defined only\nin terms of a drop of temperature is somewhat obscured by the usual\nmethod of thermodynamic analysis. In describing a Carnot engine, for\nexample, it is specified that the engine shall work between a source and\na sink so large that their temperature is not changed by the heat given\nout or absorbed by them, so that the impression is natural that heat may\nin some way be measured apart from temperature changes. This of course\nis not the case; we merely require that the source and sink be so large\nthat their temperature changes are of a different order of magnitude\nfrom those in the working substance itself, so that with respect to the\nworking substance, source and sink may be considered to be at constant\ntemperature.\n\nAssuming now that we are able to measure quantity of heat in those cases\nin which the concept has meaning, let us examine the first law of\nthermodynamics, which we write in the form:\n\n_dQ_ + _dW_ = _dE_\n\nHere _dQ_ is the heat imparted to a given body by other bodies, _dW_ is\nthe work of all kinds done on it from outside, and _dE_ is the increase\nof internal energy. Now if this equation says what appears at a na\u00efve\nfirst glance, it should say that we find experimentally that the\nrelation written always holds between the measured quantities _dQ_,\n_dW_, and _dE_. We have seen that in the general case it is not possible\nto assign a unique operational significance to _dQ_ and _dW_, and\npresumably not to their sum. We ignore for the present difficulties of\nthis kind and confine attention on _dE_; how shall we measure it? I\nbelieve it does not take much examination to convince us that there are\nno physical operations for measuring _dE_ as such, and that therefore\nthe equation expressing the first law must have a different significance\nfrom that which appears on the surface. This is often recognized in the\nstatement that the essence of the first law is that _dE_ is an exact\ndifferential determined only by the variables which fix the internal\ncondition of the body, and not a function of the path by which the body\nis carried from one condition to another. But what shall we mean by\ninternal condition, and how shall we be sure that we have found all the\nvariables required to specify it completely? Internal condition may be a\nmost complicated thing and require many variables, as shown by a piece\nof iron with a complicated magnetic history or by a piece of aluminum\nabout to undergo recrystallization after overstrain. Here again I\nbelieve there is no physical procedure by which general meaning can be\ngiven to this concept of internal condition. In specific cases we can\nstate what the variables are which determine internal condition, and the\ncriterion that we have found the correct internal variables is that _dE_\nshall be a complete differential in terms of them. The first law of\nthermodynamics properly understood is not at all a statement that energy\nis conserved, for the energy concept without conservation is\nmeaningless. The essence of the first law is contained in the statement\nthat the energy concept exists (or has meaning in terms of operations).\n\nThe first law is often thought to be one of the most general of physics,\nbut in a paradoxical sense it is the most special of all laws, because\nno general meaning can be given to the energy concept, but only specific\nmeaning in special cases. The first law owes its complete generality to\nthe fact that no specific case has yet been found of so broad a\ncharacter that it cannot be included under one or another special case.\n\nExamination will at once justify this view. Thus we find a great many\nsystems which are adequately described in terms of two variables,\npressure and temperature, in that a function of _p_ and _t_ can be found\nsuch that its differential equals _dQ_ + _dW_. There are other systems\nin which the six components of stress and _t_ completely fix the\ninternal condition in the sense that they determine a _dE_. In other\nsystems the specification of a magnetic field may be necessary, or an\nelectric, or a gravitational field. No case is known which cannot be\nhandled in terms of the action of external forces of the proper kind,\nbut there is no general procedure, and the first law owes its generality\nto the exhaustive cataloging of special cases.\n\nWe may now return to the question left in abeyance above of the\nambiguity in _dQ_ + _dW_. In all the cases in which the specific\nvariables can be found which define _dE_, _dQ_ and _dW_ also have\nmeaning. Consider, for example, a gas, the internal condition of which\nmay be characterized in terms of _t_ and _p_. The mere fact that the\ninternal condition can be specified in terms of two variables, one a\nmechanical variable, shows that the substance is mechanically\nhomogeneous. Being mechanically homogeneous, we do not have the\npossibility of ambiguous values of _dW_ varying with the scale of the\nmeasuring instruments, and in fact we know that dW = p dv. Similarly the\ngas being homogeneous and at rest as a whole allows unique values for\n_dQ_. Of course this cannot obscure the physical fact that even in such\na gas, when we go to a small enough scale, we find inhomogeneities\narising from the Brownian movement, etc. Practically our statement means\nthat the inhomogeneities are so fine grained that over a very wide range\nof scale of the measuring instruments we find the same definite results.\nThe same sort of considerations apply to more complicated systems. If\n_dE_ is a complete differential in terms of _t_ and six stress\ncomponents, this means again that the body is homogeneous, its condition\nis determined by temperature and stress, which are the same throughout\nthe body, and again there is no possible ambiguity from the scale of the\ninstruments which measure _dW_ and _dQ_. It seems in general, then, that\nif the body allows operations by which _dE_ acquires meaning, at the\nsame time _dQ_ and _dW_ are provided for. In working out this idea in\nfull detail, some care must be given to the question of order of\ndifferentials. _dQ_, for unit time and unit volume, is strictly equal to\nk\u2207^2 t, where _k_ is thermal conductivity, so that in determining _dQ_\nthe second derivatives of temperature are involved.\n\nIf the body is obviously not homogeneous, it is still a matter of\nexperience that it can be divided into small pieces, each of which are\nby themselves sufficiently homogeneous, and the first law in its usual\nform may be applied to each of the pieces.\n\nFinally, we emphasize a fact already implicitly mentioned, namely, that\nno physical significance can be directly given to flow of heat, and\nthere are no operations for measuring it. All we can measure are\ntemperature distributions and rates of rise of temperature. As at\npresent defined, a heat current is a pure invention, without physical\nreality, for any determined heat flow may always be modified by the\naddition of a solenoidal vector, with change in no measurable quantity.\nIf someone states that throughout all space there is a uniform heat\ncurrent of 10^6 cal./cm.^2 sec. in the direction of Polaris, no disproof\ncan be given, for such a stream is solenoidal, and as much heat flows\nout of every closed surface in unit time as flows in. Such a solenoidal\nflow is meaningless in terms of operations; we could give meaning to\nsuch a flow only in terms of some slight modification of the solenoidal\ncondition introduced by the measuring instrument. In all ordinary\nconditions the flow of heat given by the simple relation q = k Grad t\ncorresponds exactly to what our atomic pictures lead to expect in those\ncases where the details of the picture can be worked out. But there may\nbe cases where it is advantageous to supplement the ordinary heat flow\n(= k Grad t) by the pure fiction of a solenoidal flow, because in this\nway it may be possible to account for new phenomena which appear when\nthe solenoidal conditions are slightly departed from. Thus if in a\nconductor at uniform temperature carrying a steady electric current we\nsay that a heat current is also flowing proportional to the electric\ncurrent and therefore solenoidal, we may provide the possibility for a\nsimple correlation of phenomena found under those more complicated\nconditions when an electric current flows in a conductor of non-uniform\ntemperature in a magnetic field. If it should turn out that the heat\ncurrent is _uniquely_ determined by considerations of this character,\nthen we have taken the first step away from the pure formalism which\nthis sort of thing otherwise is in the direction of giving physical\nreality to the invention of \"heat current.\"\n\nThere are other interesting questions of a fundamental thermodynamic\ncharacter, such for example, as whether the entropy concept has any\ngeneral significance apart from the scale of our measuring instruments,\nand what is the operational significance of applying thermodynamic\nconcepts to radiation, but we shall not consider these questions here.\n\n\n\n\nELECTRICAL CONCEPTS\n\n\nWe now set ourselves the problem of finding the meaning of the various\nconcepts in terms of which we describe the behavior of electrical\nsystems, assuming that we understand what we mean by \"electrical.\" We\nstart with the simplest electrical systems, namely, those in which we\ndeal with static phenomena on a large scale. In such systems there are\nindependent physical operations by which we may find the magnitude of\nany charge, provided that it is effectively concentrated in a\ngeometrical point. The measurements involved in these operations are\nmeasurements of ordinary mechanical forces; we assume that our knowledge\nof mechanics has already taught us how to make such measurements. An\nelectrically charged body experiences forces, which may be measured by\ntying a string to it and pulling on the string with a spring balance\nhard enough to keep the body in equilibrium. Three charges are\nnumerically equal if when each is placed at unit distance from another,\nin the absence of the third (or other charge), the forces are always the\nsame. If furthermore the forces are of unit magnitude, the charges are\ndefined as unit charge. Having obtained unit charge, we define the\nmagnitude of any other charge as equal to the force which it experiences\nwhen placed at unit distance from unit charge. This of course is all\nvery trite; the important thing for us is merely that magnitude of\ncharge, or quantity of electricity, is an independent physical concept,\nand that unique operations exist for determining it. These operations\npresuppose the ability to perform certain operations of mechanics.\nHaving now learned how to measure electrical quantities, we discover\nexperimentally the inverse square law of force, and later arrive at the\nconcept of the electric field. As we have seen, the field is an\ninvention; here we shall use this concept only for the purpose for which\nthe invention was made, and shall not involve ourselves in any of the\nimplications of ascribing physical reality to the field. Notice that as\nlong as we deal only with point charges we do not have to define field\nstrength in terms of the limiting procedure of making the exploring\ncharge smaller, for the limiting small charge is necessary only to avoid\nthe reaction of the exploring charge on the positions of the charges\nwhich generate the field. All this again is trite; the important point\nis that the operations by which the inverse square law and the concept\nof the field are established presuppose that the charge is given as an\nindependent concept, since the operations involve a knowledge of\ncharges. The operations also involve the measurement of forces by the\nordinary _static_ procedure of mechanics with spring balances. With the\nmeans now at our command we establish one very important property of\nelectric charges, namely that the total amount of charge on an isolated\nbody of finite size is conserved, no matter how the charge is forced to\nrearrange itself by the motion of charges on adjacent bodies.\n\nBy procedures exactly like those outlined above, we may treat all the\ncorresponding magnetic quantities; there is formal parallelism between\nthe two sets of phenomena, but there is the physical difference that we\nhave to realize a single magnetic pole by the device of using a very\nlong slender magnet.\n\nWe now give our electrical system more freedom, in that we allow the\ncharges to be in motion with respect to each other. Perhaps the most\nimmediate question which we now have to ask is whether charge continues\nto be conserved when set in motion, or whether the total charge on an\nisolated body is a function of its velocity? To answer this question we\nmust generalize the procedure by which we assigned a numerical value to\na stationary charge. Perhaps the simplest way is to allow two unit\ncharges each to move with constant velocity, remaining at unit distance\napart, and measure with a spring balance the force required to keep them\nat constant distance apart. Now we immediately find that the force is\naltered under these conditions, so that our first impulse is to say that\nthe charge is a function of the velocity. But as we experiment further,\nwe find that the state of affairs is very complicated; the force between\nthe two charges at any moment of their motion depends not only on the\ncharges, their distance apart, and their velocities, but also on the\nangle between the line joining them and the direction of motion in the\nlines. Further experiment of other kinds yields other information; it\nrequires a force to maintain a charge in uniform motion in a magnetic\nfield, or to maintain a magnetic pole in motion in an electric field. A\nmoving electric charge exerts a force on a stationary magnetic pole, so\nthat by definition the moving charge is surrounded by a magnetic field,\nand similarly a moving magnetic pole is surrounded by an electric field.\nReturning to our two moving electric charges, we are impelled to ask\nwhether, if all these complications are possible, the numerical constant\n(unity for static charges) in the inverse square law of force is a\nfunction of velocity as well as the magnitude of the charges themselves?\nIf we broaden the question in this way, as we apparently must, our\nproblem becomes indeterminate, for we are trying to answer two different\nquestions with a single kind of measurement, namely of the force between\nmoving charges. I have had no better luck on trying other methods of\nmeasurement. Apparently the operations do not exist by which unique\nmeaning can be given to the question of whether the magnitude of a\ncharge is a function of its velocity. On realizing this situation, we\nare at first embarrassed to know how to proceed, but we reflect that the\nembarrassment is not of our own making, but corresponds to a physical\nfact. The concept of charge as a unique and independent thing\nessentially pertains only to static systems. We may extend the concept\nto moving systems if we wish, as a matter of convenience to ourselves,\nbut must recognize that such an extension is an invention of ours and\nnot a reality of nature. Now we do make such an extension, and we make\nit in the simplest possible way, that is, we define the charge on an\nisolated body in motion as that which we should find on it if we reduced\nit to rest and made measurements according to the regular static\nprocedure. That this is a convenient thing to do depends on the\nexperimental result that the charge so found is independent of the way\nin which velocity is imparted to or removed from the body; in other\nwords, whenever the body is reduced to rest, the same charge is always\nfound on it.\n\nAlthough this is pure definition on our part, it turns out to have a\nmost simple and convenient connection with experimental facts which were\ndiscovered after the decision to treat a moving charge in this way was\nmade; the discovery is of the atomic structure of electricity. If then\nwe agree to call each elementary charge a constant independent of the\nvelocity, the total charge on a body becomes merely proportional to the\ncount of the total number of atomic charges on the body, which is\ncertainly highly convenient and suggestive.\n\nHaving now fixed what we mean by the magnitude of a moving charge, we\nare ready to turn to the general problem of the behavior of any system\nof charged bodies in motion. For the present we consider only phenomena\nof the scale of everyday experience. The most general problem that has\nmeaning here is to determine all measurable properties of the system in\nterms of those data which experiment shows can be arbitrarily specified.\nNow we have already emphasized that the electromagnetic field itself is\nan invention, and is never subject to direct observation. What we\nobserve are material bodies, with or without charges (including\neventually in this category electrons), their positions, motions, and\nthe forces to which they are subject. The forces are to be measured\naccording to definition in mechanical terms, either by the strains in\nmembers of a framework if the system is in equilibrium, or in terms of\naccelerations and masses if it is not in equilibrium. The\nelectromagnetic field as such is not the final object of our\ncalculations, but the calculation of it is only an intermediate\nauxiliary step, convenient to make because our mathematical formulation\ngives so simple a connection between electromagnetic field, charges, and\nmechanical action that the latter can be calculated at once in terms of\nthe former. In fact the connection is so simple that in many cases we\nhave come to regard our problem as solved if we can compute the\nelectromagnetic field, overlooking the fact that the field has no\nimmediate meaning in terms of experience.\n\nElectromagnetic theory now presents us with a solution of the general\nproblem; this solution is contained in the four-field equations of\nMaxwell, the constitutive equations, and those additional equations\n(quite often lost sight of) which give the forces exerted by the field\non electric charges, or currents, or dielectrics. Let us inquire how we\nmay set about testing the physical correctness of these equations. We\nmay begin with one of the simplest possible tests, and inquire whether\nthe equations are correct in stating that the force acting on a charge\nmoving in an electric field is simply the product of the charge and the\nfield strength. This, on the face of it, is a surprising statement. The\nfield itself is affected by the motion of the charges which generate it,\nand it is natural to expect a converse effect. If, furthermore, we have\nsympathy with the medium point of view, it is easy to think that\nwhatever it is in the medium that gets hold of a charge and exerts a\nforce on it will find it harder to take hold when the charge is in\nmotion.\n\nIn attempting to check our statement experimentally, the only additional\ncomplication, as compared with the static case which we have already\nchecked, is afforded by the motion of the charge, for we have defined\nthe magnitude of a charge in motion, so there is no difficulty here, and\nwe may furthermore suppose that the field is generated by stationary\ncharges, so that we need not trouble to inquire whether the procedure by\nwhich the field was originally defined is here applicable. The task of\nchecking the equation then reduces to the simple physical task of\nmeasuring the force on the moving charge. How shall we do this? If the\nvelocity is low, we may tie a string to the charge and measure the force\nwith a spring balance (or its equivalent). But now an examination of the\nequations shows that in more complicated phenomena perceptible\ndeviations from the static behavior are to be expected only at much\nhigher velocities than can be attained by towing charges with a string\nand a spring balance, so that it is evidently necessary to check the\nsimple equation for the force on a moving charge also at high velocity.