FOM: Re: Mitteilungen der DMV (fwd)
rhersh at math.unm.edu
Thu Feb 26 16:54:09 EST 1998
This brief note was requested for the Notices of the German
Mathematical Society. I take the liberty of posting it here.
WHAT IS MATHEMATICS, REALLY?
Dieudonne and Cohen, among others, have noticed that the
typical mathematician is a philosophical schizophrenic, or
When he is working at his mathematics, he has no doubt that
the objects he is studying have in some sense a real, objective
existence. Whether it be N the set of all natural numbers, R the
uncountable set of real numbers, or perhaps some infinitely smooth
infinite dimensional manifold. This is, so to speak, his week-day
religion. It is a variety of "Platonism" (also often called "realism".)
However, if he should be challenged to explain where, how, in
what sense any of these invisible, intangible, infinite entities is real
or exists, he is likely to turn tail, and retreat hypocritically into
some form of formalism. That is, he drops any claim that anything in
pure math really exists; all we really are doing, he explains, is making
logical deductions from meaningless axioms. This is "formalism," so to
speak his Sunday religion.
This hypocritical schizophrenia is widely practised,
and it serves the purpose, as Dieudonne said, of getting the
philosophers off our necks and letting us do math as usual.
Nevertheless, hypocrisy and schizophrenia cannot
really be good for the soul.
I propose a way of thinking about the reality and
existence of mathematics which lets us keep our mathematical objects
really existing, really meaningful, without resort to mysticism.
The key observation is that in our world there are not two
but three main kinds of reality. Mind and matter are familiar. But they
do not help with our puzzle, because mathematical objects are not
material, and they are not mental, in the sense of being
part of anyone's private subjectivity.
But they are not the only things that are neither
mind nor matter.
For instance, your job. The money in your bank account. Your
degreee. Your career....
Catholicism, Lutheranixm, Judaism, Islam, Bolshevism, Fascism,
racism, nationalism, rationalism,. liberalism, feminism....
Beethoven's fifth symphony, Shakespeare's Hamlet, The
Bible, Darwin's theory of evolution, Einstein's theory of relativity....
This journal. Its editorial policy, its standards of
publication, its mailing list, its backlog. This mathematical
society. Its history. Its traditions....
All these things are real. They are the substance of our lives.
None of them is material, none of them is mental. What then?
I call them social-cultural-historical, or just social for short.
Once these examples are pointed out, it is hardly questionable
that there is another level of existence besides the mental and material.
Now, our problem was--what sort of existence or reality has
We have not two but three choices. Material and mental are
wrong. What about social?
I claim that social is right. Infinitely smooth infinite
dimensional manifolds exist, are real, as shared concepts, part of the
shared thinking of mathematicians. This answer does not say they
are not real. On the contrary, it explains and locates where and how
they are real.
Here is an important objection.
Aren't some mathematical concepts grounded in physical
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
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?"
It seems to be especially logicians, foundations-of-mathematics
specialists, and academic philosophers of math who find this idea
I am not trying to upset anybody. Just face the facts.
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