Subject: categories: Re: cracks and pots

From: Vaughan Pratt <[email-masked]>
Date: Thu, 16 Mar 2006 17:52:03 -0800

Category theory, and for that matter modern (as opposed to elementary)
algebra, is to mathematics as mathematics is to physics, and for that
matter to computer science.  Whereas mathematics organizes reasoning
about the phenomena studied by physicists and computer scientists,
algebra and category theory perform a similar function for mathematics.

In any setting organization is desirable, and arguably necessary on
occasion.  But the use of algebra and category theory to organize
physics and computer science is a double whammy here.  One should
therefore be doubly sympathetic of those physicists and computer
scientists who want to know what substantive contribution is being made
to their subject and can't evaluate the answers because they are one if
not two levels removed from the necessary abstractions.

Vaughan Pratt

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From: "Mamuka Jibladze" <[email-masked]>
Date: Sun, 19 Mar 2006 00:22:48 +0400

> not two levels removed from the necessary abstractions.

It just occurred to me that to justify such viewpoint we might have to look
at the point in time when mathematics began to become abstracted out from
natural sciences to see whether category theory is already in the same
position with respect to the rest of mathematics.

Although I certainly do not know enough history of science, I will still
dare to speculate that the situation now is completely different from what
it was then. I believe mathematics as a substantial part of the body of
scientific knowledge did exist and evolve long long before it began to be
considered as some separate entity which can be used to organize the rest -
in fact many people still think of mathematics as just another science on
completely equal footing with, say, biology or physics.

Whereas birth and development of category theory has been, I think, much
more deliberate, abrupt and discontinuous in comparison. If this is so, one
possible conclusion might be that probably category theorists simply want
too much too soon. Maybe they should be more patient and let their
discipline become stronger within the body of mathematics before forcibly
declaring it a new organizing force outside the rest of mathematics. This is
as if a child would be forced to care for its parents shortly after being
born.

Mamuka Jibladze

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From: James Stasheff <[email-masked]>
Date: Sat, 18 Mar 2006 10:21:24 -0500 (EST)

I think knot theory is particularly helpful here but I'll let Yetter and
Freyd reply further.

	Jim Stasheff		[email-masked]

		Home page: www.math.unc.edu/Faculty/jds