For he has to give some explanation of the nature of space and time which shall 1dent1fy these with 1mpress1ons, and at the same t1me is compelled to recognize the fact that they are not identical w1th any s1ngle 1mpress1on or set of impressions. Putting aside, then, the var1o11s obscur1t1es of terminology, such as the distinction bctween the obJects known, VIZ. ' po1nts ” or several mental states, and the 1mpress1ons themselves, which disguise the full significance of h1s conclusion, we find Hume reduced to the following as h1s theory of space and time Certain 1mpress1ons, the sensations of sight and touch, hae in themselves the element of space, for these 1mpress1ons (Hume skilfully transfers hlS statement to the poznls) have a certain order or mode of arrangement. This mode of arrangement or manner of d1spos1t1on IS common to coloured points and tangible po1nts, and, 1 ons1dered separately, IS the impression from which our 1dea of space lb 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. Th1s manner of presenting themselves IS the impression from which the 1dea of time takes 1ts r1se

lt IS almost superfluous to remark, first, that Hume here deliberately gl€S up hlS fundamental pr1nc1ple that ideas are but the fainter copies of 1mpress1ons, for it can never be maintained that order of disposition is an impression, and, secondly, that he fa1ls to offer any explanation of the mode IH which coexistence and successwn are possible elements of cognition in a conscious experience made up of isolated presentations and representations. For the consistency of h1s theory, however, it was indispensable that he should insist upon the real, 1, e. preventative 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 1n M th the manner just stated? It IS important to notice that mimi' Hume, IH regard to this problem, distinctly separates geometry from algebra and arithmetic, 1, .e. he views extensive quantity as being cognized differently from number th 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 approx1n1at1on do we gradually come near to such propositions as are laid down 1n pure geometry. He definitely repudiates a view often ascribed to him, and certainly advanced by many later emp1r1c1sts, that the data of geometry are hypothetical. The ideas of perfect lines, figures and surfaces have not, according to h1m, any existence. (See Works, 1 66, 69, 73, 97 and IV ISO) It IS 1mposs1ble 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 un1t, and adds that we are capable of considering any fact or collection of facts as a unit This manner of concewzng IS absolutely general and d1st1nct, and accord1ngiy affords the possibility of an all-comprehensive and perfect science, the science of discrete quantity. (See Works, 1 97)

(r) fn respect to the third point, the nature, extent and certainty of the elementary propositions of mathematical SCIGHCE, Hume's utterances are far fl'()lI'l clear. The principle with which he staits and from wh1ch follos h1s well known d1st1nct1on between relations of ideas and matters of fact, a distinction which Kant appears to have thought identical w1th his d1st1nct1on between analytical and synthetical Judgments, IS comparatively simple. The uleas of the quant1tat1ve 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 1mpress1ons l.o 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 w1ll he found true in any conscious experience containing coex1st1ng impressions of sense (spec1f1cally, of sight and touch), and 1n its nature successive That the propositions are hy pothet1cal 111 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 d1st1nct1on IS quite foreign to Hume, and can only be ascribed to hxm from an entire misconception of h1s view regarding the ideas of space and time (For an example of such m1sconcept1on, which is almost un1 ersal, see Riehl, Der phzlosophzsche Krztzczsmus, i 96, 97.) (d) From this po1nt onwards Hume's treatment becomes exceedingly confused The 1dent1cal relation between the ideas of space and time and the impressions corresponding to them apparently leads h1m 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 rega1d1ng the extent and truth of the Judgments founded on them. The emphatic utterances 1n the Inquzry (IV 30, 186), and even at the beginning of the relative section in the Treattse (1 95) may be cited in illustration. But 1n 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 h1m “ It IS from the idea of a triangle that we discover the relation of equality which 1ts three angles bear to two right ones; and this relation is invariable, so long as our idea remains the same ” (1. 95). If taken in isolation this passage might appear sufficient Just1f1cat1on for Kant's view that, according to Hunie, geometrical Judgments are analytical and therefore perfect. But it is to be recollected that, according to Hume, an 1dea lb actually a representation or individual picture, not a not1on 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 th1s 1nd1vidual subJect, the idea of the triangle, are, according to him, discovered by observation, and as observation, whether actual or ideal, never presents us with 1nore than the ro11gh or general appearances of geometrical quantities, the relations so discovered have only approximate exactness. “ Ask a mathematician what he means when he pronounces two quant1t1es to be equal, and he must say that the idea of equaltty IS one of those which cannot be defined, and that It is sufficient to place two equal quant1t1es before any one in o1der to suggest 1t. Now this is an appeal to the general appearances of obJects to the imagination or senses " (1v 180). “ Though it (1, e. geometry) much excels, both in unxversal1ty and exactness, the loose Judgments of the senses and imag1nat1on, yet [1t] never attains a perfect precision and exactness ” (1. 97). Any exactitude attaching to the conclusions of geometrical reasoning arises from the comparative simpl1c1ty of the data for the primary Judgments. So far, then, as geometry IS concerned, Hume's opinion 1s perfectly definite. It is an experimental or observational science, founded on primary or immediate Judgments (in h1s phraseology, perceptions), of relation between facts of 1ntu1t1on; 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 ODIHXOH, but unfortunately it is quite impossible to state IH any satisfactory fashion the grounds for it or even its full bearing He nowhere explains the origin of the notions of un1ty and number, but merely asserts that through their means we can have absolutely exact ar1thmet1cal propositions (Works, 1. 97, 98). Upon the nature of the reasoning by WhlCh 1n mathematical science we pass from data to conclusions, Hume glV€S no explicit state1nent. If we were to say that on his view the essential step must be the establishment of 1dent1t1es or equivalences, we should probably be do1ng Justice to h1s doctrine of numerical reason111g, 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 1mmed1ate 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 h1s fundamental pr1nc1ple of psychology-1s a practical condemnation of his empirical theory of perception. He has not offered even a plausible explanation of the mode by which a COHSCIOUSHQSS made up of isolated momentary 1mpress1ons and ideas can be aware of coexistence and number, or SUCCESSIOH. The relations of ideas are accepted as facts of immediate observation, as being themselves perceptions or individual elements of COHSCIOUS experience, and to all appearance they are regarded by Hume as be1ng 1n a sense analytical, because the formal cr1ter1on of identity is applicable to them. It is applicable, however, not because the predicate is contained in the subJect, but on the principle of contrad1ct1on. If these Judgments are admitted to be facts of immediate perception, the supposition of their non-existence is impossible. The ambiguity in lx1s cr1ter1on, howevei, 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 portion of his theory has been very generally overlooked or that this

1n1s1nterpreted. It does not seem necessary to endeavour 55gl':';g;d to follow his minute exam1nat1on of the pr1nc1ple of real musatian cogn1t1on with the same fulness. It w1ll probably be sufficient to 1nd1cate the problem as conceived by Hume, and the relation of the method he adopts for solving it to the fundamental doctrine of h1s theory of knowledge.

Real cogn1t1on, as Hume points out, implies transition from the present impression or feeling to something connected with it. As th1s thing can only be an impression or perception, and is not itself present, it IS represented by its cop or 1de°. Now the supreme, all-comprehensive link of connex1onibetween present feeling or impression and either past or future CXDEPICHCC is that of causation. The idea in question 1s, therefore, the idea of something connected w1th the present impression as its cause or effect. But this is expl1c1tly the idea of the said th1ng as having had or as about to have existence, -1n 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 s1gn1ficance 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.