FOM: Machover's suggestion
Reuben Hersh
rhersh at math.unm.edu
Sat Jan 24 20:19:44 EST 1998
Dear Prof. Machover,
As I take it, you propose characterizing math by its content.
A promising idea. And easy to implement. The Mathematical Reviews
of the American Math Society publishes an exhaustive list of math
specialties. Several thousand, I believe! So if a sentence of
English, Hebrew, or whatever refers to something studied in any
of these specialties, we say it's mathematics.
That seems perhaps too crude and unsophisticated.
After all, somebody decides, somehow, by some criteria, what
to put on the list. All we have to do is get those criteria
out in the open, and the job is done.
But probably these decisions are made by a committee, and
probably the committee members sometimes disagree. Certainly
the criteria change as the decades and centuries go by. Do we
want our definition of math to be so time-dependent? Even, dare
I say it, culture-dependent? No, certainly not.
Probably you would agree with many ordinary every-day
mathematicians that rectangles are a mathematical topic. So
would you say that someone who works on rectangles is doing math?
Right now on our sister list, math history, a great many
postings concern the Golden Ratio. Is it really the most
beautiful shape, as has been alleged? This is a matter of serious
concern to some members of that list. I don't think they are
doing math, I don't think you think they are doing math. It's
a matter of what questions you ask and what means you use to answer
them, not just what your topic is.
Do you know any numerical analysts? Most of them are
improving and inventing algorithms, and testing them on special
problems where the solution is already known. Rarely does
a useful, important numerical algorithm receive a rigorous
proof of convergence or stability. In fact, some purists deny
that such numerical analysis should be accepted in a serious math
department. Is numerical analysis mathematics, by your lights?
What about fluid dynamics? Do you accept that?
What about chaos and related dynamical systems, where
theorems are inextricably tangled with heuristic computations?
The content of math is not clear cut, it is fuzzy. It
is not constant, it is ever widening and expanding.
If I may bring in a related field, astrology deals with
the planets and the configurations of the stars. Does that mean
it's part of astronomy? Most astronomers say no. It's a matter
of what questions you ask and how you try to answer them, not
merely the topic you talk about.
I proposed the reproducibility or consistency of math
as the distinguishing feature that separates it from other
humanities. I don't think any of my antagonists have disputed
that.
I agree, this is not a profound discovery. Much more
profound would be the discovery of how mathematics happens,
how it is possible. I specifically declare that I am not
proposing to answer that. Who is?
I believe that it is not a philosophical question,
but an empirical one. I believe that empirical scientists
in several fields are beginning to make progress in answering it.
All the best,
Reuben Hersh
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