FOM: social construction of mathematics?
Reuben Hersh
rhersh at math.unm.edu
Thu Mar 19 11:32:16 EST 1998
Point one isn't about being around a long time. It's about
having become long ago a reflective, observant mathematician.
My dissatisfaction with the philosophies I found led me to
start reading more philosophy, and especially thinking
philosophically (reflectively, awarely, self-consciously and
others-consciously) about what was going on in my own mathematical
thinking, talking, writing and, as far as I could ascertain, in
the thinking, talkling, writing of my friends, colleagues and others.
Most mathematicians don't do this. I didn't do it in my first 20
years in math.
Yes, I found White an eye-opener. But that doesn't mean I was
completely satisfied with his paper. His explanation that
the locus of math existence is in culture as he says it, in
intersubjective consciousness of mathematicians and others as
I would say it, does seem absolutely right. But he does not
attempt to explain wherein math is different from other
cultural activities. The answer that math is uniquely
specified as that study of human ideas which has science-like
reproducibility and consensus was my own answer to that
question. It was called fast and loose by Prof Tragesser,
called unsatisfactory by you and others. The only one in
my view who actually dealt with it was Shipman, who showed it
failed to distinguish math from chess. I responded by saying
that my criterion singled out math and math-like actiities such
as chess. I left it to Shipman, as a certified chess whiz, to
figure out the difference between math and chess. I haven't
heard from him about that.
So I don't think you can say I just pick from ready made
philosophies that I find on the shelf. I don't know of
another book or author who says everything I do. There
are some who largely agree with me, but so far as I know
they do not precede me. After all, I am told that I am
"leading the charge" was it against Platonism?
But I still agree that I am hesitant to break really new ground.
After all, I'm not an fom'er, not a logician, not even a philosopher,
just a low down on the ground p.d.e. person. Who likes getting
his knuckles rapped by the cognoscenti?
At your suggestion, I'll rethink some of my thinking.
Of course it is gratifying and encouraging to me to learn that
my thinking is in the same direction as Sol Feferman, whom I
first knew in 1962 as a lowly math instructor in an august west coast
establishment.
One more thing. I don't find Platonism repulsive, disgusting, upsetting,
sickening or anything like that. I find it inconsistent with an
empiricist or (dare I say it? materialist!) or skeptical or
critical philosophical stance. It is theology. That's fine for
those who like theology. But I don't think people should pretend
it's not inconsistent with empiricism or materialism or skepticism
or any critical stance. If I may quote myself, I think Platonism
is in part a leftover from the long tradition of philosophies tying
math and religion together. Math may indeed be a thought in the
mind of God (although I don't think so.). But if you don't base your
philosophy of math on religion, Platonism is a relic. It's
like the smile on the face of Lewis Carroll's Cheshire cat.
(The holy cat vanished, yet the smile remains.)
Now about my methodology. Yes, my working premise in trying
to find a philosophy of math has been that it should come out of,
be based on, be faithful to real life, real mathematical life and
experience. I explicitly reject the opposite (sorry about the
Boolean-Aristotleian language!) method which it seems to me
is standard among the philosophical pros: decide a priori,
on the basis of tradition, or subjective preference, on some
principles that philosophy of math must conform to, and then
twist math to fit into those principles. To be specific, I
believe that what Lakatos called the foundationists (Frege,
Brouwer, Hilbert) were concerned with establishing mathematical
certainty (re-establishing it, they believed) and found three
different ways of doing that. None of which worked. I don't
agree that certainty in mathematics is indispensable, so
I don't agree with the methodology that starts out, "How
can we establish (or re-establish?) mathematical certainty."
I am more interested in looking at mathematical life, mathematical
thinking in real life, and trying to understand what's going on.
You point out that this methodology may cause me to
overlook some important aspects of math. Maybe so. I'll
think that over. But to the extent that I have achieved
some insights and clarifications about real life math
amd mathematical activity and experience, to that extent
I have done what I set out to do. Those who think the
philosophy of math should have different goals and interests
from mine probably shouldn't count on me to satisfy them.
My little story about Robin and Foebe is meant to
be an allegorical comment on my interaction with fom.
Thanks for your attention.
Reuben Hersh
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