\nSince at high velocity the spring balance method for measuring forces\nfails, we are driven to the only procedure that we have, namely a\nmeasurement in terms of the resultant acceleration, calculating the\nforce by Newton's first law of mechanics. But this involves a knowledge\nof the mass of the moving body, which we recognize in general may be a\nfunction of the velocity. Now we have already seen, in discussing the\nconcepts of mechanics, that the operations by which mechanical mass is\ndefined cannot be carried out at high velocities, so that either the\nconcept of mechanical mass becomes meaningless at high velocities, or we\nmust adopt another definition. In attempting to give this new definition\nof mass at high velocities, we are driven to a result of special\nrelativity theory, namely that all mass, mechanical or electrical, must\nbe the same function of velocity. If now electrical mass can be found in\nterms of velocity, our immediate problem is solved and we shall be in a\nposition to complete the experimental check of the equation. But as a\nmatter of fact, in order to determine electrical mass, we have to use\nthat equation which we are now engaged in trying to establish. Logically\nwe have again the vicious circle, the physical significance of which is\nthat independent operations do not exist for giving unique meaning to\nthe concept of force on a charge at high velocity.\n\nWe seemed so close to our goal a minute ago; that we may allow ourselves\nto jump the logical chasm, and assume that the equation is correct.\nElectrical mass now becomes a definite function of velocity, mechanical\nmass the same function, and we are in a position to compare the actual\nacceleration received by a charge in a field with that calculated by the\nequation. Our conviction, on the basis of all experience up to the\npresent, is that the two accelerations will be found to agree.\n\nThe equation then does somehow make correct connection with experience\nin that a consequence of the equation can be verified experimentally, in\nspite of the fact that as the equation stands it is meaningless, because\nthe operations do not exist by which meaning can be given to the\nindividual terms. At low velocities the equation really says what it\nseems to say, because the individual terms have meaning in terms of\noperations; and, what is more, what the equation says agrees with\nexperiment. At high velocities the equation does not mean at all what\nappears on the surface; by itself it has no meaning; it has meaning only\nwhen considered as a member of a system of equations, and only in so far\nas the system of equations makes by implication statements about nature\nthat have meaning in terms of operations that can be carried out\nphysically. The individual terms of the equation of the system do not\nhave meaning at high velocities, and in fact there are more terms than\nthere are independent physical operations.\n\nAn exact analysis from the operational point of view of the significance\nof the equations at high velocities has perhaps never been made, and is\nnot necessary for our immediate purpose. The discussion has brought out,\nhowever, that the number of physically independent concepts has been cut\ndown by two at least, in that we have made purely formal definitions of\nthe meaning of quantity of electricity, and of the force exerted by a\nfield on a charge at high velocity. There is no reason to think that\nthere is anything unique about this analysis, or that formal definitions\nmight not have been given to other concepts than charge and force. We\ncan only state that as far as physical content goes the equations have\nat least two degrees of freedom. It should then be possible to find\nquite different sorts of equations which agree equally well with\nexperience. In particular, since we have seen that the force on a moving\ncharge has no meaning in terms of independent operations, it should be\npossible by arbitrary definition to make this force any function of\nvelocity that we please (of course reducing to the proper value at low\nvelocities), and then to determine the other equations so that the\nentire group of equations is consistent with experiment. So far as I\nknow, no one has tried to give such a modified set of equations, and\nindeed there is no particular reason why anyone should bother to do\nthis, because the present equations are simple enough, and the modified\nequations, although perhaps differing greatly in appearance from the\npresent ones, would have no advantage in any greater or different\nphysical content.\n\nBut there is no reason to think that the present state of affairs will\nalways continue. We have seen that the decrease in the number of\nconcepts corresponds to our inability to measure as many sorts of\nphysical things at high velocities as at low. Now it is the task of the\nfuture experimenter so to refine the possibilities of measurement at\nhigh velocities as to restore these two degrees of freedom. In\nparticular, mass should be made measurable in mechanical terms at high\nvelocities. When this restoration has been made, and all the quantities\nin our equations receive independent physical meaning, the significance\nof the equations in terms of operations will be quite altered, although\nthe formal appearance will be unchanged. We must then be prepared to\nfind, as always when we change the range of phenomena, that the\nequations in their present form do not correspond to the facts at all,\nand that one of the alternative forms allowed by our present two degrees\nof freedom is the correct form. But until the new experimental facts\nhave been obtained, it seems hardly worth while to attempt to specify\nthe doubly infinite variety of forms which the equations might have\nconsistently with present experiment.[23]\n\n[Footnote 23: Since this was written, a paper has appeared by V. Bush,\nJour. of Math. and Phys., vol. V., No. 3, 1926, in which it is shown\nthat there are advantages in supposing the charge of an electron to\nchange when it is set in motion.]\n\nSo far we have discussed the extension of ordinary electric phenomena in\nonly one direction, to high velocities. There is another extension which\nis much more important physically, namely to very small scales of\nmagnitude. This extension is necessary to an understanding of the\nproperties of matter in bulk, the electrical nature of the atom having\nbeen once established. Our problem is to show how the statistical\naverage of the behavior of a large number of electrons gives the large\nscale effects which are within the reach of observation, and which are\ndescribed by the equations we have just discussed. To get this\nstatistical average we must be able to calculate at least certain\nfeatures of the behavior of the individual electrons, which means that\nwe must know the form of the equations down to dimensions of the order\nof those of an electron, or smaller. Now if one contrasts the scale of\nthe supposed dimensions of the electron with the smallest dimensions on\nwhich we can make independent experimental verification of those\nequations, he must admit that there is an enormous chance for change in\nthe type of equation beyond the limit that we can reach by direct\nexperiment, and the chances of guessing the correct extension of the\nequation to small dimensions are correspondingly almost vanishingly\nsmall. (We may perhaps say that experiments on the Brownian motion on a\nscale a good many atoms in diameter bring us the closest possible\ndirectly, which means that we are 10^6 or 10^7 fold away from electronic\ndimensions.) In spite, however, of the apparently enormous chances\nagainst it, this program of extending the field equations to small\ndimensions and following out the consequences was exactly the program\nwhich Lorentz set himself.[24] That Lorentz saw that such a program\nmight be carried through must be recognized as a vision of extraordinary\ngenius, and that he was willing to devote to it the years of arduous and\ndetailed calculation that he did is evidence of a pertinacity of purpose\nof the highest moral order.\n\n[Footnote 24: See for example, H. A. Lorentz, The Theory of Electrons,\nB. G. Teubner, 1916.]\n\nWe now have to examine critically this program and to inquire what is\nthe significance of the measure of success that Lorentz attained. The\nprecise extension of the equations that he made was very simple, for the\nlarge scale equations of Maxwell were taken over with as little change\nas possible. The equations are so familiar that it is not necessary for\nus to write them in detail; they express relations between the electric\nand magnetic force vectors (force and induction now becoming the same\nthing, the difference between them in ponderable bodies being one of the\nthings that is to be explained in terms of the electrons), the space\ndensity of electric charge, its velocity, and the force acting on\nelementary charge. We have to notice that although formally the\nequations have changed little in appearance, nevertheless the physical\ncontent, as judged by the operations, has changed a great deal.\nConsider, for instance, the meaning of charge density. In the Maxwell\nequations, _\u03c1_ was merely the number of discrete elementary charges per\nunit volume, the distances between these charges being supposed so small\ncompared with the scale of the phenomena involved that their average\neffect could be fairly represented in terms of their numbers. In the\nLorentz equations, on the other hand, _\u03c1_ has a value different from 0\nonly inside the electron; everywhere else _\u03c1_ = 0. Now an examination\nof the previous discussion, in which we questioned whether the magnitude\nof the charge might be a function of its velocity, will show that there\nare no physical operations whatever by which meaning can be given to\n_\u03c1_ at individual points inside an electron. There is a single\ncondition on this _\u03c1_, namely, that its integral throughout the total\nvolume assigned to the electron shall equal the total static charge of\nthe electron. Obviously a single scalar condition is a pretty blunt tool\nwith which to attempt to determine a point function throughout a volume.\nAgain, the equations talk about the velocity of the charge at interior\npoints of the electron; what possible physical operations are there by\nwhich meaning can be assigned to the velocity of an amorphous\nstructureless substance in regions inaccessible to experiment? Here\nagain, the concept as a detailed description of the behavior at a point\nhas become meaningless, and again there is a single integral condition,\nnamely, that the _v_ associated with every _\u03c1_ must be such that when\nintegrated over the volume of the electron it will give a total\ntransport of charge equal to that carried by the electron in its motion.\nThis again is a single condition on a function distributed through\nspace. Still again, the equations contain the electric and magnetic\nvectors at points inside the electron. What is the possible meaning of\nthese field vectors in terms of operations? Our procedure for finding\nthe field at a point involves by definition finding the force on an\nelectric charge placed at that point. But there is no charge smaller\nthan an electron, and the procedure degenerates into a fiction. Again\nthere is a single integral condition on the field vectors; the integral\nof the force on the assumed charge density when taken over the total\nvolume of the electron must give a value corresponding to experiment.\nExcept for this single condition, the concept of the field at points\ninside the electron is an invention without physical reality. Not only\nis the field concept meaningless at points inside the electron, but it\nis meaningless at points outside within a certain distance, because the\nexploring charge can never be made smaller than the electron itself, and\nso can never come closer than a certain distance.\n\nThe actual state of affairs is much worse than has already appeared. It\nwas shown in the discussion of space and time that no independent\nphysical meaning can be attached to lengths and times as small as have\nto be assumed in describing the behavior of the individual electrons.\nThe operations Div, Curl, d/dt which enter the field equations are,\ntherefore, physically meaningless as they stand; they have only a\nmathematical meaning which begins to acquire physical complexion in a\nmost complicated way when the equations are integrated over large enough\nvolumes.\n\nIt is evident, therefore, that the concepts which enter the field\nequations have entirely lost their large scale significance; they have\nbecome blurred, fused together, and fewer in number. A precise analysis\nof this situation has probably never been attempted and would obviously\nbe difficult: it would be interesting to know at least how many really\nindependent concepts there are at this order of phenomena. An attempt at\nan analysis would probably be worth while from a physical point of view\nin suggesting possible experiments by which the number of physically\nindependent concepts could be extended.\n\nSince the quantities in the field equations are meaningless in the naked\nform in which they enter the equations, it is meaningless to inquire\nwhether the equations as they stand are true or not. In our present\nstate of experimental knowledge it is also meaningless to ask whether,\nfor example, the inverse square law between electric charges continues\nto hold, or whether an accelerated charge radiates. These questions have\nmeaning only when applied to phenomena on a scale large enough to\ncorrespond to possible experiment.\n\nThere is a rather interesting obverse to the statement that it is\nmeaningless to ask whether the field equations are true, namely, that it\nmay not be meaningless to state that they are false. A statement is not\ntrue unless it is true in every particular, but it is false if it is\nfalse in a single particular. If we can show that a single consequence\nof the field equations of Lorentz, when integrated or averaged in such a\nway as to correspond to experimental possibilities, is false, then the\nequations must be false. It seems that, regarded as a complete\ndescription of physical behavior on a small scale, the equations must be\njudged false, because they contain no suggestion of quantum phenomena.\n\nEven if we have to recognize that the equations are false, there can be\nno question that they correspond to an important part of reality, and\nthat they have been of the greatest service to physics. What is the\nsignificance of the success that they have attained? It is to be noticed\nthat all the phenomena to which the Lorentz equations have been\nsuccessfully applied, although not large scale phenomena in the ordinary\nsense of the word, are nevertheless phenomena involving the co\u00f6peration\nof a number of atoms, and that the equations unquestionably fail when\napplied to phenomena involving single electrons. It appears from our\nbest present evidence that on a small scale the behavior of nature is\ngoverned by quantum principles and is therefore quite different from\nlarge scale behavior, Which we have seen is governed by the Maxwell\nequations. There must of course be a transition zone in which the\ncharacter of phenomena changes from quantum to Maxwell. Now any program\nlike that of Lorentz is almost inevitably bound to begin to give correct\nresults when we get up as far as the transition zone, for the simple\nreason that the relations of Maxwell have been put into the equations\nand are always there ready to appear as soon as the quantum relations\nbegin to give way. The physical significance of the success of the\nLorentz program seems to be that the transition from Maxwell to quantum\ntakes place at a stage pretty far down toward the individual atoms. To\nfind the precise details of the transition from Maxwell to quantum\nphenomena constitutes a large part of the program of the immediate\nfuture.\n\nAll this skepticism about the classical work of Lorentz is likely to be\nrather irritating or depressing, particularly if one attempts to imagine\nwhat other course could have been adopted. Indeed it does seem that we\nfind ourselves in a real quandary; Lorentz was practically forced,\nbecause of the character of the mathematical tools at his command, to\ntake the course that he did, in spite of any recognition of the physical\nmeaninglessness of the mathematical operations. We have already seen\nthat conventional mathematics does not correspond to the physical\nreality; it cannot easily make a qualified statement subject to\nlimitations, and it recognizes no difference between the physically big\nand the physically little and the corresponding change in the\noperational meaning of its symbols. It begins by being a most useful\nservant when dealing with phenomena of the ordinary scale of magnitude,\nbut ends by dragging us by the scruff of the neck willy nilly into the\ninside of the electron where it forces us to repeat meaningless\ngibberish. Larmor recognized this, and in his electron theory, developed\npractically contemporaneously with that of Lorentz, endeavored to treat\nelectrons as wholes, and not to make meaningless statements about their\ninsides.[25] But he was much less successful than Lorentz in making his\nanalysis give physical results, and one may suspect that it was at least\nin part due to difficulty with his tools.\n\n[Footnote 25: Joseph Larmor, \u00c6ther and Matter, Cambridge University,\nPress, 1900. In this book the electron is treated as a point singularity\nin the ether.]\n\nWhat we should like to be able to do is easy to see. The things that go\ninto our equations must have independent physical meaning, and the\ncharacter of our mathematical formulation should change to keep pace\nwith the change in the physical operations which give meaning to the\nterms. For example, electrical density has a meaning for large scale\nphenomena, but means nothing on a small scale. Our ultimate electric\nunit is the electron; when we get down to this scale of magnitude, our\nmathematics ought to be making statements about the relative behavior of\ndiscrete electrons, and not mention so much as by implication the\ndensity at points inside an electron. But this sort of thing we\napparently cannot yet do; the proper mathematical language has not been\ndeveloped. Such a language, when developed, must not only be able to\nresist the temptation to burrow inside the electron, but must also try\nto get along without the field concept, which we have seen is liable to\nso much physical abuse, and must reduce effects in complicated\nelectrical systems to the ultimate elements that have physical meaning,\nnamely, a dual action between pairs of electrical charges, with no\nimplications about physical action where the charges are not.\n\n\n\n\nTHE NATURE OF LIGHT AND THE CONCEPTS OF\nRELATIVITY\n\n\nWe have already discussed several aspects of the theory of relativity in\nconnection with the relation to it of some of our fundamental concepts.\nThere are still other topics connected with relativity which demand\nattention; most of these involve the properties of light. It will now be\nconvenient to discuss together the properties of light and these\nconcepts of relativity. We restrict our discussion of light to those\nsimple properties which bear on the theory of relativity.\n\nPractically all our thinking about optical phenomena is done in terms of\nan invention, by means of which these phenomena are assimilated to\nthose of ordinary mechanical experience, and so made easier to think\nabout. To realize that invention has been active here, we must think\nourselves back into that naive frame of mind in which experience is\ngiven directly in terms of sensation. The most elementary examination of\nwhat light means in terms of direct experience shows that we never\nexperience light itself, but our experience deals only with things\nlighted. This fundamental fact is never modified by the most complicated\nor refined physical experiments that have ever been devised; from the\npoint of view of operations, light means nothing more than things\n_lighted_. Now experience shows that these things lighted may stand to\neach other in varied relations; in attempting to reduce these relations\nto order and understandability we make a certain invention. This is\nprompted by several cardinal experimental facts: in the first place,\nthings lighted have a simple geometrical relation to each other, in that\nscreens placed on straight lines between the lighted objects may\nsuppress the illumination of one or the other and themselves become\nilluminated. This leads to the concept of rectilinear beams of light,\nwhich is no more than a description of the geometrical relation between\nlighted objects. Then we have the experimental fact of the asymmetrical\nrelation of the lighted objects, described in terms of sources and\nsinks. Finally, we have the discovery made at a much later stage, and\nnot possible until physical measurements had reached a high refinement,\nthat light has properties analogous to the velocity of material things.