{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_77011/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_77011","id":"gutenberg_77011","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/77011","summary":null,"tags":["Category: British Literature","Category: Essays, Letters & Speeches","Category: Philosophy & Ethics","Category: Teaching & Education","Education"],"text":"THE ORGANISATION\n                              OF THOUGHT\n\n\n\n\n                           THE ORGANISATION\n                              OF THOUGHT\n\n                      EDUCATIONAL AND SCIENTIFIC\n\n\n\n\n                                  BY\n                    A. N. WHITEHEAD, Sc.D., F.R.S.\n\n        FELLOW OF TRINITY COLLEGE, CAMBRIDGE, AND PROFESSOR OF\n            APPLIED MATHEMATICS AT THE IMPERIAL COLLEGE OF\n                        SCIENCE AND TECHNOLOGY\n\n\n\n\n                                LONDON\n                         WILLIAMS AND NORGATE\n               14 HENRIETTA STREET, COVENT GARDEN, W.C.2\n                                 1917\n\n\n               _The rights of translation are reserved_\n\n\n\n\n                                PREFACE\n\n\nTHE discourses included in this volume have been delivered as addresses\non various occasions which are duly noted; the only exception is\n_The Anatomy of Some Scientific Ideas_, which is now published\nfor the first time. These discourses fall into two sections, the first\nfive chapters deal with education, and the remaining three embody\ndiscussions on certain points arising in the philosophy of science.\nBut a common line of reflection extends through the whole, and the two\nsections influence each other.\n\nI have left in each chapter the reference to the particular occasion of\nits first production, and I have not sought for a verbal consistency\ncovering perplexity. But the various parts of the book were in fact\ncomposed with express reference to each other, so as to form one whole.\n\nI have to thank the Syndics of the Cambridge University Press for\npermission to republish the contents of Chapter V.\n\n_Imperial College of Science and Technology,_\n_April 1917._\n\n\n\n\n                           TABLE OF CONTENTS\n\n\nCHAP.                                                            PAGE\n\n PREFACE                                                            v\n\n I. THE AIMS OF EDUCATION--A PLEA FOR REFORM                        1\n (Presidential Address to the Mathematical Association,\n January, 1916.)\n\n II. TECHNICAL EDUCATION AND ITS RELATION TO\n SCIENCE AND LITERATURE                                            29\n (Presidential Address to the Mathematical Association,\n January, 1917.)\n\n III. A POLYTECHNIC IN WAR-TIME                                    58\n (Address at the Prize Distribution, Borough Polytechnic\n Institute, Southwark, February, 1917.)\n\n IV. THE MATHEMATICAL CURRICULUM                                   69\n (Presidential Address to the London Branch of the\n Mathematical Association, 1912.)\n\n V. THE PRINCIPLES OF MATHEMATICS IN RELATION\n TO ELEMENTARY TEACHING                                            92\n (Paper read in the Educational Section of the International\n Congress of Mathematicians, Cambridge,\n 1912.)\n\n VI. THE ORGANISATION OF THOUGHT                                  105\n (Presidential Address to Section A, British Association,\n Newcastle, 1916; also read subsequently before the\n Aristotelian Society.)\n\n VII. THE ANATOMY OF SOME SCIENTIFIC IDEAS                        134\n\n VIII. SPACE, TIME, AND RELATIVITY                                191\n (Paper read to Section A at the Manchester Meeting of\n Section A, 1915; also, with the appended Commentary,\n read subsequently before the Aristotelian\n Society.)\n\n\n\n\n                        ORGANISATION OF THOUGHT\n\n\n\n\n                               CHAPTER I\n\n               THE AIMS OF EDUCATION--A PLEA FOR REFORM\n\n   (_Presidential Address to the Mathematical Association, January,\n                                1916_)\n\n\nWHEN I had the honour of being made President of the Mathematical\nAssociation, I did not foresee the unusual responsibility which it\nentailed. It was my intention to take as the theme of a presidential\naddress the consideration of some aspect of those special subjects to\nwhich my own researches have principally been directed. Events have\nforced me to abandon that intention. It is useless to discuss abstract\nquestions in the midst of dominant practical preoccupation. We cannot\ndisregard the present crisis in European civilisation. It affects every\nfunction of life. In the harder struggle for existence which lies\nbefore the nation, all departments of national effort will be reviewed\nfor judgment. The mere necessity for economy in resources will provoke\nthis reformation.\n\nWe are concerned with education. This Association, so rich in its\nmembership of educationalists, with the conception of reform as the\nvery reason of its being, is among those bodies which must take the\nlead in guiding that educational reconstruction which by a sociological\nlaw follows every social revolution. We do not want impracticable\nideals, only to be realised beyond the clouds in\n\n    \"Some wild, weird clime,\n    Out of Space, and out of Time.\"\n\nWe require to know what is possible now in England, a nation conscious\nof its high achievements, and of great failures, shaken to its\nfoundations, distrustful of the old ways, and dreading fantastic\nnovelties.\n\nI will take my courage in both hands, and put before you an outline of\neducational principles. What I am going to say is of course entirely\nwithout your authority, and does not pledge or prejudge any action of\nthe Association. We are primarily concerned only with the intellectual\nside of education, and, as mathematicians, are naturally concerned to\nillustrate details more particularly by reference to mathematics. Thus\nmuch to explain deliberate omissions in what follows.\n\nConsider now the general and special education of two types of boys,\nnamely those in secondary schools who in after life must form the\nprofessional and directing classes in commerce, industry, and public\nadministration, and again those in junior technical schools and later\nin advanced continuation classes, who are going to form the class of\nskilled artisans and foremen of workshops. These two sets compose the\neducated strength of the nation. We must form no ideals which include\nless than these entire classes within their scope. What I shall say,\nwill in phraseology apply more directly to the secondary schools, but\nwith unessential changes it will apply equally to the other group.\n\nWhat is the first commandment to be obeyed in any educational scheme?\nIt is this: Do not teach too many subjects. The second command is this:\nWhat you teach, teach thoroughly. The devil in the scholastic world\nhas assumed the form of a general education consisting of scraps of a\nlarge number of disconnected subjects; and, with the artfulness of the\nserpent, he has entrenched himself behind the matriculation examination\nof the University of London, with a wire entanglement formed by the\nOxford and Cambridge schools' examination.\n\nCulture is activity of thought, and receptiveness to beauty, and humane\nfeeling. Scraps of information have nothing to do with it. A merely\nwell-informed man is the most useless bore on God's earth. What we\nshould aim at producing is men who possess both culture and expert\nknowledge in some special direction. Their expert knowledge will give\nthem the ground to start from, and their culture will lead them as deep\nas philosophy and as high as art. We have to remember that the valuable\nintellectual development is self-development, and that it mostly takes\nplace between the ages of sixteen and thirty. As to training, the most\nimportant part is given by mothers before the age of twelve. A saying\ndue to Archbishop Temple illustrates my meaning. Surprise was expressed\nat the success in after-life of a man, who as a boy at Rugby had been\nsomewhat undistinguished. He answered, \"It is not what they are at\neighteen, it is what they become afterwards that matters.\"\n\nIn training a child to activity of thought, above all things we must\nbeware of what I will call \"inert ideas\"--that is to say, ideas that\nare merely received into the mind without being utilised, or tested, or\nthrown into fresh combinations.\n\nIn the history of education, the most striking phenomenon is that\nschools of learning, which at one epoch are alive with a ferment of\ngenius, in a succeeding generation exhibit merely pedantry and routine.\nThe reason is, that they are overladen with inert ideas. Education\nwith inert ideas is not only useless: it is, above all things,\nharmful--_Corruptio optimi, pessima_. Except at rare intervals\nof intellectual ferment, education in the past has been radically\ninfected with inert ideas. That is the reason why uneducated clever\nwomen, who have seen much of the world, are in middle life so much\nthe most cultured part of the community. They have been saved from\nthis horrible burden of inert ideas. Every intellectual revolution\nwhich has ever stirred humanity into greatness has been a passionate\nprotest against inert ideas. Then, alas, with pathetic ignorance of\nhuman psychology, it has proceeded by some educational scheme to bind\nhumanity afresh with inert ideas of its own fashioning.\n\nLet us now ask how in our system of education we are to guard against\nthis mental dry rot. We recur to our two educational commandments,\n\"Do not teach too many subjects,\" and again, \"What you teach, teach\nthoroughly.\"\n\nThe result of teaching small parts of a large number of subjects is the\npassive reception of disconnected ideas, not illumined with any spark\nof vitality. Let the main ideas which are introduced into a child's\neducation be few and important, and let them be thrown into every\ncombination possible. The child should make them his own, and should\nunderstand their application here and now in the circumstances of his\nactual life. From the very beginning of his education, the child should\nexperience the joy of discovery. The discovery which he has to make,\nis that general ideas give an understanding of that stream of events\nwhich pours through his life, which is his life. By understanding I\nmean more than a mere logical analysis, though that is included. I mean\n\"understanding\" in the sense in which it is used in the French proverb,\n\"To understand all, is to forgive all.\" Pedants sneer at an education\nwhich is useful. But if education is not useful, what is it? Is it a\ntalent, to be hidden away in a napkin? Of course, education should be\nuseful, whatever your aim in life. It was useful to Saint Augustine\nand it was useful to Napoleon. It is useful, because understanding is\nuseful.\n\nI pass lightly over that understanding which should be given by the\nliterary side of education. It is not peculiarly the function of this\nAssociation to consider it. Nor do I wish to be supposed to pronounce\non the relative merits of a classical or a modern curriculum. I would\nonly remark that the understanding which we want is an understanding\nof an insistent present. The only use of a knowledge of the past\nis to equip us for the present. No more deadly harm can be done to\nyoung minds than by depreciation of the present. The present contains\nall that there is. It is holy ground; for it is the past, and it is\nthe future. At the same time it must be observed that an age is no\nless past if it existed two hundred years ago than if it existed two\nthousand years ago. Do not be deceived by the pedantry of dates. The\nages of Shakespeare and of Moli\u00e8re are no less past than are the ages\nof Sophocles and of Virgil. The communion of saints is a great and\ninspiring assemblage, but it has only one possible hall of meeting,\nand that is, the present; and the mere lapse of time through which any\nparticular group of saints must travel to reach that meeting-place,\nmakes very little difference.\n\nPassing now to the scientific and logical side of education, we\nremember that here also ideas which are not utilised are positively\nharmful. By utilising an idea, I mean relating it to that stream,\ncompounded of sense perceptions, feelings, hopes, desires, and of\nmental activities relating thought to thought, which forms our life.\nI can imagine a set of beings which might fortify their souls by\npassively reviewing disconnected ideas. Humanity is not built that\nway--except perhaps some editors of newspapers.\n\nIn scientific training, the first thing to do with an idea is to prove\nit. But allow me for one moment to extend the meaning of \"prove\"; I\nmean--to prove its worth. Now an idea is not worth much unless the\npropositions in which it is embodied are true. Accordingly an essential\npart of the proof of an idea is the proof, either by experiment or by\nlogic, of the truth of the propositions. But it is not essential that\nthis proof of the truth should constitute the first introduction to\nthe idea. After all, its assertion by the authority of respectable\nteachers is sufficient evidence to begin with. In our first contact\nwith a set of propositions, we commence by appreciating their\nimportance. That is what we all do in after-life. We do not attempt,\nin the strict sense, to prove or to disprove anything, unless its\nimportance makes it worthy of that honour. These two processes of\nproof, in the narrow sense, and of appreciation do not require a rigid\nseparation in time. Both can be proceeded with nearly concurrently. But\nin so far as either process must have the priority, it should be that\nof appreciation by use.\n\nFurthermore, we should not endeavour to use propositions in isolation.\nEmphatically I do not mean, a neat little set of experiments to\nillustrate Proposition I and then the proof of Proposition I, a neat\nlittle set of experiments to illustrate Proposition II and then the\nproof of Proposition II, and so on to the end of the book. Nothing\ncould be more boring. Inter-related truths are utilised _en bloc_,\nand the various propositions are employed in any order, and with any\nreiteration. Choose some important applications of your theoretical\nsubject; and study them concurrently with the systematic theoretical\nexposition. Keep the theoretical exposition short and simple, but\nlet it be strict and rigid so far as it goes. It should not be too\nlong for it easily to be known with thoroughness and accuracy. The\nconsequences of a plethora of half-digested theoretical knowledge are\ndeplorable. Also the theory should not be muddled up with the practice.\nThe child should have no doubt when it is proving and when it is\nutilising. My point is that what is proved should be utilised, and that\nwhat is utilised should--so far as is practicable--be proved. I am far\nfrom asserting that proof and utilisation are the same thing.\n\nAt this point of my discourse, I can most directly carry forward\nmy argument in the outward form of a digression. We are only just\nrealising that the art and science of education require a genius and\na study of their own; and that this genius and this science are more\nthan a bare knowledge of some branch of science or of literature. This\ntruth was partially perceived in the past generation; and headmasters,\nsomewhat crudely, were apt to supersede learning in their colleagues\nby requiring left-hand bowling and a taste for football. But culture\nis more than cricket, and more than football, and more than extent of\nknowledge.\n\nEducation is the acquisition of the art of the utilisation of\nknowledge. This is an art very difficult to impart. Whenever a\ntext-book is written of real educational worth, you may be quite\ncertain that some reviewer will say that it will be difficult to\nteach from it. Of course it will be difficult to teach from it. If it\nwere easy, the book ought to be burned; for it cannot be educational.\nIn education, as elsewhere, the broad primrose path leads to a nasty\nplace. This evil path is represented by a book or a set of lectures\nwhich will practically enable the student to learn by heart all the\nquestions likely to be asked at the next external examination. And I\nmay say in passing that no educational system is possible unless every\nquestion directly asked of a pupil at any examination is either framed\nor modified by the actual teacher of that pupil in that subject. The\nexternal assessor may report on the curriculum or on the performance\nof the pupils, but never should be allowed to ask the pupil a question\nwhich has not been strictly supervised by the actual teacher, or\nat least inspired by a long conference with him. There are a few\nexceptions to this rule, but they are exceptions, and could easily be\nallowed for under the general rule.\n\nWe now return to my previous point, that theoretical ideas should\nalways find important applications within the pupil's curriculum. This\nis not an easy doctrine to apply, but a very hard one. It contains\nwithin itself the problem of keeping knowledge alive, of preventing it\nfrom becoming inert, which is the central problem of all education.\n\nThe best procedure will depend on several factors, none of which can\nbe neglected, namely, the genius of the teacher, the intellectual type\nof the pupils, their prospects in life, the opportunities offered by\nthe immediate surroundings of the school, and allied factors of this\nsort. It is for this reason that the uniform external examination\nis so deadly. We do not denounce it because we are cranks, and like\ndenouncing established things. We are not so childish. Also, of course,\nsuch examinations have their use in testing slackness. Our reason of\ndislike is very definite and very practical. It kills the best part of\nculture. When you analyse in the light of experience the central task\nof education, you find that its successful accomplishment depends on a\ndelicate adjustment of many variable factors. The reason is that we are\ndealing with human minds, and not with dead matter. The evocation of\ncuriosity, of judgment, of the power of mastering a complicated tangle\nof circumstances, the use of theory in giving foresight in special\ncases--all these powers are not to be imparted by a set rule embodied\nin one schedule of examination subjects.\n\nI appeal to you, as practical teachers. With good discipline, it is\nalways possible to pump into the minds of a class a certain quantity of\ninert knowledge. You take a text-book and make them learn it. So far,\nso good. The child then knows how to solve a quadratic equation. But\nwhat is the point of teaching a child to solve a quadratic equation?\nThere is a traditional answer to this question. It runs thus: The\nmind is an instrument, you first sharpen it, and then use it; the\nacquisition of the power of solving a quadratic equation is part of\nthe process of sharpening the mind. Now there is just enough truth in\nthis answer to have made it live through the ages. But for all its\nhalf-truth, it embodies a radical error which bids fair to stifle the\ngenius of the modern world. I do not know who was first responsible\nfor this analogy of the mind to a dead instrument. For aught I know,\nit may have been one of the seven wise men of Greece, or a committee\nof the whole lot of them. Whoever was the originator, there can be no\ndoubt of the authority which it has acquired by the continuous approval\nwhich it has received from eminent persons. But whatever its weight of\nauthority, whatever the high approval which it can quote, I have no\nhesitation in denouncing it as one of the most fatal, erroneous, and\ndangerous conceptions ever introduced into the theory of education. The\nmind is never passive; it is a perpetual activity, delicate, receptive,\nresponsive to stimulus. You cannot postpone its life until you have\nsharpened it. Whatever interest attaches to your subject-matter, must\nbe evoked here and now; whatever powers you are strengthening in the\npupil, must be exercised here and now; whatever possibilities of mental\nlife your teaching should impart, must be exhibited here and now. That\nis the golden rule of education, and a very difficult rule to follow.\n\nThe difficulty is just this: the apprehension of general ideas,\nintellectual habits of mind, and pleasurable interest in mental\nachievement can be evoked by no form of words, however accurately\nadjusted. All practical teachers know that education is a patient\nprocess of the mastery of details, minute by minute, hour by hour, day\nby day. There is no royal road to learning through an airy path of\nbrilliant generalisations. There is a proverb about the difficulty of\nseeing the wood because of the trees. That difficulty is exactly the\npoint which I am enforcing. The problem of education is to make the\npupil see the wood by means of the trees.\n\nThe solution which I am urging, is to eradicate the fatal disconnection\nof subjects which kills the vitality of our modern curriculum.\nThere is only one subject-matter for education, and that is Life\nin all its manifestations. Instead of this single unity, we offer\nchildren--Algebra, from which nothing follows; Geometry, from which\nnothing follows; Science, from which nothing follows; History, from\nwhich nothing follows; a Couple of Languages, never mastered; and\nlastly, most dreary of all, Literature, represented by plays of\nShakespeare, with philological notes and short analyses of plot and\ncharacter to be in substance committed to memory. Can such a list be\nsaid to represent Life, as it is known in the midst of the living of\nit? The best that can be said of it is, that it is a rapid table of\ncontents which a deity might run over in his mind while he was thinking\nof creating a world, and had not yet determined how to put it together.\n\nLet us now return to quadratic equations. We still have on hand the\nunanswered question. Why should children be taught their solution?\nUnless quadratic equations fit into a connected curriculum, of course\nthere is no reason to teach anything about them. Furthermore, extensive\nas should be the place of mathematics in a complete culture, I am a\nlittle doubtful whether for many types of boys algebraic solutions of\nquadratic equations do not lie on the specialist side of mathematics.\nI may here remind you that as yet I have not said anything of the\npsychology or the content of the specialism, which is so necessary a\npart of an ideal education. But all that is an evasion of our real\nquestion, and I merely state it in order to avoid being misunderstood\nin my answer.\n\nQuadratic equations are part of algebra, and algebra is the\nintellectual instrument which has been created for rendering clear\nthe quantitative aspects of the world. There is no getting out of\nit. Through and through the world is infected with quantity. To talk\nsense, is to talk in quantities. It is no use saying that the nation\nis large,--How large? It is no use saying that radium is scarce,--How\nscarce? You cannot evade quantity. You may fly to poetry and to music,\nand quantity and number will face you in your rhythms and your octaves.\nElegant intellects which despise the theory of quantity, are but half\ndeveloped. They are more to be pitied than blamed. The scraps of\ngibberish, which in their school-days were taught to them in the name\nof algebra, deserve some contempt.\n\nThis question of the degeneration of algebra into gibberish, both in\nword and in fact, affords a pathetic instance of the uselessness of\nreforming educational schedules without a clear conception of the\nattributes which you wish to evoke in the living minds of the children.\nA few years ago there was an outcry that school algebra was in need\nof reform, but there was a general agreement that graphs would put\neverything right. So all sorts of things were extruded, and graphs were\nintroduced. So far as I can see, with no sort of idea behind them,\nbut just graphs. Now every examination paper has one or two questions\non graphs. Personally, I am an enthusiastic adherent of graphs. But I\nwonder whether as yet we have gained very much. You cannot put life\ninto any schedule of general education unless you succeed in exhibiting\nits relation to some essential characteristic of all intelligent or\nemotional perception. It is a hard saying, but it is true; and I do\nnot see how to make it any easier. In making these little formal\nalterations you are beaten by the very nature of things. You are pitted\nagainst too skilful an adversary, who will see to it that the pea is\nalways under the other thimble.\n\nReformation must begin at the other end. First, you must make up your\nmind as to those quantitative aspects of the world which are simple\nenough to be introduced into general education; then a schedule of\nalgebra should be framed which will about find its exemplification in\nthese applications. We need not fear for our pet graphs, they will\nbe there in plenty when we once begin to treat algebra as a serious\nmeans of studying the world. Some of the simplest applications will be\nfound in the quantities which occur in the simplest study of society.\nThe curves of history are more vivid and more informing than the dry\ncatalogues of names and dates which comprise the greater part of\nthat arid school study. What purpose is effected by a catalogue of\nundistinguished kings and queens? Tom, Dick, or Harry, they are all\ndead. General resurrections are failures, and are better postponed.\nThe quantitative flux of the forces of modern society are capable\nof very simple exhibition. Meanwhile, the idea of the variable, of\nthe function, of rate of change, of equations and their solution, of\nelimination, are being studied as an abstract science for their own\nsake. Not, of course, in the pompous phrases with which I am alluding\nto them here, but with that iteration of simple special cases proper to\nteaching.\n\nIf this course be followed, the route from Chaucer to the Black Death,\nfrom the Black Death to modern Labour troubles, will connect the tales\nof the medi\u00e6val pilgrims with the abstract science of algebra, both\nyielding diverse aspects of that single theme, Life. I know what most\nof you are thinking at this point. It is that the exact course which\nI have sketched out is not the particular one which you would have\nchosen, or even see how to work. I quite agree. I am not claiming that\nI could do it myself. But your objection is the precise reason why a\ncommon external examination system is fatal to education. The process\nof exhibiting the applications of knowledge must, for its success,\nessentially depend on the character of the pupils and the genius of the\nteacher. Of course I have left out the easiest applications with which\nmost of us are more at home. I mean the quantitative sides of sciences,\nsuch as mechanics and physics.\n\nMy meaning can be illustrated by looking more closely into a special\ncase of this type of application. In my rough catalogue of the sort\nof subjects which should form the schedule for algebra, I mentioned\nElimination. It was not put there by accident, for it covers a very\nimportant body of thought.\n\nIn the first place, there is the abstract process of algebraic\nelimination for suitable simple cases. The pupil acquires a firm grasp\nof this by the process, inevitable in education, of working an adequate\nnumber of examples. Again, there are the graphical solutions of the\nsame problem. Then we consider the significance in the external world.\nWe consider the velocity, time, space, acceleration diagrams. We take\nuniform acceleration; we eliminate \"_t_\" between\n\n_v_ = _u_ + _\u0192t_, and _s_ = _ut_ + \u00bd_\u0192t_^2,\n\nand eliminate \"_s_\" between\n\n_v_^2 = _u_^2 + 2_\u0192s_, and _s_ = _ut_ + \u00bd_\u0192t_^2.\n\nThen we remember that constant acceleration is a very special case,\nand we consider graphical solutions or empirically given variations of\n_v_ or of _\u0192_. In preference, we use those empirical formul\u00e6\nwhich occur in the pupil's experimental work. We compare the strong and\nweak points of the algebraic and graphical solutions.\n\nAgain, in the same connection we plot the statistics of social\nphenomena against the time. We then eliminate the time between suitable\npairs. We can speculate how far we have exhibited a real casual\nconnection, or how far a mere temporal coincidence. We notice that\nwe might have plotted against the time one set of statistics for one\ncountry and another set for another country, and thus, with suitable\nchoice of subjects, have obtained graphs which certainly exhibited mere\ncoincidence. Also other graphs exhibit obvious casual connections. We\nwonder how to discriminate. And so are drawn on as far as we will.\n\nBut in considering this description, I must beg you to remember what I\nhave been insisting on above. In the first place, one train of thought\nwill not suit all groups of children. For example, I should expect that\nartisan children will want something more concrete and, in a sense,\nswifter than I have set down here. Perhaps I am wrong, but that is\nwhat I should guess. In the second place, I am not contemplating one\nbeautiful lecture stimulating, once and for all, an admiring class.\nThat is not the way in which education proceeds. No; all the time the\npupils are hard at work solving examples, drawing graphs, and making\nexperiments, until they have a thorough hold on the whole subject.\nI am describing the interspersed explanations, the directions which\nshould be given to their thoughts. The pupils have got to be made to\nfeel that they are studying something, and are not merely executing\nintellectual minuets.\n\nIn this connection the excellence of some of the most recent text-books\non elementary algebra emanating from members of this Association,\nshould create an epoch in the teaching of the subject.\n\nFinally, if you are teaching pupils for some general examination,\nthe problem of sound teaching is greatly complicated. Have you ever\nnoticed the zig-zag moulding round a Norman arch? The ancient work is\nbeautiful, the modern work is hideous. The reason is, that the modern\nwork is done to exact measure, the ancient work is varied according\nto the idiosyncrasy of the workman. Here it is crowded, and there it\nis expanded. Now the essence of getting pupils through examinations\nis to give equal weight to all parts of the schedule. But mankind is\nnaturally specialist. One man sees a whole subject, where another can\nfind only a few detached examples. I know that it seems contradictory\nto allow for specialism in a curriculum especially designed for a\nbroad culture. Without contradictions the world would be simpler, and\nperhaps duller. But I am certain that in education wherever you exclude\nspecialism you destroy life.\n\nWe now come to the other great branch of a general mathematical\neducation, namely Geometry. The same principles apply. The theoretical\npart should be clear-cut, rigid, short, and important. Every\nproposition not absolutely necessary to exhibit the main connection\nof ideas should be cut out, but the great fundamental ideas should\nbe all there. No omission of concepts, such as those of Similarity\nand Proportion. We must remember that, owing to the aid rendered by\nthe visual presence of a figure, Geometry is a field of unequalled\nexcellence for the exercise of the deductive faculties of reasoning.\nThen, of course, there follows Geometrical Drawing, with its training\nfor the hand and eye.\n\nBut, like Algebra, Geometry and Geometrical Drawing must be extended\nbeyond the mere circle of geometrical ideas. In an industrial\nneighbourhood, machinery and workshop practice form the appropriate\nextension. For example, in the London Polytechnics this has been\nachieved with conspicuous success. For many secondary schools I suggest\nthat surveying and maps are the natural applications. In particular,\nplane-table surveying should lead pupils to a vivid apprehension of the\nimmediate application of geometric truths. Simple drawing apparatus, a\nsurveyor's chain, and a surveyor's compass, should enable the pupils\nto rise from the survey and mensuration of a field to the construction\nof the map of a small district. The best education is to be found\nin gaining the utmost information from the simplest apparatus. The\nprovision of elaborate instruments is greatly to be deprecated. To\nhave constructed the map of a small district, to have considered its\nroads, its contours, its geology, its climate, its relation to other\ndistricts, the effects on the status of its inhabitants, will teach\nmore history and geography than any knowledge of Perkin Warbeck or\nof Behren's Straits. I mean not a nebulous lecture on the subject,\nbut a serious investigation in which the real facts are definitely\nascertained by the aid of accurate theoretical knowledge. A typical\nmathematical problem should be: Survey such and such a field, draw a\nplan of it to such and such a scale, and find the area. It would be\nquite a good procedure to impart the necessary geometrical propositions\nwithout their proofs. Then, concurrently in the same term, the proofs\nof the propositions would be learnt while the survey was being made.\n\nFortunately, the specialist side of education presents an easier\nproblem than does the provision of a general culture. For this there\nare many reasons. One is that many of the principles of procedure\nto be observed are the same in both cases, and it is unnecessary\nto recapitulate. Another reason is that specialist training takes\nplace--or should take place--at a more advanced stage of the pupil's\ncourse, and thus there is easier material to work upon. But undoubtedly\nthe chief reason is that the specialist study is normally a study\nof peculiar interest to the student. He is studying it because, for\nsome reason, he wants to know it. This makes all the difference.\nThe general culture is designed to foster an activity of mind; the\nspecialist course utilises this activity. But it does not do to lay too\nmuch stress on these neat antitheses. As we have already seen, in the\ngeneral course foci of special interest will arise; and similarly in\nthe special study, the external connections of the subject drag thought\noutwards.\n\nAgain, there is not one course of study which merely gives general\nculture, and another which gives special knowledge. The subjects\npursued for the sake of a general education are special subjects\nspecially studied; and, on the other hand, one of the ways of\nencouraging general mental activity is to foster a special devotion.\nYou may not divide the seamless coat of learning. What education has\nto impart is an intimate sense for the power of ideas, for the beauty\nof ideas, and for the structure of ideas, together with a particular\nbody of knowledge which has peculiar reference to the life of the being\npossessing it.\n\nThe appreciation of the structure of ideas is that side of a cultured\nmind which can only grow under the influence of a special study. I\nmean that eye for the whole chess-board, for the bearing of one set of\nideas on another. Nothing but a special study can give any appreciation\nfor the exact formulation of general ideas, for their relations when\nformulated, for their service in the comprehension of life. A mind so\ndisciplined should be both more abstract and more concrete. It has been\ntrained in the comprehension of abstract thought and in the analysis of\nfacts.\n\nFinally, there should grow the most austere of all mental qualities; I\nmean the sense for style. It is an \u00e6sthetic sense, based on admiration\nfor the direct attainment of a foreseen end, simply and without waste.\nStyle in art, style in literature, style in science, style in logic,\nstyle in practical execution have fundamentally the same \u00e6sthetic\nqualities, namely, attainment and restraint. The love of a subject in\nitself and for itself, where it is not the sleepy pleasure of pacing a\nmental quarter-deck, is the love of style as manifested in that study.\n\nHere we are brought back to the position from which we started,\nthe utility of education. Style, in its finest sense, is the last\nacquirement of the educated mind; it is also the most useful. It\npervades the whole being. The administrator with a sense for style,\nhates waste; the engineer with a sense for style, economises his\nmaterial; the artisan with a sense for style, prefers good work. Style\nis the ultimate morality of mind.\n\nBut above style, and above knowledge, there is something, a vague\nshape like fate above the Greek gods. That something is Power. Style\nis the fashioning of power, the restraining of power. But, after all,\nthe power of attainment of the desired end is fundamental. The first\nthing is to get there. Do not bother about your style, but solve your\nproblem, justify the ways of God to man, administer your province, or\ndo whatever else is set before you.\n\nWhere, then, does style help? In this, with style the end is attained\nwithout side issues, without raising undesirable inflammations. With\nstyle you attain your end and nothing but your end. With style the\neffect of your activity is calculable, and foresight is the last gift\nof gods to men. With style your power is increased, for your mind is\nnot distracted with irrelevancies, and you are more likely to attain\nyour object. Now style is the exclusive privilege of the expert.\nWhoever heard of the style of an amateur painter, of the style of an\namateur poet? Style is always the product of specialist study, the\npeculiar contribution of specialism to culture.\n\nEnglish education in its present phase suffers from a lack of definite\naim, and from an external machinery which kills its vitality. Hitherto\nin this address I have been considering the aims which should govern\neducation. In this respect England halts between two opinions. It has\nnot decided whether to produce amateurs or experts. The profound change\nin the world which the nineteenth century has produced is that the\ngrowth of knowledge has given foresight. The amateur is essentially a\nman with appreciation and with immense versatility in mastering a given\nroutine. But he lacks the foresight which comes from special knowledge.\nThe object of this address is to suggest how to produce the expert\nwithout loss of the essential virtues of the amateur. The machinery\nof our secondary education is rigid where it should be yielding,\nand lax where it should be rigid. Every school is bound on pain of\nextinction to train its boys for a small set of definite examinations.\nNo headmaster has a free hand to develop his general education or his\nspecialist studies in accordance with the opportunities of his school,\nwhich are created by its staff, its environment, its class of boys, and\nits endowments. I suggest that no system of external tests which aims\nprimarily at examining individual scholars can result in anything but\neducational waste.\n\nPrimarily it is the schools and not the scholars which should be\ninspected. Each school should grant its own leaving certificates, based\non its own curriculum. The standards of these schools should be sampled\nand corrected. But the first requisite for educational reform is the\nschool as a unit, with its approved curriculum based on its own needs,\nand evolved by its own staff. If we fail to secure that, we simply fall\nfrom one formalism into another, from one dung-hill of inert ideas into\nanother.