{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_74732/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_74732","id":"gutenberg_74732","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/74732","summary":null,"tags":["Category: Mathematics","Category: Philosophy & Ethics","Infinite","Knowledge, Theory of","Pragmatism"],"text":"ON THE THEORY OF THE INFINITE IN MODERN THOUGHT\n\n\n\n\n\n                        ON THE THEORY OF THE\n                         INFINITE IN MODERN\n                               THOUGHT\n\n                      TWO INTRODUCTORY STUDIES\n\n                                 BY\n\n                           E. F. JOURDAIN\n\n                  DOCTOR OF THE UNIVERSITY OF PARIS\n               VICE-PRINCIPAL, ST. HUGH\u2019S HALL, OXFORD\n\n\n\n\n                       LONGMANS, GREEN AND CO.\n                     39 PATERNOSTER ROW, LONDON\n                   NEW YORK, BOMBAY, AND CALCUTTA\n                                1911\n\n                         All rights reserved\n\n\n\n\nOf the two papers here reproduced, the first was given in 1905 to\na meeting of the women science students in Oxford; the second, in\n1908, to the Philosophical Society of this College. They are printed\nby request, with the author\u2019s apologies for their incompleteness.\nThe lecture form has been retained. I am indebted to my brother, Mr.\nP. Jourdain, for help in preparing the first lecture, and for his\nrevision of the text.\n\n                                                  E. F. JOURDAIN.\n\n     ST. HUGH\u2019S HALL, OXFORD,\n         _January, 1911_.\n\n\n\n\n                              CONTENTS\n\n\n                                  I\n\n                                                    PAGE\n\n                THE PROBLEM OF THE FINITE AND\n                    THE INFINITE                       1\n\n\n                                  II\n\n                PRAGMATISM AND A THEORY OF\n                    KNOWLEDGE                         31\n\n\n\n\n                                  I\n\n             THE PROBLEM OF THE FINITE AND THE INFINITE\n\n\nThe influence of mathematics on philosophy and vice vers\u00e2 can be\ninferred from the historical progress of both studies, though it has\nnot been possible till about within the last fifteen years to give\na logical explanation for the relations between them. As long as it\nwas believed, according to the Kantian view, that the science of\nmathematics was based on intuitions of time and space, the alliance\nbetween philosophy and mathematics could not be proved to be closer\nthan that between philosophy and experimental science, although the\nhistorical fact remained that philosophy and mathematics exercised a\nmutual stimulus, and developed at the same periods of history.\n\nBut mathematics, as now defined, is independent of intuitions of\nspace and time, and also of axioms and hypotheses.[1] Mathematics,\nas now understood, is based, like formal logic, on the prerequisites\nof thought, not on the notions of space and time. Here there is no\ndefinition of number or space, but the conception of number and\nspace,[2] which is more complicated, can be derived from them. All\nother complicated mind processes can, in the same way, be reduced to\nthe simple elements of the prerequisites of thought.\n\nSuch a science might exist out of conditions of time and space as we\nknow them. It is a science of relations rather than of mere number.\nFounded, then, on the laws of symbolic logic, it is a valuable aid\nand illustration to philosophy; philosophy, on the other hand, can\nimagine lines for the exercise of the constructive power involved in\nmathematics. It is the object of this paper to show that the close\nthough apparently accidental union of philosophy and mathematics\nthroughout the history of thought can now be explained, and that the\nproblems with which pure mathematics is now concerned are those which\nlie at the core of philosophic thought and speculation. (Symbolic\nlogic has developed to meet the new demands made upon it. It does\nnot now reduce itself to the syllogism, as Aristotle thought it did;\nthe prerequisites of thought are shown to be manifold instead of\nsingle.[3])\n\nThe use of the word philosophic in this connection suggests a\nnecessity for further definition. Philosophy is held to include at\nleast two great branches--Metaphysics and Ethics. The influence of\nmathematics is most evident on the metaphysical side of philosophy;\nin fact, the grouping of mathematics and metaphysics as allied\nsciences tends to bring out the essential distinction between\nmetaphysics and ethics, and--though not by any means to imply a break\nin their real relation--to show where this has been misunderstood.\nNo philosophy has been equally strong on both sides; they represent\ndifferent forms of activity of the human mind; but it is still true,\nand from the conditions always must be, that an ethical system grows\nout of metaphysics as practice follows precept and conduct implies\nbelief. The new definition of mathematics does not touch these\nconsequences; it merely marks the limits within which philosophy on\nthe metaphysical side can submit to, or rest upon, the conclusions of\nmathematics.\n\nAs to the historical relation between metaphysics and mathematics,\nthe subject is so vast that we shall only attempt a very rapid\ngeneralisation of its results on the growth of the conception of\nthe Finite and the Infinite. (Of course, there are many other sides\nof the relation which might be studied.) The general result of the\ninquiry has been, as far as we can judge, that metaphysics has\nexercised an inspiring force on mathematics, and mathematics has\ndefined and strengthened the conceptions of metaphysics at every\ncritical stage in the history of philosophy. But where metaphysics\nhas been treated as the proof of science, where it has been laid\ndown as the foundation for exact knowledge, the results have not\ncorresponded with the truth of experience, and the quality of thought\nhas become degenerate. Progress depends on the right perception of\nthe relations between the sciences and parts of philosophy.\n\nSuch progress is especially evident in the early Greek and in the\nmodern periods, while the large period from the Christian era to the\nRenaissance gives examples of the unfortunate reversal of the parts\nof metaphysics and science and consequent confusion of thought.[4]\n\nThe problems of the metaphysician are no doubt in a sense always the\nsame; but this is equally true of the problems of any other science.\nThe methods by which the problems are attacked and the adequacy\nof the solutions they receive vary, from age to age, in close\ncorrespondence with the general development of science. Every great\nmetaphysical conception has exercised its influence on the general\nhistory of science, and in return every important movement in science\nhas affected the development of metaphysics. The metaphysician could\nnot if he would, and would not if he could, escape the duty of\nestimating the bearing of the great scientific theories of his time\nupon our ultimate conceptions of the nature of the world as a whole.\nEvery fundamental advance in science thus calls for a restatement and\nreconsideration of the old metaphysical problems in the light of the\nnew discovery.\n\n       *       *       *       *       *\n\nDuring the Greek period mathematics was the only branch of science\nwhich was at all developed, and its development coincided with\nthe age of the philosophers. Thus when Plato spoke of science he\nalways meant mathematics. And even later, when the physical sciences\nhad begun to develop, Aristotle put mathematical ideas into close\nconnection with metaphysical ones when he stated that they occupied\nthe middle term between the ideal and the sensible. Both Plato and\nAristotle referred to and depended upon mathematical proofs and\nillustrations of philosophical questions. During this Greek period\nthe conception of Infinity took shape. The pre-Platonic notion,\nreproduced again later in the decline of Socratic theory by the\nStoics, was that the Infinite was the aggregate of the Finite; the\nPlatonic and Aristotelian theory, that, namely, of the most vigorous\nmoment of Greek thought, was that the Infinite was _more_ than the\naggregate of the Finite; that it had a self-determined existence from\nwhich the Finite had been derived. Existence, as known to man, was\ntreated as a compromise between the Finite and the Infinite.