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)