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

From: categories <[email-masked]>
Date: Mon, 3 Mar 1997 13:14:11 -0400 (AST)

Date: Mon, 3 Mar 1997 10:45:55 -0500 (EST)
From: Peter Freyd <[email-masked]>

William James continues to write:

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

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

Of course they're different.

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From: categories <[email-masked]>
Date: Tue, 4 Mar 1997 22:41:13 -0400 (AST)

Date: Tue, 4 Mar 1997 18:19:51 +1030 (CST)
From: William James <[email-masked]>

> Of course they're different.

Does category theory, being mathematics, have no associated philosophy?

(I grant you the original question would have been more recognisable
 given better use of language: "...philosophies of constructive
 mathematics and *of* category theory...")

William James

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From: categories <[email-masked]>
Date: Wed, 5 Mar 1997 11:13:00 -0400 (AST)

Date: Wed, 05 Mar 1997 00:56 -0500 (EST)
From: Fred E J Linton <[email-masked]>

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

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From: categories <[email-masked]>
Date: Wed, 5 Mar 1997 11:13:50 -0400 (AST)

Date: Wed, 5 Mar 1997 10:54:19 +0000
From: Steve Vickers <[email-masked]>

> 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.

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From: categories <[email-masked]>
Date: Wed, 5 Mar 1997 11:14:35 -0400 (AST)

Date: Wed, 5 Mar 1997 12:46:26 +0000 (GMT)
From: Dusko Pavlovic <[email-masked]>

 > Does category theory, being mathematics, have no associated philosophy?

I'm afraid, William, that this presumed association of mathematics and
philosophy is actually a bit of a sad romance: while some philosophies
do like to be associated with mathematics, mathematics (it doesn't
even have a proper plural) mathematics, most of the time, can't care
less.

While philosophy spends a lot of time defining itself and its
relationship with the world, mathematics tends to be a kind of work
some people like to do, taking up the world whichever way it comes to
them: as a model of a process, as a game of signs or pictures, as a
funny language shared between them and theri colleagues... Most
mathematicians just smirk not only on philosophy, but even on category
theory, or anything else deeply concerned with its own identity. They
just like to solve their problems, and sometimes solve other people's
problems, thereby gaining everyone's respect and admiration.

At least, that's the way I have seen it. Perhaps it helps with your
questions a bit.

-- Dusko Pavlovic

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From: categories <[email-masked]>
Date: Wed, 5 Mar 1997 17:19:30 -0400 (AST)

Date: Wed, 5 Mar 1997 11:27:42 -0500 (EST)
From: James Stasheff <[email-masked]>

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		[email-masked]
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