\n\nNeo-Platonism altogether separated the Infinite from the Finite. In\nthe Alexandrine metaphysics, which represented a decadent stage of\nphilosophy and its deviation from the sciences, the conception of\nthe Infinite became less clear and logical; it diverged from the\nview which had been affected by mathematical thought, and tended\nto assimilate to itself the ideas of perfection and universality,\nwhich, philosophically speaking, are conceptions distinct from that\nof Infinity--universality referring to a common principle of unity,\nand perfection involving the moral ideal. Real progress was deferred\nby the too rapid coherence of ideas only partially analysed and\nunderstood. Thinkers passed quickly from the exclusive contemplation\nof subject to that of object and back again,[5] each new period\nnegativing all previous experience, till the result was the exclusion\nof an imperfectly analysed Relative and Finite from an insufficiently\napprehended Absolute and Infinite.\n\nAfter the Christian era Greek philosophy drifted off into\nscholasticism and lost touch of reality, the grammar of Aristotelian\nlogic replacing the vital connection of ideas. St. Anselm, it is\ntrue, attempted to find a rational proof of the existence of God, and\nidentified Him with the Infinite of Greek thought; but St. Thomas\nAquinas led away the argument to a discussion as to how far form and\nmatter, separately considered, shared in the quality of Infinity. (He\nthought form did, but not matter.) An overpowering sense of mystery,\njoined to a premature desire for definition without scientific\nanalysis, sapped the vigour of medi\u00e6val thought.\n\nThroughout the middle ages, then, we see the conditions of the Greek\nperiod reversed: philosophy during the second period is not, as\nin the first, engaged in giving a stimulus to the efforts of pure\nreason; rather the intuitions of philosophy are treated as axiomatic,\nand a false superstructure of knowledge, alien to experience and\nreality, is erected upon these foundations. Philosophy, in fact, is\nused as a general basis for science. The parts of philosophy and\nmathematics, correctly though imperfectly seen by the Greeks, are in\nthe second period exchanged, and the result is confusion of idea.\nThe notion of the Infinite, as in the Alexandrine metaphysics, is\nheld to include perfection and universality, and does not exist as a\nconception apart from these.\n\nAfter the Renaissance, the scholastic philosophy falling into disuse,\nthe attempt to find an explanation of the Cosmos, a synthesis of the\nuniverse, was abandoned, and replaced by the Cartesian idea--the\ninference of existence from thought, and the limitation of the sphere\nof inquiry to that which could be known by the ego. New scientific\nand mathematical discoveries kept pace with this new analysis and\ndevelopment of thought,[6] and the surer ground in philosophy was\ndefinitely allied with the work of the mathematical mind. The\nphilosophical thesis developed from \u201cThe Infinite is the negation\nof the Finite,\u201d to \u201cThe Infinite presupposes the Finite and does\nnot exclude it.\u201d The problem of the Finite and the Infinite became\nthe great idea of the age, and there was a reversion to the Greek\nnotion of existence as a compromise between the two, and almost the\nhint of a coming explanation of them. In the decline of Cartesian\nphilosophy, when it drifted off into Pantheism, there was only a\nvague conception of the Infinite, and we trace a tendency to identify\nthe notion of Infinity with that of the Cosmos. In medi\u00e6val thought\nthe idea of the Infinite had become confused with that of the Perfect\nand Universal; in the modern period the effort to give a concrete\nexpression to the notions of Infinity, Perfection, and Universality\ndiverted the ideas from their relation to the Creator and applied\nthem to the Creation.\n\nKant, who gave a new impulse to some parts of the Cartesian idea,\nneglected both mathematical proofs and the search for a metaphysical\nAbsolute. In avoiding the subject he helped to perpetuate the\nvague descriptions of the Finite and the Infinite, uncorrected by\nmathematical thought, which had been the currency of the philosophy\nof his age, and which corrupted the philosophy of the succeeding\ncentury. The nineteenth century produced nothing more than guesses\nat truth, which were, perhaps, not very far wrong, and which the\npresent century is engaged in correcting and substantiating. The\nsame vagueness afflicted both mathematics (theory of functions)\nand philosophy. Fichte, Schelling, and Hegel, particularly Hegel,\nidentify the metaphysical Absolute with reality, infinity, and the\nuniversal. The ideas of continuity and infinity are not separated by\nthem from those of perfection and universality, nor from one another,\nand their nature is not understood.\n\nLeaving aside the French neo-critical School (Renouvier) and the\nEnglish School (Spencer)--the first of whom deny the Infinite,\nthus acting in opposition to mathematical reasoning, while the\nsecond perpetuate Kant\u2019s error of considering the Infinite, though\nthinkable, as unknowable (Dr. Caird has pointed out that this\nposition is illogical)--we arrive at a moment in history which is\nmore fruitful in result on the mathematical side, and will, no doubt,\nhave an effect on metaphysics. For, owing to recent discoveries\nin Germany and England, mathematics is now in a position to give\ngreater support than before to the intuitions of philosophy.\nHitherto, philosophers have been reluctant to allow full value to\nthe mathematical conceptions of Infinity, and with some justice, as\nthe notion had not been sufficiently analysed. Philosophers, who\nnever attempted the analysis, have been inclined to accept certain\ncontradictions in their conception as inherent in the nature of\nInfinity. Within the last twenty-five years Cantor and Dedekind have\ncleared up the notion of continuity, and Russell has given greater\nprecision to the idea, and has applied this reasoning to philosophy.\n\nPresent-day metaphysicians seem to be divided into two groups; on\nthe one side, those who consider in philosophy the value of a theory\nof being, and, on the other, those who chiefly consider the value\nof a theory of knowledge, _i.e._ the Epistemologists. The first\ngroup, devoting themselves to psychology, evolution, and history,\nhave no _necessary_ belief in the Infinite. The Epistemologists,\nwhose work is founded on Kant, discuss the theory of knowledge and\nenumerate the conditions of knowledge. Their argument may not touch,\nbut does not exclude, the notion of the Infinite. The position of\nthe Epistemologist has been made infinitely more secure by recent\nmathematical work. That of the psychologist remains almost untouched.\nIt is necessary now to examine more closely the mathematical results\nto which reference has been made.\n\nIn general terms it may be said that mathematics has, as a study,\nled immediately from the nature of the subject to the perception\nof the Infinite, and to a knowledge of the connection between the\nInfinite and the Finite. The simplest form in which the idea can be\nput is stated by St. Augustine, who said that numbers considered\nindividually were finite, but considered as an aggregate were\ninfinite.