FOM: What is mathematics, really?, Gen Intellectual Interest, Political Agendas

[Reuben Hersh]

Since my name is occasionally being mentioned on this list, I
take the privilege of explaining myself, or rather my book, *What
Is Mathematics, Really?*

The book has two parts.  Part 1 proposes  my
answer to the title question.  I claim that the difficulty in
answering it is that the traditional choices are incomplete.  You
can say, with  Plato, Leibniz, and Frege, that math is transcendental.
In an age of materialism and empiricism,  this answer seems to many
incongrous or incredible.  You can say, with Brouwer, that math is
mental.   But 7 + 3 = 10 seems to be true, whether I know it or not.
You can say it's physical, like Quine, but FOM'ers know all too well
how far behind their math can leave physics.

	But there's a major chunk of reality that's
neither transcendental, mental or physical.  What are
money, war, religion, art, literature, music, patriotism, race hatred,
universities, department stores, language, politics, government?  Real, 
for sure.    But neither transcendental, mental, or physical.  They are
social-cultural-historic realities.

	Given this fourth choice for an answer to the title question,
I claim that this is where math fits best.   To back that up, I
tell some stories and make some explanatons of how math is actually
lived--not only on the bookshelves of libraries, but in the making.
In the Nova video about FLT, I think it was Barry Mazur who exclaimed
over the list of contributors to Wiles proof:  Shimura, Taniyama, Ribet,
Katz, Frey, and a dozen more names I forget.  Wiles
presented his first proof to an audience of mathematicians.  When
fault was found, he did it over, and again had to pass muster with his
colleagues.  Of course FLT has a long history.  The reason it was 
worth 7 years of Wiles life was the prestige that had been accorded to it 
by the math community.  All this is social.

	This proposed answer to What is Mathematics? has not been 
popular in the philosophy of 
math community.  But it isn't new.  Leslie White proposed it explicitly,
and in doing so gave credit to Emile Durkheim.

	Part 2 is a history of the philosophy of math, from Pythagoras to 
the present.  I found that the best way to present this was in two 
strands:  I called the Mainstream the thinkers from Pythagoras to the 
present who in one way or another regard mathematics as superhauman or 
inhuman.  The other strand, which regards math one way or another as 
human, I called humanists and mavericks.

	In the last chapter, almost as an after thought, I raised the 
question whether there is a correlation betweeen the humanist-Mainstream 
split, and a political split betweeen right and left.  And I found, to no 
one's surprise, that humanism tends to associate with left, Mainstream 
with right.

	This is a distinction between philosophies, not between 
mathematics!  I know nothing of aristocratic or humanitarian mathematics.

	One big mistake I made, to which I plead naivete.  I didn't 
recognize the danger of being associated with the black plague 
of postmodernism.   (Actually, after some effort I have 
failed to understand what postmodernism is.)

	I understand that the classification of mathematical reality 
as social-cultural-historical raises difficult questions.  But 
I think the chance of answering them is better than with any 
other classification of mathematical reality.