\nThis was first discovered in connection with astronomical phenomena in\nthe shift of the time of eclipse of Jupiter's satellites and in\naberration, but was later found to hold for purely terrestrial\nphenomena, in that a beam of light reflected from a distant mirror does\nnot return to the source until after the lapse of a time interval that\ncan be measured with means sufficiently refined. This property of return\nafter the lapse of time is precisely like that of material things, such\nas a messenger despatched for an answer, or a ball or a water wave\nbouncing from a wall. These various properties of light lead quite\nnaturally and almost inevitably to the invention of light as a thing\nthat _travels_, \"thing\" not necessarily connoting material thing.\n\nThe question now for us is whether we shall regard this as a mere\ninvention, made for convenience in thinking, or shall go further and\nascribe physical reality to it, that is, shall we think of light as\ncapable of independent physical existence in the space between the\nmatter that constitutes the source and the mirror? Now in spite of the\nresemblances pointed out above, there is at least one universal and\nfundamental difference between a thing that travels and light. We have\nindependent physical evidence of the continued existence of the ball,\nfor example, at all intermediate points of space; we can see it, or hear\nit, or feel the wind in the air as it passes, or even touch it. All\nthese phenomena are independent of the initial and terminal phenomena,\nand hence by our criterion for the physical reality of an invention, we\nare justified in ascribing physical reality to the ball in transit. But\nwith the beam of light it is entirely different; the only way by which\nwe can obtain physical evidence of the intermediate existence of the\nbeam is by interposing some sort of a screen, and this act destroys just\nthat part of the beam whose existence we have thereby detected. There is\nno physical phenomenon, whatever by which light may be detected apart\nfrom the phenomena of the source and the sink (understanding that a\nmirror is included in the idea of, sink); that is, no phenomenon exists\nindependent of the phenomenon which led us to the invention of a thing\ntravelling. Hence from the point of view of operations it is meaningless\nor trivial to ascribe physical reality to light in intermediate space,\nand light as a thing travelling must be recognized to be a pure\ninvention.\n\nThe status of light is exactly the same as that of an electric field;\nthere is not the slightest warrant for ascribing physical reality to\neither at points of empty space--light and field-at-a-point have no\nmeaning until we go there and make experiments with some material thing.\nOf course the electromagnetic theory of light makes this resemblance\ninevitable, provided the theory and our views of the nature of light and\nthe field are correct.\n\nIt cannot be denied that there are some phenomena which uncritically\nconsidered appear to justify thinking of light as a thing that travels;\nthese will now be discussed. Probably the argument to which most\nsignificance is usually ascribed is derived from the phenomena of\nenergy. The passing of light from source to sink is accompanied by the\ntransfer of energy. But energy is conserved, so that we have to ask\nwhere the energy is in the time interval between the emission of light\nfrom the source and its absorption by the sink. There is an obvious\nanswer: the energy is in transit, of course, somewhere in the\nintermediate space between source and sink. If we think of light as\npropagated through a medium, then the medium is such that energy may\nreside in it, as in the electromagnetic theory of light, or if light is\nmore material and ballistic in character, the thing that travels has\nitself energy. We notice in the first place that the conservation\nprinciple involves the time concept, because what we mean by\nconservation is that the total energy of the universe, at a fixed\ninstant of time, is constant. That is, we have to integrate over all\nspace the local energy at a definite instant of time, and this involves\nspreading the time concept over all space. It is further evident that\nunless we spread the time concept over space in the right way we shall\nnot get conservation. The proof that it is possible to spread the time\nconcept over space in such a way as to give conservation involves a\nknowledge of the properties of light. It would seem, then, that we ought\nnot to assume conservation in deducing the properties of light, when a\nknowledge of the properties of light is necessary to establish\nconservation. These considerations cannot be accepted as final, however,\nuntil a detailed analysis has been made, and this would be most\ncomplicated. But there is a more important consideration derived from\nour previous critique of the energy concept, namely, that there is no\nbasis for asserting that energy is localized in space at all; energy is\nnot a physical thing, but rather what we would call a property of a\nsystem as a whole. If this view of energy be granted, the whole energy\nargument for light as a thing travelling, and also for the existence of\na medium, falls. I believe that similar considerations apply to any\narguments from the conservation of momentum.\n\nThe possibility of detecting light in apparently empty space by a screen\nconstitutes perhaps the most immediate reason for considering light as a\nthing that travels. This point of view I believe is characteristic of\nthe entire attitude of Einstein in deducing the theorems of the special\ntheory of relativity. Einstein's light signal is for the purposes of the\ndeduction thought of as a simple spherical wave spreading from the\nsource and capable of being watched as it spreads by an observer outside\nthe system, in much the same way that a water wave can be watched. Of\ncourse the light signal cannot actually be watched in transit, but we\ncan come fairly close to this ideal by placing screens at any point we\nplease to make the wave visible. It is true that the mere act of showing\nthe existence of the light destroys that part of the beam whose\nexistence is detected, but the screen needs only an infinitesimal amount\nof light to make it visible, and so by the usual physical argument we\nmay suppose that the detecting screen produces only an infinitesimal\nmodification of the total original light.\n\nOur satisfaction with this picture evaporates if our present quantum\nviews of the nature of light are correct. We can no longer think of the\nspherical light pulse as of irreducible simplicity, but it is an\nexceedingly complicated thing, perhaps more complicated than a gas from\nthe point of view of kinetic theory, and simulates simplicity by some\nsort of averaging of the effects of the elementary quantum processes of\nwhich it is composed. If the principles of relativity are to continue to\nbe regarded as fundamental, or even if they are to remain intelligible,\nwe must apply our reasoning, not to spherical wavelets, but to the\nelementary process of which these wavelets are composed. Now the\nelementary quantum act is essentially a twofold thing: there is a\ndiscrete act of emission at some discrete material particle, and the act\nis consummated by another discrete act (absorption or scattering) at\nsome other discrete particle. We cannot yet fully characterize the\ndetails of this twofold process, but have to connect the place at which\nabsorption takes place with the place of emission by statistical\nconsiderations. It is evident, however, that to think of emission as\nstarting some process like a spherical wavelet travelling like a thing\nthrough space presents an entirely incorrect view, because in the wave\nthere is no hint of the discrete place which is to terminate it. We may\nsay crudely that there is no way by which the wave can know what\ndiscrete material particle is to complete the emission process. We may\nperhaps try to save the situation by remembering that a spherical wave\nis polarized and so has a unique direction associated with it; but\nfurther examination shows that this does not help, because the unique\ndirection is that of no energy flow, and absorption can take place in\nany direction _except_ this. It appears then that instead of being a\nhelp, the thing travelling point of view is a positive embarrassment\nwhen we try to picture by means of it the essentially twofold nature of\nthe elementary quantum act.\n\nAnother plausible argument for light as a thing travelling may be\ndeduced from our principle of connectivity. Imagine a dark lantern with\na shutter that can be opened or closed so as to emit a momentary flash\nof light, a distant mirror, and a receiving instrument near the source.\nOne of the properties of light that we always assume is that no\npermanent trace of the act of emission is left behind in the source. The\nmost minute examination of all the details of the lantern and its\nsurroundings at some time after the emission of the flash has not yet\nshown any phenomenon that betrays a remembrance of the emission of the\nflash, unless perhaps we measure the total energy or momentum and have\nsome way of knowing what the energy or momentum would have been if the\nflash had not been emitted, and in any event we cannot specify the\nmoment in past time when the signal was emitted. In the same way we\ncannot tell from an examination of a mirror whether it has at any time\nin the past reflected a beam of light. Consider now two systems, each\nconsisting of a source and a mirror distant 3 x 10^10 cm., identical in\nall respects, except that in one a light signal was flashed from the\nsource 1.5 seconds ago, and in the other only 0.5 seconds ago. According\nto our hypothesis, the most complete examination of source and mirror in\neither system fails to show the slightest difference, but nevertheless\nthere is something essentially different about the two systems, for in\none a light signal arrives at the screen in 0.5 seconds, and in the\nother not until 1.5 seconds. This violates what we have suggested might\nbe regarded as the cardinal and most general principle of all physics,\nthe principle of essential connectivity, which states that differences\nbetween two systems must be associated with other differences. A most\nobvious and simple way of maintaining our principle is merely to point\nout that the system really included more than we investigated: the\nsystem properly consists of source, mirror, screen, and all intermediate\nspace, so that if we had examined intermediate space we should have\nfound light there in transit in different positions in the two systems,\nthus correlating with the differences in subsequent history. This\nargument appeals to me as perhaps the strongest that can be advanced for\nthe view of light as a thing travelling. But it seems in no way\nconclusive. Our principle of essential connectivity made no mention of\nthe time concept, but we have somehow smuggled it in making the\napplication above. We sought to give a complete description of our\nsystem at some one instant of time, and this involved spreading the time\nconcept over space. This itself is a questionable operation and may be\ndone in different ways. But, more important, what is the justification\nfor supposing that the system can be completely described by giving a\ncomplete description of all the measurable parts of it at some one\nepoch? We have seen that in the most general case the principle of\nessential connectivity must recognize that the concept of \"initial\ncondition\" of a system involves all the past history, and we may have\nhere a case in point. The answer can be given only by experiment. In\ndealing with ordinary experience, when we do not have to distinguish\nbetween local and extended time, and are not dealing with optical\nphenomena, there can be little question that experience at least\napproximately justifies the expectation that future behavior is\ndetermined in terms of the present condition and that present condition\nmay be specified in terms of the results of present operations performed\nin the system. But before extending this principle to phenomena in which\nwe have to distinguish between local and extended time, we have to\nanswer just that question which we are now considering; namely, whether\nthere are physical phenomena taking place in apparently empty space, and\nwhether therefore empty space has to be included in the system. We find\nourselves again treading the vicious circle. Perhaps experience will\nshow that the extension of the principle of connectivity to optical\nphenomena involves something like this: namely, the future at any point\nin a material system is determined by a complete description of the\n_present_ state of the system in the immediate vicinity, and by a\n_history_ of the behavior at more distant points, this history extending\nover longer and longer intervals of time as the point becomes more\nremote.\n\nI believe, however, that these possibilities will not seem very\nsatisfactory, and that most physicists will discover in themselves a\nvery strong disposition to feel that the future is determined in terms\nof a complete description of some sort of instantaneous configuration,\ntime being extended in some suitable way over space. This instinctive\ndemand that the future be determined in terms of the present may easily\nbe consistent with the optical phenomena in our two systems consisting\nof source, mirror, and screen, without involving the material existence\nof light in empty space, provided that our assumption that the emission\nof light leaves behind it no permanent record in the source was\nincorrect. It may be that detailed examination of a source after\nemission will disclose permanent traces, from which the instant of\nemission may be found by extrapolation. If the conviction of determinism\nof the future is strong, the physicist may well be impelled to search\nhere for new phenomena indicating such a memory of emission.\n\nLet us now inquire how our physical structure might be affected if we\nshould give up the identification of light with a thing travelling. One\nconsequence is that light need no longer be thought of as having the\nproperty of velocity, since velocity, in terms of immediate experience,\nis a property of things moving from place to place. Giving up the\nconcept of light as a thing travelling would enable us, then, to adopt\nan alternative method of describing nature with a different concept of\nvelocity; we have seen that it is possible to define velocity in terms\nof operations different from the usual ones, in such a way as to give\nthe usual numerical results at small velocities, but different results\nat high velocities, and in particular to give an infinite velocity for\nlight.[26]\n\n[Footnote 26: No difficulty arises from the asymmetric character of\nlight in assigning an infinite velocity to light because those physical\noperations by which we discover which is the source and which the sink\nare entirely distinct from the operations by which a velocity is\nmeasured; or in other words, even in the limit, it still has meaning to\nsay that an infinite velocity has a direction associated with it.]\n\nThere is now no objection to an infinite number associated with light,\nif we no longer think of light as having physical velocity. We may, if\nwe like, continue for the sake of convenience to talk of the velocity of\nlight, clearly understanding that the infinite value which must be\nascribed to this velocity corresponds to the fact that the physical\nconcept of velocity does not apply in all respects to light. We should\nnow have to revise our process for extending the time concept over\nspace, because this was formerly so done as to give light a finite\nvelocity. We are now to make this velocity infinite, which is obviously\nto be done simply by setting a distant clock on zero at the instant it\nreceives a light signal flashed from our clock at its zero. The behavior\nof material things now takes on a simple aspect--there is no longer a\nfinite upper limit to the velocity that can be given a material thing,\nand light has no longer the paradoxical property of bearing the same\nfinite relation to each of two material systems which differ from each\nother by a finite amount (that is, the first postulate of relativity\nthat the velocity of light is 3 x 10^10 in all reference systems). Light\ninstead now bears the relation of infinitude to each of two systems\nwhich differ from each other by only a finite amount, and this is\nnatural from the mathematical point of view.\n\nHowever, all is not simplified by this change in the method of setting\nclocks, but a price has to be paid. The price is that we have to give up\nthe simple connection between the velocity of a thing and its \"go and\ncome\" time. Our changes have not affected local time; the time of\npassage of light to a distant mirror and return to the source is not\nchanged, and is therefore still finite, although we describe the\nvelocity of light as infinite. Now examination shows at once that there\nis no immediate connection between the concepts of \"velocity\" and of \"go\nand come\" time, because the operations involved are different. A\nmeasurement of a linear velocity according to our definition involves\ntwo clocks at two different places or else a clock travelling with the\nobject, while \"go and come\" time demands only a single clock at a single\nplace, and also involves necessarily a reversal of direction of motion\nin the object under measurement. We see then that, according to the\ndefinition adopted for velocity, we have the choice either of doing as\nEinstein did in the restricted theory of relativity and making \"go and\ncome\" time very simply related to velocity; or we may say that refined\nphysical measurements show that something of significance happens when\nthe direction of motion is reversed, and that phenomena are not\nsymmetrical with respect to a reversal of direction. The asymmetry which\nresults from reversing the direction of motion we may visualize as a\nsort of curvature in space and time, as of a small piece of an arc of a\ncircle bent back on itself, with the two ends diverging. This\nalternative way of treating velocity would mean that velocity can be\nmeasured simply only by a specially situated observer; this need not be\nconsidered disturbing, because in fact the operations have been defined\nonly with respect to such an observer.\n\nWhich of these two possible treatments of velocity shall be adopted is\nto a certain extent a matter of convenience, determined by the sort of\nphenomena in which we are most interested and wish most to simplify.\nEinstein's chief concern was with optical phenomena, so that the motive\nfor his choice is evident. In this choice of Einstein it is not very\nevident that the desire to make \"go and come\" time simply connected with\nvelocity played a very prominent part, but it seems rather that the\ndesire to think of light as a thing travelling, with a finite velocity,\nwas much more influential. This way of thinking of light is fundamental\nto all the treatment of restricted relativity; without this sort of\npicture all the mathematical deductions would lose their simplicity and\nconvincingness, for in all the deductions we inevitably think of\nourselves as an observer from outside, watching a thing that we call\nlight travelling back and forth like any physical thing.