Re: pragmatic foundation

[Andre.Rodin@ens.fr]

Selon Colin McLarty <colin.mclarty@case.edu>:


> I myself am also confident that people will calm down and notice that
> axiomatic categorical foundations such as ETCS and CCAF work perfectly
> well, in formal terms, and relate much more directly to practice than
> any earlier foundations.  One hundred and fifty years of explicitly
> foundational thought has made this progress possible.  By now, that
> can hardly qualify as "extraordinary"!
>
I do NOT believe that ETCS and CCAF "work perfectly well". Each of these involve
two foundational "layers", namely, the classical "bottom" and a categorical
"superstructure". By the classical bottom I mean NOT an underlying Set theory
but the "Elementary theory of categories" (ETC), i.e. a theory of categories
using the usual First-Order Logic (FOL) and relying on the standard
Hilbert-Tarski-style axiomatic method. I agree with John Mayberry and some
other people who argue that this aximatic method alone assumes a basic notion
of set or collection. Unlike Mayberry I don't think that this fact implies that
the project of categorical foundations, as a alternative to and replacement for
set-theoretic foundations, is futile. Recall that the axiomatic method we are
talking about (which is, of cause, quite different from Euclid's method and
other earlier versions of axiomatic method) emerged together with Set theory.
In order to make categorical foundations into a viable alternative of
set-theoretic foundations we still need to provide Category theory with a new
axiomatic method rather than use the older axiomatic method as do ETCS and
CCAF. Elements of this prospective axiomatic method are found in what I just
called the "categorical superstructure" of ETCS and CCAF but as far as these
theories are concerned the classical background (FOL+ETC) is indispensable.
This is why I say that ETCS and CCAF do NOT work perfectly weel as categorical
foundations.
Building of "purely categorical" foundations remains an open problem. It is not
a matter of a ideological purity but a matter of complete "rebuilding" (Manin's
word) of foundations: in my view, such a rebuilding is healthy and refreshing
in any circumstances (unless it clashes severely with practice).




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