[FOM] pathology

[Harvey Friedman]

On Thu, Aug 27, 2015 at 9:20 PM, Martin Davis <martin at eipye.com> wrote:
> Harvey asks why I say it is worthwhile to study these things,  and the
> answer is simply that it is interesting mathematics.

I don't have any feelings for "interesting mathematics" independently
of how it fits into general intellectual culture. I.e., what I used to
abbreviate as g.i.i. = general intellectual interest.

I know what complicated mathematics is, I know what technically deep
mathematics is. I know what beautiful mathematics is. But at my
increasingly advanced age, I am trying focusing only on g.i.i.

What do you mean by "interesting mathematics"? Do you take this as a
primitive notion?

> 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.

Have you seen any systematic study of what simple functional equations
have beautiful intended solutions, all others being deeply
pathological? I think this is a very rich deep interesting project.
>
> 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.

On the contrary, all the evidence shows that the brand of deep
pathology we are, or I am, talking about CAN be starkly separated from
the "tame mathematical world". That is what I am saying.

And you know who is originally responsible for fueling this stark
separation? None other than Goedel.

The conservation results are very stark. BUT perhaps it does need to
be buttoned up and stated in a newly creatively careful way:

THEOREM. Any Pi-1-4 sentence provable in ZFC can be proved in ZFC
"without reference to any deep pathology of the kind under
discussion".

THEOREM. Any Pi-1-4 sentence provable in ZFC can be proved in ZF.

"from the tame mathematical world where most mathematicians may prefer
to pitch their tents*

It would advance our discussion if you would please reflect on just
why "most mathematicians prefer to pitch their tents". Notice that I
dropped the word "may".

> 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.

Harvey does this for large cardinals, but DEMONSTRABLY cannot do this
for the kind of deep pathology we are talking about.

Notice I was careful to identify this as "the deepest pathology
mathematicians are generally familiar with - such as f(x+y) =
f(x)+f(y)".

Mathematicians are not familiar with large cardinals.
>
> 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.

Breaking the language barrier, as this deep pathology does, is in a
category all unto itself. This breaks all historical rules in a
special way.
>
> 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.)

Not "deep pathology" in the sense of breaking the language barrier.
>
> 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 love this kind of terminology. It helps a lot.
>
> I can promise one thing about future developments: they will be surprising.
>
We don't live very long by civilization standards, and we all must do
the very best with the information that we have, and the insights that
we have.

I don't spend much time preparing for the sun going supernova, or even
a much more likely event of a major comet hitting the Earth.

I will say with confidence that math and physics are, during this
century, going to be turning finitary, with the infinitary mathematics
looking more and more like a mysterious, often arcane, background
tool, known to be eliminable, but quite useful. HOWEVER, there is
emerging, well, you know what, and as that develops this century, your
comments will look more germane, not for the deep pathology breaking
the language barrier, but for large cardinals instead. That's where it
is at. Why? Because the large cardinals are DEMONSTRABLY not
eliminable.

Harvey