\n\nIn stating that the school is the true educational unit in any national\nsystem for the safe-guarding of efficiency, I have conceived the\nalternative system as being the external examination of the individual\nscholar. But every Scylla is faced by its Charybdis--or, in more homely\nlanguage, there is a ditch on both sides of the road. It will be\nequally fatal to education if we fall into the hands of a supervising\ndepartment which is under the impression that it can divide all schools\ninto two or three rigid categories, each type being forced to adopt\na rigid curriculum. When I say that the school is the educational\nunit, I mean exactly what I say, no larger unit, no smaller unit.\nEach school must have the claim to be considered in relation to its\nspecial circumstances. The classifying of schools for some purposes is\nnecessary. But no absolutely rigid curriculum, not modified by its own\nstaff, should be permissible. Exactly the same principles apply, with\nthe proper modifications, to universities and to technical colleges.\n\nWhen one considers in its length and in its breadth the importance\nof this question of the education of a nation's young, the broken\nlives, the defeated hopes, the national failures, which result from the\nfrivolous inertia with which it is treated, it is difficult to restrain\nwithin oneself a savage rage. In the conditions of modern life the ride\nis absolute, the race which does not value trained intelligence is\ndoomed. Not all your heroism, not all your social charm, not all your\nwit, not all your victories on land or at sea, can move back the finger\nof fate. To-day we maintain ourselves. To-morrow science will have\nmoved forward yet one more step, and there will be no appeal from the\njudgment which will then be pronounced on the uneducated.\n\nWe can be content with no less than the old summary of educational\nideal which has been current at any time from the dawn of our\ncivilisation. The essence of education is that it be religious.\n\nPray, what is religious education?\n\nA religious education is an education which inculcates duty and\nreverence. Duty arises from our potential control over the course\nof events. Where attainable knowledge could have changed the issue,\nignorance has the guilt of vice. And the foundation of reverence is\nthis perception, that the present holds within itself the complete sum\nof existence, backwards and forwards, that whole amplitude of time,\nwhich is eternity.\n\n\n\n\n                              CHAPTER II\n\n    TECHNICAL EDUCATION AND ITS RELATION TO SCIENCE AND LITERATURE\n\n   (_Presidential Address to the Mathematical Association, January,\n                                1917_)\n\n\nTHE subject of this address is Technical Education. I wish to examine\nits essential nature and also its relation to a liberal education. Such\nan inquiry may help us to realise the conditions for the successful\nworking of a national system of technical training. It is also a very\nburning question among mathematical teachers; for mathematics is\nincluded in most technological courses.\n\nNow it is unpractical to plunge into such a discussion without framing\nin our own minds the best ideal towards which we desire to work,\nhowever modestly we may frame our hopes as to the result which in the\nnear future is likely to be achieved.\n\nPeople are shy of ideals; and accordingly we find a formulation of the\nideal state of mankind placed by a modern dramatist[1] in the mouth of\na mad priest: \"In my dreams it is a country where the State is the\nChurch and the Church the people: three in one and one in three. It is\na commonwealth in which work is play and play is life: three in one and\none in three. It is a temple in which the priest is the worshipper and\nthe worshipper the worshipped: three in one and one in three. It is a\ngodhead in which all life is human and all humanity divine: three in\none and one in three. It is, in short, the dream of a madman.\"\n\nNow the part of this speech to which I would direct attention is\nembodied in the phrase, \"It is a commonwealth in which work is play\nand play is life.\" This is the ideal of technical education. It sounds\nvery mystical when we confront it with the actual facts, the toiling\nmillions, tired, discontented, mentally indifferent, and then the\nemployers---- I am not undertaking a social analysis, but I shall carry\nyou with me when I admit that the present facts of society are a long\nway off this ideal. Furthermore, we are agreed that an employer who\nconducted his workshop on the principle that \"work should be play\"\nwould be ruined in a week.\n\nThe curse that has been laid on humanity, in fable and in fact, is,\nthat by the sweat of its brow shall it live. But reason and moral\nintuition have seen in this curse the foundation for advance. The early\nBenedictine monks rejoiced in their labours because they conceived\nthemselves as thereby made fellow-workers with Christ.\n\nStripped of its theological trappings, the essential idea remains,\nthat work should be transfused with intellectual and moral vision and\nthereby turned into a joy, triumphing over its weariness and its pain.\nEach of us will re-state this abstract formulation in a more concrete\nshape in accordance with his private outlook. State it how you like,\nso long as you do not lose the main point in your details. However you\nphrase it, it remains the sole real hope of toiling humanity; and it\nis in the hands of technical teachers, and of those who control their\nspheres of activity, so to mould the nation that daily it may pass to\nits labours in the spirit of the monks of old.\n\nThe immediate need of the nation is a large supply of skilled workmen,\nof men with inventive genius, and of employers alert in the development\nof new ideas.\n\nThere is one--and only one--way to obtain these admirable results.\nIt is by producing workmen, men of science, and employers who enjoy\ntheir work. View the matter practically in the light of our knowledge\nof average human nature. Is it likely that a tired, bored workman,\nhowever skilful his hands, will produce a large output of first-class\nwork? He will limit his production, will scamp his work, and be an\nadept at evading inspection; he will be slow in adapting himself to\nnew methods; he will be a focus of discontent, full of unpractical\nrevolutionary ideas, controlled by no sympathetic apprehension of the\nreal working of trade conditions. If, in the troubled times which may\nbe before us, you wish appreciably to increase the chance of some\nsavage upheaval, introduce widespread technical education and ignore\nthe Benedictine ideal. Society will then get what it deserves.\n\nAgain, inventive genius requires pleasurable mental activity as a\ncondition for its vigorous exercise. \"Necessity is the mother of\ninvention\" is a silly proverb. \"Necessity is the mother of futile\ndodges\" is much nearer to the truth. The basis of the growth of modern\ninvention is science, and science is almost wholly the outgrowth of\npleasurable intellectual curiosity.\n\nThe third class are the employers, who are to be enterprising. Now\nit is to be observed that it is the successful employers who are the\nimportant people to get at, the men with business connections all over\nthe world, men who are already rich. No doubt there will always be a\ncontinuous process of rise and fall of businesses. But it is futile\nto expect flourishing trade, if in the mass the successful houses of\nbusiness are suffering from atrophy. Now if these men conceive their\nbusinesses as merely indifferent means for acquiring other disconnected\nopportunities of life, they have no spur to alertness. They are\nalready doing very well, the mere momentum of their present business\nengagements will carry them on for their time. They are not at all\nlikely to bother themselves with the doubtful chances of new methods.\nTheir real soul is in the other side of their life. Desire for money\nwill produce hard-fistedness and not enterprise. There is much more\nhope for humanity from manufacturers who enjoy their work than from\nthose who continue in irksome business with the object of founding\nhospitals.\n\nFinally, there can be no prospect of industrial peace so long as\nmasters and men in the mass conceive themselves as engaged in a\nsoulless operation of extracting money from the public. Enlarged views\nof the work performed, and of the communal service thereby rendered,\ncan be the only basis on which to found sympathetic co-operation.\n\nThe conclusion to be drawn from this discussion is, that alike for\nmasters and for men a technical or technological education, which is to\nhave any chance of satisfying the practical needs of the nation, must\nbe conceived in a liberal spirit as a real intellectual enlightenment\nin regard to principles applied and services rendered. In such an\neducation geometry and poetry are as essential as turning lathes.\n\nThe mythical figure of Plato may stand for modern liberal education\nas does that of St. Benedict for technical education. We need not\nentangle ourselves in the qualifications necessary for a balanced\nrepresentation of the actual thoughts of the actual men. They are used\nhere as symbolic figures typical of antithetical notions. We consider\nPlato in the light of the type of culture he now inspires.\n\nIn its essence a liberal education is an education for thought and\nfor \u00e6sthetic appreciation. It proceeds by imparting a knowledge of\nthe masterpieces of thought, of imaginative literature, and of art.\nThe action which it contemplates is command. It is an aristocratic\neducation implying leisure. This Platonic ideal has rendered\nimperishable services to European civilisation. It has encouraged art,\nit has fostered that spirit of disinterested curiosity which is the\norigin of science, it has maintained the dignity of mind in the face of\nmaterial force, a dignity which claims freedom of thought. Plato did\nnot, like St. Benedict, bother himself to be a fellow-worker with his\nslaves; but he must rank among the emancipators of mankind. His type\nof culture is the peculiar inspiration of the liberal aristocrat, the\nclass from which Europe derives what ordered liberty it now possesses.\nFor centuries, from Pope Nicholas V to the schools of the Jesuits, and\nfrom the Jesuits to the modern headmasters of English public schools,\nthis educational ideal has had the strenuous support of the clergy.\n\nFor certain people it is a very good education. It suits their type\nof mind and the circumstances amid which their life is passed. But\nmore has been claimed for it than this. All education has been judged\nadequate or defective according to its approximation to this sole type.\n\nThe essence of the type is a large discursive knowledge of the best\nliterature. The ideal product of the type is the man who is acquainted\nwith the best that has been written. He will have acquired the chief\nlanguages, he will have considered the histories of the rise and fall\nof nations, the poetic expression of human feeling, and have read the\ngreat dramas and novels. He will also be well grounded in the chief\nphilosophies, and have attentively read those philosophic authors who\nare distinguished for lucidity of style.\n\nIt is obvious that, except at the close of a long life, he will not\nhave much time for anything else if any approximation is to be made to\nthe fulfilment of this programme. One is reminded of the calculation\nin a dialogue of Lucian that, before a man could be justified in\npractising any one of the current ethical systems, he should have spent\na hundred and fifty years in examining their credentials.\n\nSuch ideals are not for human beings. What is meant by a liberal\nculture is nothing so ambitious as a full acquaintance with the varied\nliterary expression of civilised mankind from Asia to Europe, and from\nEurope to America. A small selection only is required; but then, as\nwe are told, it is a selection of the very best. I have my doubts of\na selection which includes Xenophon and omits Confucius, but then I\nhave read neither in the original. The ambitious programme of a liberal\neducation really shrinks to a study of some fragments of literature\nincluded in a couple of important languages.\n\nBut the expression of the human spirit is not confined to literature.\nThere are the other arts, and there are the sciences. Also education\nmust pass beyond the passive reception of the ideas of others. Powers\nof initiative must be strengthened. Unfortunately initiative does not\nmean just one acquirement--there is initiative in thought, initiative\nin action, and the imaginative initiative of art; and these three\ncategories require many subdivisions.\n\nThe field of acquirement is large, and the individual so fleeting and\nso fragmentary: classical scholars, scientists, headmasters are alike\nignoramuses.\n\nThere is a curious illusion that a more complete culture was possible\nwhen there was less to know. Surely the only gain was, that it was more\npossible to remain unconscious of ignorance. It cannot have been a gain\nto Plato to have read neither Shakespeare, nor Newton, nor Darwin.\nThe achievements of a liberal education have in recent times not been\nworsened. The change is that its pretensions have been found out.\n\nMy point is, that no course of study can claim any position of ideal\ncompleteness. Nor are the omitted factors of subordinate importance.\nThe insistence in the Platonic culture on disinterested intellectual\nappreciation is a psychological error. Action and our implication in\nthe transition of events amid the evitable bond of cause to effect\nare fundamental. An education which strives to divorce intellectual\nor \u00e6sthetic life from these fundamental facts carries with it the\ndecadence of civilisation. Essentially culture should be for action,\nand its effect should be to divest labour from the associations of\naimless toil. Art exists that we may know the deliverances of our\nsenses as good. It heightens the sense-world.\n\nDisinterested scientific curiosity is a passion for an ordered\nintellectual vision of the connection of events. But the goal of\nsuch curiosity is the marriage of action to thought. This essential\nintervention of action even in abstract science is often overlooked. No\nman of science wants merely to know. He acquires knowledge to appease\nhis passion for discovery. He does not discover in order to know, he\nknows in order to discover. The pleasure which art and science can\ngive to toil is the enjoyment which arises from successfully directed\nintention. Also it is the same pleasure which is yielded to the\nscientist and to the artist.\n\nThe antithesis between a technical and a liberal education is\nfallacious. There can be no adequate technical education which is not\nliberal, and no liberal education which is not technical: that is, no\neducation which does not impart both technique and intellectual vision.\nIn simpler language, education should turn out the pupil with something\nhe knows well and something he can do well. This intimate union of\npractice and theory aids both. The intellect does not work best in a\nvacuum. The stimulation of creative impulse requires, especially in\nthe case of a child, the quick transition to practice. Geometry and\nmechanics, followed by workshop practice, gain that reality without\nwhich mathematics is verbiage.\n\nThere are three main methods which are required in a national system of\neducation, namely, the literary curriculum, the scientific curriculum,\nthe technical curriculum. But each of these curricula should include\nthe other two. What I mean is, that every form of education should give\nthe pupil a technique, a science, an assortment of general ideas, and\n\u00e6sthetic appreciation, and that each of these sides of his training\nshould be illuminated by the others. Lack of time, even for the most\nfavoured pupil, makes it impossible to develop fully each curriculum.\nAlways there must be a dominant emphasis. The most direct \u00e6sthetic\ntraining naturally falls in the technical curriculum in those cases\nwhen the training is that requisite for some art or artistic craft. But\nit is of high importance in both a literary and a scientific education.\n\nThe educational method of the literary curriculum is the study of\nlanguage, that is, the study of our most habitual method of conveying\nto others our states of mind. The technique which should be acquired\nis the technique of verbal expression, the science is the study of the\nstructure of language and the analysis of the relations of language\nto the states of mind conveyed. Furthermore, the subtle relations of\nlanguage to feeling, and the high development of the sense organs\nto which written and spoken words appeal, lead to keen \u00e6sthetic\nappreciations being aroused by the successful employment of language.\nFinally, the wisdom of the world is preserved in the masterpieces of\nlinguistic composition.\n\nThis curriculum has the merit of homogeneity. All its various parts\nare co-ordinated and play into each other's hands. We can hardly be\nsurprised that such a curriculum, when once broadly established, should\nhave claimed the position of the sole perfect type of education. Its\ndefect is unduly to emphasise the importance of language. Indeed the\nvaried importance of verbal expression is so overwhelming that its\nsober estimation is difficult. Recent generations have been witnessing\nthe retreat of literature, and of literary forms of expression, from\ntheir position of unique importance in intellectual life. In order\ntruly to become a servant and a minister of nature something more is\nrequired than literary aptitudes.\n\nA scientific education is primarily a training in the art of observing\nnatural phenomena, and in the knowledge and deduction of laws\nconcerning the sequence of such phenomena. But here, as in the case of\na liberal education, we are met by the limitations imposed by shortness\nof time. There are many types of natural phenomena, and to each type\nthere corresponds a science with its peculiar modes of observation,\nand its peculiar types of thought employed in the deduction of laws. A\nstudy of science in general is impossible in education, all that can be\nachieved is the study of two or three allied sciences. Hence the charge\nof narrow specialism urged against any education which is primarily\nscientific. It is obvious that the charge is apt to be well-founded;\nand it is worth considering how, within the limits of a scientific\neducation and to the advantage of such an education, the danger can be\navoided.\n\nSuch a discussion requires the consideration of technical education. A\ntechnical education is in the main a training in the art of utilising\nknowledge for the manufacture of material products. Such a training\nemphasises manual skill, and the co-ordinated action of hand and\neye, and judgment in the control of the process of construction. But\njudgment necessitates knowledge of those natural processes of which the\nmanufacture is the utilisation. Thus somewhere in technical training\nan education in scientific knowledge is required. If you minimise\nthe scientific side, you will confine it to the scientific experts;\nif you maximise it, you will impart it in some measure to the men,\nand--what is of no less importance--to the directors and managers of\nthe businesses.\n\nTechnical education is not necessarily allied exclusively to science\non its mental side. It may be an education for an artist or for\napprentices to an artistic craft. In that case \u00e6sthetic appreciation\nwill have to be cultivated in connection with it.\n\nAn evil side of the Platonic culture has been its total neglect of\ntechnical education as an ingredient in the complete development\nof ideal human beings. This neglect has arisen from two disastrous\nantitheses, namely, that between mind and body, and that between\nthought and action. I will here interject, solely to avoid criticism,\nthat I am well aware that the Greeks highly valued physical beauty and\nphysical activity. They had, however, that perverted sense of values\nwhich is the nemesis of slave-owning.\n\nI lay it down as an educational axiom that in teaching you will come\nto grief as soon as you forget that your pupils have bodies. This is\nexactly the mistake of the post-renaissance Platonic curriculum. But\nnature can be kept at bay by no pitchfork; so in English education,\nbeing expelled from the classroom, she returned with a cap and bells in\nthe form of all-conquering athleticism.\n\nThe connections between intellectual activity and the body, though\ndiffused in every bodily feeling, are focussed in the eyes, the ears,\nthe voice, and the hands. There is a co-ordination of senses and\nthought, and also a reciprocal influence between brain activity and\nmaterial creative activity. In this reaction the hands are peculiarly\nimportant. It is a moot point whether the human hand created the human\nbrain, or the brain created the hand. Certainly the connection is\nintimate and reciprocal. Such deep-seated relations are not widely\natrophied by a few hundred years of disuse in exceptional families.\n\nThe disuse of hand-craft is a contributory cause to the brain-lethargy\nof aristocracies, which is only mitigated by sport where the concurrent\nbrain-activity is reduced to a minimum and the hand-craft lacks\nsubtlety. The necessity for constant writing and vocal exposition is\nsome slight stimulus to the thought-power of the professional classes.\nGreat readers, who exclude other activities, are not distinguished by\nsubtlety of brain. They tend to be timid conventional thinkers. No\ndoubt this is partly due to their excessive knowledge outrunning their\npowers of thought; but it is partly due to the lack of brain-stimulus\nfrom the productive activities of hand or voice.\n\nIn estimating the importance of technical education we must rise above\nthe exclusive association of learning with book-learning. First-hand\nknowledge is the ultimate basis of intellectual life. To a large\nextent book-learning conveys second-hand information, and as such can\nnever rise to the importance of immediate practice. Our goal is to see\nthe immediate event of our lives as instances of our general ideas.\nWhat the learned world tends to offer is one second-hand scrap of\ninformation illustrating ideas derived from another second-hand scrap\nof information. The second-handedness of the learned world is the\nsecret of its mediocrity. It is tame because it has never been scared\nby facts. The main importance of Francis Bacon's influence does not\nlie in any peculiar theory of inductive reasoning which he happened to\nexpress, but in the revolt against second-hand information of which he\nwas a leader.\n\nThe peculiar merit of a scientific education should be, that it bases\nthought upon first-hand observation; and the corresponding merit of a\ntechnical education is, that it follows our deep natural instinct to\ntranslate thought into manual skill, and manual activity into thought.\n\nWe are a Mathematical Association, and it is natural to ask: Where do\nwe come in? We come in just at this point.\n\nThe thought which science evokes is logical thought. Now logic is of\ntwo kinds: the logic of discovery and the logic of the discovered.\n\nThe logic of discovery consists in the weighing of probabilities, in\ndiscarding details deemed to be irrelevant, in divining the general\nrules according to which events occur, and in testing hypotheses by\ndevising suitable experiments. This is inductive logic.\n\nThe logic of the discovered is the deduction of the special events\nwhich, under certain circumstances, would happen in obedience to the\nassumed laws of nature. Thus when the laws are discovered or assumed,\ntheir utilisation entirely depends on deductive logic. Without\ndeductive logic science would be entirely useless. It is merely a\nbarren game to ascend from the particular to the general, unless\nafterwards we can reverse the process and descend from the general to\nthe particular, ascending and descending like the angels on Jacob's\nladder. When Newton had divined the law of gravitation he at once\nproceeded to calculate the earth's attractions on an apple at its\nsurface and on the moon. We may note in passing that inductive logic\nwould be impossible without deductive logic. Thus Newton's calculations\nwere an essential step in his inductive verification of the great law.\n\nNow mathematics is nothing else than the more complicated parts of\nthe art of deductive reasoning, especially where it concerns number,\nquantity, and space.\n\nIn the teaching of science, the art of thought should be taught:\nnamely, the art of forming clear conceptions applying to first-hand\nexperience, the art of divining the general truths which apply, the\nart of testing divinations, and the art of utilising general truths\nby reasoning to more particular cases of some peculiar importance.\nFurthermore, a power of scientific exposition is necessary, so that the\nrelevant issues from a confused mass of ideas can be stated clearly,\nwith due emphasis on important points.\n\nBy the time a science, or a small group of sciences, has been taught\nthus amply, with due regard to the general art of thought, we have gone\na long way towards correcting the specialism of science. The worst\nof a scientific education based, as necessarily must be the case, on\none or two particular branches of science, is that the teachers under\nthe influence of the examination system are apt merely to stuff\ntheir pupils with the narrow results of these special sciences. It is\nessential that the generality of the method be continually brought to\nlight and contrasted with the speciality of the particular application.\nA man who only knows his own science, as a routine peculiar to that\nscience, does not even know that. He has no fertility of thought, no\npower of quickly seizing the bearing of alien ideas. He will discover\nnothing, and be stupid in practical applications.\n\nThis exhibition of the general in the particular is extremely difficult\nto effect, especially in the case of younger pupils. The art of\neducation is never easy. To surmount its difficulties, especially those\nof elementary education, is a task worthy of the highest genius. It is\nthe training of human souls.\n\nMathematics, well taught, should be the most powerful instrument\nin gradually implanting this generality of idea. The essence of\nmathematics is perpetually to be discarding more special ideas in\nfavour of more general ideas, and special methods in favour of general\nmethods. We express the conditions of a special problem in the form of\nan equation, but that equation will serve for a hundred other problems,\nscattered through diverse sciences. The general reasoning is always\nthe powerful reasoning, because deductive cogency is the property of\nabstract form.\n\nHere, again, we must be careful. We shall ruin mathematical education\nif we use it merely to impress general truths. The general ideas are\nthe means of connecting particular results. After all, it is the\nconcrete special cases which are important. Thus in the handling of\nmathematics in your results you cannot be too concrete, and in your\nmethods you cannot be too general. The essential course of reasoning\nis to generalise what is particular, and then to particularise what is\ngeneral. Without generality there is no reasoning, without concreteness\nthere is no importance.\n\nConcreteness is the strength of technical education. I would\nremind you that truths which lack the highest generality are not\nnecessarily concrete facts. For example, _x_ + _y_ = _y_ + _x_ is an\nalgebraic truth more general than 2 + 2 = 4. But \"two and two make\nfour\" is itself a highly general proposition lacking any element of\nconcreteness. To obtain a concrete proposition immediate intuition of a\ntruth concerning particular objects is requisite; for example, \"these\ntwo apples and those apples together make four apples\" is a concrete\nproposition, if you have direct perception or immediate memory of the\napples.\n\nIn order to obtain the full realisation of truths as applying, and not\nas empty formul\u00e6, there is no alternative to technical education. Mere\npassive observation is not sufficient. In creation only is there vivid\ninsight into the properties of the object thereby produced. If you\nwant to understand anything, make it yourself, is a sound rule. Your\nfaculties will be alive, your thoughts gain vividness by an immediate\ntranslation into acts. Your ideas gain that reality which comes from\nseeing the limits of their application.\n\nIn elementary education this doctrine has long been put into practice.\nYoung children are taught to familiarise themselves with shapes and\ncolours by simple manual operations of cutting out and of sorting. But\ngood though this is, it is not quite what I mean. That is practical\nexperience before you think, experience antecedent to thought in order\nto create ideas, a very excellent discipline. But technical education\nshould be much more than that: it is creative experience while you\nthink, experience which realises your thought, experience which teaches\nyou to co-ordinate act and thought, experience leading you to associate\nthought with foresight and foresight with achievement. Technical\neducation gives theory, and a shrewd insight as to where theory fails.\n\nA technical education is not to be conceived as a maimed alternative\nto the perfect Platonic culture: namely, as a defective training\nunfortunately made necessary by cramped conditions of life. No\nhuman being can attain to anything but fragmentary knowledge and a\nfragmentary training of his capacities. There are, however, three main\nroads along which we can proceed with good hope of advancing towards\nthe best balance of intellect and character: these are the way of\nliterary culture, the way of scientific culture, the way of technical\nculture. No one of these methods can be exclusively followed without\ngrave loss of intellectual activity and of character. But a mere\nmechanical mixture of the three curricula will produce bad results in\nthe shape of scraps of information never interconnected or utilised.\nWe have already noted as one of the strong points of the traditional\nliterary culture that all its parts are co-ordinated. The problem\nof education is to retain the dominant emphasis, whether literary,\nscientific, or technical, and without loss of co-ordination to infuse\ninto each way of education something of the other two.\n\nTo make definite the problem of technical education fix attention on\ntwo ages: one thirteen, when elementary education ends; and the other\nseventeen, when technical education ends so far as it is compressed\nwithin a school curriculum. I am aware that for artisans in junior\ntechnical schools a three-years' course would be more usual. On the\nother hand, for naval officers, and for directing classes generally,\na longer time can be afforded. We want to consider the principles\nto govern a curriculum which shall land these children at the age of\nseventeen in the position of having technical skill useful to the\ncommunity.\n\nTheir technical manual training should start at thirteen, bearing a\nmodest proportion to the rest of their work, and should increase in\neach year finally to attain to a substantial proportion. Above all\nthings it should not be too specialised. Workshop finish and workshop\ndodges, adapted to one particular job, should be taught in the\ncommercial workshop, and should form no essential part of the school\ncourse. A properly trained worker would pick them up in no time. In all\neducation the main cause of failure is staleness. Technical education\nis doomed if we conceive it as a system for catching children young and\nfor giving them one highly specialised manual aptitude. The nation has\nneed of a fluidity of labour, not merely from place to place, but also\nwithin reasonable limits of allied aptitudes, from one special type of\nwork to another special type. I know that here I am on delicate ground,\nand I am not claiming that men while they are specialising on one sort\nof work should spasmodically be set to other kinds. That is a question\nof trade organisation with which educationalists have no concern. I am\nonly asserting the principles that training should be broader than the\nultimate specialisation, and that the resulting power of adaptation to\nvarying demands is advantageous to the workers, to the employers, and\nto the nation.\n\nIn considering the intellectual side of the curriculum we must be\nguided by the principle of the co-ordination of studies. In general,\nthe intellectual studies most immediately related to manual training\nwill be some branches of science. More than one branch will, in fact,\nbe concerned; and even if that be not the case, it is impossible to\nnarrow down scientific study to a single thin line of thought. It is\npossible, however, provided that we do not press the classification\ntoo far, roughly to classify technical pursuits according to the\ndominant science involved. We thus find a sixfold division, namely,\n(1) Geometrical techniques, (2) Mechanical techniques, (3) Physical\ntechniques, (4) Chemical techniques, (5) Biological techniques, (6)\nTechniques of commerce and of social service.\n\nBy this division, it is meant that apart from auxiliary sciences\nsome particular science requires emphasis in the training for most\noccupations. We can, for example, reckon carpentry, ironmongery,\nand many artistic crafts among geometrical techniques. Similarly\nagriculture is a biological technique. Probably cookery, if it includes\nfood catering, would fall midway between biological, physical, and\nchemical sciences, though of this I am not sure.\n\nThe sciences associated with commerce and social service would be\npartly algebra, including arithmetic and statistics, and partly\ngeography and history. But this section is somewhat heterogeneous in\nits scientific affinities. Anyhow the exact way in which technical\npursuits are classified in relation to science is a detail. The\nessential point is, that with some thought it is possible to find\nscientific courses which illuminate most occupations. Furthermore, the\nproblem is well understood, and has been brilliantly solved in many of\nthe schools of technology and junior technical schools throughout the\ncountry.\n\nIn passing from science to literature, in our review of the\nintellectual elements of technical education, we note that many studies\nhover between the two: for example, history and geography. They are\nboth of them very essential in education, provided that they are the\nright history and the right geography. Also books giving descriptive\naccounts of general results, and trains of thought in various sciences\nfall in the same category. Such books should be partly historical\nand partly expository of the main ideas which have finally arisen.\nProf. R. A. Gregory's recent book, _Discovery_, and the _Home\nUniversity Library_ series illustrate my meaning. Their value in\neducation depends on their quality as mental stimulants. They must not\nbe inflated with gas on the wonders of science, and must be informed\nwith a broad outlook.\n\nIt is unfortunate that the literary element in education has rarely\nbeen considered apart from grammatical study. The historical reason is,\nthat when the modern Platonic curriculum was being formed Latin and\nGreek were the sole keys which rendered great literature accessible.\nBut there is no necessary connection between literature and grammar.\nThe great age of Greek literature was already past before the arrival\nof the grammarians of Alexandria. Of all types of men to-day existing,\nclassical scholars are the most remote from the Greeks of the Periclean\ntimes.\n\nMere literary knowledge is of slight importance. The only thing that\nmatters is, how it is known. The facts related are nothing. Literature\nonly exists to express and develop that imaginative world which is our\nlife, the kingdom which is within us. It follows that the literary side\nof a technical education should consist in an effort to make the pupils\nenjoy literature. It does not matter what they know, but the enjoyment\nis vital. The great English Universities, under whose direct authority\nschool-children are examined in plays of Shakespeare, to the certain\ndestruction of their enjoyment, should be prosecuted for soul-murder.\n\nNow there are two kinds of intellectual enjoyment: the enjoyment of\ncreation, and the enjoyment of relaxation. They are not necessarily\nseparated. A change of occupation may give the full tide of happiness\nwhich comes from the concurrence of both forms of pleasure. The\nappreciation of literature is really creation. The written word, its\nmusic, and its associations, are only the stimuli. The vision which\nthey evoke is our own doing. No one, no genius other than our own,\ncan make our own life live. But except for those engaged in literary\noccupations, literature is also a relaxation. It gives exercise to that\nother side which any occupation must suppress during the working hours.\nArt also has the same function in life as has literature.\n\nTo obtain the pleasure of relaxation requires no help. The pleasure\nis merely to cease doing. Some such pure relaxation is a necessary\ncondition of health. Its dangers are notorious, and to the greater\npart of the necessary relaxation nature has affixed, not enjoyment,\nbut the oblivion of sleep. Creative enjoyment is the outcome of\nsuccessful effort and requires help for its initiation. Such enjoyment\nis necessary for high-speed work and for original achievement.\n\nTo speed up production with unrefreshed workmen is a disastrous\neconomic policy. Temporary success will be at the expense of the\nnation, which, for long years of their lives, will have to support\nworn-out artisans--unemployables. Equally disastrous is the\nalternation of spasms of effort with periods of pure relaxation.\nSuch periods are the seed-times of degeneration, unless rigorously\ncurtailed. The normal recreation should be change of activity,\nsatisfying the cravings of instincts. Games afford such activity. Their\ndisconnection emphasises the relaxation, but their excess leaves us\nempty.\n\nIt is here that literature and art should play an essential part in\na healthily organised nation. Their services to economic production\nwould be only second to those of sleep or of food. I am not now talking\nof the training of an artist, but of the use of art as a condition of\nhealthy life. It is analogous to sunshine in the physical world.\n\nWhen we have once rid our minds of the idea that knowledge is to\nbe exacted, there is no especial difficulty or expense involved in\nhelping the growth of artistic enjoyment. All school-children could\nbe sent at regular intervals to neighbouring theatres where suitable\nplays could be subsidised. Similarly for concerts and cinema films.