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Re: [HM] Getting history and philosophy right
Posted:
May 13, 2003 5:13 AM
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There seem to be three separate worries raised to Lakatos's thought that
'history of mathematics without philosophy of mathematics is blind':
1) philosophy may hold up the progress of mathematics.
2) adopting a philosophy of mathematics relevant to contemporary
mathematics would make us misunderstand the mathematics of the past.
3) concentrating on the official philosophy of mathematics of an era
would make us misunderstand the mathematics that time.
Worry (1): This is an issue about philosophy and so not Lakatos's
concern here, but his general attitude that one should always engage in
criticism (in the positive sense) is. One can point to examples where a
principle is
made explicit, e.g., 'one is not a number', and then made a piece of
dogma.
For Lakatos, the first act is beneficial but not the second.
Philosophers
of mathematics today seem so concerned about the dogma worry (and
perhaps
also the fear of appearing foolish in daring to make any comment that
would have any bearing on what mathematicians do) that they pride
themselves
in 'leaving mathematics as it is'. This was not Lakatos's way:
uncritically
generated mathematics was to be thrown out. Personally, I believe that
today
we're too far off the pace to say even as much as this, but I do think
we
can contribute to debates that flare up (or smoulder on) in math
departments. I hope my efforts to elucidate the senses of the term
'natural'
as it used by mathematicians (chap 9 of my book) will prove interesting
to them. Resources from historians (use of 'natural' in earlier times)
and
from practitioners of science studies in general are relevant to this
project.
As for worry (2): of course! Although, with sensitivity it might throw
up
a novel way of reading the past.
Worry(3): Milo Gardner's qualms may be allayed by adopting a
Collingwoodian approach, where the philosopher would interest herself in
the constellation of presuppositions underlying a practice in some
period. Then one needn't
think through the mathematics of an age merely in light of the writings
of the 'official' philosophers of the day. E.g., for the early 19th
century make sure you've read the views of Gauss, Galois, Dirichlet,
etc. on mathematics.
Replying to Elizabeth Hind:
"The past needs to be studied for its own sake and on its own terms.
That is
history, and to try and deal with mathematical texts on their own terms
we
need to understand the philosophy of the time in which they were
written.
This is not what Lakatos had in mind."
Lakatos certainly forces matters in Proofs and Refutations, and so
misses
out all the delicious part of the story of what fed into Poincare's
Analysis
Situs, e.g., Riemann's work. But compared to what was going on at the
time,
1963, P&R is a lone oasis in a very wide desert.
"In the introduction to your book you choose to quote from Hardy: "These
parts of mathematics [parts which have a utility] are, on the whole,
rather
dull; they are the parts which have the least aesthetic value." "It is
not possible to justify the life of any genuine professional
mathematician on the ground of the 'utility' of his work"
I defend anyone's right to do research which seems to have no utility.
However, I find this attitude is prevalent among people engaged in
history
of maths. Because of this I feel that history of mathematics
concentrates
on those bits which are aesthetic, what you would consider to be 'real
mathematics'. Yet, surely by doing that we are missing out on some very
important episodes in human history?"
I refuse to admit this charge. If you read on, I wasn't quoting Hardy to
agree with his notions of uselessness. In fact, if you follow up the
caveat
I mention, you'll find Hardy wasn't dogmatic on this issue. I can't
comment
on the charges you lay against historians, but as for myself the task of
understanding the relationships between what gets called pure
mathematics,
applied mathematics, mathematical physics and theoretical physics is an
essential task. And if you get to chapter 10 of my book, you'll see I
call
for work on the use of mathematics in theoretical computer science.
I'd be interested to hear what historians think about Sarah Cuomo's
allocation of space in her Ancient Mathematics between the mathematics
of the 'person-on-the-street' and that of the 'professional'
mathematician.
There's a certain relevance here to current debates in Britain as to
whether to stop having mathematics as a compulsory subject to 16,
'numeracy' being deemed what is necessary to survive in a modern
society.
David Corfield
Faculty of Philosophy
10 Merton Street
Oxford OX1 4JJ
http://users.ox.ac.uk/~sfop0076
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