\n\nNow there can be no doubt that, when choice is possible, convenience and\nsimplicity are important considerations; but I believe that there is\nanother much more important consideration, namely, the most perfect\nreproduction possible of the physical situation. It seems to me that it\nis very questionable whether Einstein, and all the rest of modern\nphysics, for that matter, have not paid too high a price for simplicity\nand mathematical tractability in choosing to treat light as a thing that\ntravels. Physically it is the essence of light that it is not a thing\nthat travels, and in choosing to treat it as a thing that does, I do not\nsee how we can expect to avoid the most serious difficulties. Of course\nthe whole problem of the nature of light is now giving the most acute\ndifficulty. The thing-travelling point of view, even as treated by\nEinstein, does not land us in a situation which is at all satisfying\nlogically. We are familiar with only two kinds of thing travelling, a\ndisturbance in a medium, and a ballistic thing like a projectile. But\nlight is not like a disturbance in a medium, for otherwise we should\nfind a different velocity when we move with respect to the medium, and\nno such phenomenon exists; neither is light like a projectile, because\nthe velocity of light with respect to the observer is independent of the\nvelocity of the source. On the other hand, in aberration we have a\nphenomenon similar to that shown by projectiles. The properties of light\nare more like those of a projectile than is perhaps commonly realized,\nas is shown in the papers of La Rosa[27] on the ballistic theory of\nlight. The properties of light remain incongruous and inconsistent when\nwe try to think of them in terms of material things.\n\n[Footnote 27: M. La Rosa, Scientia, July-August, 1924.]\n\nEinstein's restricted relativity has made a great contribution in so\ngrouping and coordinating the phenomena that they can all be embraced in\na simple mathematical formula, but he does not seem to have presented\nthem in such a light that they are simple or easy to grasp physically.\nThe explanatory aspect is completely absent from Einstein's work.\n\nIn view of all our present difficulties it would seem that we ought at\nleast to try to start over again from the beginning and devise concepts\nfor the treatment of all optical phenomena which come closer to physical\nreality. No one realizes more vividly than I that this is a most\ndifficult thing to do. If we are ever successful in carrying through\nsuch a modified treatment, it is evident that not only will the\nstructure of most of our physics be altered, but in particular the\nformal approach to those phenomena now treated by relativity theory must\nbe changed, and therefore the appearance of the entire theory altered. I\nbelieve that it is a very serious question whether we shall not\nultimately see such a change, and whether Einstein's whole formal\nstructure is not a more or less temporary affair.\n\nAlthough it is exceedingly difficult to forsee what the treatment of the\nfuture will be like, it is easy to surmise certain of its features. In\nessence the elementary process of all radiation perceived as radiation\nis twofold. There is some process at the source and some accompanying\nprocess at the sink, and nothing else, as far as we have any physical\nevidence; furthermore, the elementary act is unsymmetrical, in that the\nsource and the sink are physically differentiated from each other. This\nis the most complete expression of the physical facts; there is nowhere\nany physical evidence for the inclusion of a third element (the ether).\nTherefore all the phenomena apprehended by an observer (and this\nembraces all physical phenomena) can be determined only by the source\nand the sink and the relation to each other of source and sink, for\nthere is nothing else that has physical meaning in terms of operations.\nThis formula covers not only the possibility of such first order\nphenomena as aberration and the D\u00f6ppler effect, but also shows that\nsuch second order effects as that looked for by Michelson and Morley\nmust be non-existent. It will thus be seen that some of the consequences\nof relativity theory are implicitly contained in certain very broad\npoints of view. One interesting question that must be answered before we\ncan get very far with a new treatment is whether the elementary optical\nprocess is of _necessity_ twofold, or whether we may have emission\nwithout absorption, that is, radiation into empty space. Lewis seems to\nimply in recent papers that this is not possible.[28] The astronomers\nhave already pointed out difficulties in explaining phenomena like the\ntemperature equilibrium of the planets if we suppose this is the case.\n\n[Footnote 28: For example, in the book: G. N. Lewis, The Anatomy of\nScience, Yale University Press, 1926, p. 129.]\n\n\n\n\nOTHER RELATIVITY CONCEPTS\n\n\nWe now turn to some of the other concepts of relativity. One of the most\nimportant of these is the \"event\"; in fact this concept is made\nfundamental by Whitehead.[29] We have already discussed the concept of\n\"event\" under the \"identity\" concept with which it is closely involved.\nThe event is usually thought of by Einstein as merely an aggregate of\nfour coordinates, three of space and one of time. The principle of\ngeneral relativity, namely, that the laws of nature shall be of\ninvariant form, when formulated mathematically, involves the assumption\nthat nature may be analyzed into events, and is expressed by the\nrequirement that the mathematical relations between the coordinates of a\nchain of events shall be invariant. The same idea is also expressed by\nEinstein in another form, namely, that nature may be completely\ncharacterized in terms of space-time coincidences. In elaborating this\nidea, Einstein assumes that the results of all measurement may be given\nin terms of such coincidences.\n\nNow it appears to me most questionable whether the analysis of nature\ninto events is possible or sufficient. With regard to the coincidence\npoint of view, it seems perfectly obvious that the world of our\nimmediate _sensation_ cannot be described in terms of coincidences; how,\nfor example, shall we describe in terms of space-time coincidence the\nphotometric comparison of the intensity of two sources of illumination,\nor the comparison of the pitch of two sounds, or the location of a sound\nby the binaural effect?\n\n[Footnote 29: A. N. Whitehead, An Enquiry Concerning the Principles of\nNatural Knowledge, Cambridge University Press, 1919, Chap. V.]\n\nTo justify the coincidence point of view we apparently have to analyze\ndown to the colorless elements beyond our sense perception. It does not\nseem unreasonable, perhaps, to expect that the universe is completely\ndetermined in terms of the positions as a function of time of all the\npositive and negative electrons; but to introduce such a thesis now\ncertainly goes beyond present experimental warrant, and is contrary to\nthe general spirit of relativity, which nowhere else involves any\nreference to the small scale structure of things. Even if we were\nwilling to overlook all these objections, we would still have the fact\nthat the difference between a positive and negative electron is not\ncontained in any specification of the mere coordinates.\n\nA further very important doubt in principle as to the possibility of the\nanalysis of nature into events is afforded by the character of the\nconcept of event itself. We have seen that the idea of event involves\nthe existence of discontinuities, and that this can correspond only\napproximately to the physical fact, because discontinuities apparently\nlose their abruptness as we make our measurements more refined. The\nthesis that nature can be described in terms of discontinuities of a\nvery small scale seems much too special to be made a fundamental part of\na theory of the general pretensions of that of relativity. In fact this,\nas well as a consideration to be mentioned later, suggests that the\nargument and result of general relativity may be intrinsically\nrestricted to large scale phenomena.\n\nWe now pass from these somewhat special questions to ask why it is that\nEinstein was able in the general theory of relativity to obtain new and\nphysically correct results from general reasoning of an apparently\npurely mathematical character. We are convinced that purely mathematical\nreasoning never can yield physical results--that if anything physical\ncomes out of mathematics it must have been put in another form. Our\nproblem is to find where the physics got into the general theory.\n\nThere are two questions to be disentangled here: we have to consider in\nthe first place the significance of the fact that Einstein has been able\nto describe relations in nature in mathematical form, and in the second\nplace of the fact that he was able to arrive at the mathematical\nformulation of these physical relations by reasoning of apparently a\npurely mathematical character, from postulates of merely formal\nmathematical content (invariance of natural laws in generalized\ncoordinates). Now the theory of relativity does not seem to differ in\nthe first respect from any other branch of mathematical physics, such as\nthe classical mathematical theory of electricity and magnetism, for\ninstance, and this matter has already been touched in an earlier\nchapter. It is a fact that the behavior of nature can in many cases be\nexpressed to a high degree of precision in mathematical language, and\nrelativity is not unique in this respect. In any event, we must not\nallow this possibility of mathematical formulation to obscure the\nessential fact that all physical knowledge is by its nature only\napproximate, so that we may expect at any time to find, when we have\ncarried our measurements to a higher degree of precision, that our\nmathematical expression of the laws was not quite exact, as seems now to\nbe the case with Newton's law of gravitation, for example. I do not\nsuppose that Einstein would claim that the statements of relativity\ndiffer in this respect from any of our other statements about nature,\nalthough apparently some of his followers see something more here. (From\nthe operational viewpoint the meaning to be attached to \"something more\"\nis somewhat obscure.)\n\nWith respect to the second question, we may stop to notice that the\nspecial theory stands in quite a different position from the general\ntheory. The special theory is much more physical throughout; its\npostulates are physical in character, and it is obvious that the physics\ngot into the results through the postulates. It seems to me without\nquestion that Einstein showed the intuitive insight of a great genius in\nrecognizing that there are mutual relations between physical phenomena\nwhich can be described in very much simplified language in terms of\nconcepts slightly modified from those already in common use. In view of\nthe remarks made on the nature of light, it is legitimate to wonder,\nhowever, whether the formulations of even the special theory will always\nstand. It seems to be true that _all_ the facts of nature, even in the\nabsence of a gravitational field, cannot be connected by the simple\nformulations of the special theory; that the physical relations are\nsimple only in a sub-group; and that if we wish to deal with _all_\noptical phenomena, we have carried our simplifications too far, for the\nemission of a light signal is not a simple event, and light is not in\nnature like a thing travelling. Just the sorts of physical thing which\nare ignored in treating light as the special theory does are coming to\nbe more and more important in the minds of physicists, and this is a\nreason for wondering whether ultimately Einstein's special theory may\nnot be regarded merely as a very convenient way of tying together a\nlarge group of important physical phenomena, but not as being by any\nmeans a full or complete statement of natural relations.\n\nWith respect to the general theory, however, I believe the situation is\nquite different. The fundamental postulate that the laws of nature are\nof invariant form in all co\u00f6rdinate systems is highly mathematical, and\nof an entirely man-made character. Of what concern of nature's is it how\nman may choose to describe her phenomena, and how can we expect the\nlimitations of our descriptive process to limit the thing described?\nFurthermore, Einstein's method of connecting his mathematical\nformulation and nature by way of coincidences of 4-events (three space,\none time coordinates) seems to be very far removed from reality, since\nit entirely leaves out the descriptive background in terms only of which\nthe 4-event takes on physical significance. Nevertheless, three definite\nconclusions about the physical universe have been taken out of the hat\nby the conjuror Einstein (shift of the perihelion of Mercury,\ndisplacement of apparent position of stars at the edge of the sun's\ndisk, and the shift toward the infra-red of spectrum lines from a source\nin a gravitational field), and the problem for us as physicists is to\ndiscover by what process these results were obtained.\n\nAn examination of what Einstein actually did in deriving his results\nwill show, I believe, that the situation is really different from that\nsuggested above. In the first place, the requirement that the laws of\nnature be of invariant form actually places no restriction, as any one\ncan see by setting himself the task of expressing, for example, an\ninverse cube law for gravitation in terms of generalized coordinates.\nThe work of expressing such a law can be attacked in a perfectly routine\nway. (The essential difference between the invariability requirements of\nthe special and general theories is to be noted; the special theory\nrequires that the velocity of light, for instance, have the same\n_numerical_ value in all allowed systems: the general theory merely that\nall laws have the same _literal_ form, but with variable numerical\ncoefficients.) But, as Einstein says, if any one actually attempts to\ncarry through the work of expressing an inverse cube law in generalized\ncoordinates, he will find the task prohibitively complicated, and will\nseek for some simpler formulation. What Einstein actually did,\ntherefore, was to require that the laws of nature be _simple_ in\ngeneralized form. Now we know that the law of gravitation as formerly\nexpressed in ordinary coordinates as an inverse square law was\napproximately exact, and was also simple. Any deviations from this law\nare small, and all experience leads us to expect that to the first order\nof small quantities the deviations can be taken care of mathematically\nin the form of small correction terms to this law. This by itself gives\nnothing, however, because a small correction term can be added to our\nequations in an infinite number of ways. If, however, we know that the\nequation must be of a certain type after the correction terms have been\nadded, the possibilities are so much restricted that the form of the\ncorrection term may be determined. In arguing as to the probable type of\nthe equation, Einstein advanced the considerations by which physics gets\ninto the situation.\n\nIn the first place, the special theory had prepared us for the\npossibility of finding that our measuring instruments might be modified\nin a gravitational field, analogously to the shortening of a meter stick\nwhen set into motion. In fact, special theory had indicated that in an\naccelerated system the modifications might be too complicated to be\ntreated by that theory. In the absence, then, of specific information we\nmust be prepared for the most general possible alteration in space-time\nin a gravitational field. In describing space-time we must therefore use\nco\u00f6rdinates adapted to handling the most general possible relations,\nand these are the generalized co\u00f6rdinates of Riemann, which had been\nalready discussed by mathematicians. Going back now to Einstein's\ncriterion that the equations are to be simple, we have the demand that\nthe equations be simple in generalized co\u00f6rdinates, and of course they\nmust also reduce to the ordinary equations (that is, the equations of\nspecial relativity) in space where there is no gravitational field. In\ndeciding the further question as to what the type of equation probably\nis, we are influenced by considerations of convenience as well as by\nphysical considerations. Practically the only type of equation that can\nbe handled mathematically is linear, so that we shall certainly try\nfirst whether this type of equation may not continue to hold. Now the\nNewtonian law of the inverse square may be expressed in terms of a\nlinear differential equation of the second order in the old Cartesian\nco\u00f6rdinates (Poisson's equation), so that our most immediate suggestion\nis that the equations remain linear and of the second order in\ngeneralized co\u00f6rdinates. As a matter of fact, this requirement turns\nout to be sufficient to determine the small correction term by which the\nordinary equations can be generalized; Einstein's papers must be studied\nto see how this works out in detail.\n\nAll this looks pretty mathematical, but as a matter of fact there is\nmuch physical content, because systems which can be described by linear\nequations of the second order have definite physical properties. The\nrequirement that the equations be linear corresponds to one of the most\nfundamental properties of our universe--the causality concept would not\nbe possible or would be much modified in a universe governed by\nnon-linear equations, for the joint effect of two causes acting together\nwould not be the sum of their effects acting separately, so that the\nanalysis of a situation into simple elements would be impossible and the\ncausality concept probably would not have arisen. Furthermore, an\nequation of the type of Poisson of the second order means that there are\npropagation phenomena, and equations of mechanics of the second order\ninvolve the existence of a scalar energy function. If, then, the\nbehavior of the universe can be described by differential equations at\nall, these equations must be linear of the second order if the universe\nis to have the broadest physical characteristics of our own universe.\nWhat Einstein really did, therefore, was to demand that even when\nspace-time is warped by the presence of a gravitational field, those\nphysical phenomena which can be described in terms of differential\nequations continue to be described by linear differential equations of\nthe second order; that is, that nature continues to be describable in\nterms of a causality concept, with propagation phenomena, and a simple\nenergy function. The consequences of a guess like this about the\nproperties of nature appeal to our physical intuition as being worth\nfollowing out, and of course we know the experimental justification.\n\nSeveral general comments may now be made on the structure reached in\nthis way. In the first place, the whole structure is only descriptive in\ncharacter; we find certain correlations in nature which we describe with\nconsiderable completeness in mathematical equations, without introducing\nany new element of explanation or of mechanism. We have seen that as we\nincrease our range from the realm of ordinary phenomena to phenomena of\ndifferent character we arrive at a stage where for a time the process of\nexplanation apparently halts, and we have to be satisfied with a\nstatement of mere correlation between elements; later, however, these\nelements may be accepted as the ultimates in a broadened scheme of\nexplanation, and the explanatory process resumed. Are we at such a stage\nnow with the general relativity theory, and may later a new scheme of\nexplanation be established based on the correlations of Einstein? This\nis of course a matter of individual judgment; I personally question\nwhether the elements of Einstein's formulation, such as curvature of\nspace-time, are closely enough connected with immediate physical\nexperience ever to be accepted as an ultimate in a scheme of\nexplanation, and I very much feel the need for a formulation in more\nintimate physical terms.