Re: Intuitionism's Limits

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

Date: Mon, 3 Mar 1997 18:40:30 +1030 (CST)
From: William James <wjames@arts.adelaide.edu.au>

> Intuitionism's Limits: if C is a category sufficiently complex to
> demonstrate that some C-arrow f:a-->b is monic and B is a subcategory
> of C containing just f (and the requisite identity arrows), do we
> still know that f is monic? Should we? (Or, in other words, which
> view *should* dominate: Intuitionism, Realism, the category
> theoretic...?) What if C is something (semi?)fundamental like a
> category of all sets and functions, or a category of categories?
>

Whoops! The question is trivialised by using monicity as the relevant
property. Reconsider it in terms of say f as an isomorphism, or of
f holding some property in C that B lacks the resources to demonstrate.

I'm thinking aloud on this question: constructive maths should say that
of f in B there is no demonstration forthcoming, so judgment will be
withheld on whether or not f has the property; a category theorist might
say that category theory does not dwell on elements and that, in context,
B is no different from any isomorph of 2, so there positively is no
further property of f to be had other than that which can be
demonstrated in any isomorph of 2. This is more than Intuitionism will
allow.

Might I, then, go on to say that the philosophies of constructive
mathematics and category theory really are different?

William James (if I'm digging a hole, I want it to be big)