CHAPTER LII¶
Part VI — Space — Kant’s Theory of Space (§§ 432–436, pp. 454–461)
§ 432. The present work is diametrically opposed to Kant¶
432. In the present chapter I do not propose to undertake a minute or textual examination of Kant’s opinions; this has been done elsewhere, and notably in Vaihinger’s monumental commentary, so well that it need not be done over again here. It is only the broad outlines of the Kantian doctrine that I wish to discuss. This doctrine, more or less modified, has held the field for over a century, and has won a nearly universal acceptance. As my views are, on almost every point of mathematical theory, diametrically opposed to those of Kant, it becomes necessary explicitly to defend the opinions in which I differ from him [1]. In this I shall pay special attention to what Kant calls the transcendental arguments, i.e. those derived from the nature of mathematics.
§ 433. Summary of Kant’s theory¶
433. Broadly speaking, the way in which Kant seeks to deduce his theory of space from mathematics (especially in the Prolegomena) is as follows. Starting from the question: “How is pure mathematics possible?” Kant first points out that all the propositions of mathematics are synthetic. He infers hence that these propositions cannot, as Leibniz had hoped, be proved by means of a logical calculus; on the contrary, they require, he says, certain synthetic à priori propositions, which may be called axioms, and even then (it would seem) the reasoning employed in deductions from the axioms is different from that of pure logic. Now Kant was not willing to admit that knowledge of the external world could be obtained otherwise than by experience; hence he concluded that the propositions of mathematics all deal with something subjective, which he calls a form of intuition. Of these forms there are two, space and time; time is the source of Arithmetic, space of Geometry. It is only in the forms of time and space that objects can be experienced by a subject; and thus pure mathematics must be applicable to all experience. What is essential, from the logical point of view, is, that the à priori intuitions supply methods of reasoning and inference which formal logic does not admit; and these methods, we are told, make the figure (which may of course be merely imagined) essential to all geometrical proofs. The opinion that time and space are subjective is reinforced by the antinomies, where Kant endeavours to prove that, if they be anything more than forms of experience, they must be definitely self-contradictory.
In the above outline I have omitted everything not relevant to the philosophy of mathematics. The questions of chief importance to us, as regards the Kantian theory, are two, namely, (1) are the reasonings in mathematics in any way different from those of Formal Logic? (2) are there any contradictions in the notions of time and space? If these two pillars of the Kantian edifice can be pulled down, we shall have successfully played the part of Samson towards his disciples.
§ 434. Mathematical reasoning requires no extra-logical element¶
434. The question of the nature of mathematical reasoning was obscured in Kant’s day by several causes. In the first place, Kant never doubted for a moment that the propositions of logic are analytic, whereas he rightly perceived that those of mathematics are synthetic. It has since appeared that logic is just as synthetic as all other kinds of truth; but this is a purely philosophical question, which I shall here pass by [2]. In the second place, formal logic was, in Kant’s day, in a very much more backward state than at present. It was still possible to hold, as Kant did, that no great advance had been made since Aristotle, and that none, therefore, was likely to occur in the future. The syllogism still remained the one type of formally correct reasoning; and the syllogism was certainly inadequate for mathematics. But now, thanks mainly to the mathematical logicians, formal logic is enriched by several forms of reasoning not reducible to the syllogism [3], and by means of these all mathematics can be, and large parts of mathematics actually have been, developed strictly according to the rules. In the third place, in Kant’s day, mathematics itself was, logically, very inferior to what it is now. It is perfectly true, for example, that any one who attempts, without the use of the figure, to deduce Euclid’s seventh proposition from Euclid’s axioms, will find the task impossible; and there probably did not exist, in the eighteenth century, any single logically correct piece of mathematical reasoning, that is to say, any reasoning which correctly deduced its result from the explicit premisses laid down by the author. Since the correctness of the result seemed indubitable, it was natural to suppose that mathematical proof was something different from logical proof. But the fact is, that the whole difference lay in the fact that mathematical proofs were simply unsound. On closer examination, it has been found that many of the propositions which, to Kant, were undoubted truths, are as a matter of fact demonstrably false [4]. A still larger class of propositions—for instance, Euclid’s seventh proposition mentioned above—can be rigidly deduced from certain premisses, but it is quite doubtful whether the premisses themselves are true or false. Thus the supposed peculiarity of mathematical reasoning has disappeared.
