FOM: Anti-formalizationism and anti-foundationalism

[Joe Shipman]

Simpson, 5 May 1999 12:14:11 :
> The tyrannical force that Conway wants to liberate mathematicians from
is none other than f.o.m.

Shipman, 5 May 1999 12:32:09 :
> You are being unfair to Conway here.  Please read "On Numbers and
Games" before accusing him of anti-foundationalism.

Simpson, 1 Jun 1999 14:48:48
> Both Holmes and Shipman accuse me of accusing Conway of
anti-foundationalism, but I never did so.

Steve, I think my interpretation of your remark of May 5th as accusing
Conway of anti-foundationalism is reasonable; unless you are making a
distinction between being against "f.o.m." and against
"foundationalism".  Anyway, I think we can agree that it is more
accurate to accuse Conway of being against formalization (in certain
contexts) than against foundationalism.  Of course, ONAG is quite
rigorously formal for a mathematics book; but the formal system involved
is not ZFC and Conway merely sketches in informal remarks how one might
reformulate the book in pure ZFC.  Conway builds his formal system using
an informal metatheory which is clearly based on Set Theory, and indeed
refers to axioms of Choice and Foundation later in his development; but
his "Numbers" are NOT SETS, although his numbers are associated with
ordered pairs of sets of other numbers (and his ordered pairs are not
Kuratowski set-theoretical ordered pairs).

-- Joe Shipman