FOM: Hersh's characterization of mathematics
Reuben Hersh
rhersh at math.unm.edu
Thu Jan 22 01:29:24 EST 1998
> X On 20 Jan J SHIPMAN wrote:
>
> > To assail Hersh, we need a definition of
> > mathematics better than his. If his
> > definition calls the same things "mathematics" as ours,
he must explain why there should be a uniquely objective area
of human thought. To refute him we must find math his
definition misses or non-math it includes.
>
Yes, I agree. I say that there is human mental activity, concerned
with human mental constructs (ideas) which has the science-like
property of reproducibility, or consistency if you prefer, and which
consequently achieves a high consensus comparable to that achieved
by experimental sciences like physics or chemistry.
This type of science-like mental activity deserves to have a name, and
in fact we have given it a name--mathematics.
But the traditional philosophical question is, how is mathematics possible?
This I do not attempt to answer. I take as given that there is mathematics,
and try to understand how it is done, how it works, what is its nature in
real life.
What about the more difficult question, how is this kind of mental
activity possible? How is mathematics possible?
That was Kant's question. He answered it by postulating or inventing
the twin intuitions of time and space.
I have not detected any neo-Kantians on this list, so perhaps it's
unnnecessary to discuss why Kant's answers don't suffice for us today.
The three old foundationist schools can be thought of as explaining how
mathematics is possible. If mathematics is just logic, that is a kind
of answer.
If mathematics is just formal calculation, that also is a kind of answer.
And if mathematics is the solitary, languageless contemplation of some
Creative Intelligence, that also is a kind of answer.
Since no logicists, formalists or intuitionists have announced themselves
since I have been on the list, perhaps it's unnecessary to explain what
is inadequate, incomplete, and in contradiction to everyday mathematical
experience of each of these answers.
Philip Kitcher, in his important book the Nature of Mathematical Knowledge,
gives a careful detailed account of how mathematics grows from an empirical
root in counting and measuring. But a questioner asking How is
mathematics possible? in the spirit of Kant and his successors
might feel that Kitchers' empirical, historical answer is, as I
have been told in a related connection, IRRELEVANT.
In my opinion, this question, how is mathematics possible, is a fascinating
question, and an important question--but not only a philosophical questioh.
It's not a matter of what "trade union" an investigator belongs to. It's
a question of whether this question can be investigated by philosophical
methods, and whether it can be investigated empirically.
As a matter of fact, empirical investigation is beginning to yield
interesting information. I recommend the work of George Lakoff on
embodied metaphor. I recommend the book The Number Sense by Stanislas
Dehaene (Oxford.)
If I may make an analogy, consider music. Like mathematics, a universal
feature of human life. Of course, it's entirely different from math.
Nevertheless, I can imagine a Philosopher of Music asking, "How is music
possible? What features of the universe, or of the general nature of
man or of mind, can account for this remarkable, singular phenomenon?"
I've heard some people say music originates in the rhythm of the heart
beat or of breathing. Such questions can be approached by
studying the nervous system--maybe by harmless electrodes implanted in a
singer's brain--and by studying how children learn to respond to music, and
by seeing whether anything like music is made by chimpanzees and gorillas.
I know nothing about music. I'm just
trying to get across the idea that to learn how mathematics is
possible, we can call on developmental psychology and comparative
psychology and experimental neuroscience. We may never understand
it completely, but we are beginning to understand it somewhat.
Is there a philosophical method to explain how mathematics
is possible? I don't know of any. Kant and Russell and Hilbert and
Brouwer had a shot at it. They were all great geniuses, I agree,
and I'm not heaping scorn on anyone. But I think this question
may not be appropriate for philosophizing.
The Heideggerians in the audience may
remember their master's beautiful problem: "Why should anything
exist? Why sould there be something rather than nothing?"
To my knowledge, few physicists or philosophers have beeen
willing to waste their time on that one.
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