FOM: In defense of Conway

[Julio Gonzalez Cabillon]

Dear Colleagues,

A bit of history:

On May 05, 1999 [09:55:03 -0400] Joe Shipman <shipman at savera.com>
interested in applications of category theory to topology, number theory,
and so on, and so forth, asked (almost rhetorically):

     "[I]s there a single application of, say, the Yoneda embedding
     theorem which can't straightforwardly be redone in ZFC?"

and added:

     "If not, why isn't there a metatheorem to this effect?"

This insightful question encouraged Stephen G Simpson <simpson at math.psu.edu>,
a few hours later, to wonder about the rewards (and drawbacks!) for looking
into metatheorems in category theory, as follows:

     "One aspect of this kind of question to keep in mind is: What
     would be the *reward* for an f.o.m. researcher who took the
     trouble to formulate and prove and publish such a metatheorem?"

and linked his question with a warning:

     "You have to consider the possibility that the 'reward'
     might amount to a kick in the teeth."

which (if it were the case) undoubtedly would be an anti-f.o.m. attitude.

And next Steve Simpson goes on to name names:

     "Mathematicians like Conway, Johnstone, McLarty, Whiteley,
     Tragesser, et al could be expected to unjustly dismiss the
     metatheorem as irrelevant, pedantic, not sufficiently attuned
     to the 'mysterious dimension', etc etc, yatta yatta yatta.
     This expectation is based on their past behavior."

I am not at all sure what this *last* statement is referring to!

     "Don't forget that, according to Johnstone, Conway is the
     founder of the Mathematician's Liberation Movement.  The
     tyrannical force that Conway wants to liberate mathematicians
     from is none other than f.o.m."

And, yes, my understanding is that there's some truth in that (if "f.o.m."
is read, for instance, as "an axiomatized set theory such as ZF"), which
in Conway's case, as far as I know (and I do), "is not arrogantly
dismissive of any and all f.o.m. insights, even when these insights are
directly relevant to mathematics". Conway is a revolutionary, indeed, but
perhaps the most pacific of revolutionaries, and I'm certain he will be
very happy (as once told me) to let those who want to keep their shackles
do so. It obviously does not sound pro-f.o.m. tone (but does not mean an
anti-f.o.m attitude).

Joe Shipman's reaction to Steve's words did not wait much [May 05, 1999,
12:32:09 -0400]:

     "You are being unfair to Conway here.  Please read _On Numbers
     and Games_ [ONAG] before accusing him of anti-foundationalism.
     Conway clearly appreciates the importance of proper foundations
     and knows exactly what he is talking about.  He wants to liberate
     mathematicians from excessive fussing about formalization in
     works of ordinary mathematics, and explicitly states that the
     foundational work of proving metatheorems justifying the
     alternatives to the standard foundations one is creating still
     needs to be done, but that it can be done once and for all."

an opinion backed up by Walter Felscher <walter.felscher at uni-tuebingen.de>
on May 6, 1999 [11:16:12 +0200]

     "I want to join in the defense of Mr. Conway. From personal
     acquaintance, I can assure you that he is not prejudiced against
     foundational issues; it just happens that he is not interested
     in pursuing them with the logician's machinery. Also, should you
     ever take the time to write down the details already of the early
     proofs in Numbers And Games, you will - I hope to your pleasure -
     notice that the techniques required remind you of those used when
     carrying out the construction of ordinal notations in proof theory."

And Joe Shipman ended his previous posting with a suggestion:

     "See _Appendix to Part Zero_, ONAG pp. 64-67, which is
     foundationally well-informed and deals with exactly the
     distinctions you are accusing him of ignoring."


On May 17, 1999 [21:15:38 -0400] Stephen Simpson returned on this issue:

     "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."

I agree also that Conway shouldn't have said "even for foundational studies".

But...

I do NOT think "This is completely crazy" is a formal sentence. I do NOT
think the right/wrong binary discourse is the scholarly variation. If
history teaches us anything, it is that history does NOT teach us anything
[otherwise look *either* at ethnic cleansing OR bombs on TV stations - I
live in Montevideo and am not involved in one issue or another. SORRY!]

I strongly support John Pais <paisj at medicine.wustl.edu> message to this
forum. On May 18, 1999 [10:33:54 -0700] he provides, in my (humble)
opinion - am I also showing my stripes? ;-) - a balanced posting, in which
he describes Conway's book, _On Numbers And Games_, as "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'".

