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 For he has to give some explanation of the nature of space and time which shall identify these with impressions, and at the same time is compelled to recognize the fact that they are not identical with any single impression or set of impressions. Putting aside, then, the various obscurities of terminology, such as the distinction between the objects known, viz. “points” or several mental states, and the impressions themselves, which disguise the full significance of his conclusion, we find Hume reduced to the following as his theory of space and time. Certain impressions, the sensations of sight and touch, have in themselves the element of space, for these impressions (Hume skilfully transfers his statement to the points) have a certain order or mode of arrangement. This mode of arrangement or manner of disposition is common to coloured points and tangible points, and, considered separately, is the impression from which our idea of space is taken. All impressions and all ideas are received, or form parts of a mental experience only when received, in a certain order, the order of succession. This manner of presenting themselves is the impression from which the idea of time takes its rise.

It is almost superfluous to remark, first, that Hume here deliberately gives up his fundamental principle that ideas are but the fainter copies of impressions, for it can never be maintained that order of disposition is an impression, and, secondly, that he fails to offer any explanation of the mode in which coexistence and succession are possible elements of cognition in a conscious experience made up of isolated presentations and representations. For the consistency of his theory, however, it was indispensable that he should insist upon the real, i.e. presentative character of the ultimate units of space and time.

(b) How then are the primary data of mathematical cognition to be derived from an experience containing space and time relations in the manner just stated? It is important to notice that Hume, in regard to this problem, distinctly separates geometry from algebra and arithmetic, i.e. he views

extensive quantity as being cognized differently from number. With regard to geometry, he holds emphatically that it is an empirical doctrine, a science founded on observation of concrete facts. The rough appearances of physical facts, their outlines, surfaces and so on, are the data of observation, and only by a method of approximation do we gradually come near to such propositions as are laid down in pure geometry. He definitely repudiates a view often ascribed to him, and certainly advanced by many later empiricists, that the data of geometry are hypothetical. The ideas of perfect lines, figures and surfaces have not, according to him, any existence. (See Works, i. 66, 69, 73, 97 and iv. 180.) It is impossible to give any consistent account of his doctrine regarding number. He holds, apparently, that the foundation of all the science of number is the fact that each element of conscious experience is presented as a unit, and adds that we are capable of considering any fact or collection of facts as a unit. This manner of conceiving is absolutely general and distinct, and accordingly affords the possibility of an all-comprehensive and perfect science, the science of discrete quantity. (See Works, i. 97.)

(c) In respect to the third point, the nature, extent and certainty of the elementary propositions of mathematical science, Hume’s utterances are far from clear. The principle with which he starts and from which follows his well-known distinction between relations of ideas and matters of fact, a distinction which Kant appears to have thought identical with his distinction between analytical and synthetical judgments, is comparatively simple. The ideas of the quantitative aspects of phenomena are exact representations of these aspects or quantitative impressions; consequently, whatever is found true by consideration of the ideas may be asserted regarding the real impressions. No question arises regarding the existence of the fact represented by the idea, and in so far, at least, mathematical judgments may be described as hypothetical. For they simply assert what will be found true in any conscious experience containing coexisting impressions of sense (specifically, of sight and touch), and in its nature successive. That the propositions are hypothetical in this fashion does not imply any distinction between the abstract truth of the ideal judgments and the imperfect correspondence of concrete material with these abstract relations. Such distinction is quite foreign to Hume, and can only be ascribed to him from an entire misconception of his view regarding the ideas of space and time. (For an example of such misconception, which is almost universal, see Riehl, Der philosophische Kriticismus, i. 96, 97.)