\nPictures are more doubtful in their popular attraction; but interesting\nrepresentations of scenes or ideas which the children have read about\nwould probably appeal. The pupils themselves should be encouraged\nin artistic efforts. Above all the art of reading aloud should be\ncultivated. The Roger de Coverley essays of Addison are perfect\nexamples of readable prose.\n\nArt and literature have not merely an indirect effect on the main\nenergies of life. Directly, they give vision. The world spreads wide\nbeyond the deliverances of material sense, with subtleties of reaction\nand with pulses of emotion. Vision is the necessary antecedent to\ncontrol and to direction. In the contest of races which in its final\nissues will be decided in the workshops and not on the battle-field,\nthe victory will belong to those who are masters of stores of trained\nnervous energy, working under conditions favourable to growth. One such\nessential condition is Art.\n\nIf there had been time, there are other things which I should like\nto have said: for example, to advocate the inclusion of one foreign\nlanguage in all education. From direct observation I know this to be\npossible for artisan children. But enough has been put before you,\nto make plain the principles with which we should undertake national\neducation.\n\nIn conclusion, I recur to the thought of the Benedictines, who\nsaved for mankind the vanishing civilisation of the ancient world\nby linking together knowledge, labour, and moral energy. Our danger\nis to conceive practical affairs as the kingdom of evil, in which\nsuccess is only possible by the extrusion of ideal aims. I believe\nthat such a conception is a fallacy directly negatived by practical\nexperience. In education this error takes the form of a mean view of\ntechnical training. Our forefathers in the dark ages saved themselves\nby embodying high ideals in great organisations. It is our task,\nwithout servile imitation, boldly to exercise our creative energies,\nremembering amid discouragements that the coldest hour immediately\nprecedes the dawn.\n\n\nFOOTNOTES:\n\n[Footnote 1: _Cf._ BERNARD SHAW: _John Bull's Other\nIsland_.]\n\n\n\n\n                              CHAPTER III\n\n                       A POLYTECHNIC IN WAR-TIME\n\n  _Address at the Prize Distribution, Borough Polytechnic Institute,\n                    Southwark, 16th February, 1917_\n\n\nI WILL commence by drawing your attention to some of the satisfactory\nfeatures of the Principal's report on the work of the Institute during\nthe past year. It has been a year of great difficulties. Some of our\nstaff are serving with the colours, and our classes have been depleted.\nBut in spite of everything, we have done very well. First, the average\nresult in the examinations has been good, surprisingly good in view of\nthe present circumstances. The Governors attach great importance to\nthe maintenance of a high average result; it is the best single test\nof efficiency. Again, our individual successes have been notable. We\nhave gained--I say _we_ because we are all one in our pleasure at\nthese successes--we have gained two \u00a380 L.C.C. scholarships, nineteen\nexhibitions, in addition to a first-place, and medals, prizes and\ncertificates. All this is very satisfactory. It tells of efficient\nteaching, and of hard work and regular attendance on the part of the\nstudents. We know that we are keeping up the standard of efficiency\nwhich in the past has been a source of pride to every one connected\nwith this Institute.\n\nNow all this good work does not come about by itself without any one\nmaking an effort. Such a record requires our skilled staff of teachers\nand organisers. They have worked very hard during the last session\nunder great difficulties, in order to create the successful result\nwhich we are here to celebrate. I know something about teaching. It\nis very exacting work, and can be made successful only by continual\ndevotion. I am sure that I am voicing your feelings, and I know that\nI am expressing those of the Governors, when I thank the ladies and\ngentlemen of the staff very heartily for their services during the last\nsession.\n\nPrize-givings are always pleasant occasions. We have come here to think\nabout our successes, and to congratulate our students. There is no more\nsatisfactory Governors' Meeting in the course of the year than when we\nmeet on this occasion, and face our friends and tell them how pleased\nwe are at the successful result of their hard work. This evening I am\nin a doubly happy position, for my colleagues have asked me to be their\nspokesman in tendering our good wishes to the prize-winners. You have\nworked hard and you have done well, and I am sure that you all deserve\nyour successes; they are a pleasure not only to you, but in your homes\nand to your companions and fellow-students.\n\nSuccessful work here will enable you to acquire skill in your trades,\nand thereby the better to earn your living. Earning a living is on the\naverage no bad test of service rendered to the community. A man who\nhas made himself skilful in his trade and has done well for himself in\nhis walk of life, has in general good reason to believe that he is a\ncitizen who has benefited his country. It is an evil day for a nation\nwhen it loses respect for success in industry.\n\nBut if you steer your lives by the compass which points steadily to\nthe North Pole of personal success, you will have missed your greatest\nchances in life. The genial climate is in the south.\n\nWhat I mean is this: you must make up your mind to find the best part\nof your happiness in kindly helpful relations with others. It should be\nour ambition to leave our own small corner of the world a little tidier\nand a little happier than when we entered it. I am well aware that\nthis is an old story; but old stories are sometimes true, and this is\nthe biggest truth in the whole world. The warm kindly feelings are the\nhappy feelings. The fortunate people are those whose minds are filled\nwith thoughts in which they forget themselves and remember others. It\nis not true that nature is a mere scene of struggle in which every one\ncompetes with his neighbour. Those communities thrive best and last\nlongest which are filled with a spirit of mutual help.\n\nThe future of the country lies with you. The crown of your success\nis the promise of future work, often unrecognised work, done under\ndiscouragement, but done steadily and cheerfully. It is on you that\nthe country depends for the maintenance and the growth of those ideals\nwithout which a race withers. Do not be discouraged by difficulties\nwhich seem unsurmountable. The conditions of life which mould us all\nare modified by our will, by our energy, and by the purity of our\nintentions.\n\nIf we may judge of intensity of feeling by length of memory, the\nenjoyment of receiving a prize bites very deep. Across the space of\nmore than forty years, before many of your parents were born, or when\nthey were being carried about in long clothes, I can remember, as if it\nwere yesterday, the occasion when I received my first prize at school.\nI can see the medi\u00e6val school-room, the headmaster, and my companions.\nPerhaps some of you, when a generation has passed by, will remember the\nscene to-night--this Stanley Gymnasium recalling the memory of Miss\nMaude Stanley, who devoted to our welfare so much of her energy and\nher thought--the adjoining Edric Hall associated with the name of Mr.\nEdric Bayley, the Father of the Institute; Mr. Millis and Miss Smith,\nthe first Principal and the first Lady Superintendent, the architects\nof our prosperity; Mr. Leonard Spicer, our Chairman and member of a\nfamily and of a firm known throughout the world, and respected in\nproportion as they are known. And the cause why to-night we are a small\ngathering is one more reason why this assembly can never slip from your\nmemory. We meet at a moment when England stands in as deadly a peril as\nin any previous moment of her history--such peril as when the Spanish\nships of the Armada rode in the English Channel, or when Napoleon\nwatched our coast across the Strait of Dover. The present danger can be\novercome only by the same courage as that which saved our freedom in\nthose former times.\n\nTherefore, to-night, in recalling the activities of the various\nsections of our society which form this great Polytechnic Institute,\nour thoughts go further afield. They travel by land and by sea, till\nthey bring before our minds the gallant band whom this Institute has\nsent to the Front--more than 800 of our members are with the Colours.\nWhat our fighting men have done for us, for the world in general, and\nfor the future of England, is so overwhelming that words cannot praise\nthem enough. I will just say one thing to you: When you read of great\ndeeds done in past times, of perils encountered, of adventures, of\nundaunted courage, of patriotism, of self-sacrifice, of suffering\nendured for noble cause, you each can say--I, too, have known such\nheroes; they are among my countrymen, they are among my fellow-workers,\nthey are among my fellow-students and companions, they are among the\ndear inmates of my home. And for those who have fallen, it is for us\nto erect a monument sufficient to transmit to future ages the memory\nof their sacrifice. For this purpose there is only one memorial which\ncan suffice, namely, the cause for which they died. The greatness of\nEngland, the future of England, has been left by them to our keeping.\nGuard it well.\n\nThe greatness of a country is nothing else than the greatness of the\nlives of the men and of the women who compose it. Do not look round and\nthink who ought to be great Englishmen--be great yourselves--you are\nthe people to achieve it, you who are sitting here to-night. There can\nbe no substitute service for this purpose. It is the collective energy\nof the whole people that will be needed to fashion a new England worthy\nof the sufferings which for its sake have been endured.\n\nA few days ago I asked a man who has worked in Egypt for many years\nunder Lord Kitchener, what he would pick out as the best sign of Lord\nKitchener's greatness. He answered, whatever Kitchener set himself to\ndo, thereby became important. Now that is the secret of it all--take\nhold of your opportunities and make them important.\n\nHere we are in this Borough Polytechnic. What an opportunity it\nrepresents. This Institute is a centre for social meeting, a centre\nfor recreation, a centre for education, a centre for discussion. We\nwill not sacrifice any one of our sides. They must all be part of the\ngreatness which we claim. Make them all first-rate.\n\nConsider first the social and recreative sides. For heaven's sake don't\nthink that you must be dull in order to be great. There is no finer\ntest of a nation than the way in which it fashions its amusements.\nThree centuries ago after the Armada we made a good start in Southwark.\nShakespeare had his theatre here and wrote his plays to be acted in\nthis borough. He has walked these streets, and if you had met him in\nWestminster he would, quite likely, have told you that he was going\ndown to the \"Elephant.\" And even now the performances given at the\n\"Old Vic\" are among the best in London for the purpose of seeing his\nplays properly acted. What Southwark has done for the drama, she can\ndo for the other arts, by using this Institute as the instrument for\nher energies. Why should we not be a centre for artistic enterprise--I\nmean for our own art and our own enterprise, thought of by ourselves\nand enjoyed by ourselves and carried through by ourselves? We shall not\nalways enjoy each others' creations, but the great point is to make our\nown efforts. Of course all efforts require preparation and stimulus and\nknowledge of what others are doing.\n\nAt the present time--interrupted for the moment by the war--a great\nrevolution in the art of painting is in progress throughout the world.\nIts centres are Paris and Italy and London and Munich, and its origin\nin the far east, in China and Japan. There are two sides as in every\nrevolution, the Conservatives and the Revolutionists. Our own frescoes\nin a neighbouring room represent an early stage of the movement in\nLondon. Why should we not know all about it--obtain loans of pictures\nwhich illustrate its phases and its cross currents, and compare these\nwith examples of the old style?\n\nBut pictures are only one phase of art, and not the sort of art which\nwe ourselves can produce most easily. There are music, dancing,\nrecitation, literature, carving and modelling, and the various\ndecorative arts, such as embroidery, bookbinding, dress-making and\nupholstery. This list, incomplete as it is, tells us two great\ntruths--you cannot separate art and recreation, and you cannot\nseparate art and business. The list includes items which we consider\nas amusements, and items which we think of as business. We began with\ndancing and ended with upholstery. Make them all beautiful.\n\nBeautiful things have dignity. Enjoy the rhythm of your dancing and\nadmire the beauty of your embroidery or your bookbinding. In whatever\nyou do, have an ideal of excellence. Any separation between art and\nwork is not only an error, but it is very bad business. Our brave\nallies, the French, have made Paris the art centre of the world. They\nhave built up and maintain their large and lucrative trade in the\ndecorative products of France, mainly by reason of three qualities\nwhich they possess. In the first place, they enjoy art themselves, and\nreverence it. In the second place, they have a tremendous power of hard\nwork. And in the third place, every Frenchman, and still more every\nFrenchwoman, have within them an immense fund of common sense. The\nthreefold secret is, Love of Art, Industry, and Common Sense.\n\nTo make available our industry and common sense in the trades where\nthey are wanted, rigorous training in schools of design and technique\nare necessary. We have such departments here. But all such training of\nyou will be a failure unless you yourselves enjoy art and beauty as\na natural recreation. A technical school of training is like a deep,\nnarrow well, sunk with careful labour to tap the underground river of\nwater which flows below the surface of our natures. But your well will\nbe dry unless the bright warm sun has first drawn up the vapour from\nthe wide ocean, and the free untrammelled breezes have carried the\nclouds hither and thither, until at length they break, as it were by\nchance over the distant hills and soak the land with their downpour.\n\nWhat I have said about art is a parable which applies to other\noccupations and other studies. It is more than a parable; it is the\nliteral truth. The love of art is the love of excellence, it is the\nenjoyment of the triumph of design over the shapeless products of\nchance forces. An engineer, who is worth his salt, loves the beauty of\nhis machines, shown in their adjustment of parts and in their swift,\nsmooth motions. He loves also the sense of foresight and of insight\nwhich knowledge can give him. People say that machinery and commerce\nare driving beauty out of the modern world. I do not believe it. A new\nbeauty is being added, a more intellectual beauty, appealing to the\nunderstanding as much as to the eye.\n\nThe wonder of London ever takes the mind with fresh astonishment. The\ncity possesses parks, and palaces, and cathedrals. But no other parts\nof it surpass in wonder its houses of business and its workshops and\nits factories.\n\nIn the next few years the future of the world will be decided for\ncenturies to come. The battles of this war are only the first part of\nthe contest between races, and between the ways of life for which\nthose races stand. We believe that England, with its various peoples\nand communities scattered in islands and continents beyond the seas,\nstands for ways of life infinitely precious, the way of humanity, the\nway of liberty, the way of self-government, the way of good order based\non toleration and kindly feeling, the way of peaceful industry. The\nfinal decision in this struggle will be found in the workshops of the\nworld. It lies in your hands. Statesmen and emperors will only register\nthe results which you have achieved. Your weapons will be skill, and\nenergy, and knowledge. You will require a sane understanding of your\nown rights, and a sane understanding of the rights and the difficulties\nof other classes. The greatness of England will be your greatness, and\nits success your success.\n\nThe arsenal for war is at Woolwich. This Polytechnic Institute is an\narsenal for peace, where you can find the weapons for the conquest of\nyour lives.\n\n\n\n\n                              CHAPTER IV\n\n                      THE MATHEMATICAL CURRICULUM\n\n    (_Presidential Address to the London Branch of the Mathematical\n                          Association, 1912_)\n\n\nTHE situation in regard to education at the present time cannot find\nits parallel without going back for some centuries to the break-up of\nthe medi\u00e6val traditions of learning. Then, as now, the traditional\nintellectual outlook, despite the authority which it had justly\nacquired from its notable triumphs, had grown to be too narrow for the\ninterests of mankind. The result of this shifting of human interest\nwas a demand for a parallel shifting of the basis of education, so\nas to fit the pupils for the ideas which later in life would in fact\noccupy their minds. Any serious fundamental change in the intellectual\noutlook of human society must necessarily be followed by an educational\nrevolution. It may be delayed for a generation by vested interests or\nby the passionate attachment of some leaders of thought to the cycle\nof ideas within which they received their own mental stimulus at an\nimpressionable age. But the law is inexorable that education to be\nliving and effective must be directed to informing pupils with those\nideas, and to creating for them those capacities which will enable them\nto appreciate the current thought of their epoch.\n\nThere is no such thing as a successful system of education in a vacuum,\nthat is to say, a system which is divorced from immediate contact with\nthe existing intellectual atmosphere. Education which is not modern\nshares the fate of all organic things which are kept too long.\n\nBut the blessed word \"modern\" does not really solve our difficulties.\nWhat we mean is, relevant to modern thought, either in the ideas\nimparted or in the aptitudes produced. Something found out only\nyesterday may not really be modern in this sense. It may belong to\nsome bygone system of thought prevalent in a previous age, or, what\nis very much more likely, it may be too recondite. When we demand\nthat education should be relevant to modern thought, we are referring\nto thoughts broadly spread throughout cultivated society. It is this\nquestion of the unfitness of recondite subjects for use in general\neducation which I wish to make the keynote of my address this afternoon.\n\nIt is in fact rather a delicate subject for us mathematicians.\nOutsiders are apt to accuse our subject of being recondite. Let us\ngrasp the nettle at once and frankly admit that in general opinion it\nis the very typical example of reconditeness. By this word I do not\nmean difficulty, but that the ideas involved are of highly special\napplication, and rarely influence thought.\n\nThis liability to reconditeness is the characteristic evil which is\napt to destroy the utility of mathematics in liberal education. So\nfar as it clings to the educational use of the subject, so far we\nmust acquiesce in a miserably low level of mathematical attainment\namong cultivated people in general. I yield to no one in my anxiety to\nincrease the educational scope of mathematics. The way to achieve this\nend is not by a mere blind demand for more mathematics. We must face\nthe real difficulty which obstructs its extended use.\n\nIs the subject recondite? Now, viewed as a whole, I think it is.\n_Securus judicat orbis terrarum_--the general judgment of mankind\nis sure.\n\nThe subject as it exists in the minds and in the books of students of\nmathematics _is_ recondite. It proceeds by deducing innumerable\nspecial results from general ideas, each result more recondite than the\npreceding. It is not my task this afternoon to defend mathematics as a\nsubject for profound study. It can very well take care of itself. What\nI want to emphasise is, that the very reasons which make this science\na delight to its students are reasons which obstruct its use as an\neducational instrument--namely, the boundless wealth of deductions from\nthe interplay of general theorems, their complication, their apparent\nremoteness from the ideas from which the argument started, the variety\nof methods, and their purely abstract character which brings, as its\ngift, eternal truth.\n\nOf course, all these characteristics are of priceless value to\nstudents; for ages they have fascinated some of the keenest intellects.\nMy only remark is that, except for a highly selected class, they are\nfatal in education. The pupils are bewildered by a multiplicity of\ndetail, without apparent relevance either to great ideas or to ordinary\nthoughts. The extension of this sort of training in the direction\nof acquiring more detail is the last measure to be desired in the\ninterests of education.\n\nThe conclusion at which we arrive is, that mathematics, if it is\nto be used in general education, must be subjected to a rigorous\nprocess of selection and adaptation. I do not mean, what is of course\nobvious, that however much time we devote to the subject the average\npupil will not get very far. But that, however limited the progress,\ncertain characteristics of the subject, natural at any stage, must be\nrigorously excluded. The science as presented to young pupils must\nlose its aspect of reconditeness. It must, on the face of it, deal\ndirectly and simply with a few general ideas of far-reaching importance.\n\nNow, in this matter of the reform of mathematical instruction, the\npresent generation of teachers may take a very legitimate pride in\nits achievements. It has shown immense energy in reform, and has\naccomplished more than would have been thought possible in so short a\ntime. It is not always recognised how difficult is the task of changing\na well-established curriculum entrenched behind public examinations.\n\nBut for all that, great progress has been made, and, to put the matter\nat its lowest, the old dead tradition has been broken up. I want to\nindicate this afternoon the guiding idea which should direct our\nefforts at reconstruction. I have already summed it up in a phrase,\nnamely, we must aim at the elimination of reconditeness from the\neducational use of the subject.\n\nOur courses of instruction should be planned to illustrate simply a\nsuccession of ideas of obvious importance. All pretty divagations\nshould be rigorously excluded. The goal to be aimed at is that the\npupil should acquire familiarity with abstract thought, should realise\nhow it applies to particular concrete circumstances, and should know\nhow to apply general methods to its logical investigation. With this\neducational ideal nothing can be worse than the aimless accretion\nof theorems in our text-books, which acquire their position merely\nbecause the children can be made to learn them and examiners can set\nneat questions on them. The bookwork to be learnt should all be very\nimportant as illustrating ideas. The examples set--and let there\nbe as many examples as teachers find necessary--should be direct\nillustrations of the theorems, either by way of abstract particular\ncases or by way of application to concrete phenomena. Here it is worth\nremarking that it is quite useless to simplify the bookwork, if the\nexamples set in examinations in fact require an extended knowledge of\nrecondite details. There is a mistaken idea that problems test ability\nand genius, and that bookwork tests cram. This is not my experience.\nOnly boys who have been specially crammed for scholarships can ever\ndo a problem paper successfully. Bookwork properly set, not in mere\nsnippets according to the usual bad plan, is a far better test of\nability, provided that it is supplemented by direct examples. But this\nis a digression on the bad influence of examinations on teaching.\n\nThe main ideas which lie at the base of mathematics are not at all\nrecondite. They are abstract. But one of the main objects of the\ninclusion of mathematics in a liberal education is to train the pupils\nto handle abstract ideas. The science constitutes the first large\ngroup of abstract ideas which naturally occur to the mind in any\nprecise form. For the purposes of education, mathematics consists of\nthe relations of number, the relations of quantity, and the relations\nof space. This is not a general definition of mathematics, which, in\nmy opinion, is a much more general science. But we are now discussing\nthe use of mathematics in education. These three groups of relations,\nconcerning number, quantity, and space, are interconnected.\n\nNow, in education we proceed from the particular to the general.\nAccordingly, children should be taught the use of these ideas by\npractice among simple examples. My point is this: The goal should be,\nnot an aimless accumulation of special mathematical theorems, but the\nfinal recognition that the preceding years of work have illustrated\nthose relations of number, and of quantity, and of space, which are of\nfundamental importance. Such a training should lie at the base of all\nphilosophical thought. In fact elementary mathematics rightly conceived\nwould give just that philosophical discipline of which the ordinary\nmind is capable. But what at all costs we ought to avoid, is the\npointless accumulation of details. As many examples as you like; let\nthe children work at them for terms, or for years. But these examples\nshould be direct illustrations of the main ideas. In this way, and\nthis only, can the fatal reconditeness be avoided.\n\nI am not now speaking in particular of those who are to be professional\nmathematicians, or of those who for professional reasons require a\nknowledge of certain mathematical details. We are considering the\nliberal education of all students, including these two classes. This\ngeneral use of mathematics should be the simple study of a few general\ntruths, well illustrated by practical examples. This study should\nbe conceived by itself, and completely separated in idea from the\nprofessional study mentioned above, for which it would make a most\nexcellent preparation. Its final stage should be the recognition of\nthe general truths which the work done has illustrated. As far as I\ncan make out, at present the final stage is the proof of some property\nof circles connected with triangles. Such properties are immensely\ninteresting to mathematicians. But are they not rather recondite,\nand what is the precise relation of such theorems to the ideal of\na liberal education? The end of all the grammatical studies of the\nstudent in classics is to read Virgil and Horace--the greatest thoughts\nof the greatest men. Are we content, when pleading for the adequate\nrepresentation in education of our own science, to say that the end of\na mathematical training is that the student should know the properties\nof the nine-point circle? I ask you frankly, is it not rather a \"come\ndown\"?\n\nThis generation of mathematical teachers has done so much strenuous\nwork in the way of reorganising mathematical instruction that there is\nno need to despair of its being able to elaborate a curriculum which\nshall leave in the minds of the pupils something even nobler than \"the\nambiguous case.\"\n\nLet us think how this final review, closing the elementary course,\nmight be conducted for the more intelligent pupils. Partly no doubt\nit requires a general oversight of the whole work done, considered\nwithout undue detail so as to emphasise the general ideas used, and\ntheir possibilities of importance when subjected to further study. Also\nthe analytical and geometrical ideas find immediate application in the\nphysical laboratory where a course of simple experimental mechanics\nshould have been worked through. Here the point of view is twofold, the\nphysical ideas and the mathematical ideas illustrate each other.\n\nThe mathematical ideas are essential to the precise formulation of the\nmechanical laws. The idea of a precise law of nature, the extent to\nwhich such laws are in fact verified in our experience, and the r\u00f4le of\nabstract thought in their formulation, then become practically apparent\nto the pupil. The whole topic of course requires detailed development\nwith full particular illustration, and is not suggested as requiring\nmerely a few bare abstract statements.\n\nIt would, however, be a grave error to put too much emphasis on the\nmere process of direct explanation of the previous work by way of\nfinal review. My point is, that the latter end of the course should\nbe so selected that in fact the general ideas underlying all the\nprevious mathematical work should be brought into prominence. This may\nwell be done by apparently entering on a new subject. For example,\nthe ideas of quantity and the ideas of number are fundamental to\nall precise thought. In the previous stages they will not have been\nsharply separated; and children are, rightly enough, pushed on to\nalgebra without too much bother and quantity. But the more intelligent\namong them at the end of their curriculum would gain immensely by a\ncareful consideration of those fundamental properties of quantity in\ngeneral which lead to the introduction of numerical measurement. This\nis a topic which also has the advantage that the necessary books are\nactually to hand. Euclid's fifth book is regarded by those qualified\nto judge as one of the triumphs of Greek mathematics. It deals with\nthis very point. Nothing can be more characteristic of the hopelessly\nilliberal character of the traditional mathematical education than the\nfact that this book has always been omitted. It deals with ideas,\nand therefore was ostracised. Of course a careful selection of the\nmore important propositions and a careful revision of the argument are\nrequired. This also is to hand in the publications of my immediate\npredecessor in the office of president, Prof. Hill. Furthermore, in Sir\nT. L. Heath's complete edition of Euclid, there is a full commentary\nembodying most of what has been said and thought on the point. Thus\nit is perfectly easy for teachers to inform themselves generally\non the topic. The whole book would not be wanted, but just the few\npropositions which embody the fundamental ideas. The subject is not\nfit for backward pupils; but certainly it could be made interesting to\nthe more advanced class. There would be great scope for interesting\ndiscussion as to the nature of quantity, and the tests which we should\napply to ascertain when we are dealing with quantities. The work\nwould not be at all in the air, but would be illustrated at every\nstage by reference to actual examples of cases where the quantitative\ncharacter is absent, or obscure, or doubtful, or evident. Temperature,\nheat, electricity, pleasure and pain, mass and distance could all be\nconsidered.\n\nAnother idea which requires illustration is that of functionality.\nA function in analysis is the counterpart of a law in the physical\nuniverse, and of a curve in geometry. Children have studied the\nrelations of functions to curves from the first beginning of their\nstudy of algebra, namely in drawing graphs. Of recent years there has\nbeen a great reform in respect to graphs. But at its present stage it\nhas either gone too far or not far enough. It is not enough merely\nto draw a graph. The idea behind the graph--like the man behind the\ngun--is essential in order to make it effective. At present there is\nsome tendency merely to set the children to draw curves, and there to\nleave the whole question.\n\nIn the study of simple algebraic functions and of trigonometrical\nfunctions we are initiating the study of the precise expression of\nphysical laws. Curves are another way of representing these laws. The\nsimple fundamental laws--such as the inverse square and the direct\ndistance--should be passed under review, and the applications of the\nsimple functions to express important concrete cases of physical laws\nconsidered. I cannot help thinking that the final review of this topic\nmight well take the form of a study of some of the main ideas of the\ndifferential calculus applied to simple curves. There is nothing\nparticularly difficult about the conception of a rate of change; and\nthe differentiation of a few powers of _x_, such as _x_^2,\n_x_^3, etc., could easily be effected; perhaps by the aid of\ngeometry even sin _x_ and cos _x_ could be differentiated.\nIf we once abandon our fatal habit of cramming the children with\ntheorems which they do not understand, and will never use, there will\nbe plenty of time to concentrate their attention on really important\ntopics. We can give them familiarity with conceptions which really\ninfluence thought.\n\nBefore leaving this topic of physical laws and mathematical functions,\nthere are other points to be noticed. The fact that the precise law is\nnever really verified by observation in its full precision is capable\nof easy illustration and of affording excellent examples. Again,\nstatistical laws, namely laws which are only satisfied on the average\nby large numbers, can easily be studied and illustrated. In fact a\nslight study of statistical methods and their application to social\nphenomena affords one of the simplest examples of the application of\nalgebraic ideas.\n\nAnother way in which the students' ideas can be generalised is by the\nuse of the History of Mathematics, conceived not as a mere assemblage\nof the dates and names of men, but as an exposition of the general\ncurrent of thought which occasioned the subjects to be objects of\ninterest at the time of their first elaboration. The use of the History\nof Mathematics is to be considered at a later stage of our proceedings\nthis afternoon. Accordingly I merely draw attention to it now, to\npoint out that perhaps it is the very subject which may best obtain the\nresults for which I am pleading.\n\nWe have indicated two main topics, namely general ideas of quantity\nand of laws of nature, which should be an object of study in the\nmathematical curriculum of a liberal education. But there is another\nside to mathematics which must not be overlooked. It is the chief\ninstrument for discipline in logical method.\n\nNow, what is logical method, and how can any one be trained in it?\n\nLogical method is more than the mere knowledge of valid types of\nreasoning and practice in the concentration of mind necessary to\nfollow them. If it were only this, it would still be very important;\nfor the human mind was not evolved in the bygone ages for the sake of\nreasoning, but merely to enable mankind with more art to hunt between\nmeals for fresh food supplies. Accordingly few people can follow close\nreasoning without considerable practice.\n\nMore than this is wanted to make a good reasoner, or even to enlighten\nordinary people with knowledge of what constitutes the essence of the\nart. The art of reasoning consists in getting hold of the subject at\nthe right end, of seizing on the few general ideas which illuminate\nthe whole, and of persistently marshalling all subsidiary facts round\nthem. Nobody can be a good reasoner unless by constant practice he\nhas realised the importance of getting hold of the big ideas and of\nhanging on to them like grim death. For this sort of training geometry\nis, I think, better than algebra. The field of thought of algebra is\nrather obscure, whereas space is an obvious insistent thing evident to\nall. Then the process of simplification, or abstraction, by which all\nirrelevant properties of matter, such as colour, taste, and weight, are\nput aside is an education in itself. Again, the definitions and the\npropositions assumed without proof illustrate the necessity of forming\nclear notions of the fundamental facts of the subject-matter and of the\nrelations between them. All this belongs to the mere prolegomena of the\nsubject. When we come to its development, its excellence increases. The\nlearner is not initially confronted with any symbolism which bothers\nthe memory by its rules, however simple they may be. Also, from the\nvery beginning the reasoning, if properly conducted, is dominated by\nwell-marked ideas which guide each stage of development. Accordingly\nthe essence of logical method receives immediate exemplification.\n\nLet us now put aside for the moment the limitations introduced by\nthe dullness of average pupils and the pressure on time due to other\nsubjects, and consider what geometry has to offer in the way of a\nliberal education. I will indicate some stages in the subject,\nwithout meaning that necessarily they are to be studied in this\nexclusive order. The first stage is the study of _congruence_.\nOur perception of congruence is in practice dependent on our judgments\nof the invariability of the intrinsic properties of bodies when their\nexternal circumstances are varying. But however it arises, congruence\nis in essence the correlation of two regions of space, point by point,\nso that all homologous distances and all homologous angles are equal.\nIt is to be noticed that the definition of the equality of lengths and\nangles is their congruence, and all tests of equality, such as the use\nof the yard measure, are merely devices for making immediate judgments\nof congruence easy. I make these remarks to suggest that apart from the\nreasoning connected with it, congruence, both as an example of a larger\nand very far-reaching idea and also for its own sake, is well worthy\nof attentive consideration. The propositions concerning it elucidate\nthe elementary properties of the triangle, the parallelogram, and the\ncircle, and of the relations of two planes to each other. It is very\ndesirable to restrict the proved propositions of this part within the\nnarrowest bounds, partly by assuming redundant axiomatic propositions,\nand partly by introducing only those propositions of absolutely\nfundamental importance.\n\nThe second stage is the study of similarity. This can be reduced to\nthree or four fundamental propositions. Similarity is an enlargement of\nthe idea of congruence, and, like that idea, is another example of a\none-to-one correlation of points of spaces. Any extension of study of\nthis subject might well be in the direction of the investigation of one\nor two simple properties of similar and similarly situated rectilinear\nfigures. The whole subject receives its immediate applications in plans\nand maps. It is important, however, to remember that trigonometry is\nreally the method by which the main theorems are made available for use.