FOM: naive or brainwashed?

Reuben Hersh rhersh at
Wed Mar 18 16:54:25 EST 1998

"What determines what math is?"
Do you think this is a clearly formulated question?

I gave a criterion that distinguishes math from other 
fields of academic study.  The fact that mathematicians can check
each other and come to agreement on their conclusions is
an essential, non-ignorable feature that makes
mathematics what it is.

However, you say that is external, and you want something internal.

There is certainly much more to be said to describe
or characterize or distinguish math.  In fact, the principal
purpose of the Mathematical Experience, which you read and even
enjoyed, was exactly that.  It does take a whole book,
or several books, to characterize math in any amount of thouroughness
or detail.  Since my previous book tried to do that, I didn't
realize I had to do it again in my new book.

The only way I see to respond to your request is concretely, by
real examples.  There are situations when one must
decide, "Is this mathematics?"  Such situations are
extreme cases of situations where one must evaluate a  person,
a book, a manuscript, or whatever.  Is it correct?  Is it interesting?
Is it nonsense?  Is it promising?  Is it math?  Is it applied math?
Is it (as you propose) astrology?

I thought of answering you by describing what I do in writing a
math paper.  Would that help?

I could tell you what I (or any other mathematician) does in refereeing
a journal submission.  Is it interesting?  Is it reasonably correct?
Is it more or less new?  Is this the right journal to publish it?
All that, of course, includes the elementary question, what is it?
Is it math?  Is it garbage?  Is it, as you propose, astrology?
I could report some of the thinking that goes on in answering
these refereeing questions.  Would that be on your point?

But your question doesn't concern correctness, novelty,
or interest.  Only, "is it math?"

A plausible looking young man comes to me with a method for
trisecting any angle by ruler and compass.  He knows that
in the books this is proved impossible.  But he wants me
to find his mistake in his proof.  The proof is pages and
pages of trigonometry.  I would call this incorrect or
confused or incompetent math.  Would you say, no, it's not math?

I could tell you how I would decide if I walked into a random
seminar room in a random university, whether the seminar topic
was math or not-math.  Would that be pertinent?

I'm afraid, though that you are demanding something 
much less interesting:  a 
one- or two-paragraph stipulation about mathematical proof, or
about shapes and numbers, or about sets, which could never be made 
precise or inclusive.

Is applied math math?  Some say yes, some say no.

Is the computer calculation of 8 billion digits of PI math?

Is fom math?

Is fuzzy logic math?

Is operations research math?

Is category theory math?

Is theoretical computer science math?

Is classical mechanics math?

Is statistics math?

Is theoretical fluid dynamics math?

    To all these questions, you will 
find some to say yes and some to say no.

The answer "math is what mathematicians do" is not much of a joke.
It becomes meaningful if I actually try to describe what mathematicians
actually do.  As I said, this was a principal purpose of the Math Experience.

Another kind of answer can be given genetically, see the Nature
of Mathematical Knowledge by Philip Kitcher (briefly described in
What Is Mathematics Really?)  Mathematics there is characterized
by its historic descent from arithemetic and visual geometry.

Obviously anything as complex, dynamic, many-faceted as
math can be characterized in more than one way.  The
question is really, what do you want to know?  What is the purpose of 
your question?

Reuben Hersh

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