Re: Intuitionism's (read "Philosophy's") Limits

[categories <cat-dist@mta.ca>]

Date: Wed, 5 Mar 1997 11:27:42 -0500 (EST)
From: James Stasheff <jds@math.unc.edu>

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
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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
> 
> 
>