\n\nThe third stage is the study of the elements of trigonometry. This is\nthe study of the periodicity introduced by rotation and of properties\npreserved in a correlation of similar figures. Here for the first time\nwe introduce a slight use of the algebraic analysis founded on the\nstudy of number and quantity. The importance of the periodic character\nof the functions requires full illustration. The simplest properties of\nthe functions are the only ones required for the solution of triangles,\nand the consequent applications to surveying. The wealth of formul\u00e6,\noften important in themselves, but entirely useless for this type of\nstudy, which crowd our books should be rigorously excluded, except\nso far as they are capable of being proved by the pupils as direct\nexamples of the bookwork.\n\nThis question of the exclusion of formul\u00e6 is best illustrated by\nconsidering this example of Trigonometry, though of course I may well\nhave hit on an unfortunate case in which my judgment is at fault. A\ngreat part of the educational advantage of the subject can be obtained\nby confining study to Trigonometry of one angle and by exclusion of the\naddition formul\u00e6 for the sine and cosine of the sum of two angles. The\nfunctions can be graphed, and the solution of triangles effected. Thus\nthe aspects of the science as (1) embodying analytically the immediate\nresults of some of the theorems deduced from congruence and similarity,\n(2) as a solution of the main problem of surveying, (3) as a study of\nthe fundamental functions required to express periodicity and wave\nmotion, will all be impressed on the pupils' minds both by bookwork and\nexample.\n\nIf it be desired to extend this course, the addition formul\u00e6 should\nbe added. But great care should be taken to exclude specialising\nthe pupils in the wealth of formul\u00e6 which comes in their train. By\n\"exclude\" is meant that the pupils should not have spent time or energy\nin acquiring any facility in their deduction. The teacher may find\nit interesting to work a few such examples before a class. But such\nresults are not among those which learners need retain. Also, I would\nexclude the whole subject of circumscribed and inscribed circles both\nfrom Trigonometry and from the previous geometrical courses. It is\nall very pretty, but I do not understand what its function is in an\nelementary non-professional curriculum.\n\nAccordingly, the actual bookwork of the subject is reduced to very\nmanageable proportions. I was told the other day of an American college\nwhere the students are expected to know by heart ninety formul\u00e6 or\nresults in Trigonometry alone. We are not quite so bad as that. In\nfact, in Trigonometry we have nearly approached the ideal here sketched\nout as far as our elementary courses are concerned.\n\nThe fourth stage introduces Analytical Geometry. The study of graphs in\nalgebra has already employed the fundamental notions, and all that is\nnow required is a rigorously pruned course on the straight line, the\ncircle, and the three types of conic sections, defined by the forms of\ntheir equations. At this point there are two remarks to be made. It is\noften desirable to give our pupils mathematical information which we do\nnot prove. For example, in co-ordinate geometry, the reduction of the\ngeneral equation of the second degree is probably beyond the capacities\nof most of the type of students whom we are considering. But that need\nnot prevent us from explaining the fundamental position of conics, as\nexhausting the possible types of such curves.\n\nThe second remark is to advocate the entire sweeping away of\ngeometrical conics as a separate subject. Naturally, on suitable\noccasions the analysis of analytical geometry will be lightened by\nthe use of direct deduction from some simple figure. But geometrical\nconics, as developed from the definition of a conic section by the\nfocus and directrix property, suffers from glaring defects. It is\nhopelessly recondite. The fundamental definition of a conic, _SP_ = _e_\n\u00b7 _PM_, usual in this subject at this stage, is thoroughly bad. It is\nvery recondite, and has no obvious importance. Why should such curves\nbe studied at all, any more than those defined by an indefinite number\nof other formul\u00e6? But when we have commenced the study of the Cartesian\nmethods, the equations of the first and second degrees are naturally\nthe first things to think about.\n\nIn this ideal course of Geometry, the fifth stage is occupied with\nthe elements of Projective Geometry. The general ideas of cross ratio\nand of projection are here fundamental. Projection is yet a more\ngeneral instance of that one-to-one correlation which we have already\nconsidered under congruence and similarity. Here again we must avoid\nthe danger of being led into a bewildering wealth of detail.\n\nThe intellectual idea which projective geometry is to illustrate is\nthe importance in reasoning of the correlation of all cases which\ncan be proved to possess in common certain identical properties. The\npreservation of the projective properties in projection is the one\nimportant educational idea of the subject. Cross ratio only enters\nas the fundamental metrical property which is preserved. The few\npropositions considered are selected to illustrate the two allied\nprocesses which are made possible by this procedure. One is proof\nby simplification. Here the simplification is psychological and not\nlogical--for the general case is logically the simplest. What is meant\nis: Proof by considering the case which is in fact the most familiar to\nus, or the easiest to think about. The other procedure is the deduction\nof particular cases from known general truths, as soon as we have a\nmeans of discovering such cases or a criterion for testing them.\n\nThe projective definition of conic sections and the identity of the\nresults obtained with the curves derived from the general equation\nof the second degree are capable of simple exposition, but lie on\nthe border-line of the subject. It is the sort of topic on which\ninformation can be given, and the proofs suppressed.\n\nThe course of geometry as here conceived in its complete ideal--and\nideals can never be realised--is not a long one. The actual amount\nof mathematical deduction at each stage in the form of bookwork is\nvery slight. But much more explanation would be given, the importance\nof each proposition being illustrated by examples, either worked out\nor for students to work, so selected as to indicate the fields of\nthought to which it applies. By such a course the student would gain an\nanalysis of the leading properties of space, and of the chief methods\nby which they are investigated.\n\nThe study of the elements of mathematics, conceived in this spirit,\nwould constitute a training in logical method together with an\nacquisition of the precise ideas which lie at the base of the\nscientific and philosophical investigations of the universe. Would it\nbe easy to continue the excellent reforms in mathematical instruction\nwhich this generation has already achieved, so as to include in the\ncurriculum this wider and more philosophic spirit? Frankly, I think\nthat this result would be very hard to achieve as the result of\nsingle individual efforts. For reasons which I have already briefly\nindicated, all reforms in education are very difficult to effect. But\nthe continued pressure of combined effort, provided that the ideal is\nreally present in the minds of the mass of teachers, can do much, and\neffects in the end surprising modification. Gradually the requisite\nbooks get written, still more gradually the examinations are reformed\nso as to give weight to the less technical aspects of the subject, and\nthen all recent experience has shown that the majority of teachers are\nonly too ready to welcome any practicable means of rescuing the subject\nfrom the reproach of being a mechanical discipline.\n\n\n\n\n                               CHAPTER V\n\n   THE PRINCIPLES OF MATHEMATICS IN RELATION TO ELEMENTARY TEACHING\n\n    (_International Congress of Mathematicians, Cambridge, August,\n                                1912_)\n\n\nWE are concerned not with the advanced teaching of a few specialist\nmathematical students, but with the mathematical education of the\nmajority of boys in our secondary schools. Again these boys must be\ndivided into two sections: one section consists of those who desire to\nrestrict their mathematical education; the other section is made up of\nthose who will require some mathematical training for their subsequent\nprofessional careers, either in the form of definite mathematical\nresults or in the form of mathematically trained minds.\n\nI shall call the latter of these two sections the mathematical section,\nand the former the non-mathematical section. But I must repeat that by\nthe mathematical section is meant that large number of boys who desire\nto learn more than the minimum amount of mathematics. Furthermore, most\nof my remarks about these sections of boys apply also to elementary\nclasses of our University students.\n\nThe subject of this paper is the investigation of the place which\nshould be occupied by modern investigations respecting mathematical\nprinciples in the education of both of these sections of school-boys.\n\nTo find a foothold from which to start the inquiry, let us ask why the\nnon-mathematical section should be taught any mathematics at all beyond\nthe barest elements of arithmetic. What are the qualities of mind which\na mathematical training is designed to produce when it is employed as\nan element in a liberal education?\n\nMy answer, which applies equally to both sections of students, is\nthat there are two allied forms of mental discipline which should be\nacquired by a well-designed mathematical course. These two forms though\nclosely allied are perfectly distinct.\n\nThe first form of discipline is not in its essence logical at all.\nIt is the power of clearly grasping abstract ideas, and of relating\nthem to particular circumstances. In other words, the first use of\nmathematics is to strengthen the power of abstract thought. I repeat\nthat in its essence this has nothing to do with logic, though as a\nmatter of fact a logical discipline is the best method of producing the\ndesired effect. It is not the philosophical theory of abstract ideas\nwhich is to be acquired, but the habit and the power of using them.\nThere is one and only way of acquiring the habit and the power of using\nanything, that is by the simple common-place method of habitually using\nit. There is no other short cut. If in education we desire to produce\na certain conformation of mind, we must day by day, and year by year,\naccustom the students' minds to grow into the desired structural shape.\nThus to teach the power of grasping abstract ideas and the habit of\nusing them, we must select a group of such ideas, which are important\nand are also easy to think about because they are clear and definite.\n\nThe fundamental mathematical truths concerning geometry, ratio,\nquantity, and number, satisfy these conditions as do no others. Hence,\nthe fundamental universal position held by mathematics as an element of\na liberal education.\n\nBut what are the fundamental mathematical truths concerning geometry,\nquantity and number?\n\nAt this point we come to the great question of the relation between\nthe modern science of the principles of mathematics and a mathematical\neducation.\n\nMy answer to the question as to these fundamental mathematical truths\nis, that in any absolute sense there are none. There is no unique\nsmall body of independent primitive unproved propositions which are\nthe necessary starting points of all mathematical reasoning on these\nsubjects. In mathematical reasoning the only absolute necessary\npre-suppositions are those which make logical deduction possible.\nBetween these absolute logical truths and so-called fundamental truths\nconcerning geometry, quantity and number, there is a whole new world of\nmathematical subjects concerning the logic of propositions, of classes,\nand of relations.\n\nBut this subject is too abstract to form an elementary training ground\nin the difficult art of abstract thought.\n\nIt is for this reason that we have to make a compromise and start from\nsuch obvious general ideas as naturally occur to all men when they\nperceive objects with their senses.\n\nIn geometry, the ideas elaborated by the Greeks and presented by Euclid\nare, roughly speaking, those adapted for our purpose, namely, ideas\nof volumes, surfaces, lines, of straightness and of curvature, of\nintersection and of congruence, of greater and less, of similarity,\nshape, and scale. In fact, we use in education those general ideas of\nspatial properties which must be habitually present in the mind of any\none who is to observe the world of phenomena with understanding.\n\nThus we come back to Plato's opinion that for a liberal education,\ngeometry, as he knew it, is the queen of sciences.\n\nIn addition to geometry, there remain the ideas of quantity, ratio,\nand of number. This in practice means, elementary algebra. Here the\nprominent ideas are those of \"any number,\" in other words, the use\nof the familiar _x_, _y_, _z_, and of the dependence of variables on\neach other, or otherwise, the idea of functionality. All this is to be\ngradually acquired by the continual use of the very simplest functions\nwhich we can devise: of linear functions, graphically represented\nby straight lines; of quadratic functions, graphically represented\nby parabolas; and of those simple implicit functions, graphically\nrepresented by conic sections. Thence, with good fortune and a willing\nclass, we can advance to the ideas of rates of increase, still\nconfining ourselves to the simplest possible cases.\n\nI wish here emphatically to remind you that both in geometry and in\nalgebra a clear grasp of these general ideas is not what the pupil\nstarts from, it is the goal at which he is to arrive. The method of\nprogression is continual practice in the consideration of the simplest\nparticular cases, and the goal is not philosophical analysis but the\npower of use.\n\nBut how is he to practise himself in their use? He cannot simply sit\ndown and think of the relation _y_ = _x_ + 1, he must employ\nit in some simple obvious way.\n\nThis brings us to the second power of mind which is to be produced by\na mathematical training, namely, the power of logical reasoning. Here\nagain, it is not the knowledge of the philosophy of logic which it is\nessential to teach, but the habit of thinking logically. By logic, I\nmean deductive logic.\n\nDeductive logic is the science of certain relations, such as\nimplication, etc., between general ideas. When logic begins, definite\nparticular individual things have been banished. I cannot relate\nlogically this thing to that thing, for example this pen to that pen,\nexcept by the indirect way of relating some general idea which applies\nto this pen to some general idea which applies to that pen. And the\nindividualities of the two pens are quite irrelevant to the logical\nprocess. This process is entirely concerned with the two general ideas.\nThus the practice of logic is a certain way of employing the mind\nin the consideration of such ideas; and an elementary mathematical\ntraining is in fact nothing else but the logical use of the general\nideas respecting geometry and algebra which we have enumerated above.\nIt has therefore, as I started this paper by stating, a double\nadvantage. It makes the mind capable of abstract thought, and it\nachieves this object by training the mind in the most important kind of\nabstract thought, namely, deductive logic.\n\nI may remind you that other choices of a type of abstract thought\nmight be made. We might train the children to contemplate directly the\nbeauty of abstract moral ideas, in the hope of making them religious\nmystics. The general practice of education has decided in favour of\nlogic, as exemplified in elementary mathematics.\n\nWe have now to answer the further question, what is the r\u00f4le of\nlogical precision in the teaching of mathematics? Our general answer\nto the implied question is obvious: logical precision is one of the\ntwo objects of the teaching of mathematics, and it is the only weapon\nby which the teaching of mathematics can achieve its other object. To\nteach mathematics is to teach logical precision. A mathematical teacher\nwho has not taught that has taught nothing.\n\nBut having stated this thesis in this unqualified way, its meaning must\nbe carefully explained; for otherwise its real bearing on the problem\nwill be entirely misunderstood.\n\nLogical precision is the faculty to be acquired. It is the quality of\nmind which it is the object of the teaching to impart. Thus the habit\nof reading great literature is the goal at which a literary education\naims. But we do not expect a child to start its first lesson by reading\nfor itself Shakespeare. We recognise that reading is impossible till\nthe pupil has learnt its alphabet and can spell, and then we start it\nwith books of one syllable.\n\nIn the same way, a mathematical education should grow in logical\nprecision. It is folly to expect the same careful logical analysis at\nthe commencement of the training as would be appropriate at the end.\nIt is an entire misconception of my thesis to construe it as meaning\nthat a mathematical training should assume in the pupil a power of\nconcentrated logical thought. My thesis is in fact the exact opposite,\nnamely, that this power cannot be assumed, and has got to be acquired,\nand that a mathematical training is nothing else than the process of\nacquiring it. My whole groundwork of assumption is that this power does\nnot initially exist in a fully developed state. Of course like every\nother power which is acquired, it must be developed gradually.\n\nThe various stages of development must be guided by the judgment and\nthe genius of the teacher. But what is essential is, that the teacher\nshould keep clearly in his mind that it is just this power of logical\nprecise reasoning which is the whole object of his efforts. If his\npupils have in any measure gained this, they have gained all.\n\nWe have not yet, however, fully considered this part of our subject.\nLogical precision is the full realisation of the steps of the argument.\nBut what are the steps of the argument? The full statement of all the\nsteps is far too elaborate and difficult an operation to be introduced\ninto the mathematical reasoning of an educational curriculum. Such a\nstatement involves the introduction of abstract logical ideas which\nare very difficult to grasp, because there is so rarely any need to\nmake them explicit in ordinary thought. They are therefore not a fit\nsubject-ground for an elementary education.\n\nI do not think that it is possible to draw any theoretical line\nbetween those logical steps which form a theoretically full logical\ninvestigation, and those which are full enough for most practical\npurposes, including that of education. The question is one of\npsychology, to be solved by a process of experiment. The object to be\nattained is to gain that amount of logical alertness which will enable\nits possessors to detect fallacy and to know the types of sound logical\ndeduction. The objects of going further are partly philosophical, and\nalso partly to lay bare abstract ideas whose investigation is in itself\nimportant. But both these objects are foreign to education.\n\nMy opinion is, that, on the whole, the type of logical precision handed\ndown to us by the Greek mathematicians is, roughly speaking, what we\nwant. In geometry, this means the sort of precision which we find in\nEuclid. I do not mean that we should use his famous _Elements_ as\na text-book, nor that here and there a certain compression in his mode\nof exposition is not advisable. All this is mere detail. What I do mean\nis, that the sort of logical transition which he made explicit, we\nshould make explicit, and that the sort of transition which he omits,\nwe should omit.\n\nI doubt, however, whether it is desirable to plunge the student into\nthe full rigour of euclidean geometry without some mitigation. It is\nfor this reason that the modern habit, at least in England, of laying\ngreat stress in the initial stages on training the pupil in simple\nconstructions from numerical data is to be praised. It means that after\na slight amount of reasoning on the euclidean basis of accuracy, the\nmind of the learner is relieved by doing the things in various special\ncases, and noting by rough measurements that the desired results are\nactually attained. It is important, however, that the measurements\nbe not mistaken for the proofs. Their object is to make the beginner\napprehend what the abstract ideas really mean.\n\nAgain in algebra, the notation and the practical use of the symbols\nshould be acquired in the simplest cases, and the more theoretical\ntreatment of the symbolism reserved to a suitable later stage. My\nrule would be initially to learn the meaning of the ideas by a crude\npractice in simple ways, and to refine the logical procedure in\npreparation for an advance to greater generality. In fact the thesis of\nmy paper can be put in another way thus, the object of a mathematical\neducation is, to acquire the powers of analysis, of generalisation,\nand of reasoning. The two processes of analysis and generalisation were\nin my previous statement put together as the power of grasping abstract\nideas.\n\nBut in order to analyse and to generalise, we must commence with the\ncrude material of ideas which are to be analysed and generalised.\nAccordingly it is a positive error in education to start with the\nultimate products of this process, namely the ideas in their refined\nanalysed and generalised forms. We are thereby skipping an important\npart of the training, which is to take the ideas as they actually exist\nin the child's mind, and to exercise the child in the difficult art of\ncivilising them and clothing them.\n\nThe schoolmaster is in fact a missionary, the savages are the ideas in\nthe child's mind; and the missionary shirks his main task if he refuse\nto risk his body among the cannibals.\n\nAt this point I should like to turn your attention to those pupils\nforming the mathematical section. There is an idea, widely prevalent,\nthat it is possible to teach mathematics of a relatively advanced\ntype--such as differential calculus, for instance--in a way useful to\nphysicists and engineers without any attention to its logic or its\ntheory.\n\nThis seems to me to be a profound mistake. It implies that a merely\nmechanical knowledge without understanding of ways of arriving at\nmathematical results is useful in applied science. It is of no use\nwhatever. The results themselves can all be found stated in the\nappropriate pocket books and in other elementary works of reference. No\none when applying a result need bother himself as to why it is true. He\naccepts it and applies it. What is of supreme importance in physics and\nin engineering is a mathematically trained mind, and such a mind can\nonly be acquired by a proper mathematical discipline.\n\nI fully admit that the proper way to start such a subject as the\ndifferential calculus is to plunge quickly into the use of the notation\nin a few absurdly simple cases, with a crude explanation of the idea\nof rates of increase. The notation as thus known can then be used by\nthe lecturers in the Physical and Engineering Laboratories. But the\nmathematical training of the applied scientists consists in making\nthese ideas precise and the proofs accurate.\n\nI hope that the thesis of this paper respecting the position of logical\nprecision in the teaching of mathematics has been rendered plain. The\nhabit of logical precision with its necessary concentration of thought\nupon abstract ideas is not wholly possible in the initial stages of\nlearning. It is the ideal at which the teacher should aim. Also logical\nprecision, in the sense of logical explicitness, is not an absolute\nthing: it is an affair of more or less. Accordingly the quantity of\nexplicitness to be introduced at each stage of progress must depend\nupon the practical judgment of the teacher. Lastly, in a sense, the\ninstructed mind is less explicit; for it travels more quickly over a\nwell-remembered path, and may save the trouble of putting into words\ntrains of thought which are very obvious to it. But on the other\nhand it atones for this rapidity by a concentration on every subtle\npoint where a fallacy can lurk. The habit of logical precision is the\ninstinct for the subtle difficulty.\n\n\n\n\n                              CHAPTER VI\n\n                      THE ORGANISATION OF THOUGHT\n\n (_Presidential Address to Section A, British Association, Newcastle,\n                                1916_)\n\n\nTHE subject of this address is the organisation of thought, a topic\nevidently capable of many diverse modes of treatment. I intend more\nparticularly to give some account of that department of logical science\nwith which some of my own studies have been connected. But I am\nanxious, if I can succeed in so doing, to handle this account so as to\nexhibit the relation with certain considerations which underlie general\nscientific activities.\n\nIt is no accident that an age of science has developed into an age\nof organisation. Organised thought is the basis of organised action.\nOrganisation is the adjustment of diverse elements so that their\nmutual relations may exhibit some predetermined quality. An epic poem\nis a triumph of organisation, that is to say, it is a triumph in the\nunlikely event of its being a good epic poem. It is the successful\norganisation of multitudinous sounds of words, associations of words,\npictorial memories of diverse events and feelings ordinarily occurring\nin life, combined with a special narrative of great events: the whole\nso disposed as to excite emotions which, as defined by Milton, are\nsimple, sensuous, and passionate. The number of successful epic poems\nis commensurate, or rather, is inversely commensurate, with the obvious\ndifficulty of the task of organisation.\n\nScience is the organisation of thought. But the example of the epic\npoem warns us that science is not any organisation of thought. It is\nan organisation of a certain definite type which we will endeavour to\ndetermine.\n\nScience is a river with two sources, the practical source and the\ntheoretical source. The practical source is the desire to direct our\nactions to achieve predetermined ends. For example, the British nation,\nfighting for justice, turns to science, which teaches it the importance\nof compounds of nitrogen. The theoretical source is the desire to\nunderstand. Now I am going to emphasise the importance of theory in\nscience. But to avoid misconception I most emphatically state that I\ndo not consider one source as in any sense nobler than the other, or\nintrinsically more interesting. I cannot see why it is nobler to strive\nto understand than to busy oneself with the right ordering of one's\nactions. Both have their bad sides; there are evil ends directing\nactions, and there are ignoble curiosities of the understanding.\n\nThe importance, even in practice, of the theoretical side of science\narises from the fact that action must be immediate, and takes place\nunder circumstances which are excessively complicated. If we wait for\nthe necessities of action before we commence to arrange our ideas, in\npeace we shall have lost our trade, and in war we shall have lost the\nbattle. Success in practice depends on theorists who, led by other\nmotives of exploration, have been there before, and by some good chance\nhave hit upon the relevant ideas. By a theorist I do not mean a man who\nis up in the clouds, but a man whose motive for thought is the desire\nto formulate correctly the rules according to which events occur. A\nsuccessful theorist should be excessively interested in immediate\nevents, otherwise he is not at all likely to formulate correctly\nanything about them. Of course, both sources of science exist in all\nmen.\n\nNow, what is this thought organisation which we call science? The\nfirst aspect of modern science which struck thoughtful observers was\nits inductive character. The nature of induction, its importance, and\nthe rules of inductive logic have been considered by a long series of\nthinkers, especially English thinkers: Bacon, Herschel, J. S. Mill,\nVenn, Jevons, and others. I am not going to plunge into an analysis\nof the process of induction. Induction is the machinery and not the\nproduct, and it is the product which I want to consider. When we\nunderstand the product we shall be in a stronger position to improve\nthe machinery.\n\nFirst, there is one point which it is necessary to emphasise. There\nis a tendency in analysing scientific processes to assume a given\nassemblage of concepts applying to nature, and to imagine that the\ndiscovery of laws of nature consists in selecting by means of inductive\nlogic some one out of a definite set of possible alternative relations\nwhich may hold between the things in nature answering to these obvious\nconcepts. In a sense this assumption is fairly correct, especially\nin regard to the earlier stages of science. Mankind found itself in\npossession of certain concepts respecting nature--for example, the\nconcept of fairly permanent material bodies--and proceeded to determine\nlaws which related the corresponding percepts in nature. But the\nformulation of laws changed the concepts, sometimes gently by an added\nprecision, sometimes violently. At first this process was not much\nnoticed, or at least was felt to be a process curbed within narrow\nbounds, not touching fundamental ideas. At the stage where we now\nare, the formulation of the concepts can be seen to be as important\nas the formulation of the empirical laws connecting the events in the\nuniverse as thus conceived by us. For example, the concepts of life,\nof heredity, of a material body, of a molecule, of an atom, of an\nelectron, of energy, of space, of time, of quantity, and of number. I\nam not dogmatising about the best way of getting such ideas straight.\nCertainly it will only be done by those who have devoted themselves to\na special study of the facts in question. Success is never absolute,\nand progress in the right direction is the result of a slow, gradual\nprocess of continual comparison of ideas with facts. The criterion of\nsuccess is that we should be able to formulate empirical laws, that is,\nstatements of relations, connecting the various parts of the universe\nas thus conceived, laws with the property that we can interpret the\nactual events of our lives as being our fragmentary knowledge of this\nconceived interrelated whole.\n\nBut, for the purpose of science, what is the actual world? Has science\nto wait for the termination of the metaphysical debate till it can\ndetermine its own subject-matter? I suggest that science has a much\nmore homely starting-ground. Its task is the discovery of the relations\nwhich exist within that flux of perceptions, sensations, and emotions\nwhich forms our experience of life. The panorama yielded by sight,\nsound, taste, smell, touch, and by more inchoate sensible feelings,\nis the sole field of activity. It is in this way that science is the\nthought organisation of experience. The most obvious aspect of this\nfield of actual experience is its disorderly character. It is for\neach person a _continuum_, fragmentary, and with elements not\nclearly differentiated. The comparison of the sensible experiences of\ndiverse people brings its own difficulties. I insist on the radically\nuntidy, ill-adjusted character of the fields of actual experience from\nwhich science starts. To grasp this fundamental truth is the first\nstep in wisdom, when constructing a philosophy of science. This fact\nis concealed by the influence of language, moulded by science, which\nfoists on us exact concepts as though they represented the immediate\ndeliverances of experience. The result is, that we imagine that we have\nimmediate experience of a world of perfectly defined objects implicated\nin perfectly defined events which, as known to us by the direct\ndeliverance of our senses, happen at exact instants of time, in a space\nformed by exact points, without parts and without magnitude: the neat,\ntrim, tidy, exact world which is the goal of scientific thought.\n\nMy contention is, that this world is a world of ideas, and that its\ninternal relations are relations between abstract concepts, and that\nthe elucidation of the precise connection between this world and the\nfeelings of actual experience is the fundamental question of scientific\nphilosophy. The question which I am inviting you to consider is this:\nHow does exact thought apply to the fragmentary, vague _continua_\nof experience? I am not saying that it does not apply: quite the\ncontrary. But I want to know how it applies. The solution I am\nasking for is not a phrase, however brilliant, but a solid branch of\nscience, constructed with slow patience, showing in detail how the\ncorrespondence is effected.\n\nThe first great steps in the organisation of thought were due\nexclusively to the practical source of scientific activity, without\nany admixture of theoretical impulse. Their slow accomplishment was\nthe cause and also the effect of the gradual evolution of moderately\nrational beings. I mean the formation of the concepts of definite\nmaterial objects, of the determinate lapse of time, of simultaneity,\nof recurrence, of definite relative position, and of analogous\nfundamental ideas, according to which the flux of our experience is\nmentally arranged for handy reference: in fact, the whole apparatus of\ncommonsense thought. Consider in your mind some definite chair. The\nconcept of that chair is simply the concept of all the interrelated\nexperiences connected with that chair--namely, of the experience of the\nfolk who made it, of the folk who sold it, of the folk who have seen\nit or used it, of the man who is now experiencing a comfortable sense\nof support, combined with our expectations of an analogous future,\nterminated finally by a different set of experiences when the chair\ncollapses and becomes firewood. The formation of that type of concept\nwas a tremendous job, and zoologists and geologists tell us that it\ntook many tens of millions of years. I can well believe it.\n\nI now emphasise two points. In the first place, science is rooted in\nwhat I have just called the whole apparatus of commonsense thought.\nThat is the _datum_ from which it starts, and to which it\nmust recur. We may speculate, if it amuses us, of other beings in\nother planets who have arranged analogous experiences according to\nan entirely different conceptual code--namely, who have directed\ntheir chief attention to different relations between their various\nexperiences. But the task is too complex, too gigantic, to be revised\nin its main outlines. You may polish up commonsense, you may contradict\nit in detail, you may surprise it. But ultimately your whole task is to\nsatisfy it.\n\nIn the second place, neither commonsense nor science can proceed with\ntheir task of thought organisation without departing in some respect\nfrom the strict consideration of what is actual in experience. Think\nagain of the chair. Among the experiences upon which its concept is\nbased, I included our expectations of its future history. I should\nhave gone further and included our imagination of all the possible\nexperiences which in ordinary language we should call perceptions of\nthe chair which might have occurred. This is a difficult question, and\nI do not see my way through it. But, at present, in the construction of\na theory of space and of time there seem insuperable difficulties if we\nrefuse to admit ideal experiences.\n\nThis imaginative perception of experiences, which, if they occurred,\nwould be coherent with our actual experiences, seems fundamental in our\nlives. It is neither wholly arbitrary, nor yet fully determined. It\nis a vague background which is only made in part definite by isolated\nactivities of thought. Consider, for example, our thoughts of the\nunseen flora of Brazil.\n\nIdeal experiences are closely connected with our imaginative\nreproduction of the actual experiences of other people, and also\nwith our almost inevitable conception of ourselves as receiving our\nimpressions from an external complex reality beyond ourselves. It\nmay be that an adequate analysis of every source and every type of\nexperience yields demonstrative proof of such a reality and of its\nnature. Indeed, it is hardly to be doubted that this is the case. The\nprecise elucidation of this question is the problem of metaphysics. One\nof the points which I am urging in this address is, that the basis of\nscience does not depend on the assumption of any of the conclusions of\nmetaphysics; but that both science and metaphysics start from the same\ngiven groundwork of immediate experience, and in the main proceed in\nopposite directions on their diverse tasks.\n\nFor example, metaphysics inquires how our perceptions of the chair\nrelate us to some true reality. Science gathers up these perceptions\ninto a determinate class, adds to them ideal perceptions of analogous\nsort, which under assignable circumstances would be obtained, and this\nsingle concept of that set of perceptions is all that science needs;\nunless indeed you prefer that thought find its origin in some legend of\nthose great twin brethren, the Cock and Bull.\n\nMy immediate problem is to inquire into the nature of the texture of\nscience. Science is essentially logical. The nexus between its concepts\nis a logical nexus, and the grounds for its detailed assertions\nare logical grounds. King James said, \"No bishops, no king.\" With\ngreater confidence we can say, \"No logic, no science.\" The reason for\nthe instinctive dislike which most men of science feel towards the\nrecognition of this truth is, I think, the barren failure of logical\ntheory during the past three or four centuries. We may trace this\nfailure back to the worship of authority, which in some respects\nincreased in the learned world at the time of the Renaissance. Mankind\nthen changed its authority, and this fact temporally acted as an\nemancipation. But the main fact, and we can find complaints[2] of it\nat the very commencement of the modern movement, was the establishment\nof a reverential attitude towards any statement made by a classical\nauthor. Scholars became commentators on truths too fragile to bear\ntranslation. A science which hesitates to forget its founders is lost.\nTo this hesitation I ascribe the barrenness of logic. Another reason\nfor distrust of logical theory and of mathematics is the belief that\ndeductive reasoning can give you nothing new. Your conclusions are\ncontained in your premises, which by hypothesis are known to you.\n\nIn the first place this last condemnation of logic neglects the\nfragmentary, disconnected character of human knowledge. To know one\npremise on Monday, and another premise on Tuesday, is useless to you\non Wednesday. Science is a permanent record of premises, deductions,\nand conclusions, verified all along the line by its correspondence\nwith facts. Secondly, it is untrue that when we know the premises we\nalso know the conclusions. In arithmetic, for example, mankind are not\ncalculating boys. Any theory which proves that they are conversant with\nthe consequences of their assumptions must be wrong. We can imagine\nbeings who possess such insight. But we are not such creatures.