\n\nIn the second place, we must repeat the comment already made in\ndiscussing time, namely, that there is still a very wide gap between the\ntheory and its physical application, in that we have no way of\nidentifying our physical clocks and our physical measures of time with\nthe thing called time in the formulas. This gap must be filled by a\nspecification of the physical structure of a clock.\n\nIt has always been very puzzling to understand why Einstein has so\nstrenuously insisted that the shift toward the infra-red is an integral\npart of the general theory, and that if the shift is not found, the\ntheory must fall. In other words, Einstein insists that the assumption\nthat an atom is a clock is an integral part of his theory. I believe\nthat this attitude may be due to a realization by Einstein of that very\nflaw in the logical structure which we are now emphasizing. In the\nabsence of any method of specifying the details of construction of at\nleast one clock, relativity becomes a purely academic affair, unless\nthere exist in nature concrete things which may serve as clocks.\nEinstein _must_ either be able to tell how to construct a clock, or else\nbe able to point to a specific example of a clock. He chose the atom as\nthe specific thing. Doubtless the reason was the apparent simplicity of\nthe vibrating mechanism of an atom, as shown by the precise equality of\nthe frequencies emitted by all atoms of the same element. If the atom is\nnot a clock, where in nature can one be found? But in the last few years\nwe have come to appreciate the exceedingly complicated quantum structure\nof an atom, and Einstein's thesis loses much of its instinctive appeal.\n\nSince Einstein created the theory of relativity, it is perhaps\nungracious to question his right to stipulate that the assumption that\nthe atom is a clock is an integral part of the theory. This, however,\ndegenerates to a mere matter of language, and does not touch the\narbitrary nature of the procedure. It does not prevent us from having a\nsecond brand of relativity theory, that of X instead of Einstein,\nexactly like that of Einstein except that perhaps now the \"clock\" is\nconstructed in terms of the life period of a radioactively\ndisintegrating element. The only way to eliminate the arbitrariness\nseems to be to postulate that _all_ natural processes, which run\nnaturally of themselves independently of what we may do, may equally\nwell serve as clocks and give the same results. But in answering the\nquestion of the operational meaning of \"independently of what we may do\"\nwe shall effectively have to answer the question of what is a clock.\nThis point of view may possibly, however, get us a little nearer to our\ngoal of finding how to specify the structure of a clock.\n\nFinally, the general theory is not completely general, but applies only\nto a certain range of phenomena, just as we saw that the special theory\ndoes not embrace all optical phenomena. The general theory applies only\nto those phenomena which can be described in terms of differential\nequations, that is, par excellence, to large scale phenomena. If quantum\nphenomena cannot be described by differential equations,[30] as\napparently now they cannot, general relativity cannot by its very nature\nbe applicable. General relativity does not give us a comprehensive\nformulation of the behavior of all nature, and as far as we can see, we\nare still as far as ever from such a general formulation.\n\n[Footnote 30: This statement now takes on a very questionable aspect in\nview of the new quantum wave mechanics (March, 1927).]\n\n\n\n\nROTATIONAL MOTION AND RELATIVITY\n\n\nPhysically there is a great difference between the behavior of systems\nin uniform relative rectilinear motion and those in uniform relative\nrotation. The special theory of relativity states that there is a triply\ninfinite number of systems with all possible uniform rectilinear\nvelocities with respect to each other, in all of which physical\nphenomena have exactly the same mutual relations, that is, natural laws\nare the same. Now the mere formulation of the principle suggests the\nsense in which \"system\" is here used. It is obvious that \"system\" refers\nonly to a part of the universe; we are not making a hopelessly academic\nstatement about what would happen if we had an infinite number of\nuniverses to experiment with, but are talking about operations that may\nbe approximately realized in our own universe. The \"system\" of the\nformulation we may think of as a completely equipped laboratory, out in\nempty space, so far from the heavenly bodies that they can have no\neffect. The different systems of the formulation are different\nlaboratories, all built to exactly the same architectural blue prints.\nThe phenomenon to which the postulates of relativity apply are phenomena\nwhich pertain entirely only to one or another of these laboratories. The\nmeaning of this restriction is not completely definite and has, in any\nspecial case, to be judged partly by the context. Obviously, to see from\nthe window of a laboratory another laboratory passing with a certain\nvelocity cannot be counted as one of the allowed phenomena. Still less\nis it one of the allowed phenomena to observe that the center of gravity\nof the entire stellar universe has a certain velocity of translation\nwith respect to the laboratory. The special principle of relativity\ncontains by implication therefore the statement that certain very large\nand important classes of physical phenomena may be isolated and treated\nas taking place unaffected by the rest of the universe. Granted now the\npossibility of isolation, we have a second statement, which is usually\ntreated as if it were the entire statement of the restricted principle,\nnamely, that there is a triply infinite set of systems in which these\nphenomena run in the same way independent of the relative motion of the\nsystems with respect to each other. When once the significance of the\nobservation is grasped that absolute motion has no meaning in terms of\noperations, we see that this last statement takes immediately a most\nsimple and satisfying aspect, in fact, so simple and inevitable that we\nare inclined to see in this the complete essence of the situation and\nregard the meaninglessness of absolute motion as affording peremptory\nproof of the restricted principle.\n\nWith this bias we now turn to examine the facts of rotary motion, and\nare disconcerted to find them quite different. No meaning in terms of\nmeasuring operations can be given to absolute rotary motion any more\nthan to absolute translation, but nevertheless phenomena are obviously\nentirely different in different systems in relative rotary motion\n(phenomena of rupture, for example), so that apparently there are\nphysical phenomena by which the concept of absolute rotary motion might\nbe given a certain physical significance. Given two worlds like our own\nin empty space, but surrounded by impenetrable clouds, and each provided\nwith a Foucault pendulum, then we believe that it is physically possible\nthat we may find on one of these worlds the plane of rotation of the\npendulum gradually changing in direction, while on the other it remains\nstationary. This difference we regard as possible without other\naccompanying physical phenomena which are causally related to the\nrotation of the pendulum (of course we have to make the two worlds of\ninfinitely rigid material and eliminate other phenomena which we regard\nas purely incidental), so that we apparently have here a contradiction\nof our cardinal physical principle of essential connectivity. We are\ncertainly not inclined to give up our principle, and we believe that as\na physical fact, if the clouds could be evaporated, an observer in one\nworld would find that he was rotating with respect to the system of the\nfixed stars, whereas the corresponding observer on the other world would\nfind that he was stationary. Our principle of essential connectivity is\ntherefore maintained, in that the rotation of the plane of the pendulum\nis connected with a rotation with respect to the rest of the universe of\nthe entire world in which the pendulum is mounted. As far as I am aware,\nno other way of maintaining our principle has ever been suggested. But\nthis demands that we give up our physical hypothesis of the possibility\nof isolating a system. There is here no question of limiting behavior;\nwe believe that no matter how far our rotating world gets from the rest\nof the universe the Foucault pendulum would always behave in the same\nway; the system can never be isolated, but such local phenomena as the\ninvariance of the plane of the pendulum are always essentially\ndetermined by the rest of the universe.\n\nIf now our system cannot be isolated, we must return to the phenomena of\ntranslational motion. In principle the act of isolation cannot be\nperformed, the rest of the universe cannot be disregarded, and we should\nexpect that different states of translational motion as well as\ndifferent states of rotational motion with respect to the rest of the\nuniverse would have an effect on phenomena. We set ourselves the problem\nof understanding this apparent enormous difference between phenomena of\ntranslation and rotation. We remark that what apparently is a difference\nin principle may, in virtue of the approximate character of all\nmeasurement, be only a difference in magnitude, and that translational\neffects may exist too small to detect. A physical basis for such a\ndifference may be found in the enormously different numerical values of\ntranslational and rotational velocities with respect to the rest of the\nuniverse attainable in practice. In describing phenomena of cosmic\nmagnitude, we may plausibly measure the phenomena in units commensurable\nwith the scale of the phenomena. Thus in measuring linear distances, we\nmay perhaps choose as the unit of length the diameter of the stellar\nuniverse, and in measuring rotation, a complete reversal of direction\nwith respect to the entire universe. This last means a change of angular\norientation of 2 \u03c0, the first means a length of the order of 10^6 light\nyears. Measured in such cosmic units the angular velocities attainable\nin practice are incomparably greater than linear velocities. We now see\nthat it is possible that the real state of affairs is as follows:\nnamely, phenomena in any system are affected by motion with respect to\nthe entire universe, whether that motion is of translation or of\nrotation, and the magnitude of the effect is connected with the velocity\nof the motion by a factor which is of the general order of unity when\nvelocity is measured in cosmic units. This last is merely an application\nof the argument so often made in physics as to the order of magnitude of\nunknown numerical factors, and will be found expanded on page 88 of my\nbook on _Dimensional Analysis_. The linear velocities attainable in\npractice are now so exceedingly low that their effect has not yet been\ndetected experimentally, but angular velocities are high, and the effect\nis easily demonstrable. In this light the special principle of\nrelativity is no different in character from any other physical law; it\nis only approximate, and some day our measurements may become refined\nenough to detect its limitations.\n\nWe have made a hypothesis here, which we may call the hypothesis of the\nimmanence of the entire universe, namely, that isolation is impossible,\nor that the rest of the universe, no matter how distant, always has a\nlocal effect on at least some phenomena. This is essentially the\nhypothesis of Mach,[31] and leads to a situation which can, I think, be\ncontemplated with logical equanimity, although it has always seemed to\nmany physicists most highly antiphysical in character.\n\n[Footnote 31: E. Mach, The Science of Mechanics, translated by\nMcCormack, The Open Court Publishing Co., Chicago, 1893. See especially\np. 235.]\n\nIt must certainly be admitted that most physical experience justifies us\nin thinking that effects may be made as small as we please by getting\nfar enough away from the cause of the effect. But if we accept the\nconsiderations of the preceding pages, we must be prepared to admit that\nas phenomena change in range their character may change, and that in\nthese new realms we must, at first at least, be satisfied with a mere\nstatement of correlations. Certainly we have very strong physical\nevidence of a formal correlation between the Foucault pendulum and the\nrest of the universe. But a correlation of this sort may be without\nsignificance because of its very breadth; we never can prove the\nsignificance of the correlation by performing an experiment with the\nrest of the universe absent. Have we really done anything more than\nmerely get things into such a formal situation that they cannot be\nassailed, a possibility which the mere laws of our thinking seem always\nto leave open, as has been suggested, or is there any physical content\nto what we have done? We have seen that if our correlation is also\nsuggested by other phenomena, then we may accept it as having physical\ncontent. Now there is just a glimmer of a suggestion that our hypothesis\nof the immanence of the universe may be needed in other ways. The\ngravitational constant and the velocity of light are always treated as\narbitrary magnitudes thrust on the universe from outside with no\nconnection with other phenomena. Nevertheless, I suppose that no one\nregards this situation as ultimately satisfactory and does not entertain\nthe hope that some day we may be able to give some sort of account of\nthe numerical magnitude of these constants. We have not hitherto\nsucceeded in finding any connection between these constants and small\nscale phenomena such as the charge on the electron, its mass, etc., so\nthat there is some plausibility in expecting that a connection may be\nsometime found with cosmic things; indeed general relativity theory\nalready prepares us for exactly this possibility. Now the velocity of\nlight and the gravitational constant control small scale experiments,\nfor of course these two constants can be measured by local experiments,\nso that if the cosmic connection is found, we should have a control of\nlocal behavior by cosmic things, and therefore another example of the\nimmanence of the entire universe. There is no need for me to waste time\nin apologizing for the highly speculative character of all this. It is\nworth while to emphasize, however, that our general considerations on\nthe meaning of \"explanation\" have prepared us to admit as reasonable\njust the sort of explanation contained in the hypothesis of the\nimmanence of the universe, and therefore to reserve a place in our\nphysical thinking for possibilities of this sort, in spite of the fact\nthat such considerations are not usually entertained, and may seem to\nmany opposed to the spirit of physics.\n\n\n\n\nQUANTUM CONCEPTS[32]\n\n\nThe history of quantum theory up to the present is a repetition in many\nrespects of that of the early theories of electricity, in that all our\nthinking has been in mechanical terms. As far as we now know, quantum\nphenomena are always associated with atoms. We make for the atom a\nmental model with all the properties of the mechanisms of the ordinary\nscale of magnitude and with a few impressed properties in addition which\nrepresent the new quantum relations. As we now think of it, the atom has\na massive core about which electrons revolve under an inverse square\nlaw, the connection between the mass of the electron, its acceleration,\nand the force acting on it being that usual in Newtonian mechanics.\n\n[Footnote 32: This section was written early in 1926 without access to\nrecent literature. Our attitude toward quantum phenomena has been so\nmuch changed since then by the \"new\" quantum mechanics, that a number of\nthe following statements are superseded as a statement of present\nopinion. However it has seemed worth while to let the section stand as\nwritten, because many of the developments actually taken in the new\nmechanics follow the lines that it is here urged they ought to take, and\nin so far afford interesting confirmation of the point of view of this\nessay.]\n\nThe space in which the electron circulates is thought of as Euclidean,\nand the motion is described in time, which may be measured with clocks\nin the usual way. The general equations of electrodynamics do not apply;\nthere are no propagation effects inside the atom, the motion of the\nelectrons does not produce a magnetic field, and there is no radiation\nwhen the electron is in one of its possible stable states, in spite of\nthe acceleration. We may, if we please, in working out the character of\nthe motion, entirely neglect the electrical origin of the inverse square\nlaw, and treat this merely as an impressed force without further\nimplications. Superposed on the ordinary spatial, temporal, and\nmechanical characteristics of the model are additional quantum\nproperties, one which determines the particular orbit in which the\nelectron moves [\u222b pdq = nh], and another which determines the\nfrequency of the radiation emitted when the electron passes from one\nallowed orbit to another. No mechanism is suggested to account for these\nquantum conditions, although the conditions are formulated in mechanical\nterms.\n\nWe now have to ask what is the meaning in terms of operations of our\nusual concepts of space-time and mechanics when applied to phenomena of\nthis order. It is of course evident, as has already been emphasized,\nthat the concepts have entirely changed in character, because we do not\nmeasure an electron orbit, for example, by stepping off the diameter\nwith meter sticks, or by measuring the time required for light to travel\nacross the diameter. The particular feature of immediate interest in\nthis changed situation is the change in number of our concepts on the\natomic level. I shall not attempt to find by an exact analysis the\nnumber of independent concepts at this level; probably such an analysis\nis not possible. We may, however, make an approximate suggestion.\nApparently the most important concept in describing relations inside a\nquantum system corresponds to that of energy on the ordinary scale.\nChanges of energy determine the frequency of emitted radiation, as well\nas the relations during collisions of atoms and electrons; these\ncollisional relations make direct connection with experiment through the\nvoltages applied to electrons in collision experiments. The analogue of\nthe momentum concept also seems to have independent significance, as\nshown by the Compton effect. The frequency of emitted radiation is also\nsomething with independent experimental significance. I believe that\nthese three things are all that have direct significance for quantum\nexperiments made up to the present time. In any event, it is perfectly\nevident that on the quantum level the concepts which at present have\noperational significance are considerably fewer than on the level of\nordinary experience.\n\nApart from the question of convenience, there may be justification in\ncontinuing to use our old mechanical forms of thought if new\nexperimental relations are thereby suggested. That a very large number\nof such as yet undiscovered relations may be suggested in some such way\nis at once evident. Thus we have no present knowledge of any phenomenon\nassociated with what the electron does when passing from one energy\nlevel to another. How long does it take to make the passage? What is its\npath during passage? Is it subject to the ordinary laws of\nelectrodynamics during passage? When and where is the radiation emitted\nthat corresponds to passage? When the electron leaves one stable orbit\nis the orbit on which it will eventually land already determined? Does\nthe radiation train emitted during a change from one energy level to\nanother have a definite length in space, or may it have a variable\nlength and correspondingly something that corresponds to variable\namplitude? What happens to the radiation when the electron passages are\ninterfered with before the emission of a quantum has been completed?