FOM: A pro cat-fom argument
S Riis
pmtsr at amsta.leeds.ac.uk
Sun Feb 15 04:08:57 EST 1998
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A pro cat-fom argument
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In a recent issue of the Mathematical intelligencer
(Vol 20 Num 1 1998) there is an interview with Pierre
Cartier who was a member of Bourbaki from around 1951
to 1983.
On page 26 line 15 from below he says:
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Pierre Cartier: Most people agree now that you do need
general foundations for mathematics at least if you
believe in the unity of mathematics.
But I believe this unity should be organic, while Bourbaki
advocated a structural point of view.
I accordance with Hilbert's views, set theory was thought
by Bourbaki to provide that badly needed general framework.
If you need some logical foundations, categories are a
more flexible tool than set theory. The point is that
categories offer both a general philosophical foundation -
that is the encyclopedic, or taxomomic part - and a very
efficient mathematical tool, to be used in mathematical
situations. That set theory and structures are, by
contrast, more rigid can be seen by reading the final
chapter in Bourbaki set theory, with a monstrous endeavor
to formulate categories without categories.
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Pierre Cartier ends the interview with the following
interesting statement:
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Pierre Cartier: The purpose of mathematics, in the fifties
and sixties, was that, to create a new era of normal
science. Now we are again at the beginning of a new
revolution. Mathematics is undergoing major changes. We don't
know exactly where it will go. It is not yet time to make
a synthesis of all these things - maybe in twenty or thirty
years it will be time for a new Bourbaki.
I consider myself very fortunate to have had two lives, a
life of normal science and a life of scientific revolution.
------------------
Can anyone elaborate? I am very interested in understanding
this better.
Concerning the cat-fom topic this interview shows that the
idea of cat-fom has to be taken very serious (whether we like
it or not) and that we are forced to accept that some
mathematicians see Cat-fom as MORE than just an alternative
to the traditional ZFC-fom. Let me present a line of argument
which I think is quite powerful and which definitely have to
be addressed. I do not accept the argument (partly because I
am uncertain whether it actually represent an accurate picture)
and it does thus NOT represent my view.
Rather it represent the most SERIOUS argument supporting
cat-fom I can think of. I am actually surprised none of the
cat-fom advocates have tried this line of argument:
Category theory concerns the "essence" of mathematical facts.
Take any theorem from mathematics and generalize it to it
reaches it most general form. It is irrelevant whether the
most general and ultimative formulation is useful or not.
The point is that the ultimative generalization always seems
to take a certain form!! A major discovery of Category theory
has been that the essence (often expressed as the most general
version) of virtually any mathematical result (i.e. theorem
plus proof) is best expressed and understood in terms of
Category Theory.
>From this view its hardly surprising that the underlying logic
is intuitionistic rather than classic. The essence of many
mathematical facts are intuitionistic. When we push a fact and
try to express it in its most general formulation, the excluded
middle might be irrelevant in the given context.
Now generalizing results to their most general form is usually
seen as bad style. This however is irrelevant, because it is
the principle which is counts. It is for example usually also
considered bad style to present a proof as a formal ZFC-proof.
>From a traditional fom perspective it is the principle (that
virtually any mathematical argument can be formalized within
ZFC) which is the crucial thing. It is the principle which
counts. Similary from a cat-fom perspective it is the principle
(that all mathematical facts in their most general form seems
best expressed in terms of cat-theory) which makes Category
Theory a foundation for mathematics. On this account Category
Theory is much more than just organisation of mathematics.
The claim is that the "essence" of virtually any mathematical
fact is best captured by category theory. This have the nature
of a discovery. Many working mathematicians might find this
irrelevant because they have no interests in reaching the
"ultimate essence" of an established fact. But what does this
show? Not more than for example the fact the most working
mathematician are completely oblivious to which axioms of Set
Theory are needed for their results.
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Any comments concerning this argument?
Soren Riis
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