\nBoth these answers are, I think, true and relevant. But they are\nnot satisfactory. They are too much in the nature of bludgeons,\ntoo external. We want something more explanatory of the very real\ndifficulty which the question suggests. In fact, the true answer is\nembedded in the discussion of our main problem of the relation of logic\nto natural science.\n\nIt will be necessary to sketch in broad outline some relevant features\nof modern logic. In doing so I shall try to avoid the profound general\ndiscussions and the minute technical classifications which occupy the\nmain part of traditional logic. It is characteristic of a science in\nits earlier stages--and logic has become fossilised in such a stage--to\nbe both ambitiously profound in its aims and trivial in its handling of\ndetails.\n\nWe can discern four departments of logical theory. By an analogy\nwhich is not so very remote I will call these departments or sections\nthe arithmetic section, the algebraic section, the section of\ngeneral-function theory, the analytical section. I do not mean that\narithmetic arises in the first section, algebra in the second section,\nand so on; but the names are suggestive of certain qualities of\nthought in each section which are reminiscent of analogous qualities\nin arithmetic, in algebra, in the general theory of a mathematical\nfunction, and in the mathematical analysis of the properties of\nparticular functions.\n\nThe first section--namely, the arithmetic stage--deals with the\nrelations of definite propositions to each other, just as arithmetic\ndeals with definite numbers. Consider any definite proposition; call\nit \"_p_.\" We conceive that there is always another proposition which\nis the direct contradictory to \"_p_\"; call it \"not-_p_.\" When we have\ngot two propositions, _p_ and _q_, we can form derivative propositions\nfrom them, and from their contradictories. We can say, \"At last one of\n_p_ or _q_ is true, and perhaps both.\" Let us call this proposition\n\"_p_ or _q_.\" I may mention as an aside that one of the greatest living\nphilosophers has stated that this use of the word \"or\"--namely, \"_p_ or\n_q_\" in the sense that either or both may be true--makes him despair of\nexact expression. We must brave his wrath, which is unintelligible to\nme.\n\nWe have thus got hold of four new propositions, namely, \"_p_ or _q_,\"\nand \"not-_p_ or _q_,\" and \"_p_ or not-_q_,\" and \"not-_p_ or not-_q_.\"\nCall these the set of disjunctive derivatives. There are, so far,\nin all eight propositions, _p_, not-_p_, _q_, not-_q_, and the four\ndisjunctive derivatives. Any pair of these eight propositions can be\ntaken, and substituted for _p_ and _q_ in the foregoing treatment.\nThus each pair yields eight propositions, some of which may have been\nobtained before. By proceeding in this way we arrive at an unending set\nof propositions of growing complexity, ultimately derived from the two\noriginal propositions _p_ or _q_. Of course, only a few are important.\nSimilarly we can start from three propositions, _p_, _q_, _r_, or\nfrom four propositions, _p_, _q_, _r_, _s_, and so on. Any one of the\npropositions of these aggregates may be true or false. It has no other\nalternative. Whichever it is, true or false, call it the \"truth-value\"\nof the proposition.\n\nThe first section of logical inquiry is to settle what we know of the\ntruth-values of these propositions, when we know the truth-values of\nsome of them. The inquiry, so far as it is worth while carrying it,\nis not very abstruse, and the best way of expressing its results is a\ndetail which I will not now consider. This inquiry forms the arithmetic\nstage.\n\nThe next section of logic is the algebraic stage. Now, the difference\nbetween arithmetic and algebra is, that in arithmetic definite\nnumbers are considered, and in algebra symbols--namely, letters--are\nintroduced which stand for any numbers. The idea of a number is also\nenlarged. These letters, standing for any numbers, are called sometimes\nvariables and sometimes parameters. Their essential characteristic is\nthat they are undetermined, unless, indeed, the algebraic conditions\nwhich they satisfy implicitly determine them. Then they are sometimes\ncalled unknowns. An algebraic formula with letters is a blank form. It\nbecomes a determinate arithmetic statement when definite numbers are\nsubstituted for the letters. The importance of algebra is a tribute to\nthe study of form. Consider now the following proposition--\n\n The specific heat of mercury is 0\u00b7033.\n\nThis is a definite proposition which, with certain limitations, is\ntrue. But the truth-value of the proposition does not immediately\nconcern us. Instead of mercury put a mere letter which is the name of\nsome undetermined thing: we get--\n\n The specific heat of _x_ is 0\u00b7033.\n\nThis is not a proposition; it has been called by Russell a\npropositional function. It is the logical analogy of an algebraic\nexpression. Let us write \u0192(_x_) for any propositional function.\n\nWe could also generalise still further, and say,\n\n The specific heat of _x_ is _y_.\n\nWe thus get another propositional function, F(_x_, _y_),\nof two arguments _x_ and _y_, and so on for any number of\narguments.\n\nNow, consider \u0192(_x_). There is the range of values of _x_,\nfor which \u0192(_x_) is a proposition, true or false. For values of\n_x_ outside this range, \u0192(_x_) is not a proposition at all,\nand is neither true nor false. It may have vague suggestions for us,\nbut it has no unit meaning of definite assertion. For example,\n\n The specific heat of water is 0\u00b7033\n\nis a proposition which is false; and--\n\n The specific heat of virtue is 0\u00b7033\n\nis, I should imagine, not a proposition at all; so that it is neither\ntrue nor false, though its component parts raise various associations\nin our minds. This range of values, for which \u0192(_x_) has sense, is\ncalled the \"type\" of the argument _x_.\n\nBut there is also a range of values of _x_ for which \u0192(_x_)\nis a true proposition. This is the class of those values of the\nargument which _satisfy_ \u0192(_x_). This class may have no\nmembers, or, in the other extreme, the class may be the whole type of\nthe arguments.\n\nWe thus conceive two general propositions respecting the indefinite\nnumber of propositions which share in the same logical form, that is,\nwhich are values of the same propositional function. One of these\npropositions is,\n\n \u0192(_x_) yields a true proposition for each value of _x_ of\n the proper type;\n\nthe other proposition is,\n\n There is a value of _x_ for which \u0192(_x_) is true.\n\nGiven two, or more, propositional functions \u0192(_x_) and \u03d5(_x_)\nwith the same argument _x_, we form derivative propositional\nfunctions, namely,\n\n\u0192(_x_) or \u03d5(_x_), \u0192(_x_) or not-\u03d5(_x_),\n\nand so on with the contradictories, obtaining, as in the arithmetical\nstage, an unending aggregate of propositional functions. Also each\npropositional function yields two general propositions. The theory\nof the interconnection between the truth-values of the general\npropositions arising from any such aggregate of propositional functions\nforms a simple and elegant chapter of mathematical logic.\n\nIn this algebraic section of logic the theory of types crops up, as we\nhave already noted. It cannot be neglected without the introduction of\nerror. Its theory has to be settled at least by some safe hypothesis,\neven if it does not go to the philosophic basis of the question. This\npart of the subject is obscure and difficult, and has not been finally\nelucidated, though Russell's brilliant work has opened out the subject.\n\nThe final impulse to modern logic comes from the independent discovery\nof the importance of the logic variable by Frege and Peano. Frege went\nfurther than Peano, but by an unfortunate symbolism rendered his work\nso obscure that no one fully recognised his meaning who had not found\nit out for himself. But the movement has a large history reaching back\nto Leibniz and even to Aristotle. Among English contributors are De\nMorgan, Boole, and Sir Alfred Kempe; their work is of the first rank.\n\nThe third logical section is the stage of general-function theory.\nIn logical language, we perform in this stage the transition from\nintension to extension, and investigate the theory of denotation. Take\nthe propositional function, \u0192(_x_). There is the class, or range\nof values for _x_, whose members satisfy \u0192(_x_). But the same\nrange may be the class whose members satisfy another propositional\nfunction \u03d5(_x_). It is necessary to investigate how to indicate\nthe class by a way which is indifferent as between the various\npropositional functions which are satisfied by any member of it, and of\nit only. What has to be done is to analyse the nature of propositions\nabout a class--namely, those propositions whose truth-values depend on\nthe class itself and not on the particular meaning by which the class\nis indicated.\n\nFurthermore, there are propositions about alleged individuals\nindicated by descriptive phrases: for example, propositions about \"the\npresent King of England,\" who does exist, and \"the present Emperor\nof Brazil,\" who does not exist. More complicated, but analogous,\nquestions involving propositional functions of two variables involve\nthe notion of \"correlation,\" just as functions of one argument involve\nclasses. Similarly functions of three arguments yield three-cornered\ncorrelations, and so on. This logical section is one which Russell has\nmade peculiarly his own by work which must always remain fundamental.\nI have called this the section of functional theory, because its\nideas are essential to the construction of logical denoting functions\nwhich include as a special case ordinary mathematical functions, such\nas sine, logarithm, etc. In each of these three stages it will be\nnecessary gradually to introduce an appropriate symbolism, if we are to\npass on to the fourth stage.\n\nThe fourth logical section, the analytic stage, is concerned with the\ninvestigation of the properties of special logical constructions,\nthat is, of classes and correlations of special sorts. The whole of\nmathematics is included here. So the section is a large one. In fact,\nit is mathematics, neither more nor less, but it includes an analysis\nof mathematical ideas not hitherto included in the scope of that\nscience, nor, indeed, contemplated at all. The essence of this stage is\nconstruction. It is by means of suitable constructions that the great\nframework of applied mathematics, comprising the theories of number,\nquantity, time, and space, is elaborated.\n\nIt is impossible, even in brief outline, to explain how mathematics\nis developed from the concepts of class and correlation, including\nmany-cornered correlations, which are established in the third section.\nI can only allude to the headings of the process, which is fully\ndeveloped in the work, _Principia Mathematica_, by Mr. Russell\nand myself. There are in this process of development seven special\nsorts of correlations which are of peculiar interest. The first sort\ncomprises one-to-many, many-to-one, and one-to-one correlations.\nThe second sort comprises serial relations, that is, correlations\nby which the members of some field are arranged in serial order, so\nthat, in the sense defined by the relation, any member of the field\nis either before or after any other member. The third class comprises\ninductive relations, that is, correlations on which the theory of\nmathematical induction depends. The fourth class comprises selective\nrelations, which are required for the general theory of arithmetic\noperations, and elsewhere. It is in connection with such relations that\nthe famous multiplicative axiom arises for consideration. The fifth\nclass comprises vector relations, from which the theory of quantity\narises. The sixth class comprises ratio relations, which interconnect\nnumber and quantity. The seventh class comprises three-cornered and\nfour-cornered relations which occur in geometry.\n\nA bare enumeration of technical names, such as the above, is not very\nilluminating, though it may help to a comprehension of the demarcations\nof the subject. Please remember that the names are technical names,\nmeant, no doubt, to be suggestive, but used in strictly defined\nsenses. We have suffered much from critics who consider it sufficient\nto criticise our procedure on the slender basis of a knowledge of\nthe dictionary meanings of such terms. For example, a one-to-one\ncorrelation depends on the notion of a class with only one member, and\nthis notion is defined without appeal to the concept of the number one.\nThe notion of diversity is all that is wanted. Thus the class \u03b1 has\nonly one member, if (1) the class of values of _x_ which satisfies\nthe propositional function,\n\n _x_ is not a member of \u03b1,\n\nis not the whole type of relevant values of _x_, and if (2) the\npropositional function,\n\n _x_ and _y_ are members of \u03b1, and _x_ is diverse from\n _y_\n\nis false, whatever be the values of _x_ and _y_ in the\nrelevant type.\n\nAnalogous procedures are obviously possible for higher finite cardinal\nmembers. Thus, step by step, the whole cycle of current mathematical\nideas is capable of logical definition. The process is detailed and\nlaborious, and, like all science, knows nothing of a royal road of\nairy phrases. The essence of the process is, first, to construct\nthe notion in terms of the forms of propositions, that is, in terms\nof the relevant propositional functions, and secondly, to prove the\nfundamental truths which hold about the notion by reference to the\nresults obtained in the algebraic section of logic.\n\nIt will be seen that in this process the whole apparatus of special\nindefinable mathematical concepts, and special _a priori_\nmathematical premises, respecting number, quantity, and space, has\nvanished. Mathematics is merely an apparatus for analysing the\ndeductions which can be drawn from any particular premises, supplied\nby commonsense, or by more refined scientific observation, so far as\nthese deductions depend on the forms of the propositions. Propositions\nof certain forms are continually occurring in thought. Our existing\nmathematics is the analysis of deductions which concern those forms and\nin some way are important, either from practical utility or theoretical\ninterest. Here I am speaking of the science as it in fact exists. A\ntheoretical definition of mathematics must include in its scope any\ndeductions depending on the mere forms of propositions. But, of course\nno one would wish to develop that part of mathematics which in no sense\nis of importance.\n\nThis hasty summary of logical ideas suggests some reflections. The\nquestion arises, How many forms of propositions are there? The\nanswer is, An unending number. The reason for the supposed sterility\nof logical science can thus be discerned. Aristotle founded the\nscience by conceiving the idea of the form of a proposition, and by\nconceiving deduction as taking place in virtue of the forms. But he\nconfined propositions to four forms, now named A, I, E, O. So long as\nlogicians were obsessed by this unfortunate restriction, real progress\nwas impossible. Again, in their theory of form, both Aristotle and\nsubsequent logicians came very near to the theory of the logical\nvariable. But to come very near to a true theory, and to grasp its\nprecise application, are two very different things, as the history of\nscience teaches us. Everything of importance has been said before by\nsomebody who did not discover it.\n\nAgain, one reason why logical deductions are not obvious is, that\nlogical form is not a subject which ordinarily enters into thought.\nCommonsense deduction probably moves by blind instinct from concrete\nproposition to concrete proposition, guided by some habitual\nassociation of ideas. Thus commonsense fails in the presence of a\nwealth of material.\n\nA more important question is the relation of induction, based on\nobservation, to deductive logic. There is a tradition of opposition\nbetween adherents of induction and of deduction. In my view, it\nwould be just as sensible for the two ends of a worm to quarrel.\nBoth observation and deduction are necessary for any knowledge worth\nhaving. We cannot get at an inductive law without having recourse to a\npropositional function. For example, take the statement of observed\nfact,\n\n This body is mercury, and its specific heat is 0\u00b7033.\n\nThe propositional function is formed,\n\n Either _x_ is not mercury, or its specific heat is 0\u00b7033.\n\nThe inductive law is the assumption of the truth of the general\nproposition, that the above propositional function is true for every\nvalue of _x_ in the relevant type.\n\nBut it is objected that this process and its consequences are so simple\nthat an elaborate science is out of place. In the same way, a British\nsailor knows the salt sea when he sails over it. What, then, is the use\nof an elaborate chemical analysis of sea-water? There is the general\nanswer, that you cannot know too much of methods which you always\nemploy; and there is the special answer, that logical forms and logical\nimplications are not so very simple, and that the whole of mathematics\nis evidence to this effect.\n\nOne great use of the study of logical method is not in the region of\nelaborate deduction, but to guide us in the study of the formation of\nthe main concepts of science. Consider geometry, for example. What are\nthe points which compose space? Euclid tells us that they are without\nparts and without magnitude. But how is the notion of a point derived\nfrom the sense-perceptions from which science starts? Certainly points\nare not direct deliverances of the senses. Here and there we may see or\nunpleasantly feel something suggestive of a point. But this is a rare\nphenomenon, and certainly does not warrant the conception of space as\ncomposed of points. Our knowledge of space properties is not based on\nany observations of relations between points. It arises from experience\nof relations between bodies. Now a fundamental space-relation between\nbodies is that one body may be part of another. We are tempted to\ndefine the \"whole and part\" relation by saying that the points occupied\nby the part are some of the points occupied by the whole. But \"whole\nand part\" being more fundamental than the notion of \"point,\" this\ndefinition is really circular and vicious.\n\nWe accordingly ask whether any other definition of \"spatial whole and\npart\" can be given. I think that it can be done in this way, though,\nif I be mistaken, it is unessential to my general argument. We have\ncome to the conclusion that an extended body is nothing else than the\nclass of perception of it by all its percipients, actual or ideal. Of\ncourse, it is not any class of perceptions, but a certain definite sort\nof class which I have not defined here, except by the vicious method\nof saying that they are perceptions of body. Now, the perceptions of a\npart of a body are among the perceptions which compose the whole body.\nThus two bodies _a_ and _b_ are both classes of perceptions; and _b_ is\npart of _a_ when the class which is _b_ is contained in the class which\nis _a_. It immediately follows from the logical form of this definition\nthat if _b_ is part of _a_, and _c_ is part of _b_, then _c_ is part of\n_a_. Thus the relation \"whole to part\" is transitive. Again, it will\nbe convenient to allow that a body is part of itself. This is a mere\nquestion of how you draw the definition. With this understanding, the\nrelation is reflexive. Finally, if _a_ is part of _b_, and _b_ is part\nof _a_, then _a_ and _b_ must be identical. These properties of \"whole\nand part\" are not fresh assumptions, they follow from the logical form\nof our definition.\n\nOne assumption has to be made if we assume the ideal infinite\ndivisibility of space. Namely, we assume that every class of\nperceptions which is an extended body contains other classes of\nperceptions which are extended bodies diverse from itself. This\nassumption makes rather a large draft on the theory of ideal\nperceptions. Geometry vanishes unless in some form you make it. The\nassumption is not peculiar to my exposition.\n\nIt is then possible to define what we mean by a point. A point is the\nclass of extended objects which, in ordinary language, contain that\npoint. The definition, without presupposing the idea of a point, is\nrather elaborate, and I have not now time for its statement.\n\nThe advantage of introducing points into geometry is the simplicity\nof the logical expression of their mutual relations. For science,\nsimplicity of definition is of slight importance, but simplicity of\nmutual relations is essential. Another example of this law is the way\nphysicists and chemists have dissolved the simple idea of an extended\nbody, say of a chair, which a child understands, into a bewildering\nnotion of a complex dance of molecules and atoms and electrons and\nwaves of light. They have thereby gained notions with simpler logical\nrelations.\n\nSpace as thus conceived is the exact formulation of the properties of\nthe apparent space of the commonsense world of experience. It is not\nnecessarily the best mode of conceiving the space of the physicist.\nThe one essential requisite is that the correspondence between the\ncommonsense world in its space and the physicists' world in its space\nshould be definite and reciprocal.\n\nI will now break off the exposition of the function of logic in\nconnection with the science of natural phenomena. I have endeavoured\nto exhibit it as the organising principle, analysing the derivation\nof the concepts from the immediate phenomena, examining the structure\nof the general propositions which are the assumed laws of nature,\nestablishing their relations to each other in respect to reciprocal\nimplications, deducing the phenomena we may expect under given\ncircumstances.\n\nLogic, properly used, does not shackle thought. It gives freedom, and\nabove all, boldness. Illogical thought hesitates to draw conclusions,\nbecause it never knows either what it means, or what it assumes, or\nhow far it trusts its own assumptions, or what will be the effect of\nany modification of assumptions. Also the mind untrained in that part\nof constructive logic which is relevant to the subject in hand will be\nignorant of the sort of conclusions which follow from various sorts of\nassumptions, and will be correspondingly dull in divining the inductive\nlaws. The fundamental training in this relevant logic is, undoubtedly,\nto ponder with an active mind over the known facts of the case,\ndirectly observed. But where elaborate deductions are possible, this\nmental activity requires for its full exercise the direct study of the\nabstract logical relations. This is applied mathematics.\n\nNeither logic without observation, nor observation without logic, can\nmove one step in the formation of science. We may conceive humanity\nas engaged in an internecine conflict between youth and age. Youth is\nnot defined by years but by the creative impulse to make something. The\naged are those who, before all things, desire not to make a mistake.\nLogic is the olive branch from the old to the young, the wand which in\nthe hands of youth has the magic property of creating science.\n\n\nFOOTNOTES:\n\n[Footnote 2: _E.g._ in 1551 by Italian schoolmen; cf. Sarpi's\n_History of the Council of Trent_, under that date.]\n\n\n\n\n                              CHAPTER VII\n\n                 THE ANATOMY OF SOME SCIENTIFIC IDEAS\n\n\n                               _I. Fact_\n\nTHE characteristic of physical science is, that it ignores all\njudgments of value: for example, \u00e6sthetic or moral judgments. It is\npurely matter-of-fact, and this is the sense in which we must interpret\nthe sonorous phrase, \"Man, the servant and the minister of Nature.\"\n\nThe sphere of thought which is thus left is even then too wide for\nphysical science. It would include Ontology, namely, the determination\nof the nature of what truly exists; in other words, Metaphysics. From\nan abstract point of view this exclusion of metaphysical inquiry\nis a pity. Such an inquiry is a necessary critique of the worth of\nscience, to tell us what it all comes to. The reasons for its careful\nseparation from scientific thought are purely practical; namely,\nbecause we can agree about science--after due debate--whereas in\nrespect to metaphysics debate has hitherto accentuated disagreement.\nThese characteristics of science and metaphysics were unexpected in the\nearly days of civilised thought. The Greeks thought that metaphysics\nwas easier than physics, and tended to deduce scientific principles\nfrom _a priori_ conceptions of the nature of things. They were\nrestrained in this disastrous tendency by their vivid naturalism, their\ndelight in first-hand perception. Medi\u00e6val Europe shared the tendency\nwithout the restraint. It is possible that some distant generations\nmay arrive at unanimous conclusions on ontological questions, whereas\nscientific progress may have led to ingrained opposing veins of\nthought which can neither be reconciled nor abandoned. In such times\nmetaphysics and physical science will exchange their r\u00f4les. Meanwhile\nwe must take the case as we find it.\n\nBut a problem remains. How can mankind agree about science without\na preliminary determination of what really is? The answer must be\nfound in an analysis of the facts which form the field of scientific\nactivity. Mankind perceives, and finds itself thinking about its\nperceptions. It is the thought that matters and not that element of\nperception which is not thought. When the immediate judgment has been\nformed--Hullo, red!--it does not matter if we can imagine that in other\ncircumstances--in better circumstances, perhaps--the judgment would\nhave been--Hullo, blue!--or even--Hullo, nothing! For all intents and\npurposes, at the time it was red. Everything else is hypothetical\nreconstruction. The field of physical science is composed of these\nprimary thoughts, and of thoughts about these thoughts.\n\nBut--to avoid confusion--a false simplicity has been introduced above\ninto the example given of a primary perceptive thought. \"Hullo, red!\"\nis not really a primary perceptive thought, though it often is the\nfirst thought which finds verbal expression even silently in the mind.\nNothing is in isolation. The perception of red is of a red object in\nits relations to the whole content of the perceiving consciousness.\n\nAmong the most easily analysed of such relations are the space\nrelations. Again the red object is in immediate perception nothing else\nthan a red object. It is better termed an \"object of redness.\" Thus a\nbetter approximation to an immediate perceptive judgment is, \"Hullo,\nobject of redness there!\" But, of course, in this formulation other\nmore complex relations are omitted.\n\nThis tendency towards a false simplicity in scientific analysis, to\nan excessive abstraction, to an over-universalising of universals,\nis derived from the earlier metaphysical stage. It arises from the\nimplicit belief that we are endeavouring to qualify the real with\nappropriate adjectives. In conformity with this tendency we think,\n\"this real thing is red.\" Whereas our true goal is to make explicit\nour perception of the apparent in terms of its relations. What we\nperceive is redness related to other apparents. Our object is the\nanalysis of the relations.\n\nOne aim of science is the harmony of thought, that is, to secure\nthat judgments which are logical contraries should not be\nthought-expressions of consciousness. Another aim is the extension of\nsuch harmonised thought.\n\nSome thoughts arise directly from sense-presentation, and are part of\nthe state of consciousness which is perception. Such a thought is, \"An\nobject of redness is there.\" But in general the thought is not verbal,\nbut is a direct apprehension of qualities and relations within the\ncontent of consciousness.\n\nAmid such thoughts there can be no lack of harmony. For direct\napprehension is in its essence unique, and it is impossible to\napprehend an object as both red and blue. Subsequently it may be judged\nthat if other elements of the consciousness had been different, the\napprehension would have been of a blue object. Then--under certain\ncircumstances--the original apprehension will be called an error. But\nfor all that the fact remains, there was an apprehension of a red\nobject.\n\nWhen we speak of sense-presentation, we mean these primary thoughts\nessentially involved in its perception. But there are thoughts about\nthoughts, and thoughts derived from other thoughts. These are secondary\nthoughts. At this point it is well explicitly to discriminate between\nan actual thought-expression, namely, a judgment actually made, and a\nmere proposition which is a hypothetical thought-expression, namely,\nan imagined possibility of thought-expression. Note that the actual\ncomplete thought-content of the consciousness is explicitly neither\naffirmed or denied. It is just what _is_ thought. Thus, to think\n\"two and two make four\" is distinct from affirming that two and two\nmake four. In the first case the proposition is the thought-expression,\nin the second case the affirmation of the proposition is the\nthought-expression, and the proposition has been degraded to a mere\nproposition, namely, to a hypothetical thought-expression which is\nreflected upon.\n\nA distinction is sometimes made between facts and thoughts. So far as\nphysical science is concerned, the facts are thoughts, and thoughts\nare facts. Namely, the facts of sense-presentation as they affect\nscience are those elements in the immediate apprehensions which are\nthoughts. Also, actual thought-expressions, primary or secondary, are\nthe material facts which science interprets.\n\nThe distinction that facts are given, but thoughts are free, is\nnot absolute. We can select and modify our sense-presentation, so\nthat facts--in the narrower sense of immediate apprehension of\nsense-presentation--are to some degree subject to volition. Again, our\nstream of thought-expression is only partially modified by explicit\nvolition. We can choose our physical experience, and we find ourselves\nthinking; namely, on the one hand there is selection amid the dominant\nnecessity of sense, and on the other hand, the thought-content of\nconsciousness (so far as secondary thoughts are concerned) is not\nwholly constituted by the selection of will.\n\nThus, on the whole there is a large primary region of secondary\nthought, as well as of the primary thoughts of sense-presentation,\nwhich is given in type. That is the way in which we do think of things,\nnot wholly from any abstract necessity, so far as we know, but because\nwe have inherited the method from an environment. It is the way we find\nourselves thinking, a way which can only be fundamentally laid aside by\nan immense effort, and then only for isolated short periods of time.\nThis is what I have called the \"whole apparatus of commonsense thought.\"\n\nIt is this body of thought which is assumed in science. It is a way\nof thinking rather than a set of axioms. It is, in fact, the set of\nconcepts which commonsense has found useful in sorting out human\nexperience. It is modified in detail, but assumed in gross. The\nexplanations of science are directed to finding conceptions and\npropositions concerning nature which explain the importance of these\ncommon sense notions. For example, a chair is a common sense notion,\nmolecules and electrons explain our vision of chairs.\n\nNow science aims at harmonising our reflective and derivative thoughts\nwith the primary thoughts involved in the immediate apprehension of\nsense-presentation. It also aims at producing such derivative thoughts,\nlogically knit together. This is scientific theory; and the harmony to\nbe achieved is the agreement of theory with observation, which is the\napprehension of sense-presentation.\n\nThus there is a twofold scientific aim: (1) the production of theory\nwhich agrees with experience; and (2) the explanation of commonsense\nconcepts of nature, at least in their main outlines. This explanation\nconsists in the preservation of the concepts in a scientific theory of\nharmonised thought.\n\nIt is not asserted that this is what scientists in the past meant to\nachieve, or thought that they could achieve. It is suggested as the\nactual result of scientific effort, so far as that effort has had\nany measure of success. In short, we are here discussing the natural\nhistory of ideas and not volitions of scientists.\n\n\n                             _II. Objects_\n\nWe perceive things in space. For example, among such things are dogs,\nchairs, curtains, drops of water, gusts of air, flames, rainbows,\nchimes of bells, odours, aches and pains. There is a scientific\nexplanation of the origin of these perceptions. This explanation\nis given in terms of molecules, atoms, electrons, and their mutual\nrelations, in particular of their space-relations, and waves of\ndisturbance of these space-relations which are propagated through\nspace. The primary elements of the scientific explanation--molecules,\netc.--are not the things directly perceived. For example, we do not\nperceive a wave of light; the sensation of sight is the resultant\neffect of the impact of millions of such waves through a stretch of\ntime. Thus the object directly perceived corresponds to a series of\nevents in the physical world, events which are prolonged through\na stretch of time. Nor is it true that a perceived object always\ncorresponds to the same group of molecules. After a few years we\nrecognise the same cat, but we are thereby related to different\nmolecules.\n\nAgain, neglecting for a moment the scientific explanation, the\nperceived object is largely the supposition of our imagination. When\nwe recognised the cat, we also recognised that it was glad to see\nus. But we merely heard its mewing, saw it arch its back, and felt it\nrubbing itself against us. We must distinguish, therefore, between the\nmany direct objects of sense, and the single indirect object of thought\nwhich is the cat.\n\nThus, when we say that we perceived the cat and understood its\nfeelings, we mean that we heard a sense-object of sound, that we saw a\nsense-object of sight, that we felt a sense-object of touch, and that\nwe thought of a cat and imagined its feelings.\n\nSense-objects are correlated by time-relations and space-relations.\nThree simultaneous sense-objects which are also spatially coincident,\nare combined by thought into the perception of one cat. Such\ncombination of sense-objects is an instinctive immediate judgment in\ngeneral without effort of reasoning. Sometimes only one sense-object\nis present. For example, we hear mewing and say there must be a cat\nin the room. The transition from the sense-object to the cat has\nthen been made, by deliberate ratiocination. Even the concurrence of\nsense-objects may provoke such a self-conscious effort. For example, in\nthe dark we feel something, and hear mewing from the same place, and\nthink, Surely this is a cat. Sight is more bold; when we see a cat, we\ndo not think further. We identify the sight with the cat, whereas the\ncat and the mew are separate. But such immediate identification of\na sight object and an object of thought may lead to error; the birds\npecked at the grapes of Apelles.\n\nA single sense-object is a complex entity. The sight-object of a\ntile on the hearth may remain unchanged as we watch it in a steady\nlight, remaining ourselves unchanged in position. Even then it is\nprolonged in time, and has parts in space. Also it is somewhat\narbitrarily distinguished from a larger whole of which it forms part.\nBut the glancing fire-light and a change in our position alters the\nsight-object. We judge that the tile thought-object remains unchanged.\nThe sight-object of the coal on the fire gradually modifies, though\nwithin short intervals it remains unchanged. We judge that the coal\nthought-object is changing. The flame is never the same, and its shape\nis only vaguely distinguishable.\n\nWe conclude that a single self-identical sight-object is already a\nphantasy of thought. Consider the unchanging sight-object of the tile,\nas we remain still in a steady light. Now a sense-object perceived at\none time is a distinct object from a sense-object seen at another time.\nThus the sight of the tile at noon is distinct from its sight at 12.30.\nBut there is no such thing as a sense-object at an instant. As we stare\nat the tile, a minute, or a second, or a tenth of a second, has flown\nby: essentially there is a duration. There is a stream of sight, and\nwe can distinguish its parts. But the parts also are streams, and\nit is only in thought that the stream separates into a succession of\nelements. The stream may be \"steady\" as in the case of the unchanging\nsight-tile, or may be \"turbulent\" as in the case of the glancing\nsight-flame. In either case a sight-object is some arbitrarily small\npart of the stream.\n\nAgain, the stream which forms the succession of sight-tiles is merely a\ndistinguishable part of the whole stream of sight-presentation.\n\nSo, finally, we conceive ourselves each experiencing a complete\ntime-flux (or stream) of sense-presentation. This stream is\ndistinguishable into parts. The grounds of distinction are differences\nof sense--including within that term, differences of types of sense,\nand differences of quality and of intensity within the same type\nof sense--and differences of time-relations, and differences of\nspace-relations. Also the parts are not mutually exclusive and exist in\nunbounded variety.\n\nThe time-relation between the parts raises the questions of memory and\nrecognition, subjects too complex for discussion here. One remark must\nbe made. If it be admitted, as stated above, that we live in durations\nand not in instants, namely, that the present essentially occupies\na stretch of time, the distinction between memory and immediate\npresentation cannot be quite fundamental; for always we have with us\nthe fading present as it becomes the immediate past. This region of our\nconsciousness is neither pure memory nor pure immediate presentation.\nAnyhow, memory is also a presentation in consciousness.\n\nAnother point is to be noted in connection with memory. There is no\ndirectly perceived time-relation between a present event and a past\nevent. The present event is only related to the memory of the past\nevent. But the memory of a past event is itself a present element\nin consciousness. We assert the principle that directly perceived\nrelations can only exist between elements of consciousness, both in\nthat present during which the perception occurs. All other relations\nbetween elements of perception are inferential constructions. It thus\nbecomes necessary to explain how the time stream of events establishes\nitself in thought, and how the apparent world fails to collapse into\none single present. The solution of the difficulty is arrived at\nby observing that the present is itself a duration, and therefore\nincludes directly perceived time-relations between events contained\nwithin it. In other words we put the present on the same footing as\nthe past and the future in respect to the inclusion within it of\nantecedent and succeeding events, so that past, present, and future\nare in this respect exactly analogous ideas. Thus there will be two\nevents _a_ and _b_, both in the same present, but the event\n_a_ will be directly perceived to precede the event _b_.