[FOM] reply to Harvey

[Martin Davis]

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)

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

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.

Martin
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