[FOM] FLT Decisive by Normal Math Standards

[David Fernandez Breton]

Arnon,


> FLT is not (only) a proposition of algebraic geometry, and it is
> definitely not the property of the community of algebraic geometry.
> Once this community has claimed to prove it, they should follow
> the standard, eternal crfiteria of mathematics (at least mathematics
> that deserves that name.) For example: making explicit in a very precise
> way what are the assumptions that underline the alleged proof.
>

I'm not so sure about the existence of "standard, eternal criteria of
mathematics", and in any case, if you declare that only mathematics that
adhere to that standard deserve the name, then I suspect that the
overwhelming majority of mathematicians would beg to differ.

As I've learned throughout the years, there are at least two kinds of
mathematicians: those (like me, and I suspect also like you) who were first
attracted to mathematics because of its very specific epistemic situation
(because of the fact that, once proved, a mathematical statement seems to
be much more certain than most everyday statements, and even than
scientific statements), and those for whom the main appeal of mathematics
was not the certainty, but something else (possibly what Cohen once called
"the combinatorial thinking, or to put it more bluntly, the cleverness"
(I'm paraphrasing here), the thrill of solving a problem, the degree of
abstraction, etc.). Mathematicians from the first category tend to end up
working on FOM-ish topics, but the overwhelming majority of mathematicians
seem to belong primarily to the second category. As such, many of them
might not even know what the ZFC axioms are, or might have learned them and
long forgotten. They don't think that what they do is to work within an
axiom system, and they would probably deny that this is even the main point
of what they do. Their viewpoint is that they are just "doing mathematics"
(an admittedly vague statement), as opposed to following some rigid
axiomatics (the usual caveats apply here: I apologize for generalizing to
such a large group of people, which is not a monolithic group, and clearly
the general attitute might differ between individuals, etc; I'm really just
trying to present a simplified version of an opinion that I've heard and
observed multiple times). And to a certain extent, they are right: the
mathematical objects with which they work tend to be very "safe", in the
sense that you can just manipulate them naively without any risk of running
into any serious set-theoretic subtleties. This is, I think, largely the
reason why people from arithmetic geometry never worried too much about
this whole "axiom of universes" business. If anyone should worry about
this, it should be the people working on foundations (which is the way
things actually happened, since it was McLarty rather than a number
theorist the one who clarified all this issue).


> > This is the way the math community operates.
>
> This might indeed be what the math community has deteriorated
> to in this post-modernist world.


This obviously relates to my previous point above, but also: I think you're
getting this backwards. Math has not always been this "paradise of rigour",
which has just stopped being so in recent times. On the contrary, for the
longest time (with possibly the only exception of Euclid's elements),
mathematics was an activity where the primary objective was to obtain
results, rather than to be rigorous. Looking at most proofs from two or
three centuries ago, they look much more like the kind of intuitive
arguments that physicists do than like modern times proofs. The whole
concern about foundational issues is relatively recent, its embryo probably
first arose about halfway through the XIX century, and it only took full
force in the early XX century. Considering that math has been around for
millenia, I'd say that being careful about axiom systems, and what
assumptions are being made, and about completely rigorous proofs, and the
like, is more the exception, rather than the rule.

     David
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