[FOM] pathology

Martin Davis martin at eipye.com
Thu Aug 27 21:20:18 EDT 2015

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.

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