\nAgain time flows on, and the event _a_ fades into the past, and\nin the new present duration events _b_ and _c_ occur, event\n_b_ preceding event _c_, also in the same present duration\nthere is the memory of the time-relation between _a_ and _b_.\nThen by an inferential construction the event _a_ in the past\nprecedes the event _c_ in the present. By proceeding according to\nthis principle the time-relations between elements of consciousness,\nnot in the same present, are established. The method of procedure\nhere explained is a first example of what we will call the Principle\nof Aggregation. This is one of the fundamental principles of mental\nconstruction according to which our conception of the external physical\nworld is constructed. Other examples will later on be met with.\n\nThe space-relations between the parts are confused and fluctuating,\nand in general lack determinate precision. The master-key by which\nwe confine our attention to such parts as possess mutual relations\nsufficiently simple for our intellects to consider is the principle of\nconvergence to simplicity with diminution of extent. We will call it\nthe \"principle of convergence.\" This principle extends throughout the\nwhole field of sense-presentation.\n\nThe first application of the principle occurs in respect to time.\nThe shorter the stretch of time, the simpler are the aspects of the\nsense-presentation contained within it. The perplexing effects of\nchange are diminished and in many cases can be neglected. Nature has\nrestricted the acts of thought which endeavour to realise the content\nof the present, to stretches of time sufficiently short to secure this\nstatic simplicity over the greater part of the sense-stream.\n\nSpatial relations become simplified within the approximately static\nsense-world of the short time. A further simplicity is gained by\npartitioning this static world into parts of restricted space-content.\nThe various parts thus obtained have simpler mutual space-relations,\nand again the principle of convergence holds.\n\nFinally, the last simplicity is obtained by partitioning the\nparts, already restricted as to space and time, into further parts\ncharacterised by homogeneity in type of sense, and homogeneity in\nquality and intensity of sense. These three processes of restriction\nyield, finally, the sense-objects which have been mentioned above. Thus\nthe sense-object is the result of an active process of discrimination\nmade in virtue of the principle of convergence. It is the result of\nthe quest for simplicity of relations within the complete stream of\nsense-presentation.\n\nThe thought-objects of perception are instances of a fundamental\nlaw of nature, the law of objective stability. It is the law of the\ncoherence of sense-objects. This law of stability has an application\nto time and an application to space; also it must be applied in\nconjunction with that other law, the principle of convergence to\nsimplicity from which sense-objects are derived.\n\nSome composite partial streams of sense-presentation can be\ndistinguished with the following characteristics: (1) the\ntime-succession of sense-objects, belonging to a single sense, involved\nin any such a composite partial stream, is composed of very similar\nobjects whose modifications increase only gradually, and thus forms\na homogeneous component stream within the composite stream; (2) the\nspace-relations of those sense-objects (of various senses) of such\na composite stream which are confined within any sufficiently short\ntime are identical so far as they are definitely apprehended, and thus\nthese various component streams, each homogeneous, \"cohere\" to form the\nwhole composite partial stream; (3) there are other sense-presentations\noccurring in association with that composite partial stream which\ncan be determined by rules derived from analogous composite partial\nstreams, with other space and time relations, provided that the analogy\nbe sufficiently close. Call these the \"associated sense-presentations.\"\nA partial stream of this sort, viewed as a whole, is here called a\n\"first crude thought-object of perception.\"\n\nFor example, we look at an orange for half a minute, handle it, and\nsmell it, note its position in the fruit-basket, and then turn away.\nThe stream of sense-presentation of the orange during that half-minute\nis a first crude thought-object of perception. Among the associated\nsense presentations are those of the fruit-basket which we conceive as\nsupporting the orange.\n\nThe essential ground of the association of sense-objects of\nvarious types, perceived within one short duration, into a first\ncrude thought-object of perception is the coincidence of their\nspace-relations, that is, in general an approximate coincidence of\nsuch relations perhaps only vaguely apprehended. Thus coincident\nspace-relations associate sense-objects into a first crude\nthought-object, and diverse space-relations dissociate sense-objects\nfrom aggregation into a first crude thought-object. In respect to\nsome groups of sense-objects the association may be an immediate\njudgment devoid of all inference, so that the primary perceptual\nthought is that of the first crude thought-object, and the separate\nsense-objects are the result of reflective analysis acting on memory.\nFor example sense-objects of sight and sense-objects of touch are\noften thus primarily associated and only secondarily dissociated in\nthought. But sometimes the association is wavering and indeterminate,\nfor example, that between the sound-object of the mew of the cat and\nthe sight-object of the cat. Thus to sum up, the partial stream of\nsense-perceptions coalesces into that first crude thought-object of\nperception which is the momentary cat because the sense-perceptions\nbelonging to this stream are in the same place, but equally it would\nbe true to say that they are in the same place because they belong\nto the same momentary cat. This analysis of the complete stream of\nsense-presentation in any small present duration into a variety of\nfirst crude thought-objects only partially fits the facts; for one\nreason because many sense-objects, such as sound for instance, have\nvague and indeterminate space-relations, for example vaguely those\nspace-relations which we associate with our organs of sense and also\nvaguely those of the origin from which (in the scientific explanation)\nthey proceed.\n\nThe procedure by which the orange of half a minute is elaborated into\nthe orange in the ordinary sense of the term involves in addition the\ntwo principles of aggregation and of hypothetical sense-presentation.\n\nThe principle of aggregation, as here employed, takes the form that\nmany distinct first crude thought-objects of perception are conceived\nas one thought-object of perception, if the many partial streams\nforming these objects are sufficiently analogous, if their times of\noccurrence are distinct, and if the associated sense-presentations are\nsufficiently analogous.\n\nFor example, after leaving the orange, in five minutes we return.\nA new first crude thought-object of perception presents itself to\nus, indistinguishable from the half-minute orange we previously\nexperienced; it is in the same fruit-basket. We aggregate the two\npresentations of an orange into the same orange. By such aggregations\nwe obtain \"second crude thought-objects of perception.\" But however far\nwe can proceed with aggregation of this type, the orange is more than\nthat. For example, what do we mean when we say, The orange is in the\ncupboard, if Tom has not eaten it?\n\nThe world of present fact is more than a stream of sense-presentation.\nWe find ourselves with emotions, volitions, imaginations, conceptions,\nand judgments. No factor which enters into consciousness is by itself\nor even can exist in isolation. We are analysing certain relations\nbetween sense-presentation and other factors of consciousness.\nHitherto we have taken into account merely the factors of concept and\njudgment. Imagination is necessary to complete the orange, namely,\nthe imagination of hypothetical sense-presentations. It is beside\nthe point to argue whether we ought to have such imaginations, or to\ndiscuss what are the metaphysical truths concerning reality to which\nthey correspond. We are here only concerned with the fact that such\nimaginations exist and essentially enter into the formation of the\nconcepts of the thought-objects of perception which are the first\ndata of science. We conceive the orange as a permanent collection of\nsense-presentations existing as if they were an actual element in our\nconsciousness, which they are not. The orange is thus conceived as\nin the cupboard with its shape, odour, colour, and other qualities.\nNamely, we imagine hypothetical possibilities of sense-presentation,\nand conceive their want of actuality in our consciousness as immaterial\nto their existence in fact. The fact which is essential for science is\nour conception; its meaning in regard to the metaphysics of reality is\nof no scientific importance, so far as physical science is concerned.\n\nThe orange completed in this way is the thought-object of perception.\n\nIt must be remembered that the judgments and concepts arising in the\nformation of thought-objects of perception are in the main instinctive\njudgments, and instinctive concepts, and are not concepts and judgments\nconsciously sought for and consciously criticised before adoption.\nTheir adoption is facilitated by and interwoven with the expectation of\nthe future in which the hypothetical passes into the actual, and also\nwith the further judgment of the existence of other consciousnesses,\nso that much that is hypothetical to one consciousness is judged to be\nactual to others.\n\nThe thought-object of perception is, in fact, a device to make plain\nto our reflective consciousness relations which hold within the\ncomplete stream of sense-presentation. Concerning the utility of this\nweapon there can be no question; it is the rock upon which the whole\nstructure of commonsense thought is erected. But when we consider the\nlimits of its application the evidence is confused. A great part of our\nsense-presentation can be construed as perception of various persistent\nthought-objects. But hardly at any time can the sense-presentations\nbe construed wholly in that way. Sights lend themselves easily to\nthis construction, but sight can be baffled: for example, consider\nreflections in looking-glasses, apparently bent sticks half in and half\nout of water, rainbows, brilliant patches of light which conceal the\nobject from which they emanate, and many analogous phenomena. Sound\nis more difficult; it tends largely to disengage itself from any such\nobject. For example, we see the bell, but we hear the sound which\ncomes from the bell; yet we also say that we hear the bell. Again, a\ntoothache is largely by itself, and is only indirectly a perception\nof the nerve of the tooth. Illustrations to the same effect can be\naccumulated from every type of sensation.\n\nAnother difficulty arises from the fact of change. The thought-object\nis conceived as one thing, wholly actual at each instant. But since\nthe meat has been bought it has been cooked, the grass grows and then\nwithers, the coal burns in the fire, the pyramids of Egypt remain\nunchanged for ages, but even the pyramids are not wholly unchanged. The\ndifficulty of change is merely evaded by affixing a technical Latin\nname to a supposed logical fallacy. A slight cooking leaves the meat\nthe same object, but two days in the oven burns it to a cinder. When\ndoes the meat cease to be? Now the chief use of the thought-object\nis the concept of it as one thing, here and now, which later can be\nrecognised, there and then. This concept applies sufficiently well to\nmost things for short times, and to many things for long times. But\nsense-presentation as a whole entirely refuses to be patient of the\nconcept.\n\nWe have now come to the reflective region of explanation, which is\nscience.\n\nA great part of the difficulty is at once removed by applying the\nprinciple of convergence to simplicity. We habitually make our\nthought-objects too large; we should think in smaller parts. For\nexample, the Sphinx has changed by its nose becoming chipped, but by\nproper inquiry we could find the missing part in some private house\nof Western Europe or Northern America. Thus, either part, the rest\nof the Sphinx or the chip, regains its permanence. Furthermore, we\nenlarge this explanation by conceiving parts so small that they can\nonly be observed under the most favourable circumstances. This is a\nwide extension of the principle of convergence in its application to\nnature; but it is a principle amply supported by the history of exact\nobservation.\n\nThus, change in thought-objects of perception is largely explained as\na disintegration into smaller parts, themselves thought-objects of\nperception. The thought-objects of perception which are presupposed in\nthe common thought of civilised beings are almost wholly hypothetical.\nThe material universe is largely a concept of the imagination which\nrests on a slender basis of direct sense-presentation. But none the\nless it is a fact; for it is a fact that actually we imagine it. Thus\nit is actual in our consciousness just as sense-presentation also is\nactual there. The effort of reflective criticism is to make these two\nfactors in our consciousness agree where they are related, namely,\nto construe our sense-presentation as actual realisation of the\nhypothetical thought-objects of perception.\n\nThe wholesale employment of purely hypothetical thought-objects of\nperception enables science to explain some of the stray sense-objects\nwhich cannot be construed as perceptions of a thought-object of\nperception: for example, sounds. But the phenomena as a whole defy\nexplanation on these lines until a further fundamental step is taken,\nwhich transforms the whole concept of the material universe. Namely,\nthe thought-object of perception is superseded by the thought-object of\nscience.\n\nThe thought-objects of science are molecules, atoms, and electrons. The\npeculiarity of these objects is that they have shed all the qualities\nwhich are capable of direct sense-representation in consciousness.\nThey are known to us only by their associated phenomena, namely,\nseries of events in which they are implicated are represented in our\nconsciousness by sense-presentations. In this way, the thought-objects\nof science are conceived as the causes of sense-representation. The\ntransition from thought-objects of perception to thought-objects of\nscience is decently veiled by an elaborate theory concerning primary\nand secondary qualities of bodies.\n\nThis device, by which sense-presentations are represented in thought\nas our perception of events in which thought-objects of science are\nimplicated, is the fundamental means by which a bridge is formed\nbetween the fluid vagueness of sense and the exact definition of\nthought. In thought a proposition is either true or false, an entity\nis exactly what it is, and relations between entities are expressible\n(in idea) by definite propositions about distinctly conceived entities.\nSense-perception knows none of these things, except by courtesy.\nAccuracy essentially collapses at some stage of inquiry.\n\n\n                         _III. Time and Space_\n\n_Recapitulation._--Relations of time and relations of space\nhold between sense-objects of perception. These sense-objects are\ndistinguished as separate objects by the recognition of either (1)\ndifferences of sense-content, or (2) time-relations between them\nother than simultaneity, or (3) space-relations between them other\nthan coincidence. Thus sense-objects arise from the recognition of\ncontrast within the complete stream of sense-presentation, namely, from\nthe recognition of the objects as related terms, by relations which\ncontrast them. Differences of sense-content are infinitely complex in\ntheir variety. Their analysis under the heading of general ideas is the\nunending task of physical science. Time-relations and space-relations\nare comparatively simple, and the general ideas according to which\ntheir analysis should proceed are obvious.\n\nThis simplicity of time and space is perhaps the reason why thought\nchooses them as the permanent ground for objectival distinction,\nthrowing the various sense-objects thus obtainable into one heap, as\na first crude thought-object of perception, and thence, as described\nabove, obtaining a thought-object of perception. Thus a thought-object\nof perception conceived as in the present of a short duration is a\nfirst crude thought-object of perception either actual or hypothetical.\nSuch a thought-object of perception, confined within a short duration,\ntakes on the space-relations of its component sense-objects within that\nsame duration. Accordingly thought-objects of perception, conceived in\ntheir whole extents, have to each other the time-relationships of their\ncomplete existences, and within any small duration have to each other\nthe space-relationships of their component sense-objects which lie\nwithin that duration.\n\nRelations bind together: thus thought-objects of perception are\nconnected in time and in space. The genesis of the objectival analysis\nof sense-presentation is the recognition of sense-objects as distinct\nterms in time-relations and space-relations: thus thought-objects of\nperception are separated by time and by space.\n\n_Whole and Part._--A sense-object is part of the complete stream\nof presentation. This concept of being a part is merely the statement\nof the relation of the sense-object to the complete sense-presentation\nfor that consciousness. Also a sense-object can be part of another\nsense-object. It can be a part in two ways, namely, a part in time\nand a part in space. It seems probable that both these concepts of\ntime-part and space-part are fundamental; that is, are concepts\nexpressing relations which are directly presented to us, and are not\nconcepts about concepts. In that case no further definition of the\nactual presentation is possible. It may even then be possible to\ndefine an adequate criterion of the occurrence of such a presentation.\nFor example, adopting for the moment a realist metaphysic as to the\nexistence of the physical world of molecules and electrons, the vision\nof a chair as occurring for some definite person at some definite\ntime is essentially indefinable. It is his vision, though each of us\nguesses that it must be uncommonly like our vision under analogous\ncircumstances. But the existence of the definable molecules and\nwaves of light in certain definable relations to his bodily organs\nof sense, his body also being in a certain definable state, forms an\nadequate criterion of the occurrence of the vision, a criterion which\nis accepted in Courts of Law and for physical science is tacitly\nsubstituted for the vision.\n\nThe connection between the relations \"whole and part\" and \"all and\nsome\" is intimate. It can be explained thus so far as concerns directly\npresented sense-objects. Call two sense-objects \"separated\" if there\nis no third sense-object which is a part of both of them. Then an\nobject A is composed of the two objects B and C, if (1) B and C are\nboth parts of A, (2) B and C are separated, and (3) there is no part of\nA which is separated both from B and from C. In such a case the class\n\u03b1 which is composed of the two objects B and C is often substituted\nin thought for the sense-object A. But this process presupposes the\nfundamental relation \"whole and part.\" Conversely the objects B and C\nmay be actual sense-objects, but the sense-object A which corresponds\nto the class \u03b1 may remain hypothetical. For example, the round world\non which we live remains a conception corresponding to no single\nsense-object at any time presented in any human being's consciousness.\n\nIt is possible, however, that some mode of conceiving the\nwhole-and-part relation between extended objects as the all-and-some\nrelation of logical classes can be found. But in this case the\nextended objects as here conceived cannot be the true sense-objects\nwhich are present to consciousness. For as here conceived a part of a\nsense-object is another sense-object of the same type; and therefore\none sense-object cannot be a class of other sense-objects, just as\na tea-spoon cannot be a class of other tea-spoons. The ordinary way\nin thought by which whole-and-part is reduced to all-and-some is by\nthe device of points, namely, the part of an object occupies some\nof the points occupied by the whole object. If any one holds that\nin his consciousness the sense-presentation is a presentation of\npoint-objects, and that an extended object is merely a class of such\npoint-objects collected together in thought, then this ordinary method\nis completely satisfactory. We shall proceed on the assumption that\nthis conception of directly perceived point-objects has no relation to\nthe facts.\n\nIn the preceding address on \"The Organisation of Thought,\" another mode\nis suggested. But this method would apply only to the thought-object\nof perception, and has no reference to the primary sense-objects here\nconsidered. Accordingly it must reckon as a subordinate device for a\nlater stage of thought.\n\nThus the point-object in time and the point-object in space, and the\ndouble point-object both in time and space, must be conceived as\nintellectual constructions. The fundamental fact is the sense-object,\nextended both in time and space, with the fundamental relation of\nwhole-to-part to other such objects, and subject to the law of\nconvergence to simplicity as we proceed in thought through a series of\nsuccessively contained parts.\n\nThe relation whole-to-part is a temporal or spatial relation, and\nis therefore primarily a relation holding between sense-objects of\nperception, and it is only derivatively ascribed to the thought-objects\nof perception of which they are components. More generally, space and\ntime relations hold primarily between sense-objects of perception and\nderivatively between thought-objects of perception.\n\n_Definition of Points._--The genesis of points of time and of\nspace can now be studied. We must distinguish (1) sense-time and\nsense-space, and (2) thought-time of perception and thought-space of\nperception.\n\nSense-time and sense-space are the actually observed time-relations\nand space-relations between sense-objects. Sense-time and sense-space\nhave no points except, perhaps, a few sparse instances, sufficient\nto suggest the logical idea; also, sense-time and sense-space are\ndiscontinuous and fragmentary.\n\nThought-time of perception and thought-space of perception are the time\nand space relations which hold between thought-objects of perception.\nThought-time of perception and thought-space of perception are each\ncontinuous. By \"continuous\" is here meant that all thought-objects of\nperception have to each other a time (or space) relation.\n\nThe origin of points is the effort to take full advantage of\nthe principle of convergence to simplicity. In so far as this\nprinciple does not apply, a point is merely a cumbrous way of\ndirecting attention to a set of relations between a certain set of\nthought-objects of perception, which set of relations, though actual so\nfar as a thought-object is actual, is (under this supposition) of no\nparticular importance. Thus the proved importance in physical science\nof the concepts of points in time and points in space is a tribute to\nthe wide applicability of this principle of convergence.\n\nEuclid defines a point as without parts and without magnitude. In\nmodern language a point is often described as an ideal limit by\nindefinitely continuing the process of diminishing a volume (or\narea). Points as thus conceived are often called convenient fictions.\nThis language is ambiguous. What is meant by a fiction? If it means\na conception which does not correspond to any fact, there is some\ndifficulty in understanding how it can be of any use in physical\nscience. For example, the fiction of a red man in a green coat\ninhabiting the moon can never be of the slightest scientific service,\nsimply because--as we may presume--it corresponds to no fact. By\ncalling the concept of points a convenient fiction, it must be meant\nthat the concept does correspond to some important facts. It is, then,\nrequisite, in the place of such vague allusiveness, to explain exactly\nwhat are the facts to which the concept corresponds.\n\nWe are not much helped by explaining that a point is an ideal limit.\nWhat is a limit? The idea of a limit has a precise meaning in the\ntheory of series, and in the theory of the values of functions; but\nneither of these meanings apply here. It may be observed that, before\nthe ordinary mathematical meanings of limit had received a precise\nexplanation, the idea of a point as a limit might be considered as\none among other examples of an idea only to be apprehended by direct\nintuition. This view is not now open to us. Thus, again, we are\nconfronted with the question: What are the precise properties meant\nwhen a point is described as an ideal limit? The discussion which now\nfollows is an attempt to express the concept of a point in terms of\nthought-objects of perception related together by the whole-and-part\nrelation, considered either as a time-relation or as a space-relation.\nIf it is so preferred, it may be considered that the discussion is\ndirected towards a precise elucidation of the term \"ideal limit\" as\noften used in this connection.\n\nThe subsequent explanations can be made easier to follow by a small\npiece of symbolism: Let _aEb_ mean that \"_b_ is part of _a_.\" We need\nnot decide whether we are talking of time-parts or space-parts, but\nwhichever choice is supposed to be made must be conceived as adhered to\nthroughout any connected discussion. The symbol _E_ may be considered\nas the initial letter of \"encloses,\" so we read \"_aEb_\" as \"_a_\nencloses _b_.\" Again the \"field of _E_\" is the set of things which\neither enclose or are enclosed, _i. e._ everything \"_a_,\" which is such\nthat _x_ can be found so that either _aEx_ or _xEa_. A member of the\nfield of _E_ is called \"an enclosure-object.\"\n\nNow, we assume that this relation of whole-to-part, which in the future\nwe will call \"enclosure,\" always satisfies the conditions in that the\nrelation _E_ is (1) transitive, (2) asymmetrical, and (3) with its\ndomain including its converse domain.\n\nThese four conditions deserve some slight consideration; only the first\ntwo of them embody hypotheses which enter vitally into the reasoning.\n\nCondition (1) may be stated as the condition that _aEb_ and _bEc_\nalways implies _aEc_. The fact that an entity _b_ can be found such\nthat _aEb_ and _bEc_ may be conceived as a relation between _a_ and\n_c_. It is natural to write _E_^2 for this relation. Thus the condition\nis now written: If _aE_^2_c_, then _aEc_. This can be still otherwise\nexpressed by saying that the relation _E_^2 implies, whenever it holds,\nthat the relation _E_ also holds.\n\nCondition (2) is partly a mere question of trivial definition, and\npartly a substantial assumption. The asymmetrical relation (_E_) is\nsuch that _aEb_ and _bEa_ can never hold simultaneously. This property\nsplits up into two parts: (1) that no instance of _aEb_ and _bEa_ and\n\"_a_ diverse from _b_,\" can occur, and (2) that _aEa_ cannot occur.\nThe first part is a substantial assumption, the second part (so far as\nwe are concerned) reduces to the trivial convention that we shall not\nconsider an object as part of itself, but will confine attention to\n\"proper parts.\"\n\nCondition (3) means that _aEb_ always implies that _c_ can be\nfound such that _bEc_. This condition, taken in conjunction with\nthe fact that we are only considering proper parts, is the assertion of\nthe principle of the indefinite divisibility of extended objects, both\nin space and in time.\n\nAn indivisible part will lack duration in time, and extension in\nspace, and is thus an entity of essentially a different character\nto a divisible part. If we admit such indivisibles as the only true\nsense-objects, our subsequent procedure is an unnecessary elaboration.\n\nIt will be found that a fourth condition is necessary owing to\nlogical difficulties connected with the theory of an infinite number\nof choices. It will not be necessary for us to enter further on\nthis question, which involves difficult considerations of abstract\nlogic. The outcome is, that apart from hypothesis we cannot prove the\nexistence of the sets, each containing an infinite number of objects,\nwhich are here called points, as will be explained immediately.\n\nNow consider a set of enclosure objects which is such that (1)\nof any two of its members one encloses the other, and (2) there\nis no member which is enclosed by all the others, and (3) there\nis no enclosure-object, not a member of the set which is enclosed\nby every member of the set. Call such a set a \"convergent set of\nenclosure-objects.\" As we pass along the series from larger to smaller\nmembers, evidently we converge towards an ideal simplicity to any\ndegree of approximation to which we like to proceed, and the series as\na whole embodies the complete ideal along that route of approximation.\nIn fact, to repeat, the series is a _route of approximation_.\n\nWe have now to inquire if the principle of convergence to simplicity\nmay be expected to yield the same type of simplicity for every such\nconvergent route. The answer is, as we might expect, namely, that this\ndepends upon the nature of the properties which are to be simplified.\n\nFor example, consider the application to time. Now, time is\none-dimensional; so when this property of one-dimensionality has been\nexpressed by the proper conditions, not here stated, a convergent set\nof enclosure-objects must, considered as a route of approximation,\nexhibit the properties of one unique instant of time, as ordinarily\nconceived by the euclidean definition. Accordingly, whatever simplicity\nis to be achieved by the application to time of the principle of\nconvergence to simplicity must be exhibited among the properties of\nany such route of approximation.\n\nFor space, different considerations arise. Owing to its multiple\ndimensions, we can show that different convergent sets of\nenclosure-objects, indicating different routes of approximation, may\nexhibit convergence to different types of simplicity, some more complex\nthan others.\n\nFor example, consider a rectangular box of height _h_ ft., breadth\n_b_ ft., and thickness _c_ ft. Now, keep _h_ and _b_ constant, and\nlet the central plane (height _h_, breadth _b_) perpendicular to the\nthickness be fixed, then make _c_ diminish indefinitely. We thus obtain\na convergent series of an indefinitely large number of boxes, and there\nis no smallest box. Thus this convergent series exhibits the route of\napproximation towards the type of simplicity expressed as being a plane\narea of height _h_, breadth _b_, and no thickness.\n\nAgain, by keeping the central line of height _h_ fixed, and by making\n_b_ and _c_ diminish indefinitely, the series converges to the segment\nof a straight line of length _h_.\n\nFinally, by keeping only the central point fixed, and by making _h_,\n_b_, and _c_ diminish indefinitely, the series converges to a point.\n\nFurthermore, we have introduced as yet no concept which would prevent\nan enclosure-object being formed of detached fragments in space. Thus\nwe can easily imagine a convergent set which converges to a number of\npoints in space. For example, each object of the set might be formed of\ntwo not overlapping spheres of radius _r_, with centres _A_ and _B_.\nThen by diminishing _r_ indefinitely, and keeping _A_ and _B_ fixed, we\nhave convergence to the pair of points _A_ and _B_.\n\nIt remains now to consider how those convergent sets which converge to\na single point can be discriminated from all the other types of such\nsets, merely by utilising concepts founded on the relation of enclosure.\n\nLet us name convergent sets by Greek letters; by proceeding \"forward\"\nalong any such set let us understand the process of continually passing\nfrom the larger to the smaller enclosure-objects which form the set.\n\nThe convergent set \u03b1 will be said to \"cover\" the convergent set \u03b2, if\nevery member of \u03b1 encloses some members of \u03b2. We notice that if an\nenclosure-object _x_ encloses any member (_y_) of \u03b2, then every member\nof the \"tail-end\" of \u03b2, found by proceeding forward along \u03b2 from _y_,\nmust be enclosed by _x_. Thus if \u03b1 covers \u03b2, every member of \u03b1 encloses\nevery member of the tail-end of \u03b2, starting from the largest member of\n\u03b2 which is enclosed by that member of \u03b1.\n\nIt is possible for each of two convergent sets to cover the other. For\nexample, let one set (\u03b1) be a set of concentric spheres converging to\ntheir centre _A_, and the other set (\u03b2) be a set of concentric\ncubes, similarly situated, converging to the same centre _A_. Then\n\u03b1 and \u03b2 will each cover the other.\n\nLet two convergent sets which are such that each covers the other be\ncalled \"equal.\"\n\nThen it is a sufficient condition to secure that a convergent set\n\u03b1 possesses the point type of convergence, if every convergent set\ncovered by it is also equal to it, namely, \u03b1 is a convergent set with\nthe punctual type of convergence, if \"\u03b1 covers \u03b2\" always implies that \u03b2\ncovers \u03b1.\n\nIt can easily be seen by simple examples that the other types of\nconvergence to surfaces or lines or sets of points cannot possess this\nproperty. Consider, for example, the three convergent sets of boxes in\nthe preceding illustration, which converge respectively to a central\nplane, a central line in the central plane, and the central point in\nthe central line. The first set covers the second and third sets, and\nthe second set covers the third set, but no two of the sets are equal.\n\nIt is a more difficult question to determine whether the condition here\nindicated as sufficient to secure the punctual type of convergence\nis also necessary. The question turns on how far thought-objects\nof perception possess exact boundaries prior to the elaboration of\nexact mathematical concepts of space. If they are to be conceived as\npossessing such exact boundaries, then convergent sets converging to\npoints on such boundaries must be allowed for. The procedure necessary\nfor the specification of the complete punctual condition becomes then\nvery elaborate,[3] and will not be considered here.\n\nBut such exact determination as is involved in the conception of\nan exact spatial boundary does not seem to belong to the true\nthought-object of perception. The ascription of an exact boundary\nreally belongs to the transition stage of thought as it passes from\nthe thought-object of perception to the thought-object of science.\nThe transition from the sense-object immediately presented to the\nthought-object of perception is historically made in a wavering\nindeterminate line of thought. The definite stages here marked out\nsimply serve to prove that a logically explicable transition is\npossible.\n\nWe accordingly assume that the condition laid down above to secure the\npunctual convergence of a convergent set of enclosure-objects is not\nonly sufficient, but necessary.\n\nIt can be proved that, if two convergent sets of enclosure-objects are\nboth equal to a third convergent set, they are equal to each other.\nConsider now any punctual convergent set (\u03b1). We want to define the\n\"point\" to which \u03b1 is a route of approximation in a way which is\nneutral between \u03b1 and all the convergent sets which are equal to \u03b1.\nEach of these sets is a route of approximation to the same \"point\"\nas \u03b1. This definition is secured if we define the point as the class\nformed by all the enclosure-objects which belong either to \u03b1 or to\nany convergent set which is equal to \u03b1. Let _P_ be this class\nof enclosure-objects. Then any convergent set (\u03b2) which consists of\nenclosure-objects entirely selected from members of the class _P_\nmust be a route of approximation to the same \"point\" as does the\noriginal punctual set \u03b1; namely, provided that we choose a small enough\nenclosure-object in \u03b2, we can always find a member of \u03b1 which encloses\nit; and provided that we choose a small enough enclosure-object in\n\u03b1, we can always find a member of \u03b2 which encloses it. Thus _P_\nonly includes convergent sets of the punctual type, and the route\nof approximation indicated by any two convergent sets selected from\n_P_ converges to identical results.\n\n_The Uses of Points._--The sole use of points is to facilitate the\nemployment of the principle of Convergence to Simplicity. By this\nprinciple some simple relations in appropriate circumstances become\ntrue, when objects are considered which are sufficiently restricted in\ntime or in space. The introduction of points enables this principle to\nbe carried through to its ideal limit. For example, suppose _g_ (_a_,\n_b_, _c_) represents some statement concerning three enclosure-objects,\n_a_, _b_, _c_, which may be true if the objects are sufficiently\nrestricted in extent. Let _A_, _B_, _C_ be three given points,\nthen we define _g_ (_A_, _B_, _C_) to mean that _whatever_ three\nenclosure-objects _a_, _b_, _c_ are chosen, such that _a_ is a member\nof _A_, _b_ of _B_, and _c_ of _C_, it is _always possible_ to find\nthree other members of _A_, _B_, _C_, namely, _x_ a member of _A_, _y_\nof _B_, and _z_ of _C_, such that _aEx_, _bEy_, _cEz_, and _g_ (_x_,\n_y_, _z_). So by going far enough down in the tail-ends of _A_, _B_,\n_C_ we can always secure three objects _x_, _y_, _z_ for which _g_\n(_x_, _y_, _z_) is true.\n\nFor example, let _g_ (_A_, _B_, _C_) mean \"_A_, _B_, _C_ are three\npoints in a linear row.\" This must be construed to mean that whatever\nthree objects _a_, _b_, _c_ we choose, members of _A_, _B_, _C_\nrespectively, we can always find three objects _x_, _y_, _z_, also\nmembers of _A_, _B_, _C_ respectively, and such that _a_ encloses _x_,\n_b_ encloses _y_, _c_ encloses _z_, and also such that _x_, _y_, _z_\nare in a linear row.\n\nSometimes a double convergence is necessary, namely, a convergence of\nconditions as well as a convergence of objects. For example, consider\nthe statement, \"the points _A_ and _B_ are two feet apart.