\nWhat is the mechanism by which the quantum conditions are imposed? Is it\nnot possible that part of the clew to the riddle of the manner of\ntransition from purely quantum behavior to the behavior of classical\nmechanics may be found in the behavior of the electron during passage\nfrom one energy level to another? Certainly we have a tendency to the\nclassical behavior under those conditions, such as at high temperature\nor in strongly condensed systems, in which the time occupied in passage\nmight be expected to become a more important part of the total time.\n\nCorresponding to these questions there should be many as yet\nundiscovered phenomena, and the mechanical point of view therefore has\nits value in suggesting experiments to detect such effects. It is of\ncourse too early to see what the final result will be here; we cannot\ntell whether eventually enough new experimental kinds of behavior will\nbe found to restore the number of independent concepts to that of the\nlevel of ordinary experience or not, or whether indeed it will turn out\nthat a greater number of concepts is required. It is contrary to our\ninstincts to expect a greater number, and a smaller number now seems to\nus not unnatural, but the considerations of this essay should prepare us\nfor either possibility.\n\nIt is often said that quantum phenomena are inconsistent with ordinary\nmechanics, and proofs of this assertion are often offered. I believe\nthat no such proof, in the spirit in which the attempt is usually made,\ncan be correct, for it seems to me that the remark of Poincar\u00e9 applies,\nnamely, that any sort of behavior can be imitated by a mechanical\nsystem, provided it is only complicated enough. A peremptory proof of\nthis can be given to any one who is not a believer in vitalism. If a\nsentient being can be regarded as a mechanical system, we merely have to\nstation inside each atom a Maxwell demon, with instructions to make the\natom react according to quantum rules. Opposed to the spirit of this\nsort of reduction of quantum phenomena to mechanical terms, we have to\nremember that it makes sense to talk about the character of our\nconceptual structure only when the number of concepts is reduced to the\nnumber that have independent operational significance, that is, to the\nminimum number.\n\nIn the meantime let us examine what may be the significance in the light\nof present experiment of statements like those ascribed to Bohr that our\nusual concepts of space and time may be inapplicable in dealing with\nquantum phenomena. This idea is often given the more explicit form that\nspace and time may be essentially discontinuous at the quantum level.\nFrom the operational point of view, it is most difficult to see exactly\nwhat this more explicit statement means, at least in terms of those\noperations by which length and time were originally defined. Thus if\nspace were discontinuous, it might mean that a point exists which may be\nreached by laying off a meter stick fourteen times, for example, and\nanother point by laying off sixteen times, but that no point can be\nfound with fifteen applications. Such a state of affairs seems to be\ninconsistent with our definition of the counting operation and to have\nno concern with any properties of space; for what shall we mean by\nlaying off a meter stick sixteen times if it cannot be laid off fifteen\ntimes? It is conceivable that space might end, in the sense that beyond\na certain limit there might be some irremovable physical hindrance to\nthe continued laying off of distances with a meter stick (although I\nthink that we should be inclined to describe such a state of affairs in\nterms of matter enclosing empty space rather than as the end of space),\nbut to say that space may be discontinuous seems to be meaningless. In\nthe same way, I believe it meaningless to speak of discontinuous time.\nWe may have phenomena discontinuous in space and time, but not\ndiscontinuous space or time.\n\nIt seems then that we must give up the idea that in the quantum domain\nthe usual concepts of space and time may fail, in the specific sense\nthat they may become discontinuous. What may we understand by the\nfailure of these concepts in a more general sense? No one of course\nwould expect that even eventually the concepts will have the same\noperational significance for the inside of an atom that they have on the\nordinary scale; it must be a modified sort of concept with which we are\nconcerned, such as we have already seen is given by the field equations\nof electrodynamics. If now the number of operationally independent\nconcepts on the quantum level turns out to be the same as on the level\nof ordinary experience, and if there is also the possibility of\ncontinuous transition from the operations of the quantum domain to those\nof ordinary experience, then it seems to me that we should say that our\nusual concepts of space and time still apply in the quantum domain. But\nif the number of operationally independent concepts is either greater or\nless than on the ordinary level, then I believe we must say that the\nordinary concepts of space and time cannot apply. One might still look\nfor the possibility of separating out from the complex of concepts on\nthe quantum level a group which might change continuously to those of\nspace and time on the ordinary level, but I think that such a\npossibility is very remote when one considers that the total number of\nconcepts changes, and that in the zone where the number changes the\ndefinitions are not unique by which one extrapolates a concept from one\ndomain to another.\n\nIf Bohr's idea is true that space and time cannot be used in describing\nultimate quantum phenomena, one of the most immediate implications in\nterms of experiment might be that phenomena corresponding to\nintermediate positions of the electron between stable orbits do not\nexist.\n\nFinally, we must comment on the general tactics of the quantum\nsituation. It would seem that there have already been a sufficient\nnumber of unsuccessful attempts to formulate quantum behavior in terms\nof ordinary mechanics to justify the expectation that ultimately\nsomething quite different must evolve. The difficulties of an unmodified\ncarrying over of ordinary mechanical notions to quantum phenomena may be\nillustrated by a simple example. Consider a particle of mass _m_\nrotating in a frictionless circular track of radius _r_. Then according\nto quantum conditions it can move stably on this track only with certain\ndefinite velocities, such that \u222b pdq = mv 2\u03c0r = nh. Suppose now the\nparticle rotating with one of the allowed velocities, and a tangential\nforce applied. If the usual mechanical notions of force are still valid,\nthe particle must respond by moving in its track with continually\nincreasing velocity. After the velocity has been increased by a small\namount, we remove the force. The motion is now no longer one of the\nallowed ones, and the particle must in some way change its velocity; it\nmust either slow down or speed up. In the first case it must either\nradiate energy, which a system of the simple mechanical properties we\nhave supposed is not capable of doing, or else the law of conservation\nof energy fails, and also Newton's first law of motion during the\nprocess of acquiring the steady condition. If, on the other hand, the\nparticle speeds up, it must increase its energy from nowhere, and again\nordinary mechanics does not apply.\n\nIt seems then a mistake to attempt to formulate the quantum conditions\nin terms of the notions of ordinary mechanics (momentum, and position\nco\u00f6rdinates in either the ordinary or the generalized Lagrangean\nsense). It would seem, on the other hand, plausible to expect that\nmechanics is not a fundamental thing, but is in some way an effect\nproduced by the aggregate action of a great many elementary quantum\nprocesses. Amplitude of radiational vibration, for example, may be such\na statistical aspect of a great many processes, in some such way as on\nthe ordinary level of experience temperature is a statistical aspect of\nthe average kinetic energy of the atoms. One possibility of this kind\nhas already been more explicitly indicated; in the elementary process of\nemission of radiation, frequency and energy are not two independently\nassignable variables, but are connected [E = h\u03bd]. That is, on the\nquantum level radiation has only a single property, which is properly\nneither energy or frequency. [We are now neglecting the polarization\naspect of radiation.] On a higher level, that of ordinary radiation, the\nsingle elementary property has expanded itself into two (energy and\nfrequency) through the additional variable of the number of elementary\nquantum processes in the complex radiation.\n\nThe program of the immediate future should be an extension of something\nof this sort, namely, to invent new concepts corresponding to the\nexperimentally independent things on the quantum level (such perhaps as\nthe resultant of the fusion of the energy and frequency concepts for\nradiation), and then to show how the ordinary concepts of mechanics (and\nvery likely those also of space and time) are generated by statistical\neffects in aggregates of great numbers. Perhaps it is yet too early for\nan attempt of this sort, because it may seem that there are still too\nmany possibilities of new experimental discoveries which might upset the\nresults of elaborate theoretical speculation. If this should really be\nfelt to be the case, I believe that physics ought for the present to\nhold in partial abeyance its theoretical activities in this field, and\ndevote itself to acquiring as rapidly as possible the necessary\nexperimental facts. We may emphasize again that the possibility of\ncarrying out this plausible program can be proved only by experiment; it\nmay be that more concepts will be required on the quantum level than for\nordinary experience.\n\nThe invention of new concepts is certainly not an easy thing, and is\nsomething which physics has always deliberately, and perhaps\njustifiably, shirked, as shown by the persistent attempts to carry the\nnotions of mechanics down into the finest structure of things. This\nshirking has not had bad results, but on the contrary good results, as\nlong as physics has been primarily concerned with phenomena near the\nrange of ordinary experience, but I believe that as we get farther and\nfarther away from ordinary experience, the invention of new concepts\nwill become an increasing necessity.\n\n\n\n\nCHAPTER IV\n\nSPECIAL VIEWS OF NATURE\n\n\nIN this last chapter we propose to discuss certain special hypotheses\nabout the structure of nature, and certain other matters that could best\nbe left until we had examined our fundamental concepts.\n\nWe have seen that in setting up the general rules which are to guide us\nin describing and correlating nature, we have to take extreme care to\nallow no special hypotheses to creep in, as otherwise we might be\nrestraining possible future experience. Even here there is no hard and\nfast line of separation of the general from the special, and one might\nentangle himself in inextricable difficulties if his ideals were too\nmeticulous. How for example is the critic to be answered who says: \"Your\nvery endeavor to formulate principles so broad as not to restrict future\nexperience means, when examined in the light of operations, that you are\nseeking for principles which _past_ experience suggests will not limit\nthe future. It is in the very nature of things impossible to escape all\nthe implications of past experience and therefore to find any completely\ngeneral principle.\" I believe that we must admit the critic is right,\nand that rigorously our goal is impossible of attainment. We may say in\npartial self-defense that all the discussion of this essay has been\nsubject to one explicit assumption, namely, that the working of our\nminds is understood, which of course involves the assumption that our\nminds continue to function in the future in the same way as in the past.\nEven with this proviso we can not rigorously avoid the implications of\nthe past, but there can be no practical question that we recognize\ncertain assumptions about the behavior of nature to be so special as to\nlimit seriously the physical possibilities, and other assumptions to be\nless restricting. In the previous discussion we had to make assumptions,\nbut I hope these assumptions will be recognized by all with physical\nexperience to be so broad as not to restrict us seriously. More special\nassumptions or hypotheses have their very great use, however, when we\nattempt to push forward the domains of experimental knowledge, because\nthey may suggest new experiments or aid in correlating information\nalready obtained. These special hypotheses may cover a very wide range\nof generality; some of them are general enough in character to be\ndiscussed here.\n\nAmong these special hypotheses there is a group which play an important\npart in the speculation of most physicists, and which have features in\ncommon. These are: the hypotheses of the simplicity of nature, of the\nfiniteness of nature in the direction of the very small, and of the\ndeterminateness of the future in terms of the present. That these views\nhave points of similarity is obvious if we consider a hypothetical\nspecial case. Suppose that no physical structure beyond the electrons\nand protons can be discovered, or is even suggested by any known\nphenomenon, so that the entire future behavior of a system can be\ndetermined by a specification of the present relations of all its\nprotons and electrons; in this case nature would be both simple and\nfinite and the future determined by the present.\n\n\n\n\nTHE SIMPLICITY OF NATURE\n\n\nOf these hypotheses, perhaps the most important is that of the\nsimplicity of nature, because of its wide spread diffusion and the\neffect it has had on physical thought. The hypothesis of simplicity\nassumes several forms; some physicists are convinced that the laws which\ngovern nature are simple, others that the ultimate stuff of which nature\nis composed is simple (perhaps protons and electrons and energy), or\nthere may be a combination of both views into the belief that ultimately\nwe shall find simple ultimate elements behaving according to simple\nlaws. In one respect it is obvious that nature is not simple, namely\nnumerically--try counting the electrons or atoms or stars!\n\nConsider now the first of these aspects of the thesis of simplicity,\nwhich may be expressed as the conviction that the behavior of the entire\nuniverse can be comprehended in a few principles of great breadth and\nsimplicity, such as the inverse square law of force, or the second law\nof thermodynamics, or perhaps still better the equality of the\nelementary positive and negative charges, which apparently holds to an\nenormous degree of precision. In explanation of a view like this there\nis in the first place the mental urge, because we can take a\nsatisfaction almost \u00e6sthetic in contemplating such a universe, and\nthere is in the second place a strong suggestion from experience.\nPractically all the history of physics is a history of the reduction of\nthe complicated to the simpler. For example, the behavior of a large\npart of the world of immediate experience can be reduced to the simple\nlaws of mechanics. The behavior of another very large group of natural\nphenomena can be reduced to thermodynamics. The behavior of the heavenly\nbodies, which at first was described in a rather complicated way in the\nPtolemaic system of astronomy, can be reduced to those same laws of\nmechanics which we find in our immediate neighborhood, with the one\naddition of the universal law of gravitation, which later refined\nexperiment discloses is really active in our immediate surroundings.\nSimilarly the laws of thermodynamics (except that part dealing with\nradiation) are reduced to the ordinary laws of mechanics through the\nadditional assumption of the atomic structure of matter. Truly a\nstupendous accomplishment that may well color our whole future outlook.\nOne may find great justification here for the belief that all nature\nwill ultimately be reduced to a similar simplicity, and, in particular,\njustification for the attempt to find the explanation of all nature in\nthe action of mechanical laws. Now, of course, as a matter of physical\nand historical fact, this program could not be carried through, but\nobdurate physical phenomena were discovered. Electric phenomena, which\nat first seemed so promising, refused to fit into the scheme, and the\nconverse attempt, to explain mechanical effects in terms of electrical\neffects, also failed. We still carry our ordinary mechanical notions\ndown into the realm of small electric effects, and still talk, for\ninstance, about non-electrical forces which hold an electron together.\nNor are there experiments affording sufficient basis for believing that\nall the mass of a positive nucleus is electrical in character. We also\nthink of electrical charges as having the property of identifiability,\nwhich involves the possession of sharp edges and a change in the law of\nforce at small distances, and this is certainly a property carried over\nfrom our large scale experience.\n\nIt seems fairly evident then that the laws of nature cannot be reduced\nto either those of mechanics or of electricity, nor probably, as is\nsuggested by quantum phenomena, to a combination of both. This of course\ndoes not preclude the possibility that the laws still may be simple when\nexpressed in other forms. An example of such a broad general law that\ngoes deeper than mechanics or electrodynamics is probably afforded by\nthe second law of thermodynamics when extended to include radiation\nphenomena.\n\nExamples of attempts to find other such simple laws are Tolman's \nPrinciple of Similitude,[33] and Lewis's theory of Ultimate Rational \nUnits,[34] and his recently enunciated principle of Complete \nReversibility.[35] The first two of these attempts I do not believe are \nsuccessful, for reasons I have stated elsewhere,[36] the third also \nseems somewhat doubtful. \n\n[Footnote 33: R. C. Tolman, Phys. Rev. 3, 244-255, 1914; 6, 219-233,\n1915; 15, 521, 1920; Jour. Amer. Chem. Soc., 43, 866-875, 1921.]\n\n[Footnote 34: G. N. Lewis. Vol. 15, 1921 of the Contributions from the\nJefferson Physical Laboratory, dedicated to Professor Hall, Cambridge,\nMass.; Phil. Mag., 49, 739-750, 1925.]\n\n[Footnote 35: G. N. Lewis, Proc. Nat. Acad. Sci., 11, 179-183, 422-428,\n1925.]\n\n[Footnote 36: P. W. Bridgman, Phys. Rev. 8, 423-431, 1916; \"Dimensional\nAnalysis,\" p. 105.]\n\nWith regard to the general question of simple laws, there are at least\ntwo attitudes; one is that there are probably simple general laws still\nundiscovered, the other is that nature has a predilection for simple\nlaws. I do not see how there can be any quarrel with the first of these\nattitudes. Let us examine the second. We have in the first place to\nnotice that \"simple\" means simple to us, when stated in terms of our\nconcepts. This is in itself sufficient to raise a presumption against\nthis general attitude. It is evident that our thinking must follow those\nlines imposed by the nature of our thinking mechanism: does it seem\nlikely that all nature accepts these same limitations? If this were the\ncase, our conceptions ought to stand in certain simple and definite\nrelations to nature. Now if our discussion has brought out any one\nthing, it is that our concepts are not well defined things, but they are\nhazy and do not fit nature exactly, and many of them fit even\napproximately only within restricted range. The task of finding concepts\nwhich shall adequately describe nature and at the same time be easily\nhandled by us, that is, be simple, is the most important and difficult\nof physics, and we never achieve more than approximate and temporary\nsuccess. Consider the example of time. The original concept of local\ntime, which for long seemed satisfactory, turns out to be inadequate,\nand has to be replaced by extended time, which is so complicated that it\nis questionable whether we shall ever be able to grasp it with the\nconfidence that we must demand in a useful concept (by \"grasp\" I mean\nintuitive command of all the implications of the operations which are\ninvolved). The concept has not yet been found which describes simply the\ntemporal relations of the universe.