FOM: Platonism v. social constructivism
Reuben Hersh
rhersh at math.unm.edu
Fri Dec 26 14:29:39 EST 1997
Dear Professor Silver,
Thank you for an enjoyable, thought-provoking letter. I will confine myself
(more or less) to direct answers to your questions, rather than take the
time and space to repeat what I said as well as I could in my book,
"What is Mathematics, Really?"
First of all, what about that "really"? As I explain in the first
few pages of the book, there is another book, a famous classic, by
Courant and Robbins, called "What is Mathematics?" Courant and
Robbins was a major influence on me, as was, later, Courant
himself. Yet I found the title of their book a little misleading,
for it consists of exposition and demonstration of beautiful mathematics,
but no direct answer to the title question. So I, somewhat jokingly,
somewhat in humble tribute, thought of my book as carrying out the
promise in their title.
To someone who has not looked at my book, all
this may smack of an "in" joke; that was not my intention.
Why do I renounce aristocratic vs. humanitarian math? Because
Prof. Tragesser, in all good will I am sure, attributed such thoughts
to me in one of his postings. I do not wish to be asked to explain
the meaning of those expressions. Of course that doesn't mean denying
that there could be two versions of math. In fact, there already are
at least two--"classical" and constructivist. The indeterminate
state of the Continuum Hypothesis (CH to FOM'ers) opens another
possibility of 2 maths. I don't know if the same can be said about
categorical foundations vs. sets. I don't see this as a problem.
Next, postmodernism. I am in shock following
a column in the New York Times last Saturday by a person namned Edward
Rothstein. He shredded me up as a multiculturalist and postmodernist!!!!
Then there was a posting by Jon Barwise where, if I recall, he said he
would be careful about social constructivism for fear of "the black
plague" of postmodernism. So I wrote that I had been naive not to foresee
such problems.
No, I certainly don't believe math (or poetry or fiction or
journalism) is no more than "text." From what I hear of what's going
on in lit crit and related areas, that idea deserves to be
called a black plague. One of the principal themes of my book is that
mathematics is a living practise of a human community, much more than
just piles or rows of journal articles and treatises. In fact, as a
rule those articles are incomprehensible to any one not initiated into the
mathematical subculture of the authors. This
is carefully explained in a section of the book called "Mathematics
Has a Front and a Back."
In my book I avoided the term "social constructivism"
while acknowledging that it's used by some authors to whom I am
sympathetic. But I did not want my thoughts to be tied to theirs.
So I adopted the name "humanism", wanting my point of view
to be judged on its own, not by association. Nevertheless, the
commentators on my book on this list always say "social constructivism."
Do I intend to reduce the study of mathematics to history and
perhaps some kind of sociology, so that mathematics becomes about--nothing?
No! Not at all! Just the opposite!
I seem to hear a notion that what isn't mental, physical,
or transcendental, is--nothing! But that's the whole point of my list
of social-cultural-historical entities. The death penalty, the baseball
pennant race, the stock market, anti-semitism, patriotism, Catholicism,
the Lubavitcher othodox community, the yen, the mark and the dollar,
and yes, multiculturalism and postmodernism--they are all real things!
None of them is--nothing. And to say mathematics is the same sort of thing
is not to say that it's nothing. Mathematical ideas, beliefs, facts,
theorems, are real objects. Their reality resides in a social consensus,
of which the consensus of mathematicians is a crucial part. When
humanity departs from the cosmos, sooner or later, all those ideas,
beliefs, theorems will disappear. This is not to ignore the roots of
mathematics in physical reality. When humanity disappears, there
will no longer be any integers as abstract objects. There will
still be 9 planets, for example. But the theorem that there are two
groups of order 9 will not "exist", because it is an idea, a conception,
that can exist only in minds as parts of a community or society.
For us, here, today, the fact that there are two groups of
order 9 is not about nothing, it is about our shared concepts of groups
and "9". It is objectively true in that sense--perhaps you would prefer
"intersubjectively."
My viewpoint is indeed on a level with and
challenges Platonism, formalism and logicism. I affirm that
mathematics is real and meaningful, by proposing a social reality
and a social meaning. If you have trouble with my question,
"What is mathematics, really?" I can rephrase it:
In what sense is mathematics real, it what sense is it meaningful?
Now, what about Platonism?
Neal Tennant is right when he says that
nothing about the actual practise of math can refute Platonism.
So far as I know, nothing can refute Platonism. It's irrefutable.
With all respect, I am compelled to compare it with the existence
of G-d. No one ever could or can prove there is no G-d. Those who
don't believe in Him/Her don't make such a claim. They merely hold
that the arguments or evidence for His/Her existence are unconvincing.
So it is with a transcendent mathematical reality. To some, it seems
obvious, undeniable, that math always existed and always will exist
in some immaterial, inhuman sense or other. To others, such a
belief seems groundless and extremely implausible. If I may repeat,
I have never taken on the responsibility of converting Platonists
to humanism. I merely undertook to explain another point of view,
that lets us see mathematics in terms of real mathematical life, and which
affirms the reality and meaningfulness of mathematics.
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