(d) From this point onwards Hume’s treatment becomes exceedingly confused. The identical relation between the ideas of space and time and the impressions corresponding to them apparently leads him to regard judgments of continuous and discrete quantity as standing on the same footing, while the ideal character of the data gives a certain colour to his inexact statements regarding the extent and truth of the judgments founded on them. The emphatic utterances in the Inquiry (iv. 30, 186), and even at the beginning of the relative section in the Treatise (i. 95) may be cited in illustration. But in both works these utterances are qualified in such a manner as to enable us to perceive the real bearings of his doctrine, and to pronounce at once that it differs widely from that commonly ascribed to him. “It is from the idea of a triangle that we discover the relation of equality which its three angles bear to two right ones; and this relation is invariable, so long as our idea remains the same” (i. 95). If taken in isolation this passage might appear sufficient justification for Kant’s view that, according to Hume, geometrical judgments are analytical and therefore perfect. But it is to be recollected that, according to Hume, an idea is actually a representation or individual picture, not a notion or even a schema, and that he never claims to be able to extract the predicate of a geometrical judgment by analysis of the subject. The properties of this individual subject, the idea of the triangle, are, according to him, discovered by observation, and as observation, whether actual or ideal, never presents us with more than the rough or general appearances of geometrical quantities, the relations so discovered have only approximate exactness. “Ask a mathematician what he means when he pronounces two quantities to be equal, and he must say that the idea of equality is one of those which cannot be defined, and that it is sufficient to place two equal quantities before any one in order to suggest it. Now this is an appeal to the general appearances of objects to the imagination or senses” (iv. 180). “Though it (i.e. geometry) much excels, both in universality and exactness, the loose judgments of the senses and imagination, yet [it] never attains a perfect precision and exactness” (i. 97). Any exactitude attaching to the conclusions of geometrical reasoning arises from the comparative simplicity of the data for the primary judgments.

So far, then, as geometry is concerned, Hume’s opinion is perfectly definite. It is an experimental or observational science, founded on primary or immediate judgments (in his phraseology, perceptions), of relation between facts of intuition; its conclusions are hypothetical only in so far as they do not imply the existence at the moment of corresponding real experience; and its propositions have no exact truth. With respect to arithmetic and algebra, the science of numbers, he expresses an equally definite opinion, but unfortunately it is quite impossible to state in any satisfactory fashion the grounds for it or even its full bearing. He nowhere explains the origin of the notions of unity and number, but merely asserts that through their means we can have absolutely exact arithmetical propositions (Works, i. 97, 98). Upon the nature of the reasoning by which in mathematical science we pass from data to conclusions, Hume gives no explicit statement. If we were to say that on his view the essential step must be the establishment of identities or equivalences, we should probably be doing justice to his doctrine of numerical reasoning, but should have some difficulty in showing the application of the method to geometrical reasoning. For in the latter case we possess, according to Hume, no standard of equivalence other than that supplied by immediate observation, and consequently transition from one premise to another by way of reasoning must be, in geometrical matters, a purely verbal process.

Hume’s theory of mathematics—the only one, perhaps, which is compatible with his fundamental principle of psychology—is a practical condemnation of his empirical theory of perception. He has not offered even a plausible explanation of the mode by which a consciousness made up of isolated momentary impressions and ideas can be aware of coexistence and number, or succession. The relations of ideas are accepted as facts of immediate observation, as being themselves perceptions or individual elements of conscious experience, and to all appearance they are regarded by Hume as being in a sense analytical, because the formal criterion of identity is applicable to them. It is applicable, however, not because the predicate is contained in the subject, but on the principle of contradiction. If these judgments are admitted to be facts of immediate perception, the supposition of their non-existence is impossible. The ambiguity in his criterion, however, seems entirely to have escaped Hume’s attention.

A somewhat detailed consideration of Hume’s doctrine with regard to mathematical science has been given for the reason that this portion of his theory has been very generally overlooked or misinterpreted. It does not seem necessary to endeavour to follow his minute examination of the principle of real

cognition with the same fulness. It will probably be sufficient to indicate the problem as conceived by Hume, and the relation of the method he adopts for solving it to the fundamental doctrine of his theory of knowledge.

Real cognition, as Hume points out, implies transition from the present impression or feeling to something connected with it. As this thing can only be an impression or perception, and is not itself present, it is represented by its copy or idea. Now the supreme, all-comprehensive link of connexion between present feeling or impression and either past or future experience is that of causation. The idea in question is, therefore, the idea of something connected with the present impression as its cause or effect. But this is explicitly the idea of the said thing as having had or as about to have existence,—in other words, belief in the existence of some matter of fact. What, for a conscious experience so constituted as Hume will admit, is the precise significance of such belief in real existence?

Clearly the real existence of a fact is not demonstrable. For whatever is may be conceived not to be. “No negation of a fact can involve a contradiction.” Existence of any fact, not present as a perception, can only be proved by arguments from cause or effect. But as each perception is in consciousness only as a contingent fact, which might not be or might be other than it is, we must admit that the mind can conceive no necessary relations or connexions among the several portions of its experience.