I agree with Pais also that there has been (on FOM) a "tendency to amplify
the emphasis given by Conway to those portions of the book that might be
construed as making foundational claims". The spirit and context of the
book, and what Conway was trying to do mathematically, was asking, as John
Pais put it, "for a little *foundational elbow room* to do the mathematics
he wants to do". What Steve omits after "...This appendix is in fact a cry
for a Mathematicians' Liberation Movement!" 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 relation.

      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 theories."


I do NOT think we should try to set the rules in mathematicians' debate
about *mathematics* from a f.o.m. perspective. Let the mathematicians do
what is right for their own temporal frame, social setting, interests,
needs, and even those cries like the Conwayan "Mathematicians' Liberation
Movement". Likewise, let the FOMers do what is proper for their own sake.

It is not up to FOMers, I think, to angry criticise MLMs, but to try
to understand the underlying reasons. It might not be amateurism to recall
that mathematics has both lived well without rigorous and axiomatic
foundations, and has developed from having them. It might not be naivety
to recall that mathematics has both lived (and lives) well with rigorous
and axiomatic foundations, and has been hindered by them.

In his "Indiscrete Thoughts", Gian-Carlo Rota wrote:

    The facts of mathematics are verified and presented by the axiomatic
    method. One must guard, however, against confusing the presentation of
    mathematics with the content of mathematics. An axiomatic presentation
    of a mathematical fact differs from the fact that is being presented
    as medicine differs from food. It is true that this particular medicine
    is necessary to keep mathematicians from self-delusions of the mind.
    Nonetheless, understanding mathematics means being able to forget the
    medicine and enjoy the food.

Is this an anti f.o.m. tone? ... In my opinion, it is not!

Let me end this note taking issue at the statement

                         "This is completely crazy"

As far as I am concerned, Conway's proposal is by no means an anti-f.o.m.,
but a practical attitude within the realm of the working mathematician
landscape. Let me pose some analogies so as to clarify Conway's "f.o.m.
amateurism" (as Steve put it):

++ Example 1:

Q: What IS a 'complex number'?...

A: It is an ordered pair of reals such and such ...

A': Well, it may also be a congruence class of polynomials modulo x^2 + 1 ...

Q: So... What *IS* a 'complex number'?...

A: Well, it COULD be any particular one of these things...

A': Yes, but there's no NEED for it to be that one, rather than any other.


++ Example 2:

Q: What IS a 'real vector space'?...

A: Well, it COULD be the collection of all n-tuples of real numbers such
and such ...

A': Indeed, but there's no NEED for it to be this, or ANY other particular
thing; all that matters is that suitable definitions of vector addition
and scalar multiplication exist.


++ Example 3:

Q: What IS an 'ordered pair'?

A: Well, it COULD be Kuratowski's object ...

A': Certainly, but there is no NEED for it to be that, or any other
particular thing. In working mathematics, all that matters is that we
have
            (a, b) = (c, d)  iff  a = c  and  b = d.



++ Example 4:

Q: What IS a 'cardinal number'?

A: Well, the most natural thing for it to be would be an equivalence
class of sets under the existence of 1-1 correspondences, but it CAN'T
be that in ZF.

A': So what it COULD be is the restriction of such a class to the first
V-alpha it intersects.  Alternatively, in ZFC, it could be an initial
number.  But there's no NEED for it to be either of those things - it
could be just "the cardinal number of a set".  

Et cetera.

So, as far as I am concerned, Conway's (rough) proposal is for a system in
which one could freely create new concepts such as "complex number", "real
vector space", "ordered pair", "cardinal", etc., without having to link them
to be any particular constructs in a formalized set theory, such as one can
construct "vector spaces" that are abstract, rather than sets of n-tuples.
This is the **mathematician's** standpoint. Others might see this as crazy.

So...

"ONAG" was not intended, I presume, as a supplement for (say) Bourbaki's
"The/orie des Ensembles". Even this booklet has been crushed. At any rate,
do not make a storm in a teacup!

Respectfully,
Julio Gonzalez Cabillon

PS A coutesy copy is being sent to JHC. I hope I've grasped Conway's
semi-formal foundational ideas in the right direction.