[FOM] reply to Harvey

[Harvey Friedman]

On Thu, Jul 21, 2016 at 2:26 PM, Martin Davis <martin at eipye.com> wrote:
> I’m pleased that Harvey has replied to my recent FOM post. I have a few
> comments.
>
> Harvey has always refused to commit himself to a particular position on the
> philosophical issues that arise in connection with foundations. He has
> described himself in rather colorful language to emphasize his readiness to
> go wherever he finds an opening for serious work.
>
> To begin with this message seems to reiterate this stance:
>
> “The ‘matter of fact absolute truth’ (MOFAT) attitude to set theory is just
> one of many equally defensible positions, no more compelling than others.”

> But then he takes what seems to me a very decided position: he writes of “an
> inevitable shift” to a “better” or “best” way “to do set theory”. It seems
> to me that this “inevitable shift” is presented as the unique alternative to
> MOFAT. Personally, I find neither compelling. (My own view is based on a
> historical study of mathematics and the infinite:
>
> https://foundationaladventures.files.wordpress.com/2012/01/platonic.pdf)

When I said that there is an inevitable shift to "better" or "best"
ways "to do set theory", I have in mind two claims. The main claim is
that this shift is inevitable in connection with the various programs
pursued for settling CH and related matters. The secondary claim,
obviously with less certainty, is that this state of affairs is going
to persist for the foreseeable future, or more speculatively, forever.
The upshot of this is that this is a) an interesting novel(?) way to
proceed, or maybe b) the most interesting and/or promising way to
proceed - for the purpose of assigning truth values to CH and related
statements - that we have today.

It appears clear from what Martin is writing that he agrees with a).
With regard to b), I am not sure whether he agrees with this. Also, I
don't really know if Martin agrees with my "main claim" above.

Incidentally, by the standards of the usual dogma with regard to isms,
I think what I am suggesting is rather mild, even though Martin refers
to it as "a very decided position".

Also incidentally, I always viewed Martin as a matter of fact absolute
truth advocate. This was on the basis of statements made in personal
discussions.I took a glance at
https://foundationaladventures.files.wordpress.com/2012/01/platonic.pdf
and see a rather involved and rather nuanced position.

Come to think of it, it might be perfectly reasonable to interpret the
historical events Martin eludes to in his
https://foundationaladventures.files.wordpress.com/2012/01/platonic.pdf
as examples of my "better" or "best" ways "to do mathematics", and not
any search for matter of fact absolute truth, so that in the end
Martin agrees completely with everything I am saying.

> The words “better, best” trouble me. It implies an opposing series
> “bad,worse, worst” and seems a departure from Harvey’s previous “anything
> goes” stance.

I don't see this as a departure. I was just saying that "a whole
spectrum of views are equally defensible". IF you are going to do
unconstrained set theory with its full generality, THEN what is the
"better" or "best" ways to proceed? I am not taking some hard nose in
principle stance against matter of fact absolute truth, but rather
observing that it appears to be unpromising in terms of a research
methodological. I am offering up a perhaps novel(?) methodology that
looks more presuming.

> Then, we are told that taking this position “sets the stage”
> for his current “hobby”, a novel approach to CH (one that I find
> fascinating).

On this, we seem to agree. So I am getting fuzzy about just what is
the nature of the disagreement.

> Of course someone who persists in a belief that CH may have a
> definite truth value can still find this work quite interesting, so there is
> really no need to "set the stage" in this way.

But a very substantial portion of researchers at various levels and in
various areas are quite dubious about CH having a definite truth value
in the sense of matter of fact absolute truth. Many of them are very
dubious about any research concerning assignment of a truth value to
statements like CH. These people would need a reason to be interested
in what I am offering up here.

> Is Harvey wanting it made
> clear that he has no such belief? Is that why this project is a mere
> “hobby”?

I call it a hobby because I want to emphasize that the vast bulk of my
f.o.m. efforts lie in Concrete Incompleteness, and I regard that
effort as substantially more important for f.o.m., mathematics,
science, philosophy, and the general intellectual environment.
>
> Finally Harvey  comments on my statement in my previous post: “because
> Pi-0-1 propositions can be expressed as a polynomial equation having no
> natural number solutions, a counter-example to a false Pi-0-1  proposition
> can be verified by a finite number of additions and multiplications of
> integers.” He comments, “Of course, this isn't sensitive to the nuances
> concerning what is meant by ‘can be verified’ here. It means computations
> that can be done using the algorithms we learned as children for adding and
> multiplying integers that will lead to the result 0. The only “nuance” I can
> imagine Harvey has in mind is that the numbers involved can be huge. But to
> take this “nuance” seriously would be to give up on number theory. If
> someone were to prove that the zeta function has a zero off the critical
> line, but the least such zero had an imaginary part greater than the number
> of atoms in the observable universe, wouldn’t we say that the Riemann
> Hypothesis had been disproved? Who would say to the contrary that a
> pragmatic Riemann Hypothesis had been proved because all zeros that we could
> hope to compute had been shown to be on the line? Similar conundrums could
> be presented to logicians who contemplate expressions in a formal system of
> arbitrary length.
>
I referred to nuances being missed here without saying much about
them. I am referring to the following. Just because a Pi01 sentence
actually has a counterexample (a statement meant to be made in the
context of a modest amount of matter of fact absolute truth), this
doesn't mean that it is knowable that the Pi01 sentence is false. The
main nuance here is that the counterexample, or maybe all
counterexamples, are totally useless since they cannot actually be
entered or referred to in any way in a refutation of the Pi01
sentence. They may be far too large.

At later, more advanced stages of the Concrete Incompleteness Program,
we would expect to have examples of mathematically interesting Pi01
sentences which are refuted in say 10 pages of mathematical
text,starting with the existence of large cardinal axioms, but in
order to be refuted in ZFC, would require at least 2^2^100 pages of
text. So the existence of counterexamples, in this interesting cases
expected to arise, is entirely useless in order to actually refute the
Pi01 statement.

The above state of affairs in no way causes anyone to "give up on
number theory" as Martin is suggesting.

Harvey Friedman