[FOM] Re: Absoluteness of Clay prize problems

Harvey Friedman friedman at math.ohio-state.edu
Mon Aug 23 18:47:00 EDT 2004

On 8/23/04 3:11 PM, "Dmytro Taranovsky" <dmytro at MIT.EDU> wrote:

> Harvey Friedman wrote:
>> In fact, in my opinion, the essence of mathematics is mostly Pi01 and
>> predominantly at most Pi02.
>
> I do not agree with this.  I think that mathematics can be roughly separated
> into three large areas:  the study of finite structures or integers, the study
> of real numbers and certain well-behaved functions on real numbers, and
> general
> or set theoretical mathematics.

We agree that the study of finite structures or integers is dominated by
Pi-0-2 statements, and in fact I claim that it is dominated by Pi-0-1
statements in the sense of my last posting, 8/18/04, 1:42AM.

For example, "there are infinitely many primes" is Pi-0-2, but the essence
of the matter is "for all n there exists a prime somewhat bigger than n",
which is Pi01.

In the same way, I believe that in your second category, "the study of real
numbers and certain well-behaved functions on real numbers", you will find
the following phenomena. When a celebrated theorem of this kind is proved,
mathematicians will quickly and fruitfully dissect it into the trivial
component and the nontrivial novel component. The trivial component will
generally involve nonarithmetical statements. However, the nontrivial novel
component will be Pi-0-1 or maybe Pi-0-2.

It would be interesting to carry out this division into components in detail
for various central theorems of core mathematics in algebra, geometry, and
analysis, etc.

I.e., divide important celebrated theorems of core mathematics into

1) a trivial nonarithmetical component;
2) a nontrivial novel component which is Pi-0-1; or perhaps
2') a nontrivial novel component which is Pi-0-2.

We require that 1),2) trivially imply the original theorem.

Such divisions may not be entirely apparent, but I believe that they are
entirely natural, worthwhile, relevant, satisfying, illuminating, etc. I
also believe that Pi-0-1 vastly outnumbers Pi-0-2.
>
> Natural mathematical theorems tend to avoid many alterating unbounded
> quantifiers, so in number theory and related areas, the vast majority of
> theorems can naturally be expressed as Pi-0-2 statements.

Pi-0-1, in fact, as in the case of "infinitely many primes".

>Commonly used
> functions whose domain is a subset of R tend to be continuous or otherwise
> Delta-0-2.  Statements about analysis tend to be expressible (after some
> coding) as Pi-1-2 statements.

But there will be a Pi-0-1 core, with the rest trivial. At least that is
what I guess...

> Many theorems in analysis--such as the theorem stating compactness of the unit
> interval--are Pi-1-2 statements and are not reducible over RCA_0 (a weak base
> theory) to arithmetical or Pi-1-1 statements.

The compactness of the unit interval, in this way of thinking, is viewed as
a student exercise with no content. One should analyze, say, the essence of
the Riemann mapping theorem, the Stone Weierstrass theorem, complex
differentiable implies infinitely complex differentiable implies Taylor
expansion, Riesz-Fischer theorem, Parseval's theorem, Radon-Nikodym theorem,
etc.

> Graph minor theorem is a Pi-1-1 statement that
> is (as far as I know) not implied by any consistent arithmetical statement
> over Pi-1-1-CA_0 (a theory that tends to suffice for ordinary as opposed to
> "set theoretical" mathematics).

Yes, I did prove:

THEOREM. Let A be an arithmetical statement (or even Sigma11 statement).
Suppose Pi11CA0 proves "A implies the graph minor theorem". Then Pi11CA0
proves notA.

Similar results involving weaker systems for Kruskal's theorem.

However, these jewels are exceptional. We have been talking about the NORM
in mathematics.

> Also, regarding the Continuum Hypothesis, the problem is an important basic
> mathematical (or, some would say, metamathematical) problem in its own right,
> so study related to CH is important even if it has no known (non Sigma-0-1)
> Pi-0-1 or commercial applications.

There are many important matters for which one cannot make any kind of case
that they are connected with the essence of mathematics. The continuum
hypothesis is one of them. Another is "is there intelligent life in the
universe?"

>We still know relatively little and
> there are many places where a compelling answer to the CH hypothesis could
> come from.

What specifically don't we know? Where are these places?

>The theory of reasonably definable using subsets of omega_1 as
> parameters sets of subsets of omega_1 is largery unexplored, and it could be
> that the only plausible theory of such sets implies the CH.

How are you going to recognize plausibility here?

>Based on my
> beliefs about unity of knowledge, human potential, and importance of
> basic study, I would conjecture that the eventual solution to the Continuum
> Hypothesis will have important practical applications.
>
Given your beliefs about ..., give me an example of something that you would
conjecture will NOT have important practical applications.

Harvey Friedman