FOM: A dual view of foundations

[Todd Wilson]

I would like to thank the FOM readers who responded to my FOM posting
of Sun, 27 Feb 2000 16:19:09 PST, "A dual view of foundations",
including Joe Shipman, Stephen Simpson, and A.R.D.Mathias, and the FOM
moderator for allowing such a discussion.  First a reply to Simpson.
In a later posting, I hope to respond to Mathias.

Stephen G Simpson:
> Wilson's Aspect 1 [what I called "ontological" --TW] seems to
> correspond closely to what I would call *interpretational richness*
> of the given foundational scheme.  For instance, ZFC is well known
> to be interpretationally rich, in the sense that a great many
> (actually, almost all) mathematical theories can be *interpreted* (a
> la Tarski/Mostowski/Robinson) in ZFC.  (Wilson speaks of
> ``mappings'' rather than interpretations, but his intention seems
> clear enough.  Perhaps Wilson could comment on whether I am reading
> him correctly.)

Yes, this is what I had in mind.  Thank you for the clarification.

> The naive idea of ``set'' is easy to think about and work with, and
> this makes the interpretation of many mathematical concepts and
> theories in ZFC almost routine.  For instance, the interpretation of
> group theory into ZFC presents no difficulty, because a group
> consists of an underlying *set* together with operations on it, etc
> etc.  The set-theoretical interpretation of certain concepts of
> analysis and geometry (real numbers, continuity, probability, etc
> etc) is more difficult, but the foundational work of certain 19th
> and early 20th century mathematicians serves as our guide, and this
> is another success story.

Despite the obvious success and utility of these "reductionist"
treatments of real numbers, continuity, probability, etc., in ZFC, I
wonder whether they, like reductionist treatments of, say, biology in
physics (via chemistry), have missed out on any important (should we
call them "emergent"?) features of the original phenomena.  Do we, for
example, declare our intuitions of "nonpunctiform infinitesimals"
(Bell, "A primer of Infinitesimal Analysis", Introduction) vague
musings that were finally and definitively clarified by the arithmetic
continuum, or is there the possibility that the arithmetization
elucidated but one aspect of the continuum, there being still others
to capture.  In particular, we know that nilpotent infinitesimals are
incompatible with the usual picture of the reals as a field, and that
invertible infinitesimals are incompatible with the usual picture of
the reals as Archimedian, so these notions become, under the usual
treatment, fictions or facons de parler rather than primal or
foundational aspects of our view of the continuum.  Is this the way it
should be, or are we perhaps putting the cart before the horse?

The recent work in category theory reported in Bell's book shows that
"worlds" are possible in which the reals contain nilpotent and
invertible infinitesimals -- simultaneously, if desired -- and that
all functions definable on the reals are continuous.  These worlds are
very nice for the development of "smooth" analysis; in fact, arguably,
they are aesthetically the proper place to do it.  Other worlds might
be the proper places to develop probability theory (for example, see
Nelson's book "Radically Elementary Probability Theory", Princeton
Univ Press, 1987, for a treatment using non-standard analysis) or the
semantics of programming languages (as with the work on synthetic
domain theory).  If we grant that, in each of these areas, our
concepts may be most pleasingly developed in worlds individually
tailored for this development, we are then left with a situation in
which we have many different foundational pictures, each addressing a
limited set of phenomena, and we are in need of some understanding of
the connections between them.

It turns out that all of the worlds described above are toposes
(including the worlds, such as where all functions on the reals are
continuous, that are incompatible with the law of excluded middle),
and the connections and mappings bewteen toposes called for above have
been studied in great detail over the last several decades, both from
an "external", set-theoretic vantage point, and from within the system
of toposes itself.  So, if such a multi-foundational approach is worth
considering at all, then topos theory is the natural place in which to
formulate it.  Thus, perhaps it's best to say that the value of topos
theory to f.o.m. is not as a single foundational scheme rivaling ZFC,
but rather as a framework for an interconnected system of foundational
views, each of which is quite extensive (though not necessarily as
extensive as ZFC, as Simpson correctly points out), but none of which
is given a universal role.

So what are the weak points of this argument?  How about these:

1.  We want a *single* foundational system.  The idea of a connected
    system of partial foundational worlds -- even if we understand
    each world separately and understand the connections between them
    -- is too complicated.

2.  (Simpson) Where is the compelling underlying "pre-mathematical
    picture" that this multi-foundational situation is giving us?
    Topos theory seems to be a "largely unmotivated generalization of
    set theory".  How is the move from (a single) set theory to (a
    multiple) topos theory an improvement?

3.  (Feferman, Simpson) We can't even adequately describe topos theory
    without reference to sets and elements.  This shows that set
    theory is prior.

4.  Topos theory doesn't seem to be able to address the higher reaches 
    of consistency (one hopes!) strength, for example the hierarchy of 
    large cardinals.

As for 1 and 2, first steps toward answering these objections, as I
have pointed out before, can be found in the Epilog of Bell's book,
"Toposes and Local Set Theories" (and the writings cited therein).
Bell makes an analogy with the emergence of relativity in physics.
Newtonian physics can be carried a long way, but when we get "near the
fringes", the notion of an absolute frame of reference breaks down,
and we are forced to accept that there is no priviledged frame of
reference (or "world") and that all we can do is describe what is
common to all frames of reference and how to negotiate our way between
them.  The flood of independence results in set theory starting in the
1960s has sometimes been taken to imply the same thing about set
theory.  In short, Bell is proposing the analogy

   Topos Theory : Set Theory  ::  Relativity : Newtonian physics

As for 3 and 4, topos theory is a first-order theory in the language
of categories (in fact, it is an "essentially algebraic theory" in the
sense of Freyd and is even purely equational in the language of graphs
-- prima facie much more simple logically than set theory), and as
such it doesn't rely on prior notions of sets and elements any more
than any first-order theory (including set theory) does.  That aside,
it *does* appear that if topos theory were to be able to include
notions similar to large cardinals, it would need to have more in the
way of "reflective" capabilities than it currently possesses.  It is
an interesting challenge to the category theory community to
investigate such possibilities.  Perhaps the work of Benabou cited
earlier by McLarty is a good first step in this direction.

> My impression is that Harvey Friedman has some other ideas about what
> would need to be done to make NF and topos theory and other
> alternative foundational schemes viable.  Perhaps he will explain if
> we ask him nicely.

Pretty please?  :-)

-- 
Todd Wilson
Computer Science Department
California State University, Fresno