FOM: Platonism v. social constructivism
rhersh at math.unm.edu
Mon Mar 2 13:52:07 EST 1998
Thanks for your encouragement.
First of all, let me throw away that "new age" label. It happens
I live in the capital of new age-ism, Santa Fe. Without troubling
to work on a precise definition of new age, let me just say that
I understand it as a misty mysticism, addicted to Eastern Religions,
channeling, wicca, "body work," and skeptical or or hostile to
hard science, math, numbers, logic, etc.
That ain't me.
On the contrary, I view Platonism as a mystical remnant of
Pythagorean theology. In my view, saying that math is what
we do together is being hard headed, down to earth, sticking
to what we really see and do in real life here on Earth.
Disagree if you like, but don't call it new agey!!!
I gave some personal history of my thinking in the introduction
to The Mathematical Experience. Sorry, I don't have an
extra copy of that to send your way, but I bet it's in the
library. If you do look into it, I would recommend also the
section on the Ideal Mathematician, and the last two chapters.
However, since I can't presume you will really go to that trouble,
I will repeat what I say there.
Many years ago, as part of my practise of teaching all kinds of
different math courses, I taught "Foundations of Math," hoping as
usual to learn the subject by teaching the course. I learned
about logicism, intuitionism, and formalism, and that alll three
had petered out and dried up, and that there was no credible answer
to the question, what is math.
I was upset, disturbed. Started reading, looking for the answer.
Didn't find it.
I did come to realize, as I said in my recent posting, that
the typical mathematician is a philosophical hypocrite, between
an unformulated, unanalyzed Platonism, and a formalism that
he knows very well falsifies his work experience.
This bothered me too. I was sure that by reflecting and examining
on my lived experience, and that of other mathematicians, I
would be able to figure out, what is mathematics.
I came upon the article by Leslie White in Vol. 4 of the World
of Mathematics (James Newman, Ed.) White explains that
mathematics is a social-cultural reality. I could see, yes
this is the answer.
You say I haven't told you what math is about. Does everything
have to be about something else? We create math ideas, from other
ideas, and also from properties or suggestions in the nonmathematical
(physical, economic, biological) worlds. Some of these ideas
are used in mathematical models of nonmathematical things. You
can say, if you like, the linear wave equation is "about" a vibrating string.
On the other hand, the infinitely smooth infinite dimensional manifold
of infinite genus doesn't have to be about anything. It just is--
as a shared concept among mathematicians. It's not a photograph,
it's not a representational painting or sculpture, it's not
a bit of reportage on TV or whatever. It ain't about nuthin. It's
just what it is--an infinitely smooth etc etc etc. What can
I say? That's the only answer there is.
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