FOM: A pro cat-fom argument

S Riis pmtsr at
Sun Feb 15 04:08:57 EST 1998

A pro cat-fom argument
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:

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.

Pierre Cartier ends the interview with the following 
interesting statement:

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.

Any comments concerning this argument? 

Soren Riis

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