On 3/29/2013 12:50 PM, Shmuel (Seymour J.) Metz wrote: > In <otadneXBRZ4yqM_MnZ2dnUVZ_oudnZ2d@giganews.com>, on 03/26/2013 > at 06:56 PM, fom <fomJUNK@nyms.net> said: > >> I think that what you want to assert as a foundation for mathematics >> -- that is, the closeness of one's sensibility to what is expressed >> by mathematical explanation -- will be found in Kant. > > Given the existence of models for non-Eucldiean Geometry in Eucldean > Geometry, I find it hard to take Kant seriously. >
Kant actually called for the development of non-Euclidean geometries. It is a small paper translated by Ewald in "From Kant to Hilbert". Ewald refers to the fact as "surprising" given what is usually presented in the historical accounts.
Also, I read somewhere recently that Kant never mentions the parallel postulate in "Critique of Pure Reason". I have not made an exhaustive search, but, I keep my eyes open for it. I have not found it yet.
Suppose one is willing to understand Kant with respect to changes in the nineteenth century as one understands Newton's theory of gravity with respect to Einstein's theories. Then Frege's retraction and Russell's analysis of projective geometry as the a priori condition of external form speak to the possibility of considering a revised version of Kant's theses.
When I sought definitive modern references to explain these matters to me, the books recommended did just the opposite. Stewart Shapiro actually calls for some sort of structural framework along the lines of a Kantian scheme. Jeffrey Boolos has a wonderful statement that none of the foundational movements of the late nineteenth and early twentieth century had disproved Kant. It would be more correct to say that he lists the various critics of Kant by name.
In any case, David had been looking for some explanation for mathematics that was "real" in the sense of one's innate knowledge of these matters. Kant's discussion of sensible intuition as a ground for mathematical insight should appeal to him. It will also provide a foundation for interpreting the history that followed.
I have been avoiding discussions of Kant precisely because I am aware of the historical context. Besides, the key to that in which I am really interested is the misrepresentation of Leibniz. I do not need Kant for that.
Here is Frege and Russell
> "The more I have thought the matter > over, the more convinced I have become > that arithmetic and geometry have > developed on the same basis -- a > geometrical one in fact -- so that > mathematics in its entirety is > really geometry" > > Frege
> "..., I shall deal first with projective > geometry. This, I shall maintain, is > necessarily true of any form of > externality, and is, since some such > form is necessary to experience, > completely a priori." > > Russell