'Hume on Space and Geometry': A Rejoinder to Flew's 'One Reservation'

66. ' HUME ON SPACE AND GEOMETRY * : A REJOINDER TO FLEW ' S 'ONE RESERVATION '. ? Flew' s reservation about my assertion that the Enquiry contains no significant revision of the Treatise conception of geometry as a body of necessary and synthetic knowledge, appears to involve two charges. Firstly, he alleges that I dismiss but offer no substantial argument against his own view that the Enquiry restores pure geometry "to its place alongside the other two elements of the trinity ". Secondly, he thinks that I have committed a serious sin of omission in failing to take into consideration a certain passage in the Enquiry (page 31, section 27) which, in his opinion, constitutes "a quite decisive reason" for supposing that Hume abandoned the Treatise position, in that it contains an account of applied mathematics, a subject on which the Treatise is silent. To the charge of omission I plead guilty, and I shall presently seek to remedy this defect and to state why I do not share Flew' s view of the significance of the said passage. To the former charge I am not so ready to plead guilty, and I shall counter-charge that Flew has failed not only to do justice to my proposed reading of Enquiry page 25, section 20, but also to provide clear and convincing support for his own. One difficulty encountered in answering Flew lies in understanding whether or not his claim about the status of geometry as a pure science in the Enquiry is put forward independently of his interpretation of E31. As originally presented in Hume's Philosophy of Belief this appears to be the case, and if so, on what evidence does it rest? 2 In his book , Flew' s statement about Hume replacing pure geometry alongside arithmetic and algebra follows directly upon a paragraph in which he comments that Hume's references to geometrical propositions at E25 signify "an important advance from the position of the Treatise" . For whereas the status of perfect precision and certainty was 67. there withheld from geometry (T71) , Hume now speaks of the certainty and evidence of the propositions of Euclidean geometry , and declares such propositions to be discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe . (E25) Now, I argued at some length in my paper (pp. 21-24) that Hume is prepared to grant scientific status to geometry in the Enquiry because he has come to regard its certainty as that proper to a mathematics of space. He has abandoned the Treatise contrast between an ideal or theoretical standard of equality, viewed therein as necessary to geometry considered as a science proper, and the merely sensible standard of equality upon which geometry as practised depends . It should not be forgotten that Hume recognises two kinds of certainty where "Relations of Ideas" are concerned, intuitive and demonstrative. Since he fails to provide any detail of the place of each of these in mathematics, the door is left open for speculation along the lines followed in pp. 28-29 of my paper. For there is nothing in ?25 which implies that the certainty of all geometrical propositions is of the same kind, or that the process of geometrical demonstration is of the same logical nature as demonstrative reasoning in arithmetic and algebra. In his comment on my position Flew makes no reference to my proposed reason why Hume classes geometry as a science in the Enquiry.· Yet, even if this reclassification does mark an advance of a kind, the fact remains that E25 does not furnish incontrovertible evidence for that advance which Flew thinks he finds there. Nor, therefore, is it conclusive against my assertion that Hume in both works regards the propositions of geometry as necessary and synthetic truths. Given that Flew does not wish to deny the inclusion of a synthetic element even in the Enquiry account of mathematics (see H. P. B. pp. 64-66), I find it difficult to see why he supposes that the logical status of the propositions of geometry is different in the Enquiry from that in the Treatise, since in the latter too they are held 68. to be truths which depend...