FOM: Re: Question
Soren Riis
sriis at fields.fields.utoronto.ca
Wed Mar 11 14:35:55 EST 1998
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FOM: Question: why is category theory viewed as relativist?
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Michael Thayer asks:
> There seems to be an understanding on fom that Category Theory offers
only a
> "relativist" f.o.m.
>
> What is the reason for this view? The only argument I have seen seems
to be
> that there is no specific topos offered as a foundation for analysis.
How
> does this differ from the discredited view that set theory does not
offer a
> foundation for analysis, because there is no distinguished notion of
natural
> number. (The tired old "is e a member of pi" objection).
First let me make one thing absolutely clear:
Category Theory is a completely objective mathematical discipline.
It has numerous merits and a very fine history of first class
mathematicians which often spoke in this language.
Despite my DISCLAIMERS ETC people mistakes my the strong critique
of "Category Theory as foundation of mathematics" with a
critique of Category Theory. To criticize Category Theory would
be as silly as to criticize group theory.
YOU QUESTION CONVINCE ME THAT I NEVER SHOULD HAVE ENTERED THIS
DEBATE. It is far to easy to be misunderstood.
On the FOM-list we have had quite a fierce debate. I my opinion Category
Theory is an discipline which HAVE to build on set theory IF it seriously
wants to contribute to the foundation of mathematics. You CAN easily
define (and this might be preferable) Categories and Topos without
reference to any full axiomatization of "the Boolean Topos SET".
But in that case your system is to weak to serve as foundation of
mathematics.
-------------------------------
In short:
-------------------------------
You have to make up you mind about the status of your
meta-theory. The Set theory vs Category Theory fom debate
can be seen as a choice between:
(1) Category Theory which might be seen as a very flexible
perspective which avoiding the "unnecessary" structure of set theory.
One might claim that the "essence" of many mathematical FACTs
(somewhat presented with proof as primitive) best is best captured
by category theory (when all "irrelevant" information is removed).
(2) A system like ZFC (axiomatizing the boolean Topos SET)
which by some is seen as rigid (I think mainly because is hard
to discard irrelevant information) and have only a few of the
merits people in Category Theory seems to be interested in.
When we do fom we MUST insists on (2). If one wants to support
cat-fom it is not satisfactory to insist on (2) which just seems to be
set theory in disguised notation.
Instead some insists on (1) but still claim this to serve
as foundation of mathematics. They are praising all the merits of (1)
but fail to recognize that the merits of (1) only can be maintained
as long as one do NOT claim Category Theory can serve as an axiomatization
of mathematics. This is where relativism enters the picture: It is
tempting (for some) to take a relativist position in conjunction with
claiming that (1) can serve as foundation of mathematics.
By insisting on (1) they are ignoring the objective fact that the usual
axiomatization of a category is insufficient as an axiomatization
of mathematics.
In effect they are smuggling in logicism (every one choose there own
assumptions). You might of course be more sensible and start with a
"Boolean Topos with a natural number object".
In that case (it seems to me) that you have already given up a
considerable part of the merit of Category Theory (certainly if you
insists that the Topos is Boolean). So it seems Category Theory only
have it's special merits (described in (1)) TO THE EXTEND CATEGORY
THEORY IS NOT TRYING TO BE FOUNDATION OF MATHEMATICS (i.e. moving
towards (2))
Soren Riis
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