Re: Could a proof really have existed?


David Bernier wrote:

Would it be fair to say that most number theorists

s/number theorists/non-logician mathematicians/

haven't much reason to care about the strength of the
axioms they use

period.

because they trust the investigations
of logicians on the subject?

If they must feel obligated to rationalize, I suppose this is a common
one. Another one might be that they've rarely had cause to consider the
axioms, except as a curiosity.

(I'm just saying this is what I see of mathematicians, not necessarily
how it should be).

Am I right that some logicians would like to have a
categorical foundation rather than a set-theoretical one?

Logically, I'm sure that there do exist some such logicians, but for
the great majority of mathematicians whose background is logic,
set-theory is it. Category theory, however useful or important in other
areas (topology, programming language semantics), is not a 'foundation'
at all.

That last impression comes to me from the "Lawvere program",
e.g.:

"Kreisel and Lawvere on Category Theory and
the Foundations of Mathematics"
by
Jean-Pierre Marquis:

http://www.math.mcgill.ca/rags/seminar/Marquis_KreiselLawvere.pdf

Oh. Kreisel, a logician, is saying more academically (or at least
Marquis is saying that Kreisel is saying this, I don't know otherwise)
that set theory and category theory are both foundations, for different
interpretations of the word 'foundations': set theory is a foundatiuon
for 'provability' and category theory is a foundation for
'organization' (actually, on closer reading of Marquis's slides,
Kreisel's play with words is that set theory supplies foundations
whereas category theory supplies organization).. In contrast, Lawvere
(originally a algebraist) says (or really, gives constructions) that
varieities of set theories can be viewed categorically.

To be noted, Marquis is presenting ancient history. That is, the
controversy occurred a long time ago. There is little controversy now:
logicians mostly work (in a technical capacity) with set theory, others
with category theory.

Has there been a unification of categorical foundations
and set-theoretical foundations, in the sense that one
can be formulated in terms of the other?

Certainly. There is the trivial old school formulation of category
theory in terms of set theory ('a category consists of a -set- of
objects and a -set- of morphisms...'. The more interesting is the
program in category theory to describe logics. look up toposes (or
topoi) and cartesian closed categories. There is lots of research going
on here (as much as the logicians are doing on their side without
categories).

Did anybody win? Well, the categorists never really convinced the
logicians (at least not as a whole). But that didn't stop the
categorists from pursuing their ideas further.

Mitch

.

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