The belief that the reasonings of Geometry are in any way peculiar has been, I hope, sufficiently refuted already by the detailed accounts which have been given of these reasonings, and especially by Chapter xlix. We have seen that all geometrical results follow, by the mere rules of logic, from the definitions of the various spaces. And as regards the opinion that Arithmetic depends upon time, this too, I hope, has been answered by our accounts of the relation of Arithmetic to Logic. Indeed, apart from any detail, it seems to be refuted by the simple observation that time must have parts, and therefore plurality, whole and part, are prior to any theory of time. All mathematics, we may say—and in proof of our assertion we have the actual development of the subject—is deducible from the primitive propositions of formal logic: these being admitted, no further assumptions are required.
But admitting the reasonings of Geometry to be purely formal, a Kantian may still maintain that an à priori intuition assures him that the definition of three-dimensional Euclidean space, alone among the definitions of possible spaces, is the definition of an existent, or at any rate of an entity having some relation to existents which other spaces do not have. This opinion is, strictly speaking, irrelevant to the philosophy of mathematics, since mathematics is throughout indifferent to the question whether its entities exist. Kant thought that the actual reasoning of mathematics was different from that of logic; the suggested emendation drops this opinion, and maintains merely a new primitive proposition, to the effect that Euclidean space is that of the actual world. Thus, although I do not believe in any immediate intuition guaranteeing any such primitive proposition, I shall not undertake the refutation of this opinion. It is enough, for my purpose, to have shown that no such intuition is relevant in any strictly mathematical proposition.
§ 435. Kant’s mathematical antinomies¶
435. It remains to discuss the mathematical antinomies. These are concerned with infinity and continuity, which Kant supposed to be specially spatio-temporal. We have already seen that this view is mistaken, since both occur in pure Arithmetic. We have seen also in Part V (especially in Chapter xlii) that the supposed antinomies of infinity and continuity, in their arithmetical form, are soluble; it remains to prove the same conclusion concerning Kant’s spatio-temporal form. The third and fourth antinomies are not relevant here, since they involve causality; only the first two, therefore, will be examined.
First Antinomy. Thesis: “The world has a beginning in time, and as regards space also is enclosed within limits.” This statement is not concerned with pure time and pure space, but with the things in them. The proof, such as it is, applies in the first instance to time only, and is effected by reductio ad absurdum. “For assume,” it says, “that the world has no beginning in time, then an eternity has passed away (abgelaufen) before every given point of time, and consequently an infinite series of conditions of the things in the world has happened. But the infinity of a series consists in this, that it can never be completed by successive synthesis. Consequently an infinite past series of things in the world (Weltreihe) is impossible, and a beginning of the world is a necessary condition of its existence, which was first to be proved.”
This argument is difficult to follow, and suggests a covert appeal to causality and the supposed necessity for a first cause. Neglecting this aspect of the argument, it would seem that, like most of the arguments against infinity, it fails to understand the use of the class-concept and the word any. It is supposed—so it would seem—that the events preceding a given event ought to be definable by extension, which, if their number is infinite, is obviously not the case. “Completion by successive synthesis” seems roughly equivalent to enumeration, and it is true that enumeration of an infinite series is practically impossible. But the series may be none the less perfectly definable, as the class of terms having a specified relation to a specified term. It then remains a question, as with all classes, whether the class is finite or infinite; and in the latter alternative, as we saw in Part V, that there is nothing self-contradictory. In fact, to elicit a contradiction, it would be necessary to state as an axiom that every class must have a finite number of terms—an axiom which can be refuted, and for which there are no grounds. It seems, however, that previous events are regarded by Kant as causes of later ones, and that the cause is supposed to be logically prior to the effect. This, no doubt, is the reason for speaking of conditions, and for confining the antinomy to events instead of moments. If the cause were logically prior to the effect, this argument would, I think, be valid; but we shall find, in Part VII, that cause and effect are on the same logical level. Thus the thesis of the first antinomy, in so far as it concerns time, must be rejected as false, and the argument concerning space, since it depends upon that regarding time, falls also.