[7] Before St. Augustine, and after him down the long\nstream of philosophic thought, the theologian and the philosopher\nhave turned to mathematics for illustrations of the infinitely great\nand infinitely little, as developed from the concrete processes of\narithmetic and geometry. The recurring decimal in arithmetic, the\nproperties of the circle and ellipse in geometry, of the cone in\nconic sections, and of the surd in algebra, all touch the problem of\nnumber and space on the side of Infinity.\n\nIn higher mathematics it is possible to start from the idea of the\nFinite and reach the conception of the Infinite; or to reverse the\nprocess, and from the Infinite to deduce the Finite. Thus in the\nfamiliar puzzle of the subdivision of the parts of a straight line\nby halving the remainder, there will be a crowding and a coalescing\nof the points of division towards one end of the line, the points of\ndivision getting infinitely nearer, but the steps will never meet.\nHere in the centre of a straight line--a limited straight line--we\nare confronted with the problem of Infinity.\n\nAgain, from a series of finite numbers we can gain the notion of an\ninfinite series. Take two series which have a correspondence with one\nanother. If for every element of the one we can choose an element of\nthe other, and of the other there is an element for the one, when at\nany point we cut off its progress to infinity, this happens:--\n\nOne series, if summed up, will give a larger numerical result than\nthe other, and therefore can be said to be greater than the second.\nLet us call the first series A, and the second B. Let us now imagine\nthe two series, though starting at a definite point, are never cut\nat the further end. Then to all infinity series B is without certain\nnumbers which series A possesses, _and as an infinite series_ is\nsmaller than series A. But, on the other hand, when neither series\nis cut, series B retains its correspondence with series A. Thus we\nattain a definition of an infinite series. It is such that the part,\nwhile being less than the whole, has yet a complete correspondence\nwith the whole. The whole is greater than the part, but take away\nthe part from the whole and that which remains corresponds to it _in\ninfinity_, because the test of summing the series (which would give\na contrary result) involves limitation, and thus cannot be applied.\nSubtraction can take place in Infinity without loss.\n\nBy reversing this process, and by starting from the theory of the\nInfinite, we may gain some idea of the discovery of the Finite. So\nDedekind and Russell define finite numbers not only in the usual way\nas those which can be reached by mathematical induction, starting\nfrom 0 and increasing by 1 at each step, but also as those of classes\nwhich are _not_ similar to the parts of themselves obtained by taking\naway single terms. That is the reversal of the process applied just\nnow. Dedekind also has deduced the Finite from the Infinite by a\nnovel process. He predicates a world of thought which we each and all\npossess, filled with thoughts and things, to each thing corresponding\na thought. There are thus two \u201ctrans-finite\u201d series in the minds of\neach and all of us; we cannot say when the series of thoughts and\nthings will end; but they have number, though it is infinite number.\n(Number exists wherever there is a correspondence, one to one,\nbetween two aggregates.) But in this _Gedankenwelt_, says Dedekind,\nthere is one thing to which there is no corresponding thought: that\nis the ego. Each man is part of his own world of thought, but\nthere is no thought of himself in his mind corresponding exactly to\nhimself, as a thought in his mind corresponds to another object.[8]\nTwo important results follow from Dedekind\u2019s theory: first, the\nexistence of a finite number one, the number of the ego, as deduced\nfrom the _Gedankenwelt_ of two infinite systems; second, by putting\ntogether all the _Gedankenwelts_ there are or may be, we get the\nnotion of series of series, which seems to transcend Infinity, and\nit gives us the conditions which are possibly gathered up in the\nAbsolute. Now the argument from the Finite to the Infinite and the\nconverse process may both be employed in mathematics (or both may\nbe neglected, as in the elementary methods of calculation used in\narithmetic). A discussion has taken place in the _Hibbert Journal_ on\nthe relative value of the two methods. Keyser, in an article called\nthe Axiom of Infinity, argued that one method, that of Dedekind,\nshould be exclusively developed. Russell answered him, stating that\nit was not necessary to hold exclusively to either. If the Finite\nand the Infinite can in turn be deduced from one another, neither\nconception can be truly called an axiom. The real axiom is existence,\nwhich includes both, and which is defined by mathematicians as that\nwhich is _not self-contradictory_.\n\nNow the problem of Infinity includes also that of continuity; in\nother words, the problem of number includes that of _cardinal_ and\n_ordinal_ number. It is time to get to the mathematical definition of\nnumber, which we have found as a conception can be attached both to\nthe Finite and to the Infinite. What is number in mathematics?\n\nTake any collection of things--we call that an _aggregate_. If an\naggregate corresponds one to one with another aggregate, they are\nboth said to have a number, and the same number. Subtract from the\nidea of an aggregate the idea of quality or kind, and order or\narrangement, what is left is its _cardinal number_. If you subtract\nquality and not order, the result is an _ordinal number_. This\nreasoning applies both to finite and infinite aggregates; in fact,\nthe Infinite may be said to possess most of the properties which\nwe attach to the Finite. Two infinite aggregates, for example, can\nhave an ordinal correspondence, and infinite aggregates submit, like\nfinite ones, to arithmetical processes.\n\nThe mathematician analyses still more closely the relation between\nthe Finite and the Infinite, as follows:--\n\nHe starts from the aggregate, which he analyses into the Finite\nand the Infinite, and the latter he analyses into the Transfinite\nand the Absolute. Of these two elements, one only has till just\nlately been the subject of mathematical treatment--the one called\nthe Transfinite. It is the transfinite subdivision of the Infinite\nto which the idea of number is applicable, and which is, therefore,\nin a sense inseparable from the Finite. Infinite numbers or series\nought then to be more correctly described as _Transfinite_. But the\nprocesses of mathematics do not end here; they reach up to the idea\nof the _Absolute Infinite_, the conception of which has been attained\nin recent years by mathematical work. The results of this work may\nnow be briefly summarised.\n\nI. The Absolute appears to have the same relation to the Transfinite\nas the Transfinite to the Finite. If the Finite deals with numbers,\nand the Transfinite with series of numbers, the Absolute deals\nwith series of series. Thus there are at least two examples of the\nInfinite within our grasp which lead up to the idea of the Absolute.\nOne is the class of all classes of propositions; the other is the\nseries of all worlds of thought, in Dedekind\u2019s sense.\n\nII. The Finite, Transfinite, and Absolute can be further defined in\nthis way. There is no greatest finite number, but there is a least\ntransfinite number, which has been called Aleph 0, and which can\nbe proved to be greater than any possible finite number, however\nlarge, because if there were a last number it must be smaller than\nthe sum of the whole series. There are unending series of Alephs or\ninfinite numbers, which are as distinct from one another in idea\nas 1 is from 0, and which can no more be derived from one another\nby a mathematical process than 1 can be derived from 0, but can\nbe reached in the same way by induction. Beyond the Transfinite we\ncannot discover in the Absolute the idea of least or of greatest.