FOM: [cxm7@po.CWRU.Edu: Re: A pro cat-fom argument]

[Colin Mclarty]

    Reply to message from Soren Riis
    
    	Soren quotes Pierre Cartier saying (in part):
    
    >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.
    >
    >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.
    
Soren says:
    
    >Can anyone elaborate? I am very interested in understanding
    >this better.
    
    	I have not read the Cartier article yet, but I suppose
his time periods come from a certain Bourbakiste perspective: The
purpose of Bourbaki in the 50s was certainly to take the viewpoint 
they had begun creating in 1935 and make it universal in math.
    
    	Cartier jumps right from the 60s to "now". If he means
some revolution has begun since say 1990 I have no idea what it 
would be. But I do know that many Bourbakistes felt their project
was breaking down by the 1960s (see Leo Corry "Nicolas Bourbaki
and the concept of mathematical structure" in SYNTHESE 92, 1992
pp.315-348). The reason Corry gives (using internal Bourbaki
documents) is largely that they could not cope with homological
methods as used by Eilenberg, Cartan, Serre, (I believe they were
all members) and increasingly Grothendieck (who went to meetings 
but would not join). Dieudonne certainly came to believe that
category theory, which he knew almost entirely through 
Grothendieck, turned out to be better than Bourbaki's theory
of structures. I think that is the new revolution Cartier 
means--certainly its effects are not at all settled yet--but 
I would not claim to know. 
    
    	Soren suggests an argument for Cat-fom, based somewhat
on his reaction to the Cartier quote, but which Soren does not
endorse:
    
    >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. 
    (Soren compares the way it is irrelevant to ZF-fom whether 
    mathematicians write proofs out in ZF primitives or not.)
    
    	I guess he is right to say that the widest 
generalizations of most theorems now known are in category
theory. I had not thought of it that way before, and I still
think it is not the key point. To me the point is that often 
you get the categorical generalization simply by dropping a 
lot of details. Sometimes you can generalize a theorem by 
complicating it, like generalizing fixed point theorems from 
the 2-dim disk to regions in Frechet spaces or something. 
Friedman and Simpson and others keep saying category theory 
is like that--set theory "in bad notation". But they are 
wrong and Cartier knows it. The theory of Abelian categories is
a simpler way to get the needed results than starting with
modules over a ring--let alone sheaves of modules over a 
sheaf of rings, or coherent sheaves of modules and so on.
This is true even though you are finally interested in certain
(sheaves of) modules over some (sheaf of) ring(s). You keep
the particulars out as long as possible and only bring in the
ones you need. Of course in practice this is a balancing act.
How much is too much? But experience keeps showing that great
amounts of ctegory theory are not too much.


    >>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. 

Indeed for many very general questions negation is irrelevant. They
arise "before" the point where classical and intuitionistic logic
diverge. Categorists explore many weak fragments of logic, for
many reasons.

    >The claim is that the "essence" of virtually any mathematical 
    >fact is best captured by category theory. This have the nature 
    >of a discovery.

I like this way of putting it--mathematicians discovered something
that was not at all obvious at the start, and is still denied by
many. Categorical tools make it natural to neglect much and focus 
on a little, in a way that actually works on things like a 300 
year old question in Diophantine equations, and that cuts so deep
with such generality that it becomes foundational.

Colin