FOM: Categorical vs set theoretical foundations
John Mayberry
J.P.Mayberry at bristol.ac.uk
Mon Jan 26 11:13:11 EST 1998
I thought I was entering a scholarly discussion/argument over
the possibility of category-theoretic foundations, not a barroom brawl.
But even in a brawl you ought to find out what side a man is on before
you try to punch him in the nose. I tried to intervene on Friedman's
side against the category theorists, and I found him throwing punches
at me. Can't he see that I am on his side?
Let me try to put my argument in a different way. McLarty and
Pratt (and MacLane, for that matter) seem to think that the claim that
set theory provides the foundations for mathematics ultimately boils
down to the claim that mathematics can be "translated into" or
"interpreted in" the formal first order axiomatic theory ZF (or ZFC).
This encourages them to suggest that there are other formal first order
theories that could also perform this office - indeed, McLarty, if I
understand him right, claims that there are lots of such theories.
Moreover, he has argued vigorously, and not by any means without
success, that this is the case. Friedman objects that these other
theories are not properly "motivated" - that they are not backed up
with a clear notion of what their primitive symbols stand for, or of
what their intended universes of discourse consist in. But Pratt
retorts (in the passage I quoted) that, in the final analysis, whatever
may be our "motivation" or intention, "ZF weaves the entire notion of
set from the whole cloth of first order logic". He is saying, I think,
that once we have converted our "motivating" ideas into a formal
axiomatic theory, formal logic takes over and tells us all we will ever
know about our primitive notions. As is sometimes said, those notions
are "defined" by those axioms.
How can this powerful and familiar general argument be
answered? Clearly it won't do simply to say that *my* axioms are more
elegant, or more economical, or better motivated, or easier to teach to
undergraduates, or more nearly in accord with engineering practice, or
whatever. Because these are clearly all matters for judgment - often
something akin to aesthetic judgement - and different people can arrive
at different judgements without any of them lapsing into obvious
irrationality.
I say that the problem is *not* one of judging whose formal
first order axioms are best "motivated". The point is that *none* of
these formal first order theories, qua formal first order theory, can,
as a matter of *logic* (taking "logic" in its broadest sense), qualify
as a foundation for mathematics. *That* was what I was trying to say in
my earlier posting. It's not that the ordinary foundational use of
non-formalized set theory (in the introductory chapters of textbooks,
for example) *motivates* us to adopt first order ZFC as our "official"
foundational theory; it is rather that when we trace our arguments and
definitions back to their ultimate presuppositions, we find that the
natural concepts and basic truths that underlie all our mathematics are
to be found in that non-formalised set theory. When that theory is
formalized as a first order theory we get ZFC. (Notice, however, that a
second order formalization of ZFC would seem more natural, if we didn't
already know that second order logic cannot be provided with a complete
proof theory.) But it is the natural, pre-formalized set theory that
provides the real foundation for mathematics, not its formalized
simulacrum.
Finally, let me address a few words to Harvey Friedman. I cannot
understand why a man with your superior gifts and reputation should find
it necessary to indulge in the kind of fierce polemics that you
directed at me. Had I been wrong, and had your criticism been of
substance, I should still not have deserved the sort public drubbing
that you attempted to administer. In fact, however, none of your
criticism hit its mark, because you have systematically, and, it seems
to me, willfully, misunderstood or misinterpreted what I was saying.
For example, you have clearly mistaken my remarks about the
completeness of systems of logical proof for first order logic as being
about the completeness of first order axiomatic theories. Of course you
know that these are different senses of "completeness". How could Godel
have thought that the completeness of first order logic was "an open
ended process", as you say, when he himself established it. He was
obviously thinking of "completeness" in the other sense.
--------------------------
John Mayberry
Leturer in Mathematics
School of Mathematics
University of Bristol
J.P.Mayberry at bristol.ac.uk
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