\n\nIII. The relation of cardinal and ordinal number also throws some\nlight on the Finite, Transfinite, and Absolute. In the Finite,\ncardinals and ordinals are parallel to one another; in the\nTransfinite they strikingly diverge; in the Absolute we cannot trace\nany connection between cardinals and ordinals, _i.e._, it is possible\nto have an ordinal series to which there can be no corresponding\ncardinal number or type.[9]\n\nIV. If arithmetical processes are applied to the Finite, Transfinite,\nor Absolute, we get interesting results. We know the effect of\naddition, multiplication, and raising to a power, on the Finite.\nThe first two processes have been applied to the Alephs; the last\nhas been formulated, but the mathematical results have not yet been\nbrought to a satisfactory conclusion. Broadly speaking, we may say\nthat the raising of an Aleph to a power may make it transcend the\nFinite and the Transfinite and melt into the Absolute. Thus all\nmathematical processes which find their goal in the Absolute would\nfind their annihilation there. No finite mathematical conception\nwould be applicable to it.\n\nNow the conception of this Absolute Infinite, of which the aggregate\nof all ordinal numbers is perhaps a symbol,[10] has been subjected to\ncriticism. Some mathematicians[11] think that it exists, but has no\nnumber. It is discovered by a logical process, but defies analysis\nand the application to it of the notion of number. All mathematical\nconceptions find in it their aim and conclusion. The importance of\nthis theory, its practical importance, lies in the very much simpler\nmathematical formul\u00e6 that can be produced now that the logical\nprocess is shown to extend from the Finite to the Absolute Infinite\n(in the same way that the labour of summing a series arithmetically\nby statement and addition is shortened by the application of\nalgebraical principles which depend on larger knowledge). Its\nphilosophical importance is great: the Absolute is here, as\nelsewhere, the goal of human thought, and is the mathematician\u2019s name\nfor the highest power discoverable by human reason.\n\nIt would be very interesting to discuss the probable attitude of a\nPascal or a Hegel to these mathematical conceptions, if they had been\naware of them. Take Pascal\u2019s puzzle of the Finite and the Infinite.\nHe thought that if the Finite could be subtracted from the Infinite,\nthe Infinite would thereby lose some of its quality of infinity. How\ndifferently would it have appeared to him had he realised that an\naggregate infinite cardinal can have subtracted from it either finite\nor transfinite terms: if transfinite terms, many different answers\nresult, giving different degrees of transfinity: if only finite terms\nare taken away, the Infinite remains in its entirety.\n\nHow, again, would Hegel have rejoiced in a definition of thought and\nexistence which would bridge over the logical gulf in his system!\nHegel asserted that thought and existence were one. He is objected\nto by many philosophers, who ask where is the _tertium quid_ which\nmakes it possible to reach from one to the other, or predicate their\nessential unity? But the mathematician defines existence as something\nwhich is not self-contradictory. Thought, then, to him is a form\nof existence, for thought is not self-contradictory; but the two,\nthought and existence, are not necessarily conterminous.[12] Hence,\nto say that non-contradiction is a fundamental condition of true\nthinking is as much as to say that it is a fundamental characteristic\nof real existence, and he identifies thought with reality.\n\nDr. Caird remarks that the secular conscience conceives of the\nInfinite as opposed to the Finite; the religious conscience treats\nthe Infinite as real, presupposed by the illusory Finite. Where does\nthe truth lie? Mathematics does not admit the necessity of adopting\neither view at the expense of the other.\n\nMetaphysics standing alone produces results that may be disproved,\nbut cannot be proved. Mathematics standing alone produces results\nthat are susceptible of proof. Both are based on logic, and rest on\nthe prerequisites of thought. Together they are a field for the best\npowers of human reason: metaphysics supplies insight, intuition,\nimagination; mathematics offers the indubitable proof and translates\nthe ideal into the actual.\n\nBut the element in philosophical thought which, employing the\npsychological method, tends to the discussion of a theory of being\nrather than that of knowledge, and thus to the realisation of an\nethical system rather than to metaphysical discovery, is averse from\naccepting these conclusions. It remains, therefore, for us to examine\nthe criticism offered by the psychological school on what they call\nthe mathematising of philosophy; and it will be found that the attack\ndeals both with the ground of the alliance and its results.\n\nA typical exponent of this school is Moisant, who, in the _Revue\nPhilosophique_ for January 1905, attacked what he considered to be\nthe characteristic of modern philosophy and also its vice. It will\nbe observed that at the outset he reverses the _r\u00f4les_ of philosophy\nand mathematics as we have apprehended them. Philosophy, he says,\nshould expect to be inspired by mathematics, but should avoid its\nmethod. Next, he connects the modern movement with the theories of\nLeibniz, who aimed at substituting general formul\u00e6 for elementary\nforms of reason and calculation. These short cuts, which seem to the\nmathematician to liberate the mind from a burden which prevents it\nfrom employing its full activity, seem to the psychologist to tend to\na mechanical method, in which the thinker is only aware of premises\nand results, and in which the mathematical concept tends to replace\nthe real idea. Then he attacks the new definition of mathematics as\nthe science of relations, asserting that it still contains notions of\nspace.[13]\n\nFinally, he comes to the real question at issue, and enters into\nthe comparison of a metaphysical and a mathematical problem. He\ntakes as his subject the argument from the known to the unknown.\nDescartes had said that argument should lead from the known to\nthe unknown, simple to complex, and had defined the first as that\nwhich could be known without the help of the second. This logical\norder of reasoning has been attributed to mathematics, but has been\nconsidered to be inapplicable to philosophy. Mathematics, in its\nrecent development, by the argument from the Finite to the Infinite\nand back again, starts from two propositions, neither of which can\nbe said to be axiomatic, because each in turn can be proved from\nthe other, but in the course of argument from either mathematics\nmakes use of the logical process. The real axiom, as has been shown,\nis that of _existence_ or _being_. A metaphysical argument has the\nsame root--that of existence--but a metaphysical problem deals with\nparadoxes, with questions which are sometimes defined as having two\nanswers, each equally correct, and sometimes as yielding no answer\nat all. The method of thesis, antithesis, and synthesis is in the\nHegelian logic applied to their solution.\n\nA mathematical and a metaphysical problem are not, then, problems of\nthe same kind to be solved by the same method; nor is the conception\nof the mathematical Absolute reached in the same way as that of\nthe metaphysical Absolute. We are even unable to say how far they\ncorrespond except in respect of their absoluteness.[14] But the\ncontention of the mathematician to-day and of the epistemologist\nschool of philosophy is not the identity of methods and results in\nthe two sciences. It is the axiom of existence on which they both\ndepend: the law of thought by which all methods are developed, and,\nabove all, the _correlative value of each science to the other_,\nwhich allows us, in developing our knowledge from the standpoint\nof the two sciences, to recognise something of the greatness of the\nAbsolute principle to which they both reach up, and in which their\nbeing consists.