\" Now, the\nexact statement \"two feet apart\" does not apply to objects. For objects\n_x_ and _y_ we must substitute the statement, \"the distance between\n_x_ and _y_ lies between the limits (2 \u00b1 _e_) feet.\" Here _e_ is some\nnumber, less than two, which we have chosen for this statement. Then\nthe points _A_ and _B_ are two feet apart; if, _however we choose the\nnumber e_, whatever enclosure-objects _a_ and _b_, members of _A_ and\n_B_ respectively, we consider, we can always find enclosure-objects _x_\nand _y_, members of _A_ and _B_ respectively, such that _a_ encloses\n_x_ and _b_ encloses _y_, and also such that the distance between _x_\nand _y_ lies between the limits (2 \u00b1 _e_) feet. It is evident, since\n_e_ can be chosen as small as we please, that this statement exactly\nexpresses the condition that _A_ and _B_ are two feet apart.\n\n_Straight Lines and Planes._--But the problem of the intellectual\nconstruction of straight lines and planes is not yet sufficiently\nanalysed. We have interpreted the meaning of the statement that three\nor more points are collinear, and can similarly see how to interpret\nthe meaning of the statement that four or more points are coplanar,\nin either case deriving the exact geometrical statements from vaguer\nstatements respecting extended objects.\n\nThis procedure only contemplates groups of finite numbers of points.\nBut straight lines and planes are conceived as containing infinite\nnumbers of points. This completion of lines and planes is obtained\nby a renewed application of the principle of aggregation, just as\na set of first crude thought-objects of perception are aggregated\ninto one complete thought-object of perception. In this way repeated\njudgments of the collinearity of sets of points are finally, when\ncertain conditions of interlacing are fulfilled, aggregated in the\nsingle judgment of all the points of the groups as forming one whole\ncollinear group. Similarly for judgments of coplanarity. This process\nof logical aggregation can be exhibited in its exact logical analysis.\nBut it is unnecessary here to proceed to such details. Thus we conceive\nour points as sorted into planes and straight lines, concerning which\nthe various axioms of geometry hold. These axioms, in so far as they\nessentially require the conception of points, are capable of being\nexhibited as the outcome of vaguer, less exact judgments respecting the\nrelations of extended objects.\n\n_Empty Space._--It must be observed that the points, hitherto\ndefined, necessarily involve thought-objects of perception, and lie\nwithin the space-extension occupied by such objects. It is true that\nsuch objects are largely hypothetical, and that we can bring into our\nhypotheses sufficient objects to complete our lines and planes. But\nevery such hypothesis weakens the connection between our scientific\nconcept of nature and the actual observed facts which are involved in\nthe actual sense-presentations.\n\nOccam's razor, _Entia non multiplicanda pr\u00e6ter necessitatem_,\nis not an arbitrary rule based on mere logical elegancy. Nor is its\napplication purely confined to metaphysical speculation. I am ignorant\nof the precise reason for its metaphysical validity, but its scientific\nvalidity is obvious, namely, every use of hypothetical entities\ndiminishes the claim of scientific reasoning to be the necessary\noutcome of a harmony between thought and sense-presentation. As\nhypothesis increases, necessity diminishes.\n\nCommonsense thought also supports this refusal to conceive of all\nspace as essentially depending on hypothetical objects which fill it.\nWe think of material objects as filling space, but we ask whether any\nobjects exist between the Earth and the Sun, between the stars, or\nbeyond the stars. For us, space is there; the only question is whether\nor not it be full. But this form of question presupposes the meaning of\nempty space, namely, of space not containing hypothetical objects.\n\nThis brings up a wider use of the concept of points, necessitating\na wider definition. Hitherto we have conceived points as indicating\nrelations of enclosure between objects. We thus arrive at what now\nwe will term \"material points.\" But the idea of points can now be\ntransformed so as to indicate the possibilities of external relations\nnot those of enclosure. This is effected by an enlargement of the\nconcept of ideal points, already known to geometers.\n\nDefine \"material lines\" to be complete collinear classes of collinear\npoints. Consider now the set of material lines which contain a certain\nmaterial point. Call such a set of lines an ideal point. This set of\nlines indicates a possibility of position, which is in fact occupied\nby that material point common to all the material lines. So this ideal\npoint is an occupied ideal point. Now consider a set of three material\nlines, such that any two are coplanar, but not the whole three, and\nfurther consider the complete set of material lines such that each is\ncoplanar with each of the three material lines first chosen. The axioms\nwhich hold for the material lines will enable us to prove that any two\nlines of this set are coplanar. Then the whole set of lines, including\nthe three original lines, forms an ideal point, according to the\ndefinition in its full generality. Such an ideal point may be occupied.\nIn that case there is a material point common to all the lines of the\nset, but it may be unoccupied. Then the ideal point merely indicates\na possibility of spatial relations which has not been realised. It is\nthe point of empty space. Thus the ideal points, which may or may not\nbe occupied, are the points of geometry viewed as an applied science.\nThese points are distributed into straight lines and planes. But any\nfurther discussion of this question will lead us into the technical\nsubject of the axioms of geometry and their immediate consequences.\nEnough has been said to show how geometry arises according to the\nrelational theory of space.\n\nSpace as thus conceived is the thought-space of the material world.\n\n\n                         _IV. Fields of Force_\n\nThe thought-objects of science are conceived as directly related to\nthis thought-space. Their spatial relations are among those indicated\nby the points of the thought-space. Their emergence in science has\nbeen merely a further development of processes already inherent in\ncommonsense thought.\n\nRelations within the complete sense-presentation were represented\nin thought by the concept of thought-objects of perception. All\nsense-presentation could not be represented in this way; also the\nchange and disappearance of thought-objects occasioned confusion\nof thought. A reduction to order of this confusion was attempted\nby the concepts of permanent matter with primary and secondary\nqualities. Finally, this has issued in the secondary qualities being\ntraced as perception of events generated by the objects, but--as\nperceived--entirely disconnected with them. Also the thought-objects\nof perception have been replaced by molecules and electrons and\nether-waves, until at length it is never the thought-object of science\nwhich is perceived, but complicated series of events in which they are\nimplicated. If science be right, nobody ever perceived a thing, but\nonly an event. The result is, that the older language of philosophy\nwhich still survives in many quarters is now thoroughly confusing\nwhen brought into connection with the modern concepts of science.\nPhilosophy--that is, the older philosophy--conceives the thing as\ndirectly perceived. According to scientific thought, the ultimate thing\nis never perceived, perception essentially issuing from a series of\nevents. It is impossible to reconcile the two points of view.\n\nThe advantage of the modern scientific concept is that it is enabled\nto \"explain\" the fluid vague outlines of sense-presentation. The\nthought-object of perception is now conceived as a fairly stable\nstate of motion of a huge group of molecules, constantly changing,\nbut preserving a certain identity of characteristics. Also stray\nsense-objects, not immediately given as part of a thought-object of\nperception, are now explicable: the dancing light-reflection, the\nvaguely heard sound, the smell. In fact, the perceived events of\nthe scientific world have the same general definition and lack of\ndefinition, and the same general stability and lack of stability,\nas the sense-objects of the complete sense-presentation or as the\nthought-objects of perception.\n\nThe thought-objects of science, namely, molecules, atoms, and\nelectrons, have gained in permanence. The events are reduced to changes\nin space-configuration. The laws determining these changes are the\nultimate laws of nature.\n\nThe laws of change in the physical universe proceed on the assumption\nthat the preceding states of the universe determine the character of\nthe change. Thus, to know the configurations and events of the universe\nup to and including any instant would involve sufficient data from\nwhich to determine the succession of events throughout all time.\n\nBut in tracing the antecedents of events, commonsense thought, dealing\nwith the world of thought-objects of perception, habitually assumes\nthat the greater number of antecedent events can be neglected as\nirrelevant. Consideration of causes is restricted to a few events\nduring a short preceding interval. Finally, in scientific thought it\nhas been assumed that the events in an arbitrarily small preceding\nduration are sufficient. Thus physical quantities and their successive\ndifferential co-efficients up to any order at the instant, but with\ntheir limiting values just before that instant, are on this theory\nsufficient to determine the state of the universe at all times after\nthe instant. More particular laws are assumed. But the search for\nthem is guided by this general principle. Also it is assumed that the\ngreater number of events in the physical universe are irrelevant to\nthe production of any particular effect, which is assumed to issue\nfrom relatively few antecedents. These assumptions have grown out of\nthe experience of mankind. The first lesson of life is to concentrate\nattention on few factors of sense-presentations, and on still fewer of\nthe universe of thought-objects of perception.\n\nThe principle by which--consciously or unconsciously--thought has been\nguided is that in searching for particular causes, remoteness in time\nand remoteness in space are evidences of comparative disconnection\nof influence. The extreme form of this principle is the denial of\nany action at a distance either in time or space. The difficulty in\naccepting this principle in its crude form is, that since there are no\ncontiguous points, only coincident bodies can act on each other. I can\nsee no answer to this difficulty--namely, either bodies have the same\nlocation and are thus coincident, or they have different locations and\nare thus at a distance and do not act on each other.\n\nThis difficulty is not evaded by the hypothesis of an ether,\ncontinuously distributed. For two reasons: in the first place, the\ncontinuity of the ether does not avoid the dilemma; and secondly, the\ndifficulty applies to time as well as to space, and the dilemma would\nprove that causation producing change is impossible, namely, no changed\ncondition could be the result of antecedent circumstances.\n\nOn the other hand, a direct interaction between two bodies separated\nin space undoubtedly offends the conception of distance as implying\nphysical disconnection as well as spatial relation. There is no logical\ndifficulty in the assumption of action at a distance as in the case\nof its denial, but it is contradictory to persistent assumptions of\nthat apparatus of commonsense thought which it is the main business\nof science to harmonise with sense-presentation, employing only the\nminimum of modification.\n\nModern science is really unconcerned with this debate. Its\n(unacknowledged) conceptions are really quite different, though the\nverbal explanations retain the form of a previous epoch. The point of\nthe change in conception is that the old thought-object of science was\nconceived as possessing a simplicity not belonging to the material\nuniverse as a whole. It was secluded within a finite region of space,\nand changes in its circumstances could only arise from forces which\nformed no essential part of its nature. An ether was called into\nexistence to explain the active relations between these passive\nthought-objects. The whole conception suffers from the logical\ndifficulties noted above. Also no clear conception can be formed of\nthe sense in which the ether is explanatory. It is to possess a type\nof activity denied to the original thought-object, namely, it carries\npotential energy, whereas the atom possessed only kinetic energy,\nthe so-called potential energy of an atom belonging really to the\nsurrounding ether. The truth is, that ether is really excepted from the\naxiom \"no action at a distance,\" and the axiom thereby is robbed of all\nits force.\n\nThe modern thought-object of science--not yet explicitly\nacknowledged--has the complexity of the whole material universe. In\nphysics, as elsewhere, the hopeless endeavour to derive complexity\nfrom simplicity has been tacitly abandoned. What is aimed at is not\nsimplicity, but persistence and regularity. In a sense regularity\nis a sort of simplicity. But it is the simplicity of stable mutual\nrelations, and not the simplicity of absence of types of internal\nstructure or of type of relation. This thought-object fills all space.\nIt is a \"field\"; that is to say, it is a certain distribution of\nscalar and vector quantities throughout space, these quantities having\neach its value for each point of space at each point of time, being\ncontinuously distributed throughout space and throughout time, possibly\nwith some exceptional discontinuities. The various types of quantity\nwhich form the field have fixed relations to each other at each point\nof time and space. These relations are the ultimate laws of nature.\n\nFor example, consider an electron. There is a scalar distribution of\nelectricity, which is what is ordinarily called the electron. This\nscalar distribution has a volume-density \u03c1 at the time _t_ at any\npoint (_x_, _y_, _z_). Thus \u03c1 is a function of (_x_, _y_, _z_, _t_),\nwhich is zero except within a restricted region. Furthermore, at\nany time _t_, as an essential adjunct, there is a continuous space\ndistribution at each point of the two vectors (_X_, _Y_, _Z_), which is\nthe electric force, and (\u03b1, \u03b2, \u03b3), which is the magnetic force. Lastly,\nindividuality is ascribed to the scalar electric distribution, so that\nin addition to its conservation of quantity--involved in the assumed\nlaws--it is also possible to assign the velocities with which the\nvarious individual parts of the distribution are moving. Let (_u_, _v_,\n_w_) be this velocity at (_x_, _y_, _z_, _t_).\n\nThis whole scheme of scalar and vector quantities, namely, \u03c1, (_X_,\n_Y_, _Z_), (\u03b1, \u03b2, \u03b3), (_u_, _v_, _w_) is interconnected by the\nelectromagnetic laws. It follows from these laws that the electron, in\nthe sense of the scalar distribution \u03c1, is to be conceived as at each\ninstant propagating from itself an emanation which travels outwards\nwith the velocity of light _in vacuo_, and from which (_X_, _Y_,\n_Z_) and (\u03b1, \u03b2, \u03b3) can be calculated, so far as they are due to that\nelectron. Thus the field, at any time, due to the electron as a whole\ndepends on the previous history of the electron, the nearer to the\nelectron the more recent being the relevant history. The whole scheme\nof such a field is one single thought-object of science: the electron\nand its emanations form one essential whole, namely one thought-object\nof science, essentially complex and essentially filling all space. The\nelectron proper, namely, the scalar distribution \u03c1, is the focus of the\nwhole, the essential focal property being that the field at any instant\nis completely determined by the previous history of the focus and of\nits space relations through all previous time. But the field and the\nfocus are not independent concepts, they are essentially correlated in\none organised unity, namely, they are essentially correlated terms in\nthe field of one relation in virtue of which the entities enter into\nour thoughts.\n\nThe fields of a group of electrons are superposed according to the\nlinear law for aggregation, namely, pure addition for analogous scalar\nquantities and the parallelogram law for analogous vectors. The changes\nin motion of each electron depend entirely on the resultant field in\nthe region it occupies. Thus a field can be viewed as a possibility of\naction, but a possibility which represents an actuality.\n\nIt is to be noted that the two alternative views of causation are\nhere both included. The complete field within any region of space\ndepends on the past histories of all the electrons, histories extending\nbackwards in proportion to their distances. Also this dependence can\nbe conceived as a transmission. But viewing the cause which effects\nchanges on the electron within that region, it is solely that field\nwithin the region, which field is coincident with that electron both in\ntime and in space.\n\nThis process of conceiving the actuality underlying a possibility is\nthe uniform process by which regularity and permanence is introduced\ninto scientific thought, namely, we proceed from the actuality of the\nfact to the actuality of possibility.\n\nIn conformity with this principle, propositions are the outgrowth\nfrom actual thought-expressions, thought-objects of perceptions from\ncrude sense-objects, hypothetical thought-objects of perception from\nactual thought-objects of perception, material points from hypothetical\ninfinite suites of hypothetical thought-objects of perception,\nideal points from material points, thought-objects of science from\nthought-objects of perception, fields of electrons from actual mutual\nreactions of actual electrons.\n\nThe process is a research for permanence, uniformity, and simplicity\nof logical relation. But it does not issue in simplicity of internal\nstructure. Each ultimate thought-object of science retains every\nquality attributed to the whole scientific universe, but retains them\nin a form characterised by permanence and uniformity.\n\n\n                            _V. Conclusion_\n\nWe commenced by excluding judgments of worth and ontological judgments.\nWe conclude by recalling them. Judgments of worth are no part of\nthe texture of physical science, but they are part of the motive of\nits production. Mankind have raised the edifice of science, because\nthey have judged it worth while. In other words, the motives involve\ninnumerable judgments of value. Again, there has been conscious\nselection of the parts of the scientific field to be cultivated, and\nthis conscious selection involves judgments of value. These values may\nbe \u00e6sthetic, or moral, or utilitarian, namely, judgments as to the\nbeauty of the structure, or as to the duty of exploring the truth, or\nas to utility in the satisfaction of physical wants. But whatever the\nmotive, without judgments of value there would have been no science.\n\nAgain, ontological judgments were not excluded by reason of any lack\nof interest. They are in fact presupposed in every act of life: in our\naffections, in our self-restraints, and in our constructive efforts.\nThey are presupposed in moral judgments. The difficulty about them is\nthe absence of agreement as to the method of harmonising the crude\njudgments of commonsense.\n\nScience does not diminish the need of a metaphysic. Where this need is\nmost insistent is in connection with what above has been termed \"the\nactuality underlying a possibility.\" A few words of explanation may\nrender the argument clearer, although they involve a rash approach\nto metaphysical heights which it is not the purpose of this paper to\nexplore.\n\nThe conception of subject and object in careless discussion covers\ntwo distinct relations. There is the relation of the whole perceiving\nconsciousness to part of its own content, for example, the relation\nof a perceiving consciousness to an object of redness apparent to it.\nThere is also the relation of a perceiving consciousness to an entity\nwhich does not exist in virtue of being part of the content of that\nconsciousness. Such a relation, so far as known to the perceiving\nconsciousness, must be an inferred relation, the inference being\nderived from an analysis of the content of the perceiving consciousness.\n\nThe bases for such inferences must be elements in consciousness\ndirectly known as transcending their immediate presentation in\nconsciousness. Such elements are universal logical truths, moral and\n\u00e6sthetic truths, and truths embodied in hypothetical propositions.\nThese are the immediate objects of perception which are other than the\nmere affections of the perceiving subject. They have the property of\nbeing parts of the immediate presentations for individual subjects and\nyet more than such parts. All other existence is inferred existence.\n\nIn this chapter we are more directly concerned with truths embodied\nin hypothetical propositions. Such truths must not be confused with\nany doubtfulness which attaches to our judgments of the future course\nof natural phenomena. A hypothetical proposition, like a categorical\njudgment, may or may not be doubtful. Also like a categorial judgment,\nit expresses a fact. This fact is twofold: as a presentation in\nconsciousness, it is just this hypothetical judgment; as expressing a\ncategorical fact, it states a relation which lies beyond consciousness,\nholding between entities thereby inferred.\n\nBut this metaphysical analysis, short though it be, is probably wrong,\nand at the best will only command very partial assent. Certainly;\nand this admission brings out the very point which I wished to make.\nPhysical science is based on elements of thought, such as judgments\nregistering actual perceptions, and judgments registering hypothetical\nperceptions which under certain circumstances would be realised.\nThese elements form the agreed content of the apparatus of commonsense\nthought. They require metaphysical analysis; but they are among the\ndata from which metaphysics starts. A metaphysic which rejects them\nhas failed, in the same way as physical science has failed when it is\nunable to harmonise them into its theory.\n\nScience only renders the metaphysical need more urgent. In itself\nit contributes little directly to the solution of the metaphysical\nproblem. But it does contribute something, namely, the exposition of\nthe fact that our experience of sensible apparent things is capable of\nbeing analysed into a scientific theory, a theory not indeed complete,\nbut giving every promise of indefinite expansion. This achievement\nemphasises the intimate relation between our logical thought and the\nfacts of sensible apprehension. Also the special form of scientific\ntheory is bound to have some influence. In the past false science\nhas been the parent of bad metaphysics. After all, science embodies\na rigorous scrutiny of one part of the whole evidence from which\nmetaphysicians deduce their conclusions.\n\n\nFOOTNOTES:\n\n[Footnote 3: Cf. _R\u00e9vue de M\u00e9taphysique et de Morale_, May 1916,\nwhere this question is dealt with by the author at the end of an\narticle, \"La th\u00e9orie relationniste de l'espace.\"]\n\n\n\n\n                             CHAPTER VIII\n\n                      SPACE, TIME, AND RELATIVITY\n\n  (_Paper read to Section A at the Manchester Meeting of the British\n    Association, 1915, and later before the Aristotelian Society_)\n\n\nFUNDAMENTAL Problems concerning space and time have been considered\nfrom the standpoints created by many different sciences. The object of\nthis paper is the humble one of bringing some of these standpoints into\nrelation with each other. This necessitates a very cursory treatment of\neach point of view.\n\nMathematical physicists have evolved their theory of relativity to\nexplain the negative results of the Morley-Michelson experiment and of\nthe Trouton experiment. Experimental psychologists have considered the\nevolution of spatial ideas from the crude sense-data of experience.\nMetaphysicians have considered the majestic uniformity of space and\ntime, without beginning and without end, without boundaries, and\nwithout exception in the truths concerning them; all these qualities\nthe more arresting to our attention from the confused accidental\nnature of the empirical universe which is conditioned by them.\nMathematicians have studied the axioms of geometry, and can now deduce\nall that is believed to be universally true of space and of time by the\nstrictest logic from a limited number of assumptions.\n\nThese various lines of thought have been evolved with surprisingly\nlittle interconnection. Perhaps it is as well. The results of science\nare never quite true. By a healthy independence of thought perhaps we\nsometimes avoid adding other people's errors to our own. But there can\nbe no doubt that the normal method of cross-fertilising thought is by\nconsidering the same, or allied problems to our own, in the form which\nthey assume in other sciences.\n\nHere I do not propose to enter into a systematic study of these various\nchapters of science. I have neither the knowledge nor the time.\n\nFirst, let us take the ultimate basis of any theory of relativity.\nAll space measurement is from stuff in space to stuff in space. The\ngeometrical entities of empty space never appear. The only geometrical\nproperties of which we have any direct knowledge are properties of\nthose shifting, changeable appearances which we call things in space.\nIt is the sun which is distant, and the ball which is round, and the\nlamp-posts which are in linear order. Wherever mankind may have got its\nidea of an infinite unchangeable space from, it is safe to say that it\nis not an immediate deliverance of direct observation.\n\nThere are two antagonistic philosophical ways of recognising this\nconclusion.\n\nOne is to affirm that space and time are conditions for sensible\nexperience, that without projection into space and time sensible\nexperience would not exist. Thus, although it may be true to say that\nour knowledge of space and time is given in experience, it is not true\nto say that it is deduced from experience in the same sense that the\nLaw of Gravitation is so deduced. It is not deduced, because in the act\nof experiencing we are necessarily made aware of space as an infinite\ngiven whole, and of time as an unending uniform succession. This\nphilosophical position is expressed by saying that space and time are\n_a priori_ forms of sensibility.\n\nThe opposed philosophical method of dealing with the question is\nto affirm that our concepts of time and space are deductions from\nexperience, in exactly the same way as the Law of Gravitation is such\na deduction. If we form exact concepts of points, lines and surfaces,\nand of successive instants of time, and assume them to be related as\nexpressed by the axioms of geometry and the axioms for time, then we\nfind that we have framed a concept which, with all the exactness of\nwhich our observations are capable, expresses the facts of experience.\n\nThese two philosophic positions are each designed to explain a\ncertain difficulty. The _a priori_ theory explains the absolute\nuniversality ascribed to the laws of space and time, a universality\nnot ascribed to any deduction from experience. The experiential theory\nexplains the derivation of the space-time concepts without introducing\nany other factors beyond those which are admittedly present in framing\nthe other concepts of physical science.\n\nBut we have not yet done with the distinctions which in any discussion\nof space or time must essentially be kept in mind. Put aside the\nabove question as to how these space-time concepts are related to\nexperience--What are they when they are formed?\n\nWe may conceive of the points of space as self-subsistent entities\nwhich have the indefinable relation of being occupied by the ultimate\nstuff (matter, I will call it) which is there. Thus, to say that the\nsun is _there_ (wherever it is) is to affirm the relation of\noccupation between the set of positive and negative electrons which\nwe call the sun and a certain set of points, the points having an\nexistence essentially independent of the sun. This is the absolute\ntheory of space. The absolute theory is not popular just now, but it\nhas very respectable authority on its side--Newton, for one--so treat\nit tenderly.\n\nThe other theory is associated with Leibniz. Our space concepts are\nconcepts of relations between things in space. Thus there is no such\nentity as a self-subsistent point. A point is merely the name for some\npeculiarity of the relations between the matter which is, in common\nlanguage, said to be in space.\n\nIt follows from the relative theory that a point should be definable in\nterms of the relations between material things. So far as I am aware,\nthis outcome of the theory has escaped the notice of mathematicians,\nwho have invariably assumed the point as the ultimate starting ground\nof their reasoning. Many years ago I explained some types of ways in\nwhich we might achieve such a definition, and more recently have added\nsome others. Similar explanations apply to time. Before the theories\nof space and time have been carried to a satisfactory conclusion on\nthe relational basis, a long and careful scrutiny of the definitions\nof points of space and instants of time will have to be undertaken,\nand many ways of effecting these definitions will have to be tried and\ncompared. This is an unwritten chapter of mathematics, in much the same\nstate as was the theory of parallels in the eighteenth century.\n\nIn this connection I should like to draw attention to the analogy\nbetween time and space. In analysing our experience we distinguish\nevents, and we also distinguish things whose changing relations form\nthe events. If I had time it would be interesting to consider more\nclosely these concepts of events and of things. It must suffice now to\npoint out that things have certain relations to each other which we\nconsider as relations between the space extensions of the things; for\nexample, one space can contain the other, or exclude it, or overlap\nit. A point in space is nothing else than a certain set of relations\nbetween spatial extensions.\n\nAnalogously, there are certain relations between events which we\nexpress by saying that they are relations between the temporal\ndurations of these events, that is, between the temporal extensions of\nthe events. [The durations of two events A and B may one precede the\nother, or may partially overlap, or may one contain the other, giving\nin all six possibilities.] The properties of the extension of an event\nin time are largely analogous to the extension of an object in space.\nSpatial extensions are expressed by relations between objects, temporal\nextensions by relations between events.\n\nThe point in time is a set of relations between temporal extensions.\nIt needs very little reflection to convince us that a point in time\nis no direct deliverance of experience. We live in durations, and not\nin points. But what community, beyond the mere name, is there between\nextension in time and extension in space? In view of the intimate\nconnection between time and space revealed by the modern theory of\nrelativity, this question has taken on a new importance.\n\nI have not thought out an answer to this question. I suggest, however,\nthat time and space embody those relations between objects on which\ndepends our judgment of their externality to ourselves. Namely,\nlocation in space and location in time both embody and perhaps\nnecessitate a judgment of externality. This suggestion is very vague,\nand I must leave it in this crude form.\n\n\n                  _Diverse Euclidean Measure Systems_\n\nTurning now to the mathematical investigations on the axioms of\ngeometry, the outcome, which is most important for us to remember,\nis the great separation which it discloses between non-metrical\nprojective geometry, and metrical geometry. Non-metrical projective\ngeometry is by far the more fundamental. Starting with the concepts\nof points, straight lines, and planes (of which not all three need\nbe taken as indefinable), and with certain very simple non-metrical\nproperties of these entities--such as, for instance, that two points\nuniquely determine a straight line--nearly the whole of geometry can\nbe constructed. Even quantitative coordinates can be introduced, to\nfacilitate the reasoning. But no mention of distance, area, or volume,\nneed have been introduced. Points will have an order on the line, but\norder does not imply any settled distance.\n\nWhen we now inquire what measurements of distance are possible, we find\nthat there are different systems of measurement all equally possible.\nThere are three main types of system: any system of one type gives\nEuclidean geometry, any system of another type gives Hyperbolic (or\nLobatchewskian) geometry, any system of the third type gives Elliptic\ngeometry. Also different beings, or the same being if he chooses,\nmay reckon in different systems of the same type, or in systems of\ndifferent types. Consider the example which will interest us later.\nTwo beings, A and B, agree to use the same three intersecting lines\nas axes of _x_, _y_, _z_. They both employ a system of measurement of\nthe Euclidean type, and (what is not necessarily the case) agree as to\nthe plane at infinity. That is, they agree as to the lines which are\nparallel. Then with the usual method of rectangular Cartesian axes,\nthey agree that the coordinates of P are the lengths ON, NM, MP. So\nfar all is harmony. A fixes on the segment OU_{1}, on O_x_, as being\nthe unit length, and B on the segment OV_{1}, on O_x_. A calls his\ncoordinates (_x_, _y_, _z_), and B calls them (X, Y, Z).\n\nThen it is found [since both systems are Euclidean] that, whatever\npoint P be taken,\n\nX = \u03b2_x_, Y = \u03b3_y_, Z = \u03b4_z_. [\u03b2 \u2260 \u03b3 \u2260 \u03b4.]\n\nThey proceed to adjust their differences, and first take the\n_x_-coordinates. Obviously they have taken different units of\nlength along O_x_. The length OU_{1}, which A calls one unit, B\ncalls \u03b2 units. B changes his unit length to OU_{1}, from its original\nlength OV_{1}, and obtains X = _x_. But now, as he must use the\nsame unit for all his measurements, his other coordinates are altered\nin the same ratio. Thus we now have\n\nX = _x_, Y = \u03b3_y_/\u03b2, Z = \u03b4_z_/\u03b2.\n\nThe fundamental divergence is now evident. A and B agree as to their\nunits along O_x_. They settled that by taking along that axis a\ngiven segment OU_{1} as having the unit length. But they cannot agree\nas to what segment along O_y_ is equal to OU_{1}. A says it is\nOU_{2}, and B that it is OU_{2}\u2032. Similarly for lengths along OZ.\n\nThe result is that A's spheres\n\n_x_^2 + _y_^2 + _z_^2 = _r_^2,\n\nare B's ellipsoids,\n\n         X^2 + \u03b2^2Y^2/\u03b3^2 + \u03b2^2Z^2/\u03b4^2 = _r_^2,\n_i. e._  X^2/\u03b2^2 + Y^2/\u03b3^2 + Z^2/\u03b4^2 = _r_^2/\u03b2^2.\n\nThus the measurement of angles by the two is hopelessly at variance.\n\nIf \u03b2 \u2260 \u03b3 \u2260 \u03b4, there is one, and only one, set of common rectangular\naxes at O, namely that from which they started. If \u03b3 = \u03b4, but \u03b2 \u2260 \u03b3,\nthen there are a singly infinite number of common rectangular axes\nfound by rotating the axes round O_x_. This is, for us, the\ninteresting case. The same phenomena are reproduced by transferring to\nany parallel axes.\n\nThe root of the difficulty is, that A's measuring rod, which for him is\na rigid invariable body, appears to B as changing in length when turned\nin different directions. Similarly all measuring rods, satisfactory\nto A, violate B's immediate judgment of invariability, and change\naccording to the same law. There is no way out of the difficulty.\nTwo rods \u03c1 and \u03c3 coincide whenever laid one on the other; \u03c1 is held\nstill, and both men agree that it does not change. But \u03c3 is turned\nround. A says it is invariable, B says it changes. To test the matter,\n\u03c1 is turned round to measure it, and exactly fits it. But while A is\nsatisfied, B declares that \u03c1 has changed in exactly the same way as did\n\u03c3. Meanwhile B has procured two material rods satisfactory to him as\ninvariable, and A makes exactly the same objections.\n\nWe shall say that A and B employ diverse Euclidean metrical systems.\n\nThe most extraordinary fact of human life is that all beings seem to\nform their judgments of spatial quantity according to the same metrical\nsystem.\n\n\n                    _Relativity in Modern Physics_\n\nOwing to the fact that points of space are incapable of direct\nrecognition, there is a difficulty--apart from any abstract question\nof the nature of space--in deciding on the motion to be ascribed to\nany body. Even if there be such a thing as absolute position, it is\nimpossible in practice to decide directly whether a body's absolute\nposition has changed. All spatial measurement is relative to matter.\n\nNewton's laws of motion in their modern dress evade this difficulty\nby asserting that a framework of axes of coordinates can be defined\nby their relations to matter such that, assuming these axes to be at\nrest, and all velocities to be measured relatively to them, the laws\nhold. The same expedient has to be employed for time, namely, the laws\nhold when the measurement of the flow of time is made by the proper\nreference to periodic events. Thus the laws assert that the framework\nand the natural clock adapted for their use have been successfully\nfound.\n\nBut, if one framework will do, an infinity of others serve equally\nwell; namely, not only--as is of course the case--all those at rest\nrelatively to the first framework, but also all those which move\nwithout relative rotation with uniform velocity relatively to the\nfirst. This whole set of frameworks is on a level in respect to\nNewton's laws. We will call them Dynamical frameworks.\n\nNow, suppose there are two observers, A and B. They agree in their\nnon-metrical projective geometry, _e.g._, what A calls a straight\nline so does B. They also both apply a Euclidean metrical system of\nmeasurement to this space. Their two metrical systems also agree in\nhaving the same plane at infinity, that is, lines which are parallel\nfor A are also parallel for B. Furthermore, they have both successfully\napplied Newton's laws to the movement of matter, and agree in having\nthe same sets of dynamical axes. But the framework (among these sets)\nwhich A chooses to regard as at rest is different from the frame (among\nthe same sets) which B so regards.\n\nWithout alteration of their respective judgment of rest, they choose\ntheir co-ordinate axes so that the origins (O for A, and O\u2032 for B) are\nin relative motion along OO\u2032, which is the axis of _x_ for both.