FOM: Conway's foundational ideas

[Stephen G Simpson]

Joe Shipman 20 May 1999 11:22:52 

Simpson:
 > I didn't find anything in it that would not make just as much sense
 > and be just as meaningful when restricted to hereditarily countable
 > numbers and hereditarily countable games.

Shipman:
 > First of all, such a restriction would be an unnecessary
 > complication.

On the contrary, it would result in a substantial simplification,
because Conway would no longer need to worry his head about
foundational issues.  He would be able to dispense with his tedious,
vague, bombastic, inappropriate comments on Universities,
Improprieties, the Mathematicians' Liberation Movement, etc etc.

 > Secondly, there is an essential use of the full power of set theory
 > in Conway's construction of the Surreal Numbers; he wants to have
 > infinitesimals

The hereditarily countable numbers also have infinitesimals.  Thus
infinitesimals are not an issue.

 > ... to ensure both infinitesimals AND completeness his inductive
 > construction needs to "go all the way".  Thus proper classes are
 > indispensable.

What does ``completeness'' mean?  I think it means that each bounded
nonempty set of surreal numbers has a least upper bound.  But since
the surreal numbers are a proper class, one needs to also ask about
completeness with respect to bounded nonempty proper classes.  Thus
there is really no stopping point that is as natural as Conway might
wish.  This is where Conway gets into foundational trouble.

Aren't the hereditarily countable surreal numbers also complete in an
appropriate sense, namely with respect to nonempty bounded *countable*
sets?  If so, does Conway make any serious use of set-completeness
that would not be covered by countable completeness?  If not, then it
seems completeness is also not a serious issue as between Conway's
set-size surreals and the subsystem consisting of the hereditarily
countable surreals.

There may be some interesting mathematical questions here.  Presumably
the hereditarily countable surreals form a countably complete real
closed ordered field of cofinality omega_1, and one could ask if these
properties, perhaps with some other elegant properties, suffice to
characterize them up to isomorphism.  Such a characterization might be
consistent with and independent of ZFC.

 > Conway is aware that the system is easy to formalize as a
 > definitional extension of ZFC+Inaccessible,

Is he?  What system?  He doesn't present any system, nor does he give
any sign of being aware of the idea of definitional extension.
Instead, he calls for a Mathematicians' Liberation Movement.  To me
this means he thinks formalization is an unnecessary burden and
mathematicians need to be liberated from it.  As an
f.o.m. professional, I find this offensive.

 > but you seem to be missing the point that Conway, Gonzales
 > Cabillon, and I keep making,

I think you guys are being way too deferential toward Conway.

 > which is that Conway is not required to do anything more than
 > REMARK that his system can be so rendered,

In view of his call for a Mathematicians' Liberation Movement,
that would be a minimal requirement, but he doesn't even meet that
requirement.  He doesn't even define the system.

 > Conway wants to be "liberated" from Kuratowski ordered pairs ...

I think that FOM has already had a mini-debate on Kuratowski pairs
(set-theoretic foundations) versus ordered pairs as a primitive
concept (Bourbaki, data types).  I think both sides of the debate have
merit.  Let me contribute my two cents worth, because it illustrates
some basic points about the role of definitional extensions in f.o.m.

We can write down the formal system ZFC.  Then we can definitionally
extend ZFC by introducing various function symbols, e.g. (-,-) defined
a la Kuratowski, rank(-) defined a la von Neumann, etc etc.  This
results in a larger formal system, sometimes called an ``inessential
extension'' of ZFC, and the well known metatheorem applies.  Then, if
we choose, we can list the properties of the new function symbols that
we require.  Formally this amounts to throwing away the defining
axioms for (-,-), rank(-), etc, and keeping only the desired
properties, all of which are logical consequences of the defining
axioms.  This is good from the ``data types'' viewpoint, because it
discards the ``implementation'' and focuses only on the properties
that are considered important or desirable.  The resulting formal
system is no longer a definitional extension of ZFC, but it is a
subsystem of one.

Among the properties considered desirable may be the following:

  (x,y) = (u,v) iff x = u and y = v

  rank((x,y)) is less than max(rank(x),rank(y)) + omega,

  rank(x) = (sup{rank(y)| y in x}) + 1

  etc.

If we want a stronger property such as

  rank((x,y)) = max(rank(x),rank(y))

we will need to go back and modify the definition of (-,-) and/or
rank(-) or modify some of the other properties, etc.  There is nothing
wrong with this.  It is part of the normal process of foundational
hygiene.

If Conway were content to simply do his mathematics and disregard
these foundational issues, as advocated in the Rota passage that
Gonzales Cabillon quoted, then I would have no objection.  But why
should Conway try to rabble-rouse by telling everybody that this kind
of formalization is an unnecessary burden imposed by f.o.m.?  Why
should he issue a half-baked call for a Mathematicians' Liberation
Movement?  The only thing I can call this is f.o.m. amateurism.

-- Steve