# FOM: Conway's foundational ideas

Stephen G Simpson simpson at math.psu.edu
Tue May 18 13:29:46 EDT 1999

```John Pais 18 May 1999 10:33:54 quotes from Conway's book ``On Numbers
and Games'':

> What is proposed is ... that we give ourselves the freedom to crete
> arbitrary mathematical theories of these kinds, but prove a
> metatheorem which ensures once and for all that any such theory
> could be formalized in terms of any of the foundational theories."

This proposal of Conway is vague, but let's try to make it precise.

One way to make it precise is in terms of the well known f.o.m. idea
of ``definitional extension of a theory''.  This very well known idea
gives you the freedom to extend a foundational theory (e.g. ZF) by
defining new predicates and new function symbols.  There is a very
well known metatheorem saying that the extended theory is conservative
over the original theory and every formula of the extended theory can
be effectively translated as a formula of the original theory.  This
very well known metatheorem is proved in standard textbooks of
mathematical logic, for example Shoenfield's very well known textbook
``Mathematical Logic'', 1967.

Unfortunately, Conway in 1976 showed no sign of being aware of this
already very well known f.o.m. idea and the accompanying metatheorem.
Instead of opening up a logic book, Conway decided to grandstand about
a ``Mathematicians' Liberation Movement''.

I call this f.o.m. amateurism.  What do you call it?

Simpson:

> > (Example: Shelah's proof of the consistency of the Proper Forcing
> > Axiom relative to a supercompact cardinal or whatever.)  Conway
> > seems unaware of this kind of thing.  ...

Pais:

> This seems reasonable since ONAG was published in 1976 and Shelah's
> book in 1982.

But Paul J. Cohen won a Fields Medal in 1964 for his work on the
independence of the continuum hypothesis.  Furthermore, by the end of
the 1960's, the famous work of Martin and Solovay on the consistency
of Martin's Axiom (a forerunner of Shelah's work on the consistency of
the Proper Forcing Axiom) and Solovay's famous work on the consistency
of ``all sets of reals are Lebesgue measurable'' relative to an
inaccessible cardinal, were very well known in f.o.m. circles.  Other
results of a similar nature were also known.

But Conway in 1976 gave no sign of being aware of this very well known
and relevant and important f.o.m. research.  Instead he preferred to
grandstand.

I call this f.o.m. amateurism.  What do you call it?

By the way, does anyone know what Conway's subsequent posture on these
issues has been?  Did he issue any kind of retraction or apology?

> "grandstand" ? Possibly, but more likely prematurely imaginative or
> retroactively naive or just surreally cute.

It seems to me that there was nothing premature about it.  It seems to
me that Conway was simply grandstanding, with no apparent awareness of
or desire to inform himself about very well known and relevant
f.o.m. concepts and results which were readily available.

Perhaps Conway's mistake was to rely on Johnstone as his source of
f.o.m. expertise.

I call this f.o.m. amateurism.  You can call it imaginativeness and
naivete or maybe cuteness, if you insist.

-- Steve

```