foundations, Atiyah, etc

[Stephen G Simpson]

It sounds as if Anand has gotten upset about some of the questions I
raised.

 > If you are going to give your own definition of foundations and "what is
 > of interest to any educated human being" and eliminate from consideration
 > that which doesn't satisfy it, that's fine, but is not the basis for a
 > conversation.

Anand, I truly don't understand this remark.  If we remove the
emotional overtone from it, what's left is that, according to you, I
committed some sort of offense by defining what I mean by
`foundations'; that I somehow thereby removed the basis for
conversation.  My answer to that charge is: Why?  What's wrong with
defining our terms, so that we can know what we are talking about?  If
we are going to talk seriously about foundations and what is
foundational and what is not, it seems to me that it would be a good
idea to have some sort of definition of `foundations' on the table, to
avoid the usual kind of worthless touchy-feely conversations, where
people talk at cross purposes and never get anywhere.  My essay at
www.math.psu.edu/simpson/Foundations.html is a serious attempt to
define `foundations.'  And I want to take this concept seriously and
use it to analyze the issues under discussion.  If you have an
alternative concept called `foundations', I'd be glad to hear it.  But
how in the world can we have a serious conversation unless we know
what we are talking about?

I'm not trying to eliminate cohomology or anything else from
discussion.  I'm just trying to be clear in what I am saying.

 > what is of general interest to an educated human being is an
 > historical phenomenon, and what is at some stage in the realm of
 > specialists can filter its way through to general mathematical
 > culture.

Of course.  This happened in the case of Cartesian analytic geometry.
In the 17th century, Cartesian geometry used to be in the realm of
specialists, but now it informs the thinking of every educated human
being.  If cohomology et al are foundational in the same sense as
Cartesian geometry, as you asserted, then that would seem to imply
that you have in mind a scenario whereby cohomology et al will become
similarly indispensable to the thought of educated persons.  Would you
care to describe such a scenario?

On the other hand, maybe you didn't really mean it when you said that
cohomology et al are in the same league with Cartesian analytic
geometry.  If so, then that's OK too; I don't want to shut off the
discussion; I just want to understand the background of what you are
saying.

 > 2) 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. This is a separate issue
 > from that of rigour and rationality. In any case, it is  not from a bunch
 > of fashionable wierdos 
  ...
 > It is from the mathematical establishment; Atiyah and co. etc.

Really?  When a theorem is announced, don't Atiyah and company insist
on proof?  And what is the starting point of the proofs, if not
axioms?  Anand, could you please explain in more detail the view that
you ascribe to Atiyah and company, whereby the derivation of theorems
from axioms is no longer of major importance?

 > Whether or not you know the meaning of the words you used
 > (postmodernism, primitivism etc.), these are the last people who
 > they are appropriate to describe.

When I used these words, I had no way of knowing that you had Atiyah
and company in mind, so obviously I was not saying that Atiyah is a
postmodernist.

I think this misunderstanding came about because I was trying to take
a general intellectual perspective, while you were taking a pure
mathematics perspective, so we were talking at cross purposes.  That's
why definitions are important: to set the context of a discussion, to
avoid misunderstanding such as this one.

Once again Anand, what is your definition of `foundations'?  Mine, as
you know, involves a certain general intellectual or philosophical
perspective.  Mine is on the web at
www.math.psu.edu/simpson/Foundations.html.  Where is yours?

 > Of course one can say that all these people are idiots, but that is
 > neither useful nor true.

As explained above, I wasn't saying that Atiyah and company are
idiots.  I have a lot of respect for Atiyah's mathematical
achievements.  But I don't know much about his views regarding axioms,
proofs, or foundational issues generally.  Anand, could you please
explain them?

Best regards,
-- Steve

Stephen G. Simpson
Department of Mathematics, Pennsylvania State University
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