FOM: Conway's foundational ideas

[Martin Davis]

At 07:25 PM 5/24/99 -0400, simpson at math.psu.edu wrote:

>
>Therefore, I hereby apologize for and retract all of my earlier
>Conway-baiting remarks.  Let me turn instead to a sober, restrained
>discussion of Conway's foundational ideas as expounded on pages 64-67
>of ``On Numbers and Games''.  I reserve for possible future FOM
>postings the difficult question of how the f.o.m. community can best
>cope with the Conway phenomenon and similar phenomena.

For myself (and I believe for others as well) I want to thank Steve very
much for this forthright apology. I would only add that it would appropriate
even if John Conway were not a "respected and influential Establishment
mathematician". No one should be "baited" in an intellectual forum; it just
lowers the level of discourse without any compensating gain.

On the substantive issues Steve is raising, I think it would be helpful to
review some familiar history. Cantor was never upset about the "paradoxes";
he even tried to make use of, what today we would call proper classes, as a
tool. But others were upset. To people like Hilbert and even the young
G"odel (not to mention Weyl) it seemed that our very basic logical
intuitions had turned out to be inconsistent. Set theoretic reasoning was
thought to be inherently unreliable, impredicative definitions leading to
damnation or worse.

What has happened? Set theoretic foundations have triumphed. Treatises on a
huge variety of subjects are implicitly or explicitly based on ZFC. Nobody
bothers to mention uses of AC in ordinary mathematical writing. This is a
great victory for f.o.m., and as fom-ers, we should be pleased and proud.

What I take Conway to be saying is that we no longer need be afraid of the
dark. It's OK. Construct what you need, and if you're halfway reasonable
about it, there will be no problem. If needbe, a metatheorem can be proved
showing that this or that construction can be carried out in ZFC, maybe
augmented by some large (or in today's terms: medium) cardinal axiom.

This is no assault on foundational research.


                           Martin Davis
                   Visiting Scholar UC Berkeley
                     Professor Emeritus, NYU
                         martin at eipye.com
                         (Add 1 and get 0)
                       http://www.eipye.com