FOM: Re:

[Martin Davis]

At 04:07 PM 1/15/98 -0700, Reuben Hersh wrote:

>I must also disagree with your accusation that I "heap scorn"
>on fom'ers.
>
>I looked thru my book, and couldn't find any scorn at all.
>
>As for fom as it is practised by the owners of this list, I was
>almost unaware of it when I finished the book, hardly interested
>in heaping schorn.
>
>It looks to me like we have verbal confusion.  In my book I use
>Lakatos term "foundatiohnism" for Frege, Hilbert and Brouwer--all
>people who sought to repair or rebuild the foundations in a philosophical
>sense, in order to recover the lost certainty of mathematical
>knowledge.
>

I never accused you of heaping scorn on fomer's. Here's what I said:

>It is the problematic character of mathematical truth at a (shifting)
boundary >of what is understood that provides the problems for workers in
FOM. It is their >scientific conquests that provide the developing consensus
that Reuben sees as
>just out there, while heaping scorn (in his "What is Mathematics
>Really") on those he refers to as foundationists.

Note that I capitalized "FOM". It had been suggested that "fom" should stand
for this list and the capitalized version should just be an abbreviation for
the three words; I was adhering to that convention.

Here's from your book, p. 249:

"Problems intractable from the foundationist or neo-Fregean viewpoint are
approachable from the humanist viewpoint. But from the foundationist
viewpoint, humanist solutions are no solutions. They're unfair. It's not
allowed to give up certainty, indubitability, timelessness, or
tenselessmness. These restrictions in philosophy of mathematics act like the
restriction to the real line in algebra. Dropping the insistence on
certainty and indubitability is like moving off the line into the complex
plane."

To my mind, it is fair to say of this passage that it is "heaping scorn on
those he refers to as foundationists".

Look at almost any current graduate textbook in whatever branch of
mathematics. It will begin with an introductory set-theoretic section on
"notation". This represents a consensus that set theory is the proper
foundation for mathematics. This consensus was hard won. And it was won by
"foundationists". Frege, Russell, Zermelo, Hilbert, Skolem, von Neumann:
they each played a role. Have you looked at Gregory Moore's excellent
historical study "Zermelo's Axiom of Choice"? It makes it very clear waht a
difficult struggle it was.

"Certainty and indubitability": naturally, we would like as much certainty
as we can get. With, for example, Lagrange's four square theorem, we have
it. Generally, of course life is hard, and we must settle for what we can
get. The Riemann hypothesis (which by the way is equivalent to a certain
Diophantine equation having no solutions) certainly looks to be true. As a
"humanist" are you prepared to assert it on the basis of current empirical
evidence? Suppose it turns out that human mathematicians remain unable to
settle it. Will you then declare it meaningless? Or will you say that it is
surely either true or false, almost surely true, but sadly, we are unable to
give a conclusive proof?

What about the much discussed on fom CH? Does humanism help us out? Sol
Feferman thinks the question is intrinsically incoherent. John Steel sees
that various apparently unrelated large cardinal axioms seem to line up in a
linear order and hopes for a resolution. This is a real difference of
opinion between experts. What does humanism tells us about this? 

To which real questions about the foundations of mathematics does humanism
hold the key? Moving to the complex plane makes it clear why the Maclaurin
expansion of 1/(1+x^2) can't have a radius of convergence >1. What new
clarity and understanding comes from the move to "humanism"?

Martin