\n\n\nFOOTNOTES:\n\n      [1] Of course, if we comprehend in our view only elementary\n          geometrical and algebraical science, it is easy to\n          show that they _do_ demand both axioms and intuitions.\n          Take, _e.g._ Euclid I. I., where in the construction\n          it is necessary to employ intuition for the assertion\n          that the arcs really cut one another. There is no\n          logical certainty that they do; in fact, in some other\n          conditions, _e.g._ in those of other space dimensions,\n          they might not.\n\n      [2] This is, of course, not the space of experience. Logic\n          and mathematics deal with implications of thought. See\n          B. Russell (_Hibbert Journal_, 1904, pp. 809-12), who\n          has shown that in all pure mathematics it is only the\n          implications that are asserted, not the premiss or the\n          consequence, as mathematicians used formerly to assume.\n\n      [3] De Morgan, Peirce, Schr\u00f6der, and B. Russell have worked\n          out the logic of relations as well as the syllogism.\n\n      [4] See Taylor, \u201cElements of Metaphysics,\u201d p. 13.\n\n      [5] See Dr. Caird, \u201cEvolution of Theology in the Greek\n          Philosophers.\u201d\n\n      [6] So Galileo, Newton, Huygens were philosophers in\n          science. Descartes, Pascal, Leibniz were mathematicians\n          as well as philosophers.\n\n      [7] See S. Augustine, _De Civitate Dei_, Book XII. ch.\n          xix.: \u201cIta vero suis quisque numerus proprietatibus\n          terminatur, ut nullus eorum par esse cuicumque alteri\n          possit. Ergo et dispares inter se atque diversi sunt,\n          et singuli quique finiti sunt, et omnes infiniti sunt.\u201d\n\n      [8] See R. Dedekind, _Was sind und was sollen die Zahlen?_\n          1893.\n\n      [9] Two transfinite aggregates can have an ordinal\n          correspondence with one another.\n\n     [10] See G. Cantor, _Zur Lehre vom Transfiniten_. 1890.\n\n     [11] _e.g._ Mr. P. Jourdain, _Philosophical Magazine_.\n          1904.\n\n     [12] The same result is hinted at by Mr. Taylor.\n          Taylor, \u201cElements of Metaphysics,\u201d p. 22.\n\n     [13] Linear order, 1, 2, 3, &c. Circular. A CD B,\n          A CD B.\u2026 The latter, it is true, involves the idea\n          of separation. But this idea can be developed from\n          those of inclusion and exclusion, which belong to the\n          fundamental laws of thought.\n\n          [14] The Absolute, according to a recent metaphysical\n          thinker, is \u201ca conscious life which embraces the\n          totality of existence, all at once, and in a\n          perfect systematic unity, as the content of its\n          experience.\u201d--Taylor, \u201cElements of Metaphysics,\u201d p. 60.\n\n\n\n\n                                 II\n\n                PRAGMATISM AND A THEORY OF KNOWLEDGE\n\n\nThe question before us is the relation of Pragmatism to a body of\nknowledge.\n\n(_a_) One question at issue between the Idealist[15] and the\nPragmatist has to do with the way in which each defines knowledge\nand gauges its ultimate aim. Both say that knowledge is relative,\nbut one school asserts that the human mind slowly and laboriously\nuncovers or discovers what Goethe calls the \u201cLiving garment of\nDeity,\u201d _i.e._ the world of nature, and comes into a heritage of\nscientific truth which increasingly corresponds to the subject of his\nfaith; the other claims that we live in a self-evolving universe in\nwhich in the course of long ages a new heaven and a new earth may be\ncreated which are not foreseen or implied in present conditions. In\nother words, the Idealist finds the Divine in human life; he finds\nin his own small corner of the universe the microcosm and symbol of\nInfinity: the Pragmatist considers that nothing _is_ which is not a\nresult of human action, and lowers the Divine element to the result\nof individual human activity. A compromise between the two ideas on\nnew and interesting lines has recently been made by Bergson. The\nChristian doctrine of Immanence and Transcendence also combines them.\n\nNow the increase of a body of knowledge would seem to depend on\nthe comparison of the successful working out of hypotheses with\nthe discrepancies from theory that from time to time appear. Taken\ntogether, proofs and discrepancies point to the evidence of a larger\nlaw. This is Hegel\u2019s logic, and the principle, so far as it is\nhere implied, is not denied in modern times, for no one wishes to\nfound a logic on a study of discrepancies as such. Even W. James\nsays, \u201cWhenever you once place yourself at the point of view of any\nhigher synthesis you see exactly how it does, in a fashion, take up\nopposites into itself.\u201d[16] In fact, without the notion of unity,\nthat of discrepancy could not exist: there must be a background on\nwhich the differences appear. The ultimate unity is symbolised in the\nIdealist doctrine of an Absolute.\n\nThe Absolute of Idealistic thought is not, however, now conceived\nof (as the Pragmatist would have us believe) as an abstract unity,\nbut as one involving a social bond, and hence relations which can\nbe described as personal, if we remember that the Personality of\nthe Absolute transcends our notion of human personality. Such a\nconception of the term Absolute, a new reading of the theory of\nthe One and the Many, has been led up to by Bradley and Royce by\nmethods of logic, and without any reference to dogma. It has been\nconveniently expressed by Taylor. The argument is briefly that\nultimate Reality must be One, Many, and Personal.\n\n\u201cFor our conclusion that mere truth cannot be the same thing as\nultimate reality was itself based upon the principle that only\nharmonious individuality is finally real, and this is the very\nprinciple employed by the intellect itself whenever it judges one\nthought-construction relatively higher or truer than another.\u201d[17]\n\nAnd again:--\n\n\u201cIf we speak of existence as a society, then we must be careful to\nremember that the individual unity of a society is just as real a\nfact of experience as the individual unity of the members which\ncompose it, and that when we call the Absolute a society rather than\na self, we do not do so with any intention of casting doubt upon its\ncomplete spiritual unity as an individual experience.\u201d[18]\n\nThe Absolute has been stated in modern thought to be One, Many, Real,\nand Personal or Social, and these terms of its qualification have\nbeen successively arrived at.\n\nW. James\u2019s words ring hollow when he attempts to dissociate such a\nconception from the reality of which it is the crown and inclusive\nsymbol, and type and essence. \u201cI personally,\u201d he says, \u201cgive up the\nAbsolute. I find it entangles me in metaphysical paradoxes that are\ninacceptable.\u201d He allows that there may be a God, though limited\nin power and goodness, \u201cone helper amongst others, _primus inter\npares_ in the midst of all the shapers of the great world\u2019s fate.\u201d\nIn such a system, as H. Jones has pointed out, \u201cthere is neither in\nthe universe nor in God any principle to inspire or guide, or in any\nway to bring about the amelioration desired. The process is guided\nby no end. The universe begins by being an aggregate of accidents,\npluralistic, discontinuous, irrational, and, of itself, cannot become\notherwise. There is nothing actual within to change its character.\u2026\nGod is himself finite, helpless to bring about this great change, a\npart, and no more, of a universe broken in fragments.\u201d\n\nAnother form, and a very scholarly one, of the argument against\nthe existence of an Absolute has been stated by Bax in the \u201cRoots\nof Reality.