this seems to ignore the distinction between neighbors (aka comrades) and parts (elements) The group of rational integers, with its non-identity automrophisms can, i thought, be distinguished from the Thom space of the
tangent bundle of some exotic manifold with its non-identity automrophisms without comparison to other sets of `numbers' or less exotic manifolds
.oooO Jim Stasheff jds@math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds May 15 - August 15: 146 Woodland Dr Lansdale PA 19446 (215)822-6707 On Wed, 5 Mar 1997, categories wrote:
Date: Wed, 05 Mar 1997 00:56 -0500 (EST) From: Fred E J Linton <0004142427@mcimail.com>
Any philosophy category theory may have would have at its core, I think, the notion that mathematical objects are known *not* in isolation but in the context of their comrades. The group of rational integers, accompanied *only* by its identity map, and the Thom space of the tangent bundle of some exotic manifold, accompanied once again *only* by its identity map, are, as categories, indistinguishable.
Plucked out of their original contexts, there is no longer any social setting where one can find any difference between them that really *makes* a difference.
According to some other views of mathematics, the group of rational integers, that particular Thom space, the real number {pi}, and my current left shoe, all have unique mathematical personalities that let them be "obviously" distinguished one from another, without any reference even to what I would call their "natural ambient environments".
From my perspective, admittedly that of a categorist, these views result from a simple failure to recognize that what passes for the "intrinsic structure" of a mathematical object is in fact nothing more (nor less) than a clear understanding of its relations with its mates, of roughly similar character, in some category (that "went without saying") they all jointly inhabit -- even the phrase "roughly similar character" is justifiable *only* by virtue of the fact that they *do* all inhabit some same category.
I hope I'm actually making myself clear, and not just preaching to the converted.
-- Fred