foundations, Atiyah, etc

Stephen G Simpson simpson at
Mon Sep 29 23:49:37 EDT 1997

Dear Anand,

You wrote:
 > About the axiomatic paradigm. The point is that a certain view:
 > that mathematics can be defined by and reduced to the activity of
 > deriving theorems from axioms, which was widely in circulation when
 > I was a student, is no longer seen necessarily as the paradigm.

Anand, I'm somewhat astonished at your use of the word `paradigm,'
inasmuch as it is a favorite postmodernist buzzword.  Thomas Kuhn's
last published opinion on the matter was that he regretted what the
postmodernists had made of that concept (or is it an anti-concept),
which he had introduced.  Was this postmodernist reference intentional
on your part?

 > This is a separate issue from that of rigour and rationality.

How is it separate?  I think the prevailing view, at least among the
educated public, is that mathematical rigor is closely bound up with
mathematical proof.  What alternative `paradigm' of rigor do you have
in mind?

 > In any case, it is not from a bunch of fashionable wierdos who are
 > trying to undermine the traditions of Western Christian civilsation
 > which we all love and hold dear, that this emanates.

In connection with `Western Christian civilization,' I would note that
Aristotle was a pagan, not a Christian.  And I'm an atheist, for that
matter.  One of the magnificent aspects of the Western scientific
tradition based on Aristotle is that it is open to people of all
ethnicities and religions, insofar as they are rational.

 > It is from the mathematical establishment; Atiyah and co. etc.

I'd really like to understand what you are hinting at, regarding
Atiyah and company.  Could you please explain?  Surely Atiyah and
company are not rejecting the idea that mathematical theorems require
proof.  So, exactly what do you mean when you hint that Atiyah and
company have rejected the `paradigm' of reducing mathematics to the
derivation of theorems from axioms?  What alternative foundational
`paradigm' have they replaced it with?  (According to Kuhn, the only
way to overthrow a paradigm is to replace it with another paradigm.)

Does anybody else here have a clue as to what Atiyah's alternative
`paradigm' might be?

By the way Anand, I assume you realize that I am trying to have a
serious discussion here.  These matters are important, don't you
agree?  You're not trying to pull my leg, are you?

Best regards,
-- Steve

PS for John Burgess: 

Could you please give us some idea of what the postmodernists have to
say about foundations of mathematics, insofar as it may be relevant to
this discussion?

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