[FOM] pathology

Fri Aug 28 12:11:52 EDT 2015

"ZFC is a conservative over ZF for Pi^1_4 sentences" is a great result, but I'd like to know how much more has been shown:

(1) For what a and b is ZFC conservative over ZF for Pi^2_a and Sigma^2_b sentences?

(2) How much further is ZFC conservative over ZF+DC than ZFC is over ZF?

(3) How much further is ZF+DC conservative over ZF than ZFC is over ZF?

-- JS

-----Original Message-----
From: Harvey Friedman <hmflogic at gmail.com>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Fri, Aug 28, 2015 11:55 am
Subject: Re: [FOM] pathology

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
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