Constructive mathematics is a philosophy. Category theory is not. The question doesn't even type-check.

Of course they're different.

Philosophy is the love of wisdom; type-checking is not. Of course category theory has its philosophy. To me it's "all things are connected" - you cannot fully describe anything purely in itself but only by the way it connects with others. Category theory makes the connections explicit (as morphisms) and then characterizes things by their universal properties. The philosophy plays a real role in categorical practice: for instance, in the idea that isomorphism between objects is more important than equality, which is not something that can be meaningful just in terms of the formal mathematics. The philosophy also yields a criterion for evaluating the theory: Is categorical structure adequate for describing the connections that we actually find? The strength of the categorical view of "connection structure" is amply confirmed by the power of the universal properties it can express (compare it with, say, graph theory); but if it does fail us anywhere, how might it advance beyond its present formalization? (There is already a plausible answer here: topology has a different way of describing the connections between a point and its neighbours, and the categorical and topological approaches combine to make topos theory.) I hesitate to try to reduce the philosophy of constructive mathematics to a single pithy phrase, not least because there are different schools of constructivism with apparently different philosopies. I shall therefore duck the question of comparing "the philosopies of constructive mathematics and category theory", but I don't believe it's a meaningless one. Steve Vickers.

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