HUME 353 merely natural or external links of connexion, the principles of association among ideas. The foundations of cognition must be discovered by observation or analysis of experience so conceived. Hume wavers somewhat in his division of the various kinds of cog nition, laying stress now upon one now upon another of the points in which mainly they differ from one another. Nor is it of the first importance, save with the view of criticizing his own consistency, that we should adopt any of the divisions implied in his exposition. For practical purposes we may regard the most important discussions in the Treatise as falling under two heads. In the first place there are certain principles of cognition which appear to rest upon and to express relations of the universal elements in conscious experi ence, viz., space and time. The propositions of mathematics seem to be independent of this or that special fact of experience, and to remain unchanged even when the concrete matter of experience varies. They are formal. In the second place, cognition, in any real sense of that term, implies connexion for the individual mind between the present fact of experience and other facts, whether past or future. It appears to involve, therefore, some real relation among the portions of experience, on the basis of which relation judgments and inferences as to matters of fact can be shown to rest. The theoretical question is consequently that of the nature of the supposed relation, and of the certainty of judgments and inferences resting on it. Hume s well-known distinction between relations of ideas and matters of fact corresponds fairly to this separation of the formal and real problems in the theory of cognition, although that dis tinction is in itself inadequate and not fully representative of Hume s own conclusions. With regard, then, to the first problem, the formal element in knowledge, Hume has to consider several questions, distinct in nature and hardly discriminated by him with sufficient precision. For a complete treatment of this portion of the theory of knowledge, there require to be taken into consideration at least the following points : (a) the exact nature and significance of the space and time relations in our experience, (b) the mode in which the primary data, facts or principles, of mathematical cognition are obtained, (c) the nature, extent, and certainty of such data, in themselves and with reference to the concrete material of experience, (d) the principle of inference from the data, however obtained. Not all of these points are discussed by Hume with the same fulness, and with regard to some of them it is difficult to state his conclusions. It will be of service, however, to attempt a summary of his treatment under these several heads, the more so as almost all expositions of his philosophy are entirely defective in the account given of this essen tial portion. The brief statement in the Inquiry, iv. , is of no value, and indeed is almost unintelligible unless taken in reference to the full discussion contained in part ii. of the Treatise. The nature of space and time as elements in conscious experience is considered by Hume in relation to a special problem, that of their supposed infinite divisibility. Evidently upon his view of con scious experience, of the world of imagination, such infinite divisibility must be a fiction. The ultimate elements of experience must be real units, capable of being represented or imagined in isolation. Whence then do these units arise ? or, if we put the problem as it was necessary Hume should put it to himself, in what orders or classes of impressions do we find the elements of space and time ? Beyond all question Hume, in endeavouring to answer this problem, is brought face to face with one of the difficulties inher ent in his conception of conscious experience. 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 impres sions (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 pre senting themselves is the impression from which the idea of time takes its rise. It is almost superfluous to remark, first, that Hume here deliber ately 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 otter any explanation of the mode in which coexistence find succession are possible elements of cognition in a conscious experience made up of isolated presentations and representations. For the consis tency of his theory, however, it was indispensable that he should insist upon the real, i. 6. , presentativc character of the ultimate units of space and time. How then are the primary data of mathematical cognition to bo 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 em phatically that it is an empirical doctrine, a science founded on ob servation 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, andiv. 180.) It is impossible to give any consistent account of his doctrine re garding 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 accord ingly affords the possibility of an all-comprehensive and perfect science, the science of discrete quantity, (See Works, i. 97.) In respect to the third point, the nature, extent, and certainty of the elementary propositions of mathematical science, Hume s utter ances 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 syn thetical 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 regard ing the real impressions. No question arises regarding the existence of the fact represented by the idea, and in so far, at least, mathe matical 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 corre spondence 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, Dcr philosopldsche Kriticismus, i. 96, 97.) 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 ex tent and truth of the judgments founded on them. The emphatic utterances in the Inquiry (iv. 30, 186), and even at the begin ning 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 tho 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 proper ties of this individual subject, the idea of the triangle, are, accord ing 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 dis covered 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 qualities be fore 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 fit] 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.
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