\n\nNot only are concepts hazy around the edges and so incapable of fitting\nnature exactly, but there is always the chance that there are concepts\nother than those which we have adopted which would fit our present\nphenomena. Finding concepts to fit nature is much like solving a\ncross-word puzzle. In the puzzle there may be some parts of the pattern\nwhich we fill completely and easily, but sometimes we find parts in\nwhich we can fill in everything except one or two obstinate definitions,\nso that we are sure we are on the right track, and rack our brains for\nthe missing words, when with a flash of inspiration we see that the\nobstinate words can be fitted in by a complete change in those which we\nhad already accepted. It may be that we are soon to witness a similar\nchange in our concept of the nature of light. An important difference\nbetween the cross-word puzzle and nature is that we can never tell when\nwe have filled in all the squares in any of the parts of nature's\npuzzle; there is always the possibility of new phenomena which our\npresent scheme does not touch.\n\nConsidering, then, the nature of our conceptual material, it seems to me\nthat the overwhelming presumption is against the laws of nature having\nany predisposition to simplicity as formulated in terms of our concepts\n(which is of course all that simplicity means), and the wonder is that\nthere are apparently so many simple laws. There is this observation to\nbe made about all the simple laws of nature that have hitherto been\nformulated; they apply only over a certain range. We have not extended\nthe laws of gravitation to small bodies, nor have we found that our\nelectrical laws will work on a cosmic scale. It does not seem so very\nsurprising that over a limited domain, in which the most important\nphenomena are of a restricted type, the conduct of nature should follow\ncomparatively simple rules.\n\nA tempting question is whether there may not be some laws of nature that\nare _really_ simple, without relation to our mode of formulation, such\nas the law of the inverse square. I leave it to the reader to decide\nwhether this question has meaning. In this connection it is possibly\nsignificant that the average physicist is strangely reluctant to tamper\nwith the inverse square law. I find in myself a lack of sympathy, which\nI cannot justify by any of the considerations of this essay, with\nattempts like the recent one of Swann,[37] for example, to explain a\nwide variety of hitherto obstinate effects by the assumption of slightly\nunequal departures from the inverse square law by the electrons and\nprotons. Of course I hope that this feeling will turn out not to be\nprejudice, but will perhaps be justified by some such general\nobservation as that a departure from the inverse square law so slight as\nby definition to be forever beyond detection by direct experiment is\nmeaningless; but of this I am not at all sure.\n\n[Footnote 37: W. F. G. Swann, Phys. Rev., 25, 253, 1925.]\n\nWe are now ready to consider the second respect in which nature may be\nsimple, namely, because the material of which it is built may reduce to\na few sorts of elements. In this discussion it will be convenient to\nconsider also at the same time the more inclusive simplicity arising\nfrom simple laws acting on simple elements. The immediate question for\nus here is one of fact: does nature seem to be getting intrinsically\nsimpler as we get toward small scale phenomena? There is much room for\ndifference of opinion here; personally I feel that this expected\nsimplicity is not in evidence, at least to the extent that we could\ndesire. For instance, the fact that the electrons must have both\nelectrical and mechanical properties is a straw in this direction.\n\nIt must also be remembered that a certain simulation of simplicity is\ninevitable as we approach the limits of experimental knowledge, whatever\nthe actual structure of nature, for the mere reason that near the limit\nour possible experimental operations become fewer in number, and our\nconcepts fewer also. The question which we are trying to answer has,\ntherefore, its real meaning only in terms of the possible future. Do we\nbelieve that if we drive in our stakes at a certain point on our present\nfrontiers, this point will gradually, as physics advances, become\npossessed of a continually richer experience, so that nature at this\npoint will appear increasingly complicated? Or do we expect a\ntermination of this process of expansion fairly soon? It seems to me\nthat as a matter of experimental fact there is no doubt that the\nuniverse at any definite level is on the average becoming increasingly\ncomplicated, and that the region of apparent simplicity continually\nrecedes. This, however, is not the opinion of all observers. Thus\nBertrand Russell, in \"What I Believe\", page 10, writes, \"Physical\nScience is then approaching the stage where it will be complete, and\ntherefore uninteresting.\"\n\nThis is perhaps a particularly favorable epoch in the history of physics\nto urge the essential complexity of nature, because all our new quantum\nphenomena indicate a vast wealth of hitherto unsuspected relations on\nthe very edge of the attainable. There is one aspect of quantum\nrelations, as also of our ideas of the nature of the structure of the\nnucleus of the atom, which is particularly significant in this respect,\nnamely, that we have to describe phenomena by statistical methods. Now a\nstatistical method is used either to conceal a vast amount of actual\nignorance, or else to smooth out the details of a vast amount of actual\nphysical complication, most of which is unessential for our purposes.\nThere can be no doubt of the amount of ignorance that the statistical\nmethod conceals when applied to these phenomena, but there are also\nstrong indications, particularly when applied to the nucleus, that it\ncovers a vast amount of actual physical complications. The nucleus of a\nradium atom becomes unstable on the average every 10^4 years, which may\nbe plausibly taken to indicate that every 10^4 years the radium nucleus\ngets itself into some particular configuration. Considering the time\nscale on which we suppose events in the atom to take place, and also\nconsidering the fact that radioactive disintegration seems unaffected by\noutside agencies, this would indicate a perfectly appalling amount of\nstructure. We are similarly driven to statistical methods in quantum\ntheory, as for example, in Einstein's analysis of the details of\nequilibrium between emitting and absorbing atoms and radiation.\n\nIn general, we cannot admit for a minute that a statistical method,\nunless used to smooth out irrelevant details, can ever mark more than a\ntemporary stage in our progress, because the assumption of events taking\nplace according to pure chance constitutes the complete negation of our\nfundamental assumption of connectivity; such statistical methods always\nindicate the presence of physical complications which it must be our aim\nto disentangle eventually.\n\nIt appears then that present experimental evidence makes very probable\nstructures beyond the electron and the quantum; we may go even further\nand say that there is no experimental evidence that the sequence of\nphenomena in nature as we go to ever smaller scales is a terminated\nsequence, or that a drop of water is not in itself essentially infinite.\n(This statement contains by implications the meaning that we attach to\ninfinite.) All the more, then, there is no evidence that nature reduces\nto _simplicity_ as we burrow down into the small scale.\n\nWhatever may be one's opinion as to the simplicity of either the laws or\nthe material structure of nature, there can be no question that the\npossessors of some such conviction have a real advantage in the race for\nphysical discovery. Doubtless there are many simple connections still to\nbe discovered, and he who has a strong conviction of the existence of\nthese connections is much more likely to find them than he who is not at\nall sure they are there, and is merely hunting for anything that may\nturn up. It is largely a matter of psychology. Everyone knows that the\nmere suggestion that a problem has a solution, or the knowledge that\nsomeone has already solved it, is often sufficient to suggest a relation\nthat otherwise might not have been noticed. The chances are, therefore,\nthat the relations between phenomena will be found by those who are\npreviously convinced that the relations exist. The observation that most\nof the discoveries are made by men with particular sorts of conviction\nnaturally strengthens the belief that their convictions are true. But\nthis picture has an obverse side. The man who is convinced that there is\na relation where none exists may waste all his time in vain seeking for\nit. Granted that nature has no particular predisposition to simple\nrelations, the conviction that there are such relations is, from the\npoint of view of any one individual, as likely to be a hindrance as a\nhelp. From the point of view of physical society, on the other hand, it\nis desirable that there be such convictions, for in such a society there\nwill be more discoveries than in a society without such convictions. We\nhave here again the old conflict between the individual and society. As\nin all other similar conflicts, society will not be able to demand\npermanently from the individual the acceptance of any conviction or\ncreed which is not true, no matter what the gain in other ways to\nsociety. If nature is not simple, physicists will not continue to\nbelieve that it is, even if such a conviction does increase the total\nnumber of discoveries. It is an impossible attitude to expect that one\ncan maintain. Does this then mean that physics is to face a drab future,\nbecoming continually more prosaic, with new discoveries ever rarer, made\nby a continually decreasing number of misguided but fortunate\nenthusiasts? There may be such a danger, but the greatest part of the\ndanger is avoided if its nature is clearly recognized. One of the\nproblems of the future is the self-conscious development of a more\npowerful technique for the discovery of new relations without the\nnecessity for preconceived opinions on the part of the observer.\n\nThere is an aspect here of our physical research that is often lost\nsight of, namely, the small proportion of successful discoveries\ncompared with the number of investigators. Certainly the number of\nunsuccessful attempts, even in the case of those fortunate individuals\nwho make the great discoveries, is very much greater than the number of\ntheir successful attempts. (Faraday's reputed satisfaction with a \u2152%\nreturn comes to mind.) This must always be taken into account in\nestimating the probable chances of correctness of any new theory. With\nso many physicists working to devise new theories, the chances are high\nthat many false theories will be found, in which a number of phenomena\nmay apparently fit together into a new relation, but which eventually\nprove to be inconsistent with other phenomena, so that the proposed\ntheory has to be abandoned. As physics advances and the number of\ninvestigators and the amount of physical material increases, one has to\nbe more and more exacting in one's requirements of a new theory. One\nmust be particularly on guard against numerical coincidences. An\ninteresting chapter might be written on numerical relations which have\nbeen hopefully published, but later had to be abandoned as without\nsignificance.\n\n\n\n\nDETERMINISM\n\n\nIf we are right in supposing that physical evidence gives no warrant for\nthe idea that nature is finite downward, we have not only repudiated the\nthesis of simplicity but we have also made a very important observation\non the other general thesis mentioned at the beginning of this chapter,\nnamely, the thesis of physical determinism. By determinism we understand\nthe belief that the future of the whole universe, or of an isolated part\nof it, is determined in terms of a complete description of its present\ncondition. [What we mean by present condition will be discussed later.]\nIt is popularly assumed that every physicist subscribes to some such\nthesis as this. But now if there is infinite structure even in a small\nisolated part of the universe, a complete description of it is\nimpossible, and the doctrine as stated must be abandoned. It seems to me\nthat all present physical evidence prepares us to admit this\npossibility. I suppose, however, that most physicists would subscribe to\nsome modification of the original thesis, perhaps along the following\nlines. Given a description of an isolated part of the physical universe\nin the most complete terms that have physical meaning, that is, down to\nthe smallest elements of which our physical operations give us\ncognizance, then the future history of the system is determined within a\ncertain penumbra of uncertainty, this penumbra growing broader as we\npenetrate to finer details of the structure of the system or as times\ngoes on, until eventually all but certain very general properties of the\noriginal system, such as its total energy, are forever lost in the haze,\nand we have a system which was unpredictable. I suppose that it is a\nfurther conviction of at least many physicists that by sufficiently\nrefining our measurements, the amount of haze at any fixed point in the\nfuture may be made indefinitely small, and many might even go further\nand hope by studying the haze (perhaps statistically) to obtain some\ninferential evidence of structure beyond that yet experienced. In fact\nit may be that this last contains the germs of the ultimate method of\ninvestigation, if we ever reach a stage when we can no longer refine our\nmethods of measurement.\n\nDeterminism to the physicist is simply a way of stating certain\nimplications of his conviction of the connectivity of nature. We have\nseen that the broadest possible statement of the thesis of connectivity\nis: Given two isolated systems with identical past histories up to a\ncertain epoch, then the future histories will also be identical. The\nthesis of the determinism of the future by the present constitutes a\nspecialization of this general thesis in that we suppose that identity\nof _all_ past history is not necessary for identity of future behavior,\nbut only identity of present condition. The general and the special\nthesis are not equivalent by any means: if past histories are identical\nthen present conditions are also identical, but the converse does not\nnecessarily hold at all.\n\nNow I believe that the general thesis (which I suppose all physicists\nwill admit, but whose truth is nevertheless subject to the verification\nof experience) gets turned into the special thesis by a feeling of\nsomewhat metaphysical content, which we may perhaps state by saying that\nwe can see no way by which the past can affect the future except through\nthe present. We do not like to think of the effect of a cause distant in\nthe past jumping over the present and affecting the future without\ntouching the present at all. It is the analogue of that attitude of mind\nto which action at a distance in space is inconceivable; just as it is\ndifficult to conceive of a body here affecting a body there without in\nsome way an action propagated through intermediate space, so we do not\nlike to think of a past cause jumping over time and producing a future\neffect without some sort of continuity in the causal chain through all\nintermediate time.\n\nSo far our discussion has been purposely loose: it is evident that what\nwe mean by \"present state\" is crying for definition. What is meant by\nthis may depend somewhat on the specific hypothesis that one adopts\nabout the structure of nature. Historically the conviction of future\ndeterminism has been most intimately associated with a mechanical\npicture of the structure of the universe, so that it may be well to\nbegin from this point of view. Suppose the simplest possible system\ncomposed of point masses without structure, as in the kinetic theory of\ngases. What sort of specifications do we believe necessary to fix the\npresent state of such a system? The mechanical view of nature gives a\ndefinite answer. By present state we mean the positions and velocities\nof all the masses. This is sufficient for the complete determination of\nany purely mechanical system, in which the forces between the elements\nare known functions of only their relative positions. By a sort of\nextension of these ideas valid for mechanical systems, it seems to be\noften thought that the present state of _any_ system is determined by a\ncomplete specification of the positions and velocities of all the\nultimate elements of the system (provided always of course that this\nnumber is finite). This principle, however, does not appear to bear the\ncheck of experiment when applied to electrical systems with radiation.\nThe theorems of the retarded potential show that such systems are\ndetermined by the present position and velocities of the charges in the\nimmediate vicinity, and by the corresponding data at remote points given\nfor proper epochs in the past; in this case, therefore, past and present\nhistory are necessary to determine the future. But if we consider the\nelectrical field as part of the system, we may fix the future in terms\nof the present positions of the charges, their velocities, and the\nvalues of the field vectors all over space, thus returning to a certain\nformal resemblance to mechanical systems, and suggesting a reason for\nascribing physical reality to the electric field. This analogy with a\nmechanical system is, however, loose; complete analogy would allow the\ninstantaneous values of the time derivatives of the field to be given\nalso, and this is not possible.\n\nHow is it that velocity can strictly be regarded as characteristic of\nthe _present_ state of the system? Certainly the usual operations for\nmeasuring velocity demand that we know the configuration of the system\nat two different times, and calculate the velocity from certain\ndifferences of the system at these two times. The velocity is defined as\na limiting result, but even in the limit the essential physical fact\ndoes not disappear that we must know the positions of the system at two\ntimes. We may now go further; if the velocity is properly included in\nthe present attributes of the system, we can see no reason for not\nincluding a specification of all the higher time derivatives also. In\nthe case of the simple gaseous system under present consideration we can\nanswer this question by examining the operations by which we actually go\nto work to determine the future of such a system. The problem of\ndetermining the future condition of such a system reduces to the problem\nof writing the differential equations of motion of all its parts. If the\nsystem is a mechanical system, as in this case, these equations are of\nthe second order in the time derivatives of the position coordinates,\nand also involve the forces, which we suppose are known in terms of the\nrelative positions of the parts of the system. Given, then, the\npositions and the way in which the forces depend on the relative\npositions of the parts, the equations of motion can be written down for\nany configuration of the system, and these equations may be integrated\n(at least approximately) in terms of the proper initial conditions. Now\nthe only boundary conditions on a second order equation are the initial\npositions and velocities. This is the reason that velocities have to be\nspecified in giving the present condition of the system, and that it is\nnot necessary to give the higher derivatives. Apparently the reason why\nwe instinctively include velocity among the present properties of the\nsystem is not because velocity is by its nature strictly a present\nproperty of the elements of the system, but rather because our wide\nexperience with mechanical systems has shown that as a matter of fact\nvelocity is necessary in such systems to determine future motion.