\u201d He appears to have reached the conclusion that the\n_telos_, the goal of human thought, is not an Absolute involving\nany notion of fixity, but that it may be conceived of as a \u201cmoving\nsynthesis.\u201d He argues that everything of which we are conscious in\nthe universe is seen against a background which itself moves, and\nis only realisable or distinguishable if it shifts upon something\nrelatively motionless behind it. He concludes, therefore, that by\nanalogy there is no Absolute, since what we perceive always implies\nsomething against which we perceive it; thus that there is no goal by\nwhich and at which the spirit of man can find rest. On his theory we\ncould never claim to reach the conception of an Absolute, though he\nadmits the progressive character of human thought, and the increasing\nreach, lucidity, and depth of the human mind. The true answer to this\nargument is that it proves exactly what it sets out to disprove.\nAs it is acknowledged that only the permanent or the relatively\npermanent can produce the phenomena of change, so _the appearance\nof the goal of thought as a moving synthesis would presuppose an\nAbsolute as a ground reality_.[19]\n\nIf in truth we were able to apprehend entirely the source of all\nlife and the background of all experience, we might say that it did\nnot exist for us _as an Absolute_, but the fact that whatever we\nperceive postulates an unending series behind it, carries with it\nthe proof of an Absolute Infinite. (This conclusion is led up to by\nthe mathematician\u2019s idea of the series of all finite and transfinite\nordinal numbers.) Some part of this argument has been already\nsuggested in Ormond\u2019s \u201cFoundations of Knowledge,\u201d and so far was used\nby Mr. Illingworth in the \u201cDoctrine of the Trinity.\u201d[20]\n\n\u201cFrom a deeper metaphysical point of view it is the concept of\nevolution itself that must submit to the determination of knowledge,\nfor it will be found that in so far as it becomes epistemologically\nnecessary to ground relative processes in an Absolute experience,\njust so far will it become necessary also to connect the evolutionary\naspect of the world itself with a ground reality that is stable,\nand involves the flux of change only as transcending and including\nit.\u201d[21]\n\nThe further answer that any judgment, even the Pragmatist\u2019s \u201cjudgment\nof value,\u201d implies an Absolute has been stated in his Oxford\nLectures[22] by Professor H. Jones.\n\n(_b_) The next point we should like to work out is the relation of\nfact to law. The Pragmatist denies scientific law and also logic,\nand makes his appeal to facts. No conclusion can be drawn from that\ndenial except by the use of logic itself. If he consistently denied\nlogic, his position would be unassailable by logic, but he uses the\nmethod he denies, and is thus open to attack. On the subject of the\nLaws of Science the Pragmatist points out truly that there is no\nactual continuity between a fact and a law. But laws are concepts,\nthe result of mental activities; they are themselves subject to the\nlaws of logic. \u201cThey were means, and you make them ends,\u201d complains\nthe Pragmatist. That is just what nature herself does. She perfects\nmeans, such as the means of supporting life, and then these become\nends. Language, again, is at first a means, and then becomes an end.\nSo does any science change its character to the onlooker. A law, too,\nthough it generalises facts, is a limit on absolute generalisation.\nIt thus stands midway between the abstraction and the fact. The\nPragmatist, however, opposes to law what he calls a new fact--what\nshould rather be called a hypothesis. He asserts that in every event,\naction, experiment, there is a margin unseen and unrecognised by\nus; that at every moment, therefore, the unknown, the unexpected,\nmay take shape and voice and denounce all our careful and reasoned\nconclusions. \u201cWhy should the sun rise to-morrow because he has\nrisen to-day and yesterday?\u201d asks the Pragmatist. \u201cWe are making\nan enormous assumption,\u201d he says, \u201cin claiming the uniformity of\nNature and the principle of causality.\u201d The Idealist answers that the\nPragmatist makes a larger assumption in doubting the truth of the\nprinciples, which though relative and not absolute, still do work\nout in practice, than the Idealist does in his act of faith. In fact,\nthe act of faith is rational as well as natural; it is the act of\ndoubting that is in this case due to a mere scholastic quibble. It is\nthe Idealist and not the Pragmatist who makes his appeal to the truth\nof facts. Each day that the sun goes on rising finds the Idealist in\na better philosophical position and the Pragmatist in a worse, except\non the assumption that the link between man and the external world\nis a false imagination. Let us emphasise:--It is the Pragmatist who\nquibbles with logic, and the Idealist who appeals to facts.\n\n(_c_) Now there are certain facts and certain deductions from facts,\nwell known to mathematicians, which we should like to quote here as\nhaving a bearing on the theory of the Absolute, because they deal\nwith aspects of Infinity, and mark a connection between the world as\nwe know it and the concepts of the philosopher. All have the support\nof science, and furnish the Idealist philosopher with examples which\nsupport his theories, and strengthen his position in the face of the\nPragmatist attack. They have to do with the theory of Infinity as\nshown in:--\n\nI. The Indefinite Regress.\n\nII. Infinite series.\n\nIII. Dimensions in space and time.\n\nBefore entering upon them we must repeat that the question of number\nand series in mathematics is independent of the assumptions of space\nand time. As a science, mathematics could exist outside them: order\nis not necessarily spatial or temporal. Our conclusions, therefore,\ncannot be attacked on the ground that they are based on Euclidean\nconceptions of space: they are based on the laws of logic.\n\n\n                      I. THE INDEFINITE REGRESS\n\nHume and, later, Kant argued that by the principle of association\nwhen we think of one quality of a thing the others are naturally\nbrought before our minds, and thus that we get into the habit of\nattributing to the notion of the thing a certain group of qualities.\nAnd it is true that we do attend to a thing all at once, including\nin the notion of it all the qualities which we know belong to it.\n\nNow experience, according to Leibniz, gives us an example of a\nunity which embraces a multiplicity of detail. Thus a thing is one\nsubstance as embodying an individual experience, and its qualities\nbelong to it in the same sense as the constituents of experience\nbelong to the single experience. These qualities are in relation.\n(The Pragmatist denies the existence of relations as part of a\nhigher unity.[23]) But they are not only relation, since relation\nalways implies something more than itself. Let us take the example\nof number. Numbers could never have been counted if there had not\nbeen things to count. Now suppose each quality could be analysed\ninto a new relation, we should still not get rid of the quality. At\neach stage there remains a quality in relation, and this goes on\nto Infinity. Such a constant subdivision perhaps results from our\nfinite experience seizing facts in a disjointed way. When we analyse\na law in its working, we always do seem to come to this Indefinite\nRegress. Now it has been the reproach against metaphysics, as uttered\nby the Pragmatist, that there is no correspondence in scientific fact\nto this road into Infinity.\n\nW. James asserts: \u201cBut in point of fact, nature doesn\u2019t make eggs by\nmaking first half an egg, then a quarter, then an eighth, &c., and\nadding them together. She either makes a whole egg at once or none\nat all, and so of all her other units. It is only in the sphere of\nchange, then, where one phase of a thing must needs come into being\nbefore another phase can come, that Zeno\u2019s paradox gives trouble.