[FOM] Fwd: invitation to comment
Andre.Rodin at ens.fr
Andre.Rodin at ens.fr
Tue May 24 13:45:19 EDT 2011
Monroe Escew wrote:
>One plausible view is that the
>changes brought to geometry by Descartes did not require an alteration
>of any of the ancient foundations or concepts. Only new machinery and
>techniques were introduced, which could all be reduced to the original
>language and techniques.
I don't find this view plausible, I think that this view is plainly wrong. This
is the subject I'm actually working on specifically. To put it in very few
words the foundational changed that happened in maths of the 17th century was
this. In his Elements Euclid treats two basic mathematical disciplines of his
time, namely geometry and arithmetics; each has its proper subject-matter,
namely geometrical magnitude(s) and number(s) correspondingly. This is, in
particular, Euclid develops the theory of proportions twice: first for geometry
and then for arithmetics - which is from today's point of view is wholly
redundant (I mean the arithmetical one is redundant). In the 17th century
algebra began to be seen as the "universal mathematics" (mathesis universalis")
that treats magnitude(s) *in general* - such a notion of magnitude being wholly
absent from Euclid and the mainstream Greek mathematics (albeit it is not
implausible that Eudox could have some pre-conception of this sort). For the
reference (to a primary source) see, for example Arnauld's Nouveaux Eléments de
géométrie. (The introductory part of this work presents what we call today the
elementary algebra - but it presents it as a general theory of magnitude that
covers the continuous, that is, geometrical magnitude and the discrete
magnitude, that is, number - after this the author turns more specifically to
geometry.) Without this drastic foundational change the modern history of maths
wouldn't be what it really is and nothing like the algebraic proof of the
impossibility of trisection wouldn't be possible.
>Further, the eventual axiomatization by
>Hilbert did no violence to Euclid's geometry. It completely respected
>the concepts and approach; it merely clarified, brought out implicit
>assumptions, and made things fully rigorous.
There is, of course, a somewhat elusive - and yet important - link between
Euclid's Elements and Hilbert's Grundlagen - but Hilbert's foundations are just
wholly different! Hilbert did not just corrected some flaws in Euclid's theory
but invented a wholly new way of building mathematical theories! Hilbert's
axiomatic method differs drastically from Euclid's method.
>Analogously, the geometer who is reluctant to accept analytic
>techniques will not therefore assert that the angle *can* be
>trisected. She will just not accept analytic proofs of the fact that
>it cannot, and consider the problem unsolved, just as you will not
>accept perfectly ordinary proofs that PA is consistent.
What do you mean by "perfectly ordinary proofs that PA is consistent"? Arnon
Avron suggests in a recent posting that the proposition "N is a model of PA" is
a mathematical proof that PA is consistent. This argument doesn't actually look
like an "ordinary" mathematical proof, does it? And it doesn't seem me
convincing for reasons that I explain in my reply to his posting. As far as
more sophisticated proofs of Con(PA) are concerned I must look better at them
to make a more qualified judgement but what I have learned about them so far
doesn't convince me that the matter is uncontroversial. Actually it is a matter
of sociological fact that Con(PA) IS controversial: different reputed
mathematicians express different opinions about it.
I don't consider myself to be a member of mathematical community - I'm
historical and philosopher of maths who is interested in today's maths too. I
certainly wouldn't object Perelman's proof on whatever philosophical or other
basis given that the mathematical community has a consensus view that this
proof is alright. But I don't see anything like a similar consensus in case of
Con(PA) among mathematicians. And the question also seems me somewhat different
in nature than (for example) the Poincaré conjecture.
The analogy would work if I would reject some new technique of proving
mathematical theorems and stick to outdated views on mathematics. I cannot see
that I do this.
Andrei
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