Antithesis. “The world has no beginning, and no limits in space, but is infinite both in respect of time and space.” The proof of this proposition assumes the infinity of pure time and space, and argues that these imply events and things to fill them. This view was rejected, as regards space, in the preceding chapter, and can be disproved, as regards time, by precisely similar arguments; it is in any case irrelevant to our contention, since no proof is offered that time and space are themselves infinite. This, in fact, seems incapable of proof, since it depends upon the merely self-evident axiom that there is a moment before any given moment, and a point beyond any given point. But as no converse proof is valid, we may, in this instance, regard the self-evident as true. Whether events had a beginning, and whether matter is bounded by empty space, are questions which, if our philosophy of space and time be sound, no argument independent of causality can decide.
Second Antinomy. Thesis: “Every complex substance in the world consists of simple parts, and nothing exists anywhere except the simple, or what is composed of simple parts.” Here, again, the argument applies to things in space and time, not to space and time themselves. We may extend it to space and time, and to all collections, whether existent or not. It is indeed obvious that the proposition, true or false, is concerned purely with whole and part, and has no special relation to space and time. Instead of a complex substance, we might consider the numbers between 1 and 2, or any other definable collection. And with this extension, the proof of the proposition must, I think, be admitted; only that terms or concepts should be substituted for substances, and that, instead of the argument that relations between substances are accidental (zufällig), we should content ourselves with saying that relations imply terms, and complexity implies relations.
Antithesis. “No complex thing in the world consists of simple parts, and nothing simple exists in it anywhere.” The proof of this proposition, as of the first antithesis, assumes, what is alone really interesting, to us, the corresponding property of space. “Space,” Kant says, “does not consist of simple parts, but of spaces.” This dogma is regarded as self-evident, though all employment of points shows that it is not universally accepted. It appears to me that the argument of the thesis, extended as I have just suggested, applies to pure space as to any other collection, and demonstrates the existence of simple points which compose space. As the dogma is not argued, we can only conjecture the grounds upon which it is held. The usual argument from infinite division is probably what influenced Kant. However many parts we divide a space into, these parts are still spaces, not points. But however many parts we divide the stretch of ratios between 1 and 2 into, the parts are still stretches, not single numbers. Thus the argument against points proves that there are no numbers, and will equally prove that there are no colours or tones. All these absurdities involve a covert use of the axiom of finitude, i.e. the axiom that, if a space does consist of points, it must consist of some finite number of points. When once this is denied, we may admit that no finite number of divisions of a space will lead to points, while yet holding every space to be composed of points. A finite space is a whole consisting of simple parts, but not of any finite number of simple parts. Exactly the same thing is true of the stretch between 1 and 2. Thus the antinomy is not specially spatial, and any answer which is applicable in Arithmetic is applicable here also. The thesis, which is an essential postulate of Logic, should be accepted, while the antithesis should be rejected.
Thus Kant’s antinomies do not specially involve space and time: any other continuous series, including that of real numbers, raises the same problems. And what is more, the properties of space and time, to which Kant appeals, are general properties of such series. Other antinomies than Kant’s—e.g. that concerning absolute and relative position, or concerning the straight line as both a relation and a collection of points—have been solved in the preceding chapters of this Part. Kant’s antinomies, which involve the difficulties of infinity, are by far the most serious, and these being essentially arithmetical, have been already solved in Part V.
§ 436. Summary of Part VI¶
436. Before proceeding to matter and motion, let us briefly recapitulate the results of this Part. Geometry, we said, is the study of series having more than one dimension; and such series arise wherever we have a series whose terms are series. This subject is important in pure mathematics, because it gives us new kinds of order and new methods of generating order. It is important in applied mathematics, because at least one series of several dimensions exists, namely, space. We found that the abstract logical method, based upon the logic of relations, which had served hitherto, was still adequate, and enabled us to define all the classes of entities which mathematicians call spaces, and to deduce from the definitions all the propositions of the corresponding Geometries. We found that the continuity and infinity of a space can always be arithmetically defined, and that no new indefinables occur in Geometry. We saw that the philosophical objections to points raised by most philosophers are all capable of being answered by an amended logic, and that Kant’s belief in the peculiarity of geometrical reasoning, and in the existence of certain antinomies peculiar to space and time, has been disproved by the modern realization of Leibniz’s universal characteristic. Thus, although we discussed no problems specially concerned with what actually exists, we incidentally answered all the arguments usually alleged against the existence of an absolute space. Since common sense affirms this existence, there seems therefore no longer any reason for denying it; and this conclusion, we shall find, will give us the greatest assistance in the philosophy of Dynamics.
Author’s Footnotes (1903 Edition)