\nAnd it gives trouble then only if the successive steps of change be\ninfinitely divisible.\u201d[24]\n\nThe sphere of change, however, one would answer, includes all nature,\nand science in its discoveries acts on the hypothesis that these\nsteps of change may be infinitely divisible. Royce held to it firmly\nthat any consistent attempt to make an orderly arrangement of the\nterms of an infinite whole must lead to the Indefinite Regress. And\nhe further shows the connection with the fact that an infinite series\ncan be adequately represented by a part of itself.\n\nIn the Boyle Lecture, delivered in Oxford in 1908, on the properties\nof radium, two facts emerged which show that the Indefinite Regress\nis now recognised in science.\n\nFirst, that in the region of experiment we become aware of groups of\nelements allied to radium, which seem, in the number of individuals\nin their groups, to follow a simple arithmetical progression.\n\nSecondly, that radio-active elements lose in activity at a certain\nrate, which always represents an exact proportion of the mass which\nremains. The tremendous disintegrating force slackens in exact\nrelation to the time which passes, so that the smaller the morsel\nthe less the relative loss of mass. Here, then, is the Indefinite\nRegress. In the world of fact as well as of ideas we are dealing with\naspects of Infinity.[25]\n\n\n                         II. INFINITE SERIES\n\nThere are other aspects of Infinity which we can get at by studying\nseries, and which in the conception of series of series give strength\nand point to the philosophic conception of an Absolute.\n\nProf. C. Keyser develops this thought, and shows (in two recent\narticles, January and April 1909, in the _Hibbert Journal_) that\ncertain theological dogmas, such as the doctrine of the Trinity,\nand certain attributes of the Divine Being, such as Omniscience and\nOmnipresence, are entirely conceivable by the human mind if regarded\nwithout the paralysing limitations of the Finite. He shows that in\nour mathematical formul\u00e6 which have to do with infinite series we\nhave the exact replica of what to the lay, non-mathematical mind seem\nto be the paradoxes of the Athanasian Creed. He first shows that in\na mathematical analogy points of view about an Infinite Being, even\nif partially discordant, may all be true if regard is had to His\nInfinity.[26]\n\nFurther, he shows that certain assumptions, such as _the whole\nis greater than its part_, are inapplicable to Infinite Being.\nThe conception of a Trinity in Unity in which \u201cnone is afore or\nafter other, none is greater or less than another, but the whole\nthree persons are co-eternal together and co-equal\u201d is rationally\nconceivable by the mathematician who is familiar with the theory of\nmanifolds.[27]\n\nWe have, he shows, three infinite manifolds:--\n\n    E of the even integers.\n\n    O of the odd ones.\n\n    F of the fractions having integers for their terms.\n\nNo two of these have a single element in common, yet the three\ntogether constitute one manifold M, that is exactly equal in wealth\nof elements to _each_ of its infinite components.\n\nAgain, there is the apparent opposition between the Omniscience of\nGod and the freedom of man. The antithesis disappears if we realise\nthat from the point of view of Infinites the dignity and power of\nOmniscience remain the same, even if some part of experience is not\nyet drawn into the sphere of Omniscience.[28]\n\nHere we have the present conceived of as a moving plane separating\nthe unknown from the known. The \u201cpast\u201d can be said to be known,\nthough its content changes every instant. This is the real answer\nto W. James\u2019s cry that he could accept an Absolute if it had even\nthe fragment of an \u201cother.\u201d There can be this \u201cother,\u201d and yet the\nAbsolute still remains an Absolute.\n\nThe doctrine of _Omnipresence_ follows from the argument of the\nContinuum (which is the aggregate of all real numbers). Thus the\nnumber of points in space of infinite dimension is no greater than\nthe number of points in any part of space as known to us. The whole\nis incarnate in every part, because to each part, in however small an\natom, corresponds a point in the universal whole, and the number of\npoints in a space of infinite dimensions is equal to the number of\npoints in a straight line however small.\n\nAnd this is true not merely of points but also of forces. \u201cThe\nUniverse is dynamic, charged throughout with innumerable modes of\nmotion. Each point, however, of any moving thing--an ion of gas, a\nvibrating fibre of brain--is represented by a corresponding point\nin S (a small typical atom), and so within the tiny sphere--indeed,\nin every room, however small--the whole dynamics of the universe\nis depicted completely and co-enacted by motion of points and\ntransformation of point configurations. There in miniature proceed at\nonce the countless play and interplay of every kind of motion, small\nand large, simple and complex, the quivering dance of the molecule,\nthe wave and swing of universal \u00e6ther.\u201d[29]\n\n\n                      III. DIMENSIONS IN SPACE\n\nThere is another argument, one relating to the theories of time\nand space, which greatly affects the conception of omnipresence.\nThis is the argument of the many dimensions, called by Keyser the\n\u201cradiant concept of hyper-space, which only a generation ago was\nregarded, even by mathematicians--most adventurous of men--as being\npurposeless and vain, but which meanwhile has advanced so rapidly to\ncommanding position that even the following statement by Poincar\u00e9,\nin his recent address before the International Mathematical Congress\nat Rome on \u2018L\u2019Avenir des Math\u00e9matiques,\u2019 is well within the limits\nof conservatism: \u2018Nous sommes aujourd\u2019hui tellement familiaris\u00e9s\navec cette notion que nous pouvons en parler, m\u00eame dans un cours\nd\u2019universit\u00e9, sans provoquer trop d\u2019\u00e9tonnement.\u2019 The fact is that the\ndoctrine already exists in a vast and rapidly growing literature,\nflourishes in all the scientific languages of the world, and in its\nessential principles has become for mathematics as orthodox as the\nmultiplication table.\u201d\n\nThe present position of the theory is briefly this: If there did\nnot exist a fourth dimension, we could not be aware of a third as\nsuch, and so on. Are we then looking out upon a third dimensional\nworld, and realising it as such because we are mentally capable of\nconceiving dimensions beyond it? Our world sensibly contains one\ndimensional and two dimensional facts--the first such as a time\nseries, for which one number is sufficient to fix a point, and the\nsecond such as a plane where position can be fixed by two numbers.\nDoes our world contain facts of other dimensions?\n\n\u201cAll particles of air are four-dimensional in magnitude when,\nin addition to their position in space, we also consider the\nvariable densities which they assume, as they are expressed\nby the different heights of the barometer in the different\nparts of the atmosphere. Similarly all conceivable spheres in\nspace are four-dimensional magnitudes, for their centres form\na three-dimensional point-aggregate, and around each centre a\none-dimensional totality of spheres, the radii of which can be\nexpressed by every numerical magnitude from zero to infinity.