FOM: Response to Hersh

[Moshe' Machover]

Hersh says:

>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.

Oh, no; we don't. The sentence:

(*)	`Prior to the American Civil War, platoons of infantry soldiers in
 	battle were formed into rectangles.'

refers to rectangles; rectangles are studied in one of the specialties in
the MR. But the proposition expressed by (*) is not mathematical; it
belongs to military history.

Hersh:

>	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.

In this case one side is right and the other wrong.

Hersh:

>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.

Obviously the term `mathematics'--like all terms--is subject to changes of
meaning. I am mainly interested in its present meaning: in characterizing
what you and I and the editors of the MR count as mathematical
propositions. There is in fact a far-reaching consensus about this, as you
would be the last to deny. I claim that this consensus must and can be
explained by an intrinsic property of the propositions that all of us would
accept as mathematical. A necessary (and arguably sufficient) condition is
that the proposition is amenable to rigorous treatment, such as proof or
refutation.
>
>	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?

No, you are mistaken. See above. I have seen in the park opposite my house
gardners working on rectangles of grass.

>
>	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 all depends on the proposition in question. The proposition that the GR
is aesthetically pleasing is not mathematical. The proposition that gives a
representation of the GR as a continued fraction is. (If the representation
is incorrect, the statement is a false mathematical statement.)


>	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?

At least in part. It depends on what proposition you are considering. The
proposition that such and such an algorithm converges (or is stable) is
undoubtedly mathematical.

In my view, the primary question is `what is a mathematical proposition?'.
The question as to what counts as *doing* maths is secondary. This question
is indeed quite fuzzy. For example, is someone who is applying a
matheamtical proposition to a physical problem (or a problem in
gardening..) *doing* maths? Well, yes and no... But it is a mistake to
*start* from this fuzzy question and try to define `mathematical
proposition' in terms of the answer to it.

>
>	What about fluid dynamics?  Do you accept that?

Ditto. Fluid dynamics is partly an empirical science and partly a branch of
maths. If you give me specific proposition of fluid dynamics, then *once I
understand it* I will be able to tell you unfailingly whether it is
mathematical or empirical. And I will be prepared to bet that my judgment
will be endorsed by any competent f.o.m.-er: even Harvey Friedman and Colin
McLarty will for once agree; :-) I suspect that you too will agree; we will
all agree. (The last statement I have made is a sociological prediction
based on my understanding of the intrinsic nature of mathematical
propositions.) If you ask me whether doing fluid dynamics is doing maths, I
cannot give you a clear-cut answer. But, I repeat, it is erroneous to try
to define `mathematical proposition' in terms of `mathematical activity'.
You cannot define a rather precise concept in terms of a fuzzy one.

You can put Decartes before Hobbes, but you can't put the cart before the
horse.

>	What about chaos and related dynamical systems, where
>theorems are inextricably tangled with heuristic computations?

Enough said.



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