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HUME ON SPACE AND GEOMETRY Hume's discussion of our ideas of space, time and mathematics in Book One of the Treatise is referred to by one recent commentator as 'the least admired part' of this work, while another finds it to be 'one of the least satis2 factory Parts'. Hume himself, it would appear, was not far from endorsing such opinions. The omission of any detailed comment on these subjects from the first Enquiry can be taken as indicating his dissatisfaction with the earlier exposition as well as a felt incompetence to offer a suitable revision. For there was certainly no loss of interest on Hume's part in the philosophical problems of mathematics. In a letter to William Strahan in 1772 he speaks of an essay prepared for publication around 1755 entitled: On the Metaphysical Principles of Geometry , which he withdrew upon the advice of his friend the mathematician Lord Stanhope, who convinced me that either there was some defeat in the argument or in its perspicuity; I forget which. . . Defects of both kinds are present in the Treatise discussion of geometry and the infinite divisibility of space, which taken together with certain remarks in the Abstract of the Treatise and in the Enquiry , suggest that the problem defying satisfactory solution was the relation between these. Hume was of the opinion that the theory of infinite divisibility constituted an affront to human reason by virtue of the paradoxes it involved. Such paradoxes, he thought, must open the door to scepticism about the certainty of mathematical knowledge, since in the field of abstract reasoning reason is thrown into a kind of amazement and suspence, which, without the suggestions of any sceptic, gives her a diffidence of herself, and of the ground on which she treads. (E157) Hume's attempted solution involved the rejection of those abstract ideas employed in mathematics and most fundamentally, that of infinite divisibility, but he never succeeded in working out a theory of geometry 2. which preserved its desired status as a body of certain knowledge and yet was at the same time consistent with his empiricism. In this paper I set out to examine the relation between Hume's view of space and his theory of geometry. Firstly I take a detailed look at Hume's arguments against infinite divisibility in an attempt to discover why they take the form they do, and to understand what Hume has in mind when he speaks of that 'minimum' idea which is an 'adequate representation" of the most minute possible part of extension. The second section deals with Hume's account of the genesis of our idea of space. I propose, contrary to some commentators , that Hume does not treat the idea of space on an exact parallel with that of time. Although both are said to be abstract ideas they differ in the relation in which they first stand to impressions. My comments on time are confined to drawing attention to this difference since fuller discussion of passages dealing exclusively with time would require a separate paper. In the final section Hume's remarks about geometry are examined. Here I argue that Hume entertains two views of geometry in the Treatise, one of which is abandoned in the Enquiry while the other remains largely unaltered. Hume opens his account of our ideas of space and time with an attack on the thesis of infinite divisibility which he regards as belonging to that body of doctrines which are greedily embrac'd by philosophers , as shewing the superiority of their science, which cou' d discover opinions so remote from vulgar conception. (T26) Two principles, boti taken to be evident, form the basis of his argument. I shal] refer to them hereafter as principles A and B. Principle A: the capacity of the mind is limited, and can never attain a full and adequate conception of infinity . 3. Principle B: whatever is capable of being divided in infinitum , must consist of an infinite number of parts, and ( ) 'tis impossible to set any bounds to the number of parts, without setting bounds at the same time to the division. (T26, 27) From these two principles Hume draws the conclusion that...


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