FOM: comments on Wilson's dual view of foundations

[Stephen G Simpson]

This is a response to Todd Wilson's thoughtful and important posting
of February 27, 2000.

Wilson focuses on two key aspects of foundational schemes.  Let me
call them Aspect 1 and Aspect 2.  (Wilson calls them ``ontological''
and ``axiomatic''.)  From these two aspects, Wilson undertakes an
analysis of the successes and failures of both (a) orthodox
foundational schemes based on ZFC set theory and the like, and (b)
alternative foundational schemes such as those based on NF and topos
theory.

Wilson's Aspect 1 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.)

Putting this point in terms of the recent ``consistency equals
existence'' thread here on FOM, one could say: A great many (actually,
almost all) systems of mathematical objects can be ``proved to exist''
in ZFC, in the sense that ZFC proves the consistency of the axioms
describing those systems.  Of course there is nothing unique about ZFC
in this respect.  For instance, ZFC plus a large cardinal axiom is
richer still, because its consistency strength is greater.

Wilson's Aspect 2 seems to refer to the presence of a compelling
*underlying picture*.  In the case of set-theoretical foundations, we
begin with a simple, naive, intuitive picture of sets, actually a
pre-mathematical or child-like concept: sets of cards on a table, sets
of marbles, etc.  We then proceed to analyze and refine and explore
the limitations of this naive concept of set.  After some exploration,
we arrive at our clean, clear, familiar picture of the cumulative
hierarchy.  This picture is no longer pre-mathematical, but it has
many nice properties, e.g., it is demonstrably categorical in a
certain sense.

Wilson asks a series of questions along the lines of whether Aspects 1
and 2 can be separated from each other, whether both of them or only
one of them is essential, etc.  These questions are part of the
following broader issue, of crucial importance for f.o.m.:

  How are foundational schemes are to be judged?  What are the
  appropriate criteria of success or failure to be used when judging
  (orthodox or alternative) foundational schemes?

My tentative answer is that Aspects 1 and 2 are both essential for the
success of a foundational scheme.  (There may also be other important
criteria.)  Aspect 2 seems especially important, because we want our
scheme to serve as a starting point, not dependent on prior
mathematical understanding.

As to Wilson's specific questions, Shipman (Feb 28, 2000) has already
offered his answers.  Let me now offer mine.

First, the orthodox foundational scheme based on ZFC is well known to
be very successful with respect to Aspects 1 and 2, as explained
above.  Furthermore, in the case of ZFC, it is not obvious how to
separate the two aspects.  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.  In addition, natural
extensions of ZFC expressed in terms of large cardinal axioms lead to
the interpretationally richest theories that are currently known.

What about alternative foundational schemes?  

First, NF.  

Aspect 1.  NF is interpretationally rich in the same sense and for the
same reasons as ZFC.  But nobody has calibrated exactly how rich it
is, in terms of the usual ordering of large cardinal axioms.  Type
theory is interpretable in NF, but it remains unknown whether ZFC is
interpretable in NF, or whether NF is interpretable in ZFC.  Much more
is known about calibrating NFU, i.e., NF with urelements.  Solovay has
calibrated NFU + infinity + choice + ``small ordinals''.  It turns out
to be almost as rich as ZFC + a weakly compact cardinal.  However,
weakly compact cardinals are relatively puny as large cardinals go,
and it is unclear (to me at least) how the beefier large cardinals fit
in with NF.  Perhaps some NFists could explain.

Aspect 2.  NF starts with the same naive picture as ZFC (marbles on a
table, etc) but ends up being much less intuitive.  For one thing, NF
does not offer any nice picture of the universe as a hierarchy of
sets, like the cumulative hierarchy of ZFC.  At every turn it seems
necessary to work with an unintuitive syntactic notion, that of
``stratified formula''.  I find this a little off-putting.  Of course
this could be due to my relative unfamiliarity with NF.  Do NFists
claim to have a picture of the NF universe which they find intuitively
appealing?  Does such a picture have any categoricity properties?

Second, topos theory.

Aspect 1.  It is known that topos theory + natural number object +
additional axioms is interpretationally relatively poor.  Mitchell's
old paper shows that it is at same level as type theory, i.e., poorer
than Zermelo set theory, which is in turn poorer than ZFC.  But it is
still rich enough to interpret some serious mathematics, and this is
of course heartening.  I think MacLane has proposed as an important
open problem, to find natural extensions of the topos axioms with the
same interpretational richness as ZFC.  There is a topos theory
version of inaccessible cardinals, called Grothendieck universes, but
so far as I know, beefier large cardinal axioms such as weakly compact
cardinals, subtle cardinals, measurable cardinals, Woodin cardinals,
and supercompact cardinals, have not yet found a place in general
topos theory.

Aspect 2.  Here is where I think topos theory falls drastically short.
I see no pre-mathematical picture which could give rise to category
theory, let alone topos theory.  There are too many different kinds of
categories, and too many different kinds of topoi.  The
Lawvere/Schanuel book (introducing category theory to undergraduates)
avoids this pedagogical problem by piggy-backing on the naive set
theory picture.  It starts out with sets, and only gradually does it
get into other kinds of topoi.  To its credit, the Lawvere/Schanuel
book does include an attempt to motivate categories foundationally, as
a general theory of sets and functions ``viewed from the outside'', as
Wilson says.  But in my opinion this attempt fails, because the
category and topos axioms omit many properties of sets and functions
which are obvious consequences of the naive picture of sets and
functions.  Thus one is left with an impression of topos theory as a
largely unmotivated generalization of set theory, i.e., generalization
for the sake of generalization.  Only after several years of algebra
at the graduate level does one grasp the reasons for such generality,
but this kind of motivation in terms of advanced mathematics cannot
compensate for the lack of *foundational* motivation, i.e., a
compelling underlying picture.

To make topos theory viable as a competitor of ZFC in the arena of
foundational schemes, it seems to me that one would have to (1) find a
place for large cardinals, (2) come up with a much better motivating
picture.

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.

I thank Harvey for some helpful comments on an advance version of this
posting.

-- Steve