[FOM] pathology

henk henk at cs.ru.nl
Fri Aug 28 03:09:42 EDT 2015

Hello Martin,

A very poetic and true contribution!

All best wishes, Henk

On 08/28/2015 03:20 AM, Martin Davis wrote:
> Harvey asks why I say it is worthwhile to study these things,  and the
> answer is simply that it is interesting mathematics. When I was a boy my
> mathematical friends and I used to call the Polish journal Fundamenta
> Mathematica: the journal of f(x+y) = f(x)+f(y). It seems that each issue
> had a new result showing that with ever weaker conditions, only the linear
> solution would satisfy it. Starting with the almost trivial case of
> continuity, there was Borel,  measurable, and others. We were interested
> because it was interesting mathematics.
> Gödel showed us that the wild infinite could not really be separated from
> the tame mathematical world where most mathematicians may prefer to pitch
> their tents. And of course Harvey has done more than anyone else to find
> compelling examples of this. But a look at the history of mathematics makes
> it clear that we have little ability to discern that some unfamiliar and
> unexpected mathematical phenomenon is not just something annoying to be
> eliminated, but rather of great importance. It is notoriously difficult to
> predict what will happen, especially, as has been said, the future.
> I looked up the example I had slightly garbled of Hermite's reaction to
> continuous non-differentiable functions. This is what he said: "Je me
> détourne avec effroi et horreur de cette plaie lamentable des fonctions
> continues qui n'ont point de dérivées." My translation: I turn myself away
> with fear and horror from this woeful affliction of continuous functions
> that do not have derivatives.
> An example I find particularly revealing was Toricelli in the 17th century
> showing that a certain infinite area when rotated about an axis formed a
> solid of revolution of finite volume. For the shocked reaction, see Paolo
> Mancosu's monograph "Philosophy of Mathematics & Mathematical Practice in
> the Seventeenth Century". This bit of deep pathology, violating all
> Aristotelian precepts about the necessary separation of the finite from the
> infinite, is today a homework problem in a freshman calculus course. (The
> area in question may be taken as formed by the curve y = 1/x, the X-axis
> and the line x=1.)
> Mathematical terminology is littered with unfortunate terminology, residue
> of what was once though to be pathological: negative numbers, imaginary
> numbers, irrational numbers, improper integrals.
> I can promise one thing about future developments: they will be surprising.
> Martin
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150828/3dd4181b/attachment.html>

More information about the FOM mailing list