\nFurther, if we imagine a measuring-stick of invariable length to\nassume every conceivable position in space, the positions so obtained\nwill constitute a five-dimensional aggregate. For in the first place\none of the extremities of the measuring-stick may be conceived to\nassume a position at every point of space, and this determines for\none extremity alone of the stick a three-dimensional totality of\nposition, and, secondly, as we have seen above, there proceeds from\nevery such position of this extremity a two-dimensional totality\nof directions, and by conceiving the measuring-stick to be placed\nlengthwise in every one of these directions, we shall obtain all the\nconceivable positions which the second extremity can assume, and\nconsequently the dimensions must be 3 + 2 or 5 \u2026\u201d &c., &c.[30]\n\nMathematicians have for long done problems in the seventh and eighth\ndimensions. They have told us that you cannot tie a knot in the\nsecond dimension, because there is no up or down, and the threads\nwould not cross--nor in the fourth, because the knot would pull out\nin a new direction and would not hold. But it has only lately been\nrealised that fourth and other dimensions may be actual fact in the\nworld round us. Of course, from the point of view of a point there\nare only three dimensions to be known, but to a line in the same\nspace there are five, to the surface probably six. Our intelligence\nat present does not go beyond the point; but if we could think of\nspace from the point of view of a solid, worlds upon worlds would\nrise before our view.\n\nOf the fourth dimension we can discover some facts by analogy. We\ncan count the edges of its typical figure, and apply thought to\ndetermining some of its conditions. But a more interesting subject\nof research is the inquiry into the light thrown by the theory of\nfour dimensions on the determination of certain atoms in chemistry,\nthat are known to be distinct elements, but could only be determined\nactually in another dimension.[31]\n\n\u201cIn chemistry, the molecules of a compound body are said to consist\nof the atoms of the elements which are contained in the body, and\nthese are supposed to be situated at certain distances from one\nanother and to be held in their relative positions by certain\nforces. At first the centres of the atoms were conceived to lie in\none and the same plane. But Wislicenus was led by researches in\nparalactic acid to explain the differences of isomeric molecules of\nthe same structural formul\u00e6 by the different positions of the atoms\nin _space_. In fact, four points can always be so arranged in space\nthat every two of them may have any distance from each other; and the\nchange of one of the six distances does not necessarily involve the\nalteration of any other.\n\n\u201cBut suppose our molecule consists of five atoms? Four of these\nmay be so placed that the distance between any two of them can be\nmade what we please. But it is no longer possible to give the fifth\natom a position such that each of the four distances by which it\nis separated from the other atoms may be what we please. On the\ncontrary, the fourth distance is dependent on the three remaining\ndistances, for the space of experience has only three dimensions. If,\ntherefore, I have a molecule which consists of five atoms, I cannot\nalter the distance between two of them without at least altering some\nsecond distance. But if we imagine the centres of the atoms placed\nin a four-dimensioned space, this can be done; all the ten distances\nwhich may be conceived to exist between the five points will then be\nindependent of one another. To reach the same result in the case of\nsix atoms we must assume a five-dimensional space, and so on.\u201d[32]\n\nHere we see that if chemistry as a science is bound to take account\nof all its facts, the scientist is confronted with a problem of\ndimensions that is really a problem of Infinity applied not, as in\nthe other cases quoted, to number, but to space.\n\nAnd there is a reason which explains why the same problem tends\nto appear in these different ways. Both time and space can be\nmost correctly thought of as _series_: the former known to us as\npossessing one direction, though possibly involving more, and the\nlatter three, though possibly involving more. Time is not a thing nor\na condition, but it is the way in which we are enabled to apprehend\nthe relations of actions to one another. The assumption of the\nPragmatist, that a different date in history is a new condition which\nmight affect a chemical experiment, is meaningless, unless by that he\nintends to say that at the different date new conditions prevailed.\n\nThe general conclusion of recent thought is then to establish\nthe Idealist position more strongly by an appeal to mathematical\nargument. This argument is strengthened by finding at the present\ntime some support in scientific fact and experiment. The Idealist\ntherefore appeals to fact, and his position rests ultimately on a\ntruth which has its aspects of conformity with scientific experiment\nand with logical or mathematical proof.\n\n\nFOOTNOTES:\n\n     [15] This word is used here in the most general and\n          inclusive sense as applying to all thinkers who accept\n          the reality of relations as part of a higher Unity.\n\n     [16] \u201cA Pluralistic Universe,\u201d p. 99.\n\n     [17] Taylor, \u201cElements of Metaphysics,\u201d p. 312.\n\n     [18] _Ibid._, p. 350.\n\n     [19] A succession of what is disconnected is not\n          change. Change is a succession within an identity:\n          if not within the identity, there is no change, only\n          analysis and re-grouping. The closer our knowledge is\n          of ourselves or anything else, the more we see that\n          _change is the expression in time of an identity_.\n\n     [20] Illingworth, \u201cThe Doctrine of the Trinity,\u201d p. 6.\n\n     [21] Ormond, \u201cFoundations of Knowledge,\u201d p. 19.\n\n     [22] 1908-10.\n\n     [23] James, \u201cA Pluralistic Universe,\u201d p. 80.\n\n     [24] \u201cA Pluralistic Universe,\u201d p. 230.\n\n     [25] See A. T. Cameron, \u201cRadio-Chemistry,\u201d p. 17: \u201cThe\n          curves illustrate two further points. They approach\n          constant value towards the end of a month, but it is\n          seen that they reach a final value only at infinite\n          time. This property is common to all such curves; it\n          illustrates the fact that the _life_ of a radio-active\n          element is infinite.\u201d It is explained in the same book\n          (p. 31) that \u201cinfinity is only a relative term; in\n          this connection it only means a longer time than we\n          can measure.\u201d\n\n     [26] His theology is not so good as his mathematics;\n          he seems to think that in the Creed we assert our\n          belief in the Incomprehensible, in the sense of\n          that which is \u201cnot capable of being seized by the\n          mind,\u201d instead of in that which is \u201cuntrammelled by\n          limitations.\u201d The word is _immensus_, best translated\n          infinite.\n\n     [27] _Hibbert Journal_, 1909, pp. 626-28.\n\n     [28] _Hibbert Journal_, 1909, p. 629.\n\n     [29] _Hibbert Journal_, 1909, p. 632.\n\n     [30] Schubert, \u201cMathematical Essays and Recreations,\u201d\n          pp. 70, 71.\n\n     [31] See Van t\u2019 Hoff, _La Chimie dans l\u2019espace_, and\n          Schubert, \u201cMathematical Essays and Recreations,\u201d pp.\n          88-89.\n\n     [32] Schubert, \u201cMathematical Essays and Recreations,\u201d\n          pp. 88-89. See also Mach, \u201cConservation of Energy\u201d\n          (trans. Open Court Publishing Co.).\n\n\n\n\n                 Printed by BALLANTYNE, HANSON & CO.\n                         Edinburgh & London\n\n\n\n\nTranscriber\u2019s Note:\n\n\nWords and phrases in italics are surrounded by underscores, _like\nthis_. Footnotes were renumbered sequentially and were moved to the\nend of the chapter. Words may have inconsistent hyphenation in the\ntext. One misspelled word was corrected.","title":"On the theory of the infinite in modern thought : $b Two introductory studies","tradition":"philosophy-ethics","translated_text":null,"word_count":10034}