\n\nFurther, since OO\u2032 is the line of symmetry of their diverse Euclidean\nsystems, we assume that the two measure-systems agree for planes\nperpendicular to OO\u2032, _i.e._, we assume a symmetry round OO\u2032.\nThen if, for A at O, the distance OO\u2032 be \u03be, the relations at any\ninstant between A's coordinates (_x_, _y_, _z_) and B's\ncoordinates (_x\u2032_, _y\u2032_, _z\u2032_) for the same point P are\ngiven by\n\n_x\u2032_ = \u03b2(_x_ - \u03be), _y\u2032_ = _y_, _z\u2032_ = _z_.\n\nAlso, according to A's clock, O\u2032 is moving forward with a uniform\nvelocity _v_, and we measure A's time from the instant of the\ncoincidence of O and O\u2032.\n\nThus\n\n\u03be = _vt_,\n\nand\n\n_x\u2032_ = \u03b2(_x_ - _vt_), _y\u2032_ = _y_, _z\u2032_ = _z_.\n\nWe now consider B's clock, and ask for the most general supposition\nwhich is consistent with the fact that their judgments as to the fact\nof uniform motion are in agreement.\n\nWe do not assume that events in various parts of space which A\nconsiders to be simultaneous are so considered by B. But we assume that\nat any point P, with coordinates (_x_, _y_, _z_) for A, there is a\ndeterminate relation between B's time T and _x_, _y_, _z_, _t_.\n\nPut\n\nT = \u0192(_x_, _y_, _z_, _t_).\n\nWrite\n\nP = \u03b4T/\u03b4_x_, Q = \u03b4T/\u03b4_y_, R = \u03b4T/\u03b4_z_, S = \u03b4T/\u03b4_t_.\n\nNow suppose that the point P is moving, and that (_u_{1},\n_u_{2}, _u_{3}) is its set of component velocities along\nthe axes according to A's \"space and clock\" system, and (U_{1}, U_{2},\nU_{3}) is its set of component velocities according to B's \"space and\nclock\" system. Then by mere differentiation it follows after a short\nmathematical deduction that\n\nU_{1} = {(_d_\u03b2/_dt_)(_x_ - _vt_) + \u03b2(_u_{1} - _v_)}/{P_u_{1} + Q_u_{2} + R_u_{3} + S},\nU_{2} = _u_{2}/{P_u_{1} + Q_u_{2} + R_u_{3} + S},\nU_{3} = _u_{3}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}.\n\nBut we have assumed that, whatever the direction of the resultant\nvelocity (_u_{1}, _u_{2}, _u_{3}), the velocities (U_{1}, U_{2}, U_{3})\nand (_u_{1}, _u_{2}, _u_{3}) are both uniform when either is uniform.\n\nHence it is easily proved that \u03b2, P, Q, R, S are independent of the\ncoordinates (_x_, _y_, _z_) and of the time _t_. In other words, they\nare constant.\n\nHence we obtain\n\nU_{1} = \u03b2(_u_{1} - _v_)/{P_u_{1} + Q_u_{2} + R_u_{3} + S},\n\nand\n\nT = P_x_ + Q_y_ + R_z_ + S_t_.\n\nBut we assumed that OO\u2032, _i.e._, O_x_, is an axis of\nsymmetry. It follows from this assumption that\n\nQ = R = 0.\n\nWe thus obtain the simplified results\n\nT = P_x_ + S_t_,                       }\nU_{1} = \u03b2(_u_{1} - _v_)/(P_u_{1} + S), }   (I)\nU_{2} = _u_{2}/(P_u_{1} + S),          }\nU_{3} = _u_{3}/(P_u_{1} + S).          }\n\nHere we remember that (_u_{1}, _u_{2}, _u_{3}) are\nthe velocities of any particle according to A's \"space and clock\"\nsystem, and that (U_{1}, U_{2}, U_{3}) are the velocities of the same\npoint according to B's \"space and clock\" system. We have obtained the\nmost general relations consistent with the facts that (1) they both\nemploy Euclidean systems, related as described above, and (2) they\nagree in their judgments on the uniformity of velocity.\n\nWe now compare their judgments on the magnitudes of velocities.\n\nLet the magnitude of the velocity of P be V according to A's judgment,\nand V\u2032 according to B's\u2019 judgment.\n\nThen\n\n V^2 = _u_{1}^2 + _u_{2}^2 + _u_{3}^2,\nV\u2032^2 = U_{1}^2 + U_{2}^2 + U_{3}^2.\n\nAlso we can put\n\n_u_{1} = _l_V, _u_{2} = _m_V, _u_{3} = _n_V,\n\nwhere (_l_, _m_, _n_) have nothing to do with the magnitude V, but\nsimply depend on the direction of motion. In fact (_l_, _m_, _n_) are\nthe \"direction cosines\" of the velocity according to A's judgment. By\nsubstituting in the above equation for V^2 we see that\n\n_l_^2 + _m_^2 + _n_^2 = 1.\n\nNow, substituting for (_u_{1}, _u_{2}, _u_{3}) in\nthe equations (I) above, and squaring and adding, and eliminating\n_m_^2 + _n_^2 by the relation just found, we at once find\n\nV\u2032^2 = ((\u03b2^2 - 1)V^2_l_^2 - 2\u03b2^2V_vl_ + \u03b2^2_v_^2 + V^2)/(PV_l_ + S)^2.\n\nIt is thus seen that in general the relation of V\u2032 to V depends on the\ndirection cosine _l_. Now _l_ is the cosine of the angle\nwhich the direction of the velocity V makes with O_x_, according\nto A's judgment.\n\nThe meaning of this relation is, that if A discharges, from guns at\nthe point P, shells with a given muzzle velocity V according to his\njudgment, B will consider that their muzzle velocities are different\nfrom each other, except in the case of pairs of guns equally inclined\nto the axis OO\u2032. Instances of this type of diversity of judgment can\nbe noted any day by any one who looks out of the window of a railway\ncarriage, and forgets that he is travelling.\n\nNow, suppose the velocity V\u2032 bears a relation to the velocity V, which\nis independent of _l_. Then _l_ must disappear from the above\nformula. There are two conditions to be satisfied\n\nOne condition is\n\nV^2 = \u03b2^2_v_^2/(\u03b2^2 - 1),\n\nor in a more convenient form\n\n\u03b2^2 = 1/(1 - _v_^2/V^2).\n\nThe meaning of this condition is, that there is one, and only one,\nmuzzle velocity V (according to A's judgment), namely, the muzzle\nvelocity given by the above formula, which can have the property that\nB will judge that all the guns are firing in their diverse directions\nwith one common muzzle velocity.\n\nLet us now suppose that V has this peculiar value: that is, if we look\non this value V as known, we must suppose that \u03b2 is given by the second\nof the above formul\u00e6.\n\nThe other condition allows P and S to be put in the forms\n\nP = -\u03b2_v_/\u03bbV^2, S = \u03b2/\u03bb,\n\nwhere\n\nV\u2032 = \u03bbV.\n\nThus we have the bundle of formul\u00e6\n\n\u03b2^2 = 1/(1 - _v_^2/V^2),\nT = \u03b2{_t_ - _vx_/V^2}/\u03bb,\nV\u2032 = \u03bbV.\n\nThe value which we give to \u03bb is purely a matter for the adjustment of\nunits. If we want A and B to agree in their judgments of the magnitude\nof this peculiar muzzle-velocity, we put \u03bb = 1.\n\nWe then get the formul\u00e6 usually adopted, namely\n\n\u03b2^2 = 1/(1 - _v_^2/V^2),  }\nT = \u03b2{_t_ - _vx_/V^2},    }   (II)\nV\u2032 = V.                   }\n\nBut if we prefer that A and B should reckon (according to A's judgment)\nin the same units of time, we put \u03bb = \u03b2, and obtain\n\n\u03b2^2 = 1/(1 - _v_^2/V^2),\nT = (_t_ - _vx_)/V^2,\nV\u2032 = \u03b2V.\n\nBut A and B are in any case in such hopeless difficulties over their\ncomparisons of time-judgments that the detail of using the same units\ndoes not help them much. Accordingly the formul\u00e6 marked (II) are those\nused. Thus A and B agree in their judgments as to the magnitude of one\nspecial velocity V, whatever may be the direction in which the entity\npossessing it is moving.\n\nIn order to reach this measure of agreement, they have to disagree as\nto their space judgments and their time judgments. The root cause of\ntheir disagreement is their diverse judgment as to which axis system is\nto be taken at rest for the purpose of measuring velocities.\n\nBefore discussing the nature of the disagreement disclosed in formula\n(II), let us ask why we should bring these difficulties on our heads by\nsupposing that two people in relative motion, who both (for the purpose\nof measuring velocities) assume that they are at rest, should agree in\ntheir judgments in respect to this special velocity V.\n\nSuch an agreement has no counterpart in any of our obvious judgments\nmade from railway carriages. Surely we can wait till the contingency\noccurs before discussing the confusion which it creates.\n\nBut the contingency has occurred. It occurs when we consider the\nvelocity of light. Perhaps I may venture to remind a philosophical\nsociety that light moves so very quickly that it is difficult to\nconsider its velocity at all. So we need not be surprised that this\npeculiar fact concerning its velocity is not more obvious.\n\nNow V being the velocity of light, unless _v_ is large, _v_/V\n(and still more _v_^2/V^2) will be quite inappreciable. The only\nvelocity ready to hand which is big enough to give _v_/V an\nappreciable value is the velocity of the earth in its orbit.\n\nMany diverse experiments have been made, and they all agree in\nconcluding that a man who assumes the earth to be at rest will find by\nmeasurement that the velocity of light is the same in all directions.\nFurthermore, when the same man turns his attention to interstellar or\ninterplanetary phenomena, and assumes the sun to be at rest, he will\nagain find the velocity of light to be the same in all directions.\nThese are well-attested experiments made at long intervals of time.\n\nThis is the exact contingency contemplated above.\n\nAgain the velocity of light _in vacuo_ has recently taken on a\nnew dignity. It used to be one among other wave velocities such as\nthe velocity of sound in air, or in water, or the velocity of surface\nwaves in water. But Clerk Maxwell discovered that all electromagnetic\ninfluences are propagated with the velocity of light, and now modern\nphysical science half suspects that electromagnetic influences are\nthe only physical influences which relate the changes in the physical\nworld. Accordingly the velocity of light becomes the fundamental\nnatural velocity, and experiment shows that our judgment of its\nmagnitude is not affected by our choice of the framework at rest, so\nlong as we keep to a set of dynamical axes. These experiments on light\nhave been confirmed by other electromagnetic experiments not involving\nlight.\n\nThus we are driven to equations (II), where V is the velocity of light.\n\nThe first conclusion to be drawn from equations (II) is that two people\nwho make different choices of bodies at rest will disagree as to\ntheir measuring rods in the way described above. There is no peculiar\ndifficulty about that. The only wonder is that all people agree so well\nin their judgments as to metrical systems. A mathematical angel would\nnaturally expect incarnate men to be in violent disagreement on this\nsubject.\n\nBut the case of time is different. For simplicity of statement we speak\nof A as at O, and B as at O\u2032. We remember that O\u2032 is moving relatively\nto O with velocity _v_ in direction OO\u2032. Suppose A and B are\nlooking in this direction; and they both measure their time from the\ninstant when they met, as O\u2032 passed over O. Then we have\n\nT = ((_t_ - _vx_)/V^2)/((1 - _v_^2)/V^2).\n\nNow, suppose we consider all the events all over space which A\nconsiders to have happened simultaneously at the time _t_. The\nevents of this set which occurred anywhere on a plane perpendicular to\nOO\u2032 at a distance _x_ in front of O (according to A's reckoning),\nwill have occurred according to B's reckoning at the time T as given\nabove. Let us fix our attention on the fact that B does not consider\nall these events to be simultaneous. For let T_{1}, and T_{2} be B's\ntimes for such events on planes _x__{1}, and _x__{2}. Then\n\nT_{1} - T_{2} = _v_(_x__{2} - _x__{1})/(V^2 - _v_^2).\n\nThus if _x__{2} be greater than _x__{1}, T_{2} is less than\nT_{1}. Thus B judges the more distant events in front of him to have\nhappened earlier than the nearer events in front of him, and _vice\nversa_ for the events behind. This disturbance of the judgment of\nsimultaneity is the fundamental fact. Obviously the measurement of time\nintervals is a detail compared to simultaneity. A may think a sermon\nlong, and B may think it short, but at least they should both agree\nthat it stopped when the clock hand pointed at the hour. The worst of\nthe matter is that so far as any test can be applied there is no method\nof discriminating between the validities of their judgments.\n\nThus we are confronted with two distinct concepts of the common world,\nA's space-time concept, and B's space-time concept. Who is right? It is\nno use staying for an answer. We must follow the example of the wise\nold Roman, and pass on to other things.\n\nThus estimates of quantity in space and time, and, to some extent, even\nestimates of order, depend on the individual observer. But what are\nthe crude deliverances of sensible experience, apart from that world\nof imaginative reconstruction which for each of us has the best claim\nto be called our real world? Here the experimental psychologist steps\nin. We cannot get away from him. I wish we could, for he is frightfully\ndifficult to understand. Also, sometimes his knowledge of the\nprinciples of mathematics is rather weak, and I sometimes suspect----\nNo, I will not say what I sometimes think: probably he, with equal\nreason, is thinking the same sort of thing of us.\n\nI will, however, venture to summarise conclusions, which are, I\nbelieve, in harmony with the experimental evidence, both physical and\npsychological, and which are certainly suggested by the materials for\nthat unwritten chapter in mathematical logic which I have already\ncommended to your notice. The concepts of space and time and of\nquantity are capable of analysis into bundles of simpler concepts. In\nany given sensible experience it is not necessary, or even usual, that\nthe whole complete bundle of such concepts apply. For example, the\nconcept of externality may apply without that of linear order, and the\nconcept of linear order may apply without that of linear distance.\n\nAgain, the abstract mathematical concept of a space-relation\nmay confuse together distinct concepts which apply to the given\nperceptions. For example, linear order in the sense of a linear\nprojection from the observer is distinct from linear order in the sense\nof a row of objects stretching across the line of sight.\n\nMathematical physics assumes a given world of definitely related\nobjects, and the various space-time systems are alternative ways of\nexpressing those relations as concepts in a form which also applies to\nthe immediate experience of observers.\n\nYet there must be one way of expressing the relations between objects\nin a common external world. Alternative methods can only arise as the\nresult of alternative standpoints; that is to say, as the result of\nleaving something added by the observer sticking (as it were) in the\nuniverse.\n\nBut this way of conceiving the world of physical science, as composed\nof hypothetical objects, leaves it as a mere fairy tale. What is really\nactual are the immediate experiences. The task of deductive science\nis to consider the concepts which apply to these data of experience,\nand then to consider the concepts relating to these concepts, and\nso on to any necessary degree of refinement. As our concepts become\nmore abstract, their logical relations become more general, and less\nliable to exception. By this logical construction we finally arrive\nat conceptions, (i) which have determinate exemplifications in the\nexperience of the individuals, and (ii) whose logical relations have\na peculiar smoothness. For example, conceptions of mathematical time,\nof mathematical space, are such smooth conceptions. No one lives in\n\"an infinite given whole,\" but in a set of fragmentary experiences.\nThe problem is to exhibit the concepts of mathematical space and time\nas the necessary outcome of these fragments by a process of logical\nbuilding up. Similarly for the other physical concepts. This process\nbuilds a common world of conceptions out of fragmentary worlds of\nexperience. The material pyramids of Egypt are a conception, what is\nactual are the fragmentary experiences of the races who have gazed on\nthem.\n\nSo far as science seeks to rid itself of hypothesis, it cannot go\nbeyond these general logical constructions. For science, as thus\nconceived, the divergent time orders considered above present no\ndifficulty. The different time systems simply register the different\nrelations of the mathematical construct to those individual\nexperiences (actual or hypothetical) which could exist as the crude\nmaterial from which the construct is elaborated.\n\nBut after all it should be possible so to elaborate the mathematical\nconstruct so as to eliminate specific reference to particular\nexperiences. Whatever be the data of experience, there must be\nsomething which can be said of them as a whole, and that something is\na statement of the general properties of the common world. It is hard\nto believe that with proper generalisation time and space will not be\nfound among such properties.\n\n\n    _Commentary added on reading the Paper before the Aristotelian\n                               Society_\n\nThe first six pages of the paper consist of a summary of ideas which\nought to be in our minds while considering problems of time and space.\nThe ideas are mostly philosophical, and the summary has been made by\nan amateur in that science; so there is no reason to ascribe to it any\nimportance except that of a modest reminder. There are only two points\nin this summary to which I would draw attention.\n\nOn pp. 192 and 193 there occurs--\n\n\"Wherever mankind ... unending uniform succession.\"\n\nIf I understand Kant rightly--which I admit to be very\nproblematical--he holds that in the act of experience we are aware\nof space and time as ingredients necessary for the occurrence of\nexperience. I would suggest--rather timidly--that this doctrine should\nbe given a different twist, which in fact turns it in the opposite\ndirection--namely, that in the act of experience we perceive a whole\nformed of related differentiated parts. The relations between these\nparts possess certain characteristics, and time and space are the\nexpressions of some of the characteristics of these relations. Then the\ngenerality and uniformity which are ascribed to time and space express\nwhat may be termed the uniformity of the texture of experience.\n\nThe success of mankind--modest though it is--in deducing uniform laws\nof nature is, so far as it goes, a testimony that this uniformity of\ntexture goes beyond those characteristics of the data of experience\nwhich are expressed as time and space. Time and space are necessary\nto experience in the sense that they are characteristics of our\nexperience; and, of course, no one can have our experience without\nrunning into them. I cannot see that Kant's deduction amounts to much\nmore than saying that \"what is, is\"--true enough, but not very helpful.\n\nBut I admit that what I have termed the \"uniformity of the texture of\nexperience\" is a most curious and arresting fact. I am quite ready to\nbelieve that it is a mere illusion; and later on in the paper I suggest\nthat this uniformity does not belong to the immediate relations of the\ncrude data of experience, but is the result of substituting for them\nmore refined logical entities, such as relations between relations, or\nclasses of relations, or classes of classes of relations. By this means\nit can be demonstrated--I think--that the uniformity which must be\nascribed to experience is of a much more abstract attenuated character\nthan is usually allowed. This process of lifting the uniform time and\nspace of the physical world into the status of logical abstractions has\nalso the advantage of recognising another fact, namely, the extremely\nfragmentary nature of all direct individual experience.\n\nMy point in this respect is that fragmentary individual experiences\nare all that we know, and that all speculation must start from these\n_disjecta membra_ as its sole datum. It is not true that we are\ndirectly aware of a smooth running world, which in our speculations we\nare to conceive as given. In my view the creation of the world is the\nfirst unconscious act of speculative thought; and the first task of a\nself-conscious philosophy is to explain how it has been done.\n\nThere are roughly two rival explanations. One is to assert the world\nas a postulate. The other way is to obtain it as a deduction, not a\ndeduction through a chain of reasoning, but a deduction through a chain\nof definitions which, in fact, lifts thought on to a more abstract\nlevel in which the logical ideas are more complex, and their relations\nare more universal. In this way the broken limited experiences sustain\nthat connected infinite world in which in our thoughts we live. There\nare three more remarks while on this point I wish to make--\n\n(i) The fact that immediate experience is capable of this deductive\nsuperstructive must mean that it itself has a certain uniformity of\ntexture. So this great fact still remains.\n\n(ii) I do not wish to deny the world as a postulate. Speaking without\nprejudice, I do not see how in our present elementary state of\nphilosophical advance we can get on without middle axioms, which, in\nfact, we habitually assume.\n\nMy position is, that by careful scrutiny we should extrude such\npostulates from every part of our organised knowledge in which it is\npossible to do without them.\n\nNow, physical science organises our knowledge of the relations between\nthe deliverances of our various senses. I hold that in this department\nof knowledge such postulates, though not entirely to be extruded, can\nbe reduced to a minimum in the way which I have described.\n\nI have not the slightest knowledge of theories respecting our emotions,\naffections, and moral sentiments, and I can well believe that in\ndealing with them further postulates are required. And in practice I\nrecognise that we all make such postulates, uncritically.\n\n(iii) The next paragraphs on pp. 193 and 194 are as follows--\n\n\"The opposed philosophical method ... physical science.\"\n\nIt will be noted that, in the light of what has just been stated, the\nfirst of these paragraphs (which, I hope, faithfully expresses the\nexperimental way of approaching the problem) really obscures the point\nwhich I have been endeavouring to make. The phrase, \"If we form the\nexact concepts of points, etc.,\" is fatally ambiguous as between the\nmethod of postulating entities with assigned relations, and the method\nof forming logical constructions, and thus reaching points, etc., as\nthe result of a chain of definitions.\n\nTurning now to pp. 194-195, we come to the following paragraphs--\n\n\"The other theory ... eighteenth century.\"\n\nWe note again that the relational theory of space from another point\nof view brings us back to the idea of the fundamental space-entities\nas being logical constructs from the relations between things. The\ndifference is, that this paragraph is written from a more developed\npoint of view, as it implicitly assumes the things in space, and\nconceives space as an expression of certain of their relations.\nCombining this paragraph with what has gone before, we see that the\nsuggested procedure is first to define \"things\" in terms of the data of\nexperience, and then to define space in terms of the relations between\nthings.\n\nThis procedure is explicitly assumed in the next short paragraph: \"In\nthis connection ... from the events.\"\n\nThe gist of the remaining paragraphs of this section is contained in\nthe paragraph at the bottom of p. 196: \"The point in time ... new\nimportance.\"\n\nThe sentence, \"We live in durations, and not in points,\" can be\namplified by the addition, \"We live in space-extensions and not in\nspace-points.\"\n\nIt must be noted that \"whole and part\" as applied to extensions in\nspace or time must be different from the \"all and some\" of logic,\nunless we admit points to be the fundamental entities. For \"spatial\nwhole and spatial part\" can only mean \"all and some\" if they really\nmean \"all the points and some of the points.\" But if extensions and\ntheir relations are more fundamental than points, this interpretation\nis precluded. I suggest that \"spatial whole and spatial part\" is\nintimately connected with the fundamental relation between things from\nwhich our space ideas spring.\n\nThe relation of space whole to space part has many formal properties\nwhich are identical with the properties of \"all and some.\" Also when\npoints have been defined, we can replace it by the conception of \"all\nthe points and some of the points.\" But the confusion between the two\nrelations is fatal to sound views on the subject.\n\n\n                  _Diverse Euclidean Measure Systems_\n\nThe next section deals with the measure systems applicable to space.\n\nA measure system is a group of congruent transformations of space\ninto itself. Consider a rigid body occupying all space. Let this body\nbe moved in any way so that the particles of the body which occupied\npoints P_{1}, P_{2}, P_{3}, etc., now occupy points Q_{1}, Q_{2},\nQ_{3}, etc. Then any point P_{1} in space is uniquely related to the\ncorresponding point Q_{1} in space by a one-to-one transformation with\ncertain characteristics. By the aid of these transformations we can\nachieve the definition of distance in a way which definitely determines\nthe distance between any two points, provided that we can define what\nwe mean by a congruent transformation without introducing the idea of\ndistance. If we introduced the idea of distance, we should simply say\nthat a congruent transformation is one which leaves all distances\nunchanged, _i. e._, if P_{1}, P_{2} are transformed into Q_{1},\nQ_{2} then the distance P_{1}P_{2} is equal to the distance Q_{1}Q_{2}.\n\nBut mathematicians have succeeded in defining congruent transformations\nwithout any reference to distance.\n\nThere are alternative groups of such congruent transformations, and\neach group gives a different measure system for space. The distance\nP_{1}P_{2} may equal the distance Q_{1}Q_{2} for one measure system,\nand will not equal it for another measure system. All these different\nmeasure systems are on the same level, equally applicable. A being\nwith a strong enough head could think of them all at once as applying\nto space. The result so far as it interests us in respect to the\ntheory of relativity is explained on pp. 197-200, ending with \"The\nmost extraordinary fact ... same metrical system.\" This final sentence\nbears on Poincar\u00e9's assertion that the measure system adopted is\npurely \"conventional.\" I presume that by \"conventional\" a certain\narbitrariness of choice is meant; and in that case, I must express\nentire dissent. It is true that within the circle of geometrical ideas\nthere is no means of giving any preference to any one measure system,\nand any one is as good as any other. But it is not true that if we\nlook at a normal carriage wheel, and at an oval curve one foot broad\nand ten feet long, we experience any arbitrariness of judgment in\ndeciding which has approximately the form of a circle. Accordingly to\nPoincar\u00e9 the choice between them, as representing a circle, is entirely\nconventional.\n\nAgain, we equally form immediate judgments as to whether a body is\napproximately rigid. We know that a paving stone is rigid, and that a\nconcertina is not rigid. This again necessitates a determinate measure\nsystem, selected from among the others.\n\nAccordingly we conclude that (i) each being does, in fact, employ a\ndeterminate measure system, which remains the same, except possibly\nfor very small variations, and (ii) the measure systems of different\nhuman beings agree, to within the limits of our observations. These\nconclusions are not the less extraordinary because no plain man has\never doubted them.\n\nIt is an interesting subject to investigate exactly what are the\nfundamental uniformities of experience which necessitate this\nconclusion. It is not so easy as it looks, since we have to divest\nourselves of all aid of scientific hypothesis if our conclusions are to\nbe demonstrative.\n\n\n                    _Relativity in Modern Physics_\n\nPp. 201-202, \"Owing to the fact ... which B so regards.\"\n\nThe fundamental formul\u00e6 for the theory of relativity are the relations\nbetween diverse co-ordinate systems given on p. 203, and formul\u00e6 II\nat the bottom of p. 207. The general explanation of one method in\nwhich these formul\u00e6 arise--namely, Einstein's method--is given on\npp. 201-211. Namely, we seek the condition that for all dynamical\naxes the velocity of light should be the same, and the same in all\ndirections. It should be noted that the experiments which, so far as\nthey go, confirm these formul\u00e6, can also be explained in another way\nwhich makes the theory of relativity unnecessary. We need only ascribe\nto the ether a certain property of contraction in the direction of\nmotion, and the thing is done. So no one need be bludgeoned into\naccepting the rather bizarre doctrine of relativity, nor indeed any\nother scientific generalisation. The good old homely ether, which we\nall know, can in this case serve the purpose. Just as an author of\ngenius, if he lives long enough, survives the inevitable accusation\nof immorality, so the ether by dint of persistence has outlived all\nreputation of extravagance. But if we detach ourselves from the glib\nphraseology concerning it, the scientific ether is uncommonly like the\nprimitive explanation of the soul, as a little man inside us, which can\nsometimes be caught escaping in the form of a butterfly. As soon as the\nether has to be patched up with special properties to explain special\nexperiments, its scientific use is problematical, and its philosophic\nuse is _nil_.\n\nPhilosophically the ether seems to me to be an ambitious attempt\nto give a complete explanation of the physical universe by making\nan elephant stand on a tortoise. Scientifically it has a perfectly\nadequate use by veiling the extremely abstract character of scientific\ngeneralisations under a myth, which enables our imaginations to work\nmore freely. I am not advocating the extrusion of ether from our\nscientific phraseology, even though at special points we have to\nabandon it.\n\nBut the key to the reasons why it is worth while to consider seriously\nthe doctrine of relativity is to be found on pp. 209, 210: \"Again the\nvelocity of light ... not involving light.\" Namely, we have begun to\nsuspect that all physical influences require time for their propagation\nin space. This generalisation is a long way from being proved.\nGravitation stands like a lion in the path. But if it be the case, then\nall idea of an immediate presentation to us of an aspect of the world\nas it in fact is, must necessarily be abandoned. What we perceive at\nany instant is already ancient history, with the dates of the various\nparts hopelessly mixed.\n\nWe must add to this the difficulty of determining what is at rest\nand what is in motion, and the further difficulty of determining a\ndefinite uniform flow of time. It is no use discussing this matter\nas though, but for the silly extravagant doctrine of relativity,\neverything would be plain sailing. It isn't. You may be quite sure\nthat when, after prolonged study, you endeavour to give the simplest\nexplanation of a grave difficulty, you will be accused of extravagance.\nI have no responsibility for the doctrine of relativity, and hold no\nbrief for it, but it has some claim to be considered as a comparatively\nsimple way out of a scientific maze.\n\nIn the first place, we use the Newtonian dynamical sets of axes,\nand the Newtonian clock to extricate ourselves partially from the\ndifficulties of rest, motion, and time. These have proved capable\nof scientific determination within the limits of our experimental\naccuracy. Thus the only thing left over is the choice of the axes\nat rest, which is a completely indeterminate problem on Newtonian\nprinciples.\n\nAgain, so far as we can at present guess by adopting the theory\nthat all metrical influence is electromagnetic, all influences\nare propagated with the velocity of light _in vacuo_. This\nelectromagnetic hypothesis is by no means established, but it gives\nthe simplest of all possible results in respect to the propagation of\ninfluence, which we therefore adopt.\n\nBut what dynamical axes are we taking as at rest? Now our practical\nchoice gives a range of relative velocities small compared to that of\nlight. So except for certain refined experiments it does not matter.\nThere are two possibilities--\n\n(i) We may assume that one set of axes are at rest, and that the others\nwill show traces of motion in respect to the velocity of light; or--\n\n(ii) That the velocity of light is the same in all directions whichever\nbe the dynamical axes assumed.\n\nThe first supposition is negatived by experiment, and hence we are\ndriven to the second supposition; which immediately lands us in the\nwhole theory of relativity.\n\nBut if we will not have this theory we must reject the earlier\nsupposition that the velocity of light _in vacuo_ is the same in\nall directions. This we do, in fact, by assuming an ether, and assuming\na certain law for its modification. Then we, in fact, adopt the first\nsupposition so far as to hold that there are dynamical axes specially\nat rest, namely, at rest relatively to the undisturbed ether. Then an\nassumed law for the modification of the ether so alters the velocity of\nlight that we explain why no dynamical axes show traces of motion.\n\nI wish now to go back to the point which I made a few minutes ago,\nthat what we perceive at any instant is ancient history with its dates\nhopelessly mixed. In the earlier part of my comments I emphasised the\npoint that our only data as to the physical world are our sensible\nperceptions. We must not slip into the fallacy of assuming that we are\ncomparing a given world with given perceptions of it. The physical\nworld is in some general sense of the term a deduced concept.\n\nOur problem is, in fact, to fit the world to our perceptions, and not\nour perceptions to the world.\n\n\n       PRINTED IN GREAT BRITAIN BY RICHARD CLAY & SONS, LIMITED,\n        BRUNSWICK ST., STAMFORD ST., S.E., AND BUNGAY, SUFFOLK.\n\n\n\n\nFrom WILLIAMS & NORGATE'S LIST\n\n\n =A SPIRITUAL PILGRIMAGE.= By the Rev. R. J. CAMPBELL.\n (The Rev. R. J. Campbell's own Story of his Religious Life.) 2nd\n Impression. Demy 8vo. =7s. 6d.= net.\n\n \"This is eminently a book for which to be thankful--simple,\n straightforward, kindly. Neither does it lack the saving grace of\n humour.\"--_Guardian._\n\n \"Mr. Campbell devotes a chapter to his re-ordination, and it must be\n admitted that his defence of this act is effective.\"--_Methodist\n Times._\n\n =EDINBURGH.= By the Right Hon. Sir HERBERT MAXWELL,\n Bart., D.C.L., Author of \"Life of the Duke of Wellington,\" \"Scottish\n Gardens,\" etc. President of the Society of Antiquaries of Scotland,\n 1910-13, and Chairman of the Royal Commission on Scottish Historical\n Monuments. With a Coloured Frontispiece and 64 pages of Illustrations\n of the past and present city. Medium 8vo. =10s. 6d.= net.\n\n A History of the Scottish Capital from the earliest times to the\n nineteenth century.\n\n =FURTHER PAGES OF MY LIFE.= By the Right Rev. W. BOYD\n CARPENTER, K.C.V.O., D.D., D.C.L., Formerly Bishop of Ripon,\n Author of \"Some Pages of My Life,\" \"The Witness of Religious\n Experience,\" etc. With Illustrations. Medium 8vo. =10s. 6d.= net.\n\n Reflections mingled with intimate reminiscences and recollections of\n Royal personages and of men eminent in many spheres.\n\n =SOME PAGES OF MY LIFE.= By the Right Rev. W. BOYD\n CARPENTER. New and Cheaper Edition. Large post 8vo. =5s.=\n net.\n\n All who have listened to his eloquent preaching will read with delight\n his musing on what life has brought the author.\n\n =RAPHAEL MELDOLA=, Hon. D.Sc. (Oxon.), Hon. LL.D. (St. And.),\n F.R.S., Professor of Chemistry in the City and Guilds of London\n Technical College. Reminiscences of his worth and work, by those who\n knew him, together with a Chronological List of his Publications\n (1868-1915). Edited by JAMES MARCHANT. With a Preface by the\n Right Hon. LORD MOULTON, K.C.B., F.R.S. With a Portrait.\n Crown 8vo. =5s.= net.\n\n \"The book is one of the most delightful biographies of scientific\n men that have been published during recent years.\"--_Chemist and\n Druggist._\n\n =THE MANUFACTURE OF HISTORICAL MATERIAL.= An Elementary Study in\n the Sources of Story. By J. W. JEUDWINE, LL.B. (Camb.), of\n Lincoln's Inn, Barrister-at-Law, Author of \"The First Twelve Centuries\n of British Story,\" and other works. Crown 8vo. =6s.= net.\n\n A consideration of the successive phases of Historical Research; it\n explains necessary processes through which all the material has to\n pass before it is placed before us as history.\n\n\n                     LONDON: WILLIAMS AND NORGATE\n\n\n\n\n                    SOME BOOKS ON RUSSIA AND POLAND\n\n\nWith a Coloured Frontispiece, 12 Photogravure Plates, 28 Illustrations\n       in the text, and 8 Maps. Demy 8vo. =7s. 6d.= net (postage\n                             =6d.= extra).\n\n\n                 =A Thousand Years of Russian History=\n\n                           By SONIA E. HOWE.\n\n \"We can recommend the volume as an excellent, careful, and\n well-written history of a great nation.\"--_Daily Telegraph._\n\n\n                      Demy 8vo. Cloth, =6s.= net.\n\n                           =The False Dmitri=\n\n  A Russian Romance and Tragedy. Described by British Eye-witnesses,\n                              1604-1612.\n\n                Edited with a Preface by SONIA E. HOWE.\n\n        Illustrated by reproduction of contemporary portraits.\n\n \"With more than a whiff of the atmosphere Hakluyt got into his famous\n volume.\"--_Northern Whig._\n\n \"Of special interest at the present time.\"--_Oxford Chronicle._\n\n\nWith 16 Plates and 28 Illustrations in the text. Demy 8vo. =7s.\n6d.= net.\n\n               =Some Russian Heroes, Saints and Sinners=\n\n A portrait gallery of outstanding figures in Russian history who were\n               typical of their times and their country.\n\n                           By SONIA E. HOWE.\n\n \"It gives a vivid study of Russia ... and interesting book.\"--_Land\n and Water._\n\n\n     Fcap. 8vo. Cloth, =1s. 3d.= net. In full leather presentation\n                        binding, =2s. 6d.= net.\n\n                  =An Outline of Russian Literature=\n\n                      By the Hon. MAURICE BARING,\n\n Author of \"With the Russians in Manchuria,\" \"A Year in Russia,\" \"The\n                         Russian People,\" etc.\n\n \u2235 Gives a clear and interesting account of the growth of Russian\n literature.\n\n\nFcap. 8vo. Cloth, =1s. 3d.= net. In full leather binding, =2s.\n6d.= net.\n\n                               =Poland=\n\n                     By W. ALISON PHILLIPS, M.A.,\n\n    Lecky Professor of Modern History in the University of Dublin.\n\n \u2235 A concise survey of the History of Poland up to 1915.\n\n\n                     LONDON: WILLIAMS AND NORGATE\n               14 HENRIETTA STREET, COVENT GARDEN, W.C.\n\n\n\n\n                         =TRANSCRIBER\u2019S NOTES=\n\nSimple typographical errors have been silently corrected; unbalanced\nquotation marks were remedied when the change was obvious, and\notherwise left unbalanced.\n\nPunctuation, hyphenation, and spelling were made consistent when a\npredominant preference was found in the original book; otherwise they\nwere not changed.","title":"The organisation of thought, educational and scientific","tradition":"philosophy-ethics","translated_text":null,"word_count":50976}