FOM: Re: Reuben Hersh: Mitteilungen der DMV (fwd)

[Ching-Tsun Chou]

Let me first say that I am neither a mathematician nor a philosopher;
I'm just an engineer.  So what I say below is definitely not up to the
professional standards.

It seems to me that it is perhaps not very interesting to ask:

         WHAT IS MATHEMATICS, REALLY? 

Instead, a better question may be:

         Why and how does mathematics work?

By "work" I mean that using the machinery of mathematics, we can reach
conclusions that can be verified empirically.  Professor Hersh listed
quite a few "social-cultural-historical" ("social" for short)
constructs.  Most of them (especially the religions and the -isms ;-)
do not seem to "work" in this sense.  If some guru tells me that if I
intone certain magical formulas in a certain way then I shall get a
raise, I will be very skeptical.  But if some physicist tells me that,
according to her calculation, if I squeeze a certain amount of a
certain type of uranium in a certain way then there will be a gigantic
explosion, I will take her much more seriously.  Indeed, if I don't
believe her assertion, she can dispel my doubts by demonstration.  I
don't think this kind of demonstration is available in most of the
"social" constructs that Professor Hersh mentioned.

It also seems to me that the line that Professor Hersh drew between
things real and things unreal is somewhat arbitrary.  He wrote:

           Here is an important objection.

           Aren't some mathematical concepts grounded in physical
   reality?

           For instance, N, the natural numbers?

           I surely have five fingers on my left hand, so "five" has
   a physical meaning.  On the other hand, N includes some very large
   numbers,  ((2 to a very high power) raised to a very high power) raised to
   a very high power.  It is questionable what physical meaning this
   big number has.  So the natural numbers as describing physical objects
   are not the same as the natural numbers in pure mathematics. 

   The fact that I have five fingers on my left hand is an empirical
   observation.  "Five" in that usage is an adjective.  There is no
   conceptual difficulty there, any more than in saying my fingers are long
   or short.  But five in pure mathematics is less than the big number I just
   defined, and is relatively prime to it, and so on.  It possesses an
   endless list of properties and relationships, not only in N, but also in
   R, in C, and beyond.  It's part of an abstract theory.
   As such, it is not a material object, not a mental object, but a shared
   concept, existing in the social consciousness of mathematicians and
   others.

I guess Professor Hersh considers the five fingers on his left hand
real, perhaps the number 5 too, but certainly not some very large
numbers.  But would he consider the electromagnetic field real?  Or
the genes?  Or the water molecules?  What is the fundamental
difference between these quite abstract objects and the five fingers
on his left hand?

I submit that there is no essential difference between them.  They are
all useful (so far at least) in constructing an intelligible account
of "the physical reality" (whatever that term means) THAT WORKS.  But
perhaps these questions about what is real and what is not, are not
very interesting after all.  Perhaps the really interesting question
is whether Professor Hersh, or anyone else, can construct a coherent
body of mathematics that does not contain those very large numbers,
BUT STILL WORKS.  Before anyone can do that, I shall consider (((2
raised to a very high power) raised to a very high power) raised to a
very high power) as real as the five fingers on my left hand.

           Locating mathematics in the social-cultural realm
   means that it is human.  For example, there is no sense to
   talking about mathematics existing before the human race existed or after
   it has vanished.

           Some people find this conclusion shocking.  One may point to
   some beautiful theorem of Lagrange and say, "Isn't it obvious that this
   was always true, before there were any humans?"

           You could just as well cry, "Listen to this wonderful symphony of
   Beethoven!  Isn't it clear that this was always beautiful, even before
   Beethoven was born?"

Indeed, I am rather doubtful whether my goldfish finds Beethoven's
symphonies beautiful.  However, if some beautiful theorem of Lagrange
has some implications about the motion of my goldfish's body, I have
very little doubt that it would indeed be the case.  Similarly, I have
very little doubt that many of the conclusions about the physical
world that we can reach using the machinery of mathematics were true
long before the rise of the humankind, are true now, and will continue
to be true long after its demise.  I wonder whether one can say the
same about the beauty of Beethoven's symphonies.

Cheers,
Ching-Tsun Chou

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  Ching-Tsun Chou                       E-mail: ctchou at mipos2.intel.com 
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