[FOM] Am I a Platonist?

[Harvey Friedman]

Reply to Davis. The interchange seems to be a very good illustration of how
you can sit a formalist (not me) and a Platonist (not me, but Davis) down,
and have them look at the exact same phenomena, and come to opposite
conclusions. 

My own view it that as far as I can tell, both interpretations of this same
phenomena are full of weak points and strong points.

It would be interesting to see how the mathematical community at large, or
segments of the mathematical community that enjoy some special distinctions,
view this phenomena.

The reaction SO FAR seems to be compatible only with some sort of

*naturalism*

as I described it when I wrote

"whatever attitudes and preoccupations that helps mathematics get on with
its main traditional goals in as smooth and nondisruptive a manner as
possible, is to be adopted"

in my posting of 10/26/03 3:51PM, re: Formalism/Platonism.

(See P. Maddy, Naturalism in Mathematics, 1997 Oxford).

In order to elicit any other reaction from the mathematical community, it
seems to be necessary to have new kinds of results where set theoretic
axioms (demonstrably) impact extremely concrete mathematics, that is also
extremely attractive.

In fact, this requirement on new results, in order to get the issue joined,
may just be a manifestation of Naturalism having taken over.

In just plain simple terms,

extremely attractive concrete mathematics

are awkward to run away from, or ignore, or suppress, perhaps because it at
least prima facie is, or is close to, or relevant to, the "main traditional
goals of mathematics".

Of course, at least initially, this will continue to be the way that
mathematicians WILL run away. E.g.,

"what does this have to do with algebraic topology, algebraic number theory,
partial differential equations, etc"?

However, even getting to

extremely attractive concrete mathematics

is just being to begin to happen- we are still in a very early stage of the
inevitable. The inevitable is

wherever there are serious finite combinatorial structures that can be
handled by mathematicians within the usual ZFC framework, there are ADJACENT
serious finite combinatorial structures that cannot be handled without large
cardinals. 

And since

everywhere in serious mathematics there are serious finite combinatorial
structures

we have that 

everywhere in serious mathematics there are serious finite combinatorial
structures that cannot be handled without large cardinals.

In fact, I conjecture the following:

***given any interesting mathematical assertion, there is a stronger, more
detailed, mathematical assertion, in the same spirit, which cannot be
handled without large cardinals***

This is even true if the given mathematical assertion is highly infinitary.
For as we see, there are clear finite forms of large cardinals. In fact, it
seems clear that there are clear finite forms of all of the usual large
cardinals. 

So this certainly applies to, say, real analysis, and pde, etc.

It appears that progress is just too slow on this for one person, and so
successors are needed to carry this out.

But only after this is carried out quite a ways beyond where we are now,
will we be able to elicit any meaningful reaction from the mathematical
community at large. Until then, there will be

foundational silence

which is consistent with some form of

naturalism. 

On 10/28/03 2:01 PM, "Martin Davis" <martin at eipye.com> wrote:

>> I do think that this can be attacked as an oversimplification. There is the
>> question of the meaning of volume - rather than any objective situation. It
>> could be said that Torricelli succeeded in showing merely that it was a good
>> idea to extend the notion of volume to more objects. Lebesgue also did that
>> later.
> 
> I think this comment misses the point in a number of directions.
> Torricelli's work occurred in a pre-calculus environment when there was no
> general notion of volume to "extend". People like Cavalieri and Torricelli
> (following the example of Archemedes) worked specific examples using a
> Euclidean framework. In modern terms the solid Torricelli considered can be
> characterized as formed by rotating the curve y = 1/x, x>=1 about the
> Y-axis. He showed that this infinite solid had the same volume as a certain
> finite cylindrical solid. (See Paolo Mancosu's fascinating monograph,
> "Philosophy of Mathematics and Mathematical Practice in the Seventeenth
> Century" Oxford 1996 where Torricelli's proof is given in detail.) The
> reason that the result (which today is a homework problem in a freshman
> calculus class) was so shocking to contemporaries was because it violated
> classical ideas going back to Anaximander and codified by Aristotle about
> the unapproachability of the infinite.

We can focus on the simpler example of the area under the curve y = 1/x^2. I
had thought that Archimedes is now credited with sophisticated thinking
about "approaching the infinite", which is now considered to have been
rather close to epsilon/delta?

> The comparison with Lebesgue is particularly misplaced in my opinion. When
> I used to sit on qualifying exam boards, one of my favorite questions was:
> "Why do analysts like to use the Lebesgue integral?" The wrong answer was:
> "Because there are Lebesgue integrable functions that are not Riemann
> integrable." Of course, analysts never felt the inability to integrate the
> characteristic function of the irrationals as a limitation.  It was the
> limit theorems enjoyed by the Lebesgue integral that accounts for its
> popularity,

You somehow want to use Torricelli (and perhaps Lebesgue in some contrasting
way?) in order to aid an argument for Platonism and against formalism.

I submit that any formalist can look at the same history and make completely
opposite use of it.
> 
>>> When Cantor began his study of transfinite
>>> numbers, it was not at all clear that this could be carried out in a
>>> satisfactory manner, and of course many refused (and continue to
> refuse) to
>>> accept it. But the work of the set theorists continue to develop this
>>> domain, and even when assumptions that transcend ZFC are used, the
>>> objective nature of their results seems compelling.
>> 
>> This may seem compelling to you, but not to many people. I don't take sides.
> 
> And yet, your own beautiful work (which I advertise whenever I can) in
> which you find finite combinatorial consequences of strongly infinitary
> principles will only gain general currency to the extent that this
> objective nature is accepted.

This certainly is wrong, if I understand what you mean by "general
currency". Formalists and Platonists will be equally interested in this
work. In addition, the expected reaction to this kind of work, from
everybody, when suitably successful, will simply be one of amazement:

**you mean to say that this trivial looking thing, so close to interesting
things that are second nature to us, has logical difficulties? Amazing.**

This reaction has absolutely nothing to do with acceptance or rejection of
large cardinals. 
> 
> ......................
> 
>> The big difference between physics and mathematics that has to be confronted
>> is as follows.
>> 
>> When Einstein's GR is confirmed over and over again, in various settings, it
>> is clear that it could have been horribly disconfirmed - to the point of
>> refutation, by these experiments.
> 
> This is true. And  yet, large cardinal axioms could still be "horribly
> disconfirmed - to the point of
> refutation" by someone proving their inconsistency with ZFC.
> 

Yes, but that's perhaps all that the confirmation/refutation idea has going
for it, at least NOW. It is very weak, and it is subject also to the issue
of why we believe that human beings are particularly good at finding
inconsistencies, even if they exist?

I am fully aware that some people argue that if something has BEAUTIFUL
consequences, more beautiful than other things, then that is an argument for
truth. But that is a matter for another time.

Harvey Friedman