FOM: Conway's foundational ideas

John Pais paisj at
Tue May 18 13:33:54 EDT 1999

Just to make sure that those, who haven't seen Conway's (1976) book "On
Numbers And Games" (ONAG), don't get the wrong idea about the content and
purpose of the book, let me say that in my opinion this book is a
masterpiece of mathematical creativity, communication, and pedagogy that
one could hand to a curious student with some preparation and say, "here's
a very good, extraordinarily readable example of how mathematician's think
about and do mathematics." The mathematical concepts are presented with an
expert light touch and with humor in such a way as to reveal that there is
a real human being behind the scenes struggling with and developing these
(fun!) technical ideas.

ONAG was mentioned on the FOM list and so there might be a slight tendency
to amplify the emphasis given by Conway to those portions of the book that
might be construed as making foundational claims. In particular, regarding
the "Appendix to Part Zero", ONAG pp.64-67 mentioned below, I suggest that
in the spirit of, and within the context of the whole book and what he is
trying to do mathematically, he is merely asking for a little
*foundational elbow room* to do the mathematics he wants to do. For
example on p. 66 what Steve omits after "...This appendix is in fact a cry
for a Mathematicians' Liberation Movement!", which I consider a playful
(even humorous) suggestion, but not politically anti-FOM, is:

 "Among the permissible kinds of construction we should have [this is
essentially all he needs/wants to make explicit]:
(i) Objects may created from earlier objects in any reasonably
constructive fashion.
(ii) Equality among the created objects can be any desired equivalence

 In particular, set theory would be such a theory, ...

 But we could also, for instance, freely create a new object ...

I hope it is clear that this proposal is not of any particular theory as
an alternative to ZF (such as a theory of categories, or of numbers or
games considered in this book). What is proposed is instead 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

Stephen G Simpson wrote:

> Joe Shipman 05 May 1999 12:32:09
>  > See "Appendix to Part Zero", ONAG pp.64-67, which is foundationally
>  > well-informed ....
> OK Joe, I am looking at it, and I don't necessarily agree that it is
> foundationally well-informed.  Yes, Conway understands how flexible ZF
> is with respect to transfinite inductive constructions, or at least he
> understands this better than most mathematicians.  But here on page 66
> is where he seems to show his f.o.m. amateur stripes:
>   It seems to us ... that mathematics has now reached the stage where
>   formalisation within some particular axiomatic set theory is
>   irrelevant, even for foundational studies.
> This is completely crazy.  Formalization within specified axiomatic
> theories is of the essence in foundational studies, and will remain so
> for the forseeable future.
>   It should be possible to specify conditions on a mathematical theory
>   which would suffice for embeddability within ZF (supplemented by
>   additional axioms of infinity if necessary) but which do not
>   otherwise restrict the possible constructions in that theory.
> Note that ``axioms of infinity'' means large cardinal axioms.  Thus
> Conway is saying that there are interesting sufficient conditions for
> a theory to be interpretable in ZF plus large cardinal axioms.  But
> this implies a solution to a major open meta-problem in f.o.m.!  Many
> famous and difficult set-theoretic consistency results have, as their
> outcome, specific interpretability results of this kind.  (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.  Clearly Conway is underestimating the difficulties
> inherent in his vague meta-conjecture.

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

>   Of course the conditions would apply to ZF itself, and to other
>   possible theories that have been proposed as suitable foundations
>   for mathematics (certain theories of categories, etc.),
> O Ye Gods.  ``Categorical foundations'' again.  By the way, Conway in
> his preface acknowledges help from, of all people, Johnstone.
>   but would not restrict us to any particular theory.  This appendix
>   is in fact a cry for a Mathematicians' Liberation Movement!

***<insert above quotation>

> I think I could more or less agree with Conway, if he dropped the bits
> about axioms of infinity and ``categorical foundations'', and if he
> stopped trying to grandstand as a liberator of mathematicians.
> -- Steve

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

John Pais


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