From: "zdiskin" <zdiskin@email.msn.com>
To: <categories@mta.ca>
Subject: Re: Michael Healy's question on math and AI
Date: Tue, 30 Jan 2001 16:21:41 -0800 [thread overview]
Message-ID: <004901c08b1b$e880d8f0$822f1b3f@cpu> (raw)
It seems that a discussion "CT vs. ST " (despite some silent resistance
at the beginning :) has nevertheless started in this list too, and is
going in a few different directions ranging from rather technical thru
methodological to philosophical aspects. Probably it's unavoidable (and
useful) when we talk about math in general and its applications but, as
Bill Lawvere pointed, too much quite different stuff is packed into the
same title of CTvsST and the type mismatch in the very discussion and
its comprehension by the (rather diverse as I could guess) audience is
very possible. (Under type mismatch I mean something like this. Suppose
we discuss the result of multiplication 3 x 3. If the alternatives are
8,9 and, say, 12, we have good chances for a reasonable discussion, and
if even the alternatives are 12, 37 and 111, type correctness is still
respected and we have chances to achieve useful results, but if the
alternatives to be discussed are 8, 9, that triangle and that <favorite
keyword>, the situation is hopeless). So, some methodological
arrangement and figuring out explicitly the relevant meanings of the
CTvsST-problem would be useful.
Here are a few contexts already touched in the discussion, where we
deal with methodologically quite different problems whose merging under
the same title CTvsST may be misleading.
a). Ways of setting/defining math structures.
The actual meaning of CTvsST here is the opposition between the
following two ways (described by Steve Vickers yesterday, I'll just
rephrase slightly his presentation).
--(1) Math structure Carrying set (of abstract elements) +
Structure over it defined in terms of the elements (so, structure
resides in elements of the carrier).
--(2) Math structure Category (collection of abstract objects and
their morphisms) + Structure over it defined in terms of the morphisms
(and then structure on a carrier object resides in morphisms adjoint to
it).
The title CTvsST may be misleading here since in the both ways we use
sets, their elements, mappings between them. To name the two ways
somehow, I'd propose to call the 1st one Boubakian (since this way of
setting math structures got its classical completion in Bourbaki's
volumes), and the 2nd one categorical. So, the actual opposition here is
CatStr vs. BrbStr (but, of course, a category with structure is itself a
Bourbakian math structure)
This opposition is of extreme great relevance for software engineering,
business modeling, knowledge representation ... but even brief outline
of the reasons needs a more detailed description of the issue. Let's
postpone that for a next posting.
b). Modeling (formalization) of set theory underlying the setting for
math structures and reasoning about them (usually referred to as
foundations).
The actual meaning of CTvsST here is "categorical set theory" vs.
"formal set theory(ies)" (ZF, NBG,...), or CatST vs. FrmST.
This context and meaning are totally irrelevant to applications i
question and asking about what is better, CT or ST, for applications in
AI,SE, ... is very much like asking what is better for applications in
mechanical engineering: the differential/integral calculi or ST; or
what is better for general theory of relativity, the tensor calculus or
ST etc.
c). Modeling (formalization) of reasoning about math structures (often
called metamathematics)
Of course, what we have here depends on which way of setting math
structures we consider: Bourbakian or categorical. Nevertheless, it's
possible and make sense to treat reasoning about Bourbakian structures
categorically and, say, to treat reasoning about categories in a
elementwise fashion.
We again have two different approaches. Historically first (originated
by Tarsky and Mal'cev) was formalization of first order logic (FOL,
including syntax and semantics) in a quite immediate way now well known
to a wide audience of quite different backgrounds including computer
scientists and philosophers. This way is often referred to as "Tarskian
formalization of the notion of truth", and let's call this entire style
Tarskian MetaMathematics, TarMMath.
The other approach was developed in CT and is usually called
categorical logic, CatLog. So, the opposition we actually have here is
CatLog vs. TarMMath, or, if you prefer, CatMMath vs. TarMMath.
This oppostion, though of course connected with that in (a), CatStr vs.
BrbStr, has its own peculiarities which are of extreme high relevance
for knowledge representation, business modeling and similar domains. So,
details here would be very useful but I'd again postpone them for a next
posting.
So, as it was said in Lawvere's note, there are a few aspects of CTvsST
(understood in the wide sense), some of them may be not relevant for
applications (for example, b) while others are of great importance and
deserve more detailed exposition (a,c). On the other hand, all the
three context we have considered are special cases of a quite general
intellectual activity usually referred to as modeling. Mathematics is
itself a special (very refined) discipline of modeling, and foundations
and meta-mathematics are nothing but modeling math by math means (well,
there are other means, say, philosophical :). It seems that to make
these contexts more understandable for a wider audience (and for myself
:), not only more details should be provided but some sketch of what
modeling in general and math modeling in particular are, would be also
useful. So, I'd ventured to sketch some general tutorial on applying
math to engineering domains and to itself but certainly it's for a next
posting.
--Zinovy Diskin
next reply other threads:[~2001-01-31 0:21 UTC|newest]
Thread overview: 9+ messages / expand[flat|nested] mbox.gz Atom feed top
2001-01-31 0:21 zdiskin [this message]
-- strict thread matches above, loose matches on Subject: below --
2001-01-30 19:54 John Duskin
2001-01-29 15:21 S.J.Vickers
[not found] <F37M5o1gXXX3kRC9QnC00001cc1@hotmail.com>
2001-01-28 0:07 ` Michael Barr
2001-01-27 10:45 Colin McLarty
2001-01-26 21:37 Peter McBurney
2001-01-24 22:26 F. William Lawvere
2001-01-26 3:08 ` Todd Wilson
2001-01-26 18:14 ` Michael Barr
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