\n\nBut now if the equations of motion of the parts of the system are not\nthose of mechanics, they will in general be much more complicated in\nappearance and will involve higher derivatives of the time than the\nsecond. Suppose for the moment that the equations contain only\nderivatives and the mutual positions of the parts of the system. Then to\nintegrate the equations and determine the motion we have to know the\ninitial positions and the initial values of all the derivatives up to an\norder one lower than the highest which occurs in the equations. The\nequations of motion of an electron are even more complicated than this,\nin that the positions of distant parts of the system have to be given\nthroughout an interval of time instead of merely an instant. It would\nseem that the feeling that the present state of a system may be\ndetermined in terms of positions and velocities does not as a matter of\nfact apply to all the systems of our experience.\n\nThe discussion up to this point has been subject to the fundamental\nassumption that the behavior of the system is entirely determined if we\ncan give the position of each part as a function of time. This\nassumption is implicitly contained in Einstein's formulation of the\ngeneral principle of relativity, namely, that there is nothing more to a\nphysical system than a set of space-time coincidences, and that the\nsystem is fixed in terms of the space time co\u00f6rdinates of all its\nparts. Already in discussing the assumption of relativity we have\nindicated reasons for dissatisfaction with this as a means of\nreproducing all experience, because in giving only the space-time\ncoordinates of events we have entirely omitted the descriptive\nbackground of the equations, which gives physical color to the system in\nquestion. This discussion also assumes that a specification of the\npositions, velocities, and higher derivatives (if necessary) of the\nelements of the system is possible, which amounts essentially to the\nassumption that the system contains only a finite number of elements.\nNow in view of the experimental fact that there is no reason for\nsupposing that the structure of the universe is finite, this conclusion\nmust be modified, but I do not believe that the necessary modification\naffects the essential argument. In view of the possible infinite\nstructure it would seem that we cannot expect more than that the future\nis determined by the present within a certain penumbra of uncertainty,\nand this penumbra may be made less important by digging down deeper into\nthe structure when specifying the present condition.\n\nWe have also slurred over the ambiguities in \"present\" condition when\nthe system is spread over space. Probably a unique ascription of meaning\nto \"present\" is not possible for an extended system, but at least one\npossibility is indicated by relativity theory. Imagine a staff of\nassistants distributed throughout space, each equipped with clocks\nsynchronized and set with the master clock by light signals in the\nconventional manner, and each fully equipped with the necessary\nmeasuring instruments. Then what we mean at this point of the argument\nby \"present\" state of the system is the aggregate of all the information\nabout the _positions_ and _velocities_ of the ultimate elements which I\ndetermine in my immediate vicinity at my origin of time plus the reports\nof similar observations made by all the assistants, each local\nobservation being made at the time origin of each local clock.\n\nGoing back now to the main argument, we have shown that the feeling that\nthe present condition of the universe may be specified in terms of\npositions and velocities arose from experience with purely mechanical\nsystems, and that the more general formulation, in which we add to the\nvelocities the higher time derivatives, applies only to systems in which\nthe ultimate elements move according to differential equations of higher\norder than the second. Furthermore, our analysis seems to have shown\nthat systems in which there is radiation do not allow a determination of\nthe future in terms of a present condition specified in terms such as\nthese. It seems, however, that the general principle of the determinism\nof the future by the present may be saved by a change in the definition\nof what we mean by the present condition of the system, ridding it of\nits mechanical and other special implications, and making more immediate\nconnection with direct experiment. Let us understand by present\ncondition of a system the aggregate of all information that can be\nobtained by any physical means whatever, with any sort of physical\ninstrument, not attempting to get out of this analysis information about\nhypothetical ultimate physical elements, with the proviso that the\nmeasurements are to be made now, extending the concept of \"now\" to\npoints distant in space in the way intimated above. With such a general\ndefinition of the meaning of \"present\" we can now deal with systems in\nwhich there is radiation, noticing that our assistant observers must be\nstationed throughout apparently empty space as well as in the\nneighborhood of matter. That this does adequately cover the case of\nradiation is suggested by considering again the two systems of dark\nlanterns with screens and distant mirrors which we have previously\nconsidered, in one system a light signal having been despatched 0.5\nsecond ago and in the other 1.5 seconds ago. Our thesis demands that\nthere be some present difference in these two systems, because their\nfuture history is different, in one of them a light signal arriving\nafter the lapse of 1.5 seconds, and in the other after only 0.5 second.\nNow there is a present difference as reported by our assistants, for the\nassistant stationed half way between lantern and mirror reports in one\nsystem a flash of light on the side of a screen which is turned toward\nthe lantern, and in the other system on the side of the screen turned\ntoward the mirror.\n\nThis more general point of view answers the question whether velocity\nmay be regarded as a present attribute of the system, for the parts of a\nsystem which are in motion have momentum, and momentum may be detected\nby placing against such parts comparatively rigid members which will\nreceive a minute deformation, so that velocity has a meaning in terms of\nphysical measurements made at a single instant of time.\n\nThere is a subtle and difficult question here, namely, whether in\ntalking about operations of measurement we can ever get rid of temporal\nimplications, and therefore, whether a condition of the system in which\ntemporal implications remain can properly be described as \"present.\" I\nshall not attempt to answer this question: there must be some\npractically satisfying answer, involving perhaps the physical analogue\nof differentials of different orders in mathematics, short of carrying\nthe analysis to such a degree of refinement that the concept of present\nbecomes meaningless, as we can see might easily happen.\n\nWith this enlarged understanding of what we mean by present state of the\nsystem, it seems to me that physical evidence is now rather favorable to\nthe view that the present determines the future, subject to\nqualification about the penumbra, at least as far as large scale\nphenomena are concerned. It appears much more doubtful when we come to\nsmall scale phenomena, and in particular it is doubtful whether the\nprinciple can be applied to the details of the quantum process, and in\nfact it is not certain that it has meaning. It is certain that if it is\ntrue an enormous amount of structure beyond any that has yet been\ndetected is implied.\n\n\n\n\nON THE POSSIBILITY OF DESCRIBING NATURE\nCOMPLETELY IN TERMS OF ANALYSIS\n\n\nThere is a certain thesis that is loosely related to the view that\nnature is finite downward, namely, that an explanation of the universe\nis possible in which we start with small scale things, and explain large\nscale phenomena in terms of their small scale constituents, the thesis,\nin other words, that all the properties of the large are contained in\nthe properties of the small and that the large may be constructed out of\nthe small. Some such thesis as this seems implied in the general\nattitude of many physicists. Let us examine the physical basis for this.\nTo maintain this thesis would demand that aggregates of things never\nacquire properties in virtue of their numbers which they do not already\npossess as individuals. Is this true? Consider, for example, the\ntwo-dimensional geometry on the surface of a sphere. This is\nnon-Euclidean. Is the geometry of the individual elements of the surface\nof the sphere non-Euclidean, or do they acquire this property in\nchanging scale? Is the kinetic energy of a number of electrons all\nmoving together in such a way as to constitute an electric current the\nsum of the kinetic energies of the individual electrons, or is there an\nadditional term? Is the mass of an electron the sum of the masses of its\nelements?\n\nA mathematical consideration is suggestive here. Those properties of a\nsystem which can be described in terms of linear differential equations\nhave the property of additivity; the effect of a number of elements is\nthe sum of the effects separately, and no new properties appear in the\naggregate which were not present in the individual elements. But if\nthere are combination terms (as in the electrical energy, which contains\nthe square of the field), then the sum is more than (or different from)\nits parts, and new effects may appear in the aggregate. Now of course\nthe linear equation is of enormous importance in describing nature, but\nmany examples of systems with other types of equation can be found, as\nthat above for electromagnetic mass. In expecting to find in nature such\nnon-additive effects, we need not commit ourselves at all to the view\nthat nature is governed by differential equations, but by analogy may\nexpect similar effects if difference equations, for instance, should\nprove to be fundamental, or even something beyond present mathematical\nformulation.\n\nIt is certainly very much easier to handle a system physically if the\ntotal action can be built up from that of its parts, because the\nanalysis which establishes the connection between the elements is easier\nto perform. It is obviously easier to show that an explanation in such\nterms is correct, because we have seen that explanation involves making\nexperiments with representative elements absent or altered, and it is\neasier to vary the small things than the large things. Those\nexplanations which involve working from the small up will therefore be\nmade first, and will appear to be of disproportionate importance. Places\nwhere I look for an explanation from the large to the small are perhaps\nin accounting for the values of the gravitational constant and the\nvelocity of light and in those phenomena which general relativity theory\nindicates may depend on all the matter in the universe, as the Foucault\npendulum experiment. We must, of course, also be prepared for such\nnon-linear effects in the domain of unexplored quantum phenomena.\n\n\n\n\nA GLIMPSE AHEAD\n\n\nSome of the general considerations of this essay may, with considerable\nplausibility, be expected to play a part in the future of both\nspeculative and experimental physics. The most important effect may be\nexpected from the clearer recognition of the operational character of\nour physical concepts. Indeed during the writing of this essay there has\nbeen a very marked increase in emphasis on the necessity of\nunderstanding in terms of physical operations such fundamental concepts\nas that of the electron, by the new quantum mechanics [the mechanics of\nHeisenberg-Born and Schr\u00f6dinger of 1925-26].\n\nWe are to expect then in the first place a more self-conscious and\ndetailed analysis of the operational structure of all our physical\nconcepts. [It has been beyond the scope of this essay even to begin to\nattempt a systematic and thoroughgoing analysis of this character.] This\nfuture analysis will show precisely how, as we extend the range of\nexperience, the physical character of the operations changes by which we\ndefine our concepts, as, for example, in mechanics the notion of force\ndisappears at high velocity and is replaced perhaps by the notion of\nmomentum. In the region of change in the nature of our concepts, special\nstudy will be made of the accuracy of our physical measurements, and new\nexperiments devised of greater accuracy, in order that we may know\nprecisely to what extent the new concepts are equivalent to the old.\nPast experience suggests that we may perhaps expect to find new\nphenomena especially in those regions where the difficulty of carrying\nout the usual operation forces us to change the operational character of\nour concepts. There will be questions of a more or less formal nature to\nanswer, as for example the _best_ way of extending concepts when there\nare several possible courses open to us.\n\nWe may expect more interesting results, however, when we get so far\nbeyond ordinary experience that the character of the possible physical\noperations has become so restricted as to result in an apparent decrease\nin the number of independent concepts. It seems plausible to expect that\nthe structure of nature is more fundamentally connected with the number\nof independent concepts necessary for a complete description than with\nthe precise details of the structure of the individual concepts, such,\nfor example, as whether space is measured optically or tactually. In\nthose regions where the number of concepts decreases, we must make the\nmost thoroughgoing experimental examination to discover if possible new\nsorts of operations by which the number of concepts may be brought back\nto normal. In searching for such new experimental operations it seems to\nme that by far the greatest promise for the immediate future is offered\nby improvements in our powers of dealing with individual atomic and\nelectronic processes, such as we now have to a limited extent in the\nvarious spinthariscope methods of counting radioactive disintegrations,\nor Wilson's \u03b2-track experiments.[38] In this self-conscious search for\nphenomena which increase the number of operationally independent\nconcepts, we may expect to find a powerful systematic method directing\nthe discovery of new and essentially important physical facts.\n\n[Footnote 38: C. T. R. Wilson, Proc, Roy. Soc., 87, 277, 1912.]\n\nWe can only conjecture whether the number of fundamental concepts will\nprove impossible of further increase or not, but present experience\nseems to give greater probability to the view that as we penetrate\ndeeper the number of fundamental concepts will always tend to become\nfewer. We have already certainly one example in that the temperature\nconcept disappears when we get to the atomic scale of magnitude, and\npossibly a second example in the building up of separate concepts for\nenergy and frequency by the combination of great numbers of that one\noperationally simple thing which characterizes the elementary quantum\nprocess in ordinary radiation.\n\nDifferent sorts of relations between concepts are conceivable in the\ntransition zone where the number changes. We may find that other\nexamples are like that of temperature, which is simply a statistical\neffect of a great many phenomena which may be described individually in\nterms of the ordinary concepts of mechanics, so that in this case the\nnumber of concepts changes merely by temperature dropping out, leaving\nthe others more or less unaffected. Or all the concepts may be more\nclosely interwoven, so that when the total number of concepts changes it\nmay not be possible to separate out a group of concepts whose defining\noperations are unchanged. In such a case we must say that the original\nconcepts are not applicable on the new level. The most immediate\napplication of this idea has been already mentioned, namely, to the\nconcepts of space and time. If the operations by which space and time\nare measured on the ordinary scale of magnitude cannot be carried down\nas a whole into the region of quantum phenomena, then we must say that\nthe ordinary concepts of space and time are not applicable to these\nphenomena.\n\nClosely connected with the sharper analysis of the operational structure\nof our concepts, we may expect in the future also a closer analysis of\nour inventions. This will take the form of a search for new physical\nfacts which shall give to our inventions the character of physical\nreality. In case prolonged search fails to disclose such phenomena (as\nis probably now the case with the field concept of electrodynamics), we\nmust then find some way of embodying explicitly in our thinking the fact\nthat we are dealing with pure inventions and not realities.\n\n\n\n\nINDEX\n\n  Absolute, 26\n  Absolute time, 4\n  Action at a distance, 46\n  Analysis of large into small, 51,\n    220\n  Arithmetic, 35\n  Atom, 59\n\n  Bell, 84\n  Birkhoff, 72\n  Black body, 112\n  Bohr, 190, 192\n  Born, 222\n  Boscovitch, 46\n  Bothe and Geiger, 116\n  Bridgman, 201\n  Brownian movement, 107, 129,\n    143\n  Bush, 142\n\n  Caloric fluid, 59\n  Campbell, 117\n  Carnot engine, 125\n  Causality, 80 ff\n  Causal train of events, 85\n  Clifford, 28\n  Clock, 70 ff, 176\n  Compton, 116, 188\n  Continuity, 94\n  Conservation of charge, 135,\n    136\n  Conservative functions, 113\n  Constructs, 53\n  Correlation, 37\n  Cosmic units, 182\n  Cross word puzzle, 202\n\n  Descriptive background, 64\n  Determinism, 114, 209 ff\n  Discontinuous space, 191\n  D\u00f6ppler effect, 166\n  ds, 72\n\n  Eddington, 93\n  Einstein, vii, 1, 2, 3, 4, 7, 8,\n    9, 12, 13, 14, 64, 100, 155,\n    162, 163, 164, 167, 169, 170,\n    171, 172, 173, 175, 176, 177,\n    206\n  Electrical concepts, 131 ff\n  Electrical explanation of\n    universe, 50\n  Electrical mass, 139\n  Electric field, 56, 133\n  Empiricism, 3\n  Energy, 108 ff, 126 ff\n  Euclidean space, 14, 15, 16, 18,\n    23, 52, 61, 67\n  Event, 95, 167\n  Explanation, 37\n  Explanatory crisis, 41\n  Extended time, 77\n\n  Faraday, 44, 57, 58, 209\n  Final explanations, 48\n  First law of thermodynamics,\n    126 ff\n  Force, 102 ff\n  Foucault pendulum, 180, 184\n  Fourth dimension, 74\n  Future, 222\n\n  Gauss, 15, 34\n  \"Go and come\" time, 112\n  Gravitational constant, 91\n\n  Haldane, 25\n  Heat flow, 130\n  Heisenberg, 222\n  Hertz, 44\n  Hoernl\u00e9, vi\n\n  Identity, 91 ff\n  Isolation, 82\n\n  Joule, 124\n\n  Kelvin, 45, 110\n  Kinetic theory of gases, 40\n\n  Lagrangean equations, 112\n  Larmor, 149\n  La Rosa, 164\n  Length, 9 ff\n  Lewis, 166, 201\n  Light, 150 ff\n  Local time, 75\n  Lorentz, 143, 147, 148, 149\n\n  Mach, 183\n  Mass, 102 ff\n  Mathematics, 60 ff\n  Maxwell, 44, 58, 112, 137, 148\n  Meaningless questions, 28 ff\n  Measurement approximate, 33\n  Mechanism, 45\n  Mercury, 105\n  Michelson, 15, 26\n  Michelson and Morley, 66\n  Models, 45, 52\n\n  Newton, 4, 110\n\n  Operational character of\n    concepts, 5\n  Operational thinking, 32\n  Optical space, 67\n  Ostwald, 109\n\n  Penumbra, 34\n  Perrin, 107\n  Physical reality, 59\n  Planck, 69\n  Poincar\u00e9, 48, 115, 116, 190\n  Pythagoras, 61\n\n  Quantum act, 156\n  Quantum theory, 40, 47, 186\n\n  Radiation and temperature, 123\n  Relative character of\n    knowledge, 25\n  Relativity, 150 ff\n  Reynolds, 93\n  Rotational motion, 178\n  Russell, 205\n\n  Schr\u00f6dinger, 222\n  Space, 66\n  Silberstein, 11\n  Simplicity of nature, 198\n  Simultaneity, 7, 8\n  Spring balance, 103\n  Statistical methods, 115, 117\n  Stress, 54\n  Swann, 204\n\n  Table top, 106\n  Tactual space, 67\n  Temperature, 118 ff\n  Thermodynamics, 117\n  Thing traveling, 101, 152, 157,\n    164\n  Time, 69\n  Tolman, 201\n  Truth, 78\n  Turbulent motion, 120, 124\n\n  Velocity, 97 ff, 213 ff\n  Velocity of light, 100\n\n  Whitehead, 167\n  Wilson, 224","title":"The logic of modern physics","tradition":"philosophy-ethics","translated_text":null,"word_count":61177}