FOM: HF's phom-questionable criticism of Cat/Topos

[Robert Tragesser]

The Cat/Set spat: This is revealing
itself as a problem of
what Sol Feferman called local foundations
rather than (as it makes a pretense of
being) global foundations. 

[1] Harvey Friedman's attack on
Categorial and Topos Theoretical foundations
are surprisingly superficial (even if
Colin Mclarty's summations of them burlesque them,
there is something wrong that they invite
 burlesque), especially in the light of his powerful
interest in,   and his (HF's) considerable
strength at,  transforming philosophical
consideration and dispute into f.o.m.
problems and theorems,  and making the 
feasibility of such a transformation a 
necessary condition on philosophical worth-whileness.
        
[2]  First call up Sol Feferman's distinction in
"Working Foundations 91" between local and global
foundations. He gives a beautiful list of typical
"foundational problems" which can arise in local
areas of mathematical work and did typically
invite a massive global reframing of mathematics for
their resolution. One example: an effective method
which is hard to understand and make rigorous (=more
ruleful than artful).  A quite beautiful example (as
reviewed by Ian Stewart) is provided by Roman-Rota,
a clarification of "the umbral calculus"-- a mysterious
but powerful method in combinatorics (vide Riordan) --
through utilization of the language of Hopf algebras.
The example is especially interesting because one
traded utility for clarity. Clarity obtained,  the problem
remained of translating out of the language of Hopf
algebra a more servicable presentation of the method, 
a kind of logical upgrading/refining of the original
language of the umbral calculus,  which translating
had some strange logico-algrbraic effects 
(vide Rota-Taylor, SiamJ.MathAnal, March 1984). 
        What the Cat/Set spat is revealing is that
we have here two problems of local foundations:
        Set_Theory_ and Category _Theory_ are respective
attempts to clarify two methods of mathematical reasoning,-
reasoning in terms of set-structure,  reasoning in terms
of category-structure.   There is a certain analogy
between the two methods:  a canonical technique for
proliferating "structure" in terms of which defintions
are made,  and inferences drawn from the definitions
by principles latent in the proliferating structure.
That is,  these are methods for clarifying mathematical
concepts through precise definitions that also fix
the reasoning/inferences made from the (definitions of
the) concepts.  In my first encounter with category
theory I was particulary entranced by the definition of
free groups through universal mapping problem --
I felt as if I were seeing the Platonic form of free
objects (though I continued to work with the gassier,
more nearly set-theoretic--conception -- this was when
I was working on the problem of the solvability of the
free group of exponent four via Markov's constructions of
rings in groups).
        We have a local foundations problem in both cases
insofar as we cannot answer this question:  for which
mathematical conceptions/phenomena are set-theoretic methods 
(resp.,category theoretic methods) peculiarly,  distinctly 
suited,  particularly cogent?
        Perhaps of set-theoretic reasoning one would say(?):
conceptions/processes/phenomena emerging from the inter-
play of cardinality and order relations. 
        And of category-theoretic reasoning one would say:???
        These questions are questions of local foundations
for set-theoretic methods resp. category-theoretic methods.
        What conceptions are peculiarly native to st-methods
vs resp ct-methods?
        UNTIL WE HAVE SETTLED THEM,  WE HAVE NO RIGHT TO
CONTEMPLATE THEM AS METHODS OF GLOBAL FOUNDATIONS (of mathematical
conceptions)!
        One argument (but a good one):  mathematics is very
generous in its powers to find parts of itself reflected or
modelled in other parts of itself.  it is perhaps not
even surprising that where there is lot of structure pro-
liferating we can "define"/model a wide range of mathematical
concepts.   But at the same time,  likely nothing like "all"
the conceptions which can be so modelled are peculiarly
native (which is to say,  potently clarified through) the
method applied.  Until we have some strong fom theorems 
characterizing the range of genuinely native concepts,  we can
have no sharp sense of the significance of modelling non-native
concepts -- no sense of what we are losing and what we are gaining
by attempting to inflate a method to the standing of global-
foundational.
        Does SET have this clarity?  I think: nothing like.  
For one,  Lavine and Friedman's problematizing the relation
between the finite and the infinite indicates that infinitary
set theory has a heart of darkness where it is supposed to
be most illuminating.  For another, the conception of the 
cumulative hierarchy is shot through with problems. 
 For a third, Friedman et al to the contrary, we have no clarity
about the relation between sets and the so-to-say natural
phenomena of collection (a point elaborately made by,e.g.,
 Max Black and myself).  For a fourth,  the paradoxes which
lick around the edges of SET are by no means "solved"(as Friedman
himself emphasizes).  qed,  more or less.
        I don't see CAT has many advantages.
        
[3] I favor the view that in SET and CAT we have two
powerful methods for clarifying concepts (in loosely
analogous ways: rigorously proliferated structure in 
terms of which concepts are defined),  and that
we have local foundational problems with both,
especially that of characterizing the range of
mathematical concepts which are most potently,  natively
clarified and reasoned through in them.  Until that
clarification is achieved,  it is nothing but religion
-- which is to say extremely philosophical pretentiousness
-- which could make either seem fitted to function as
global foundations.

[4]  As Friedman has himself remarked,  one of the
goods of topos theory is that one can model objects
(such as the non-well-pointed) which represent
alternatives to those directly represented in ZFC.
        In particular,  one can model Brouwerian 
"sets".
        It might be fine sentiment to adore the classical
real number continuum,  but I can hardly believe
that it is Good Mathematics.  The point about
topos theory is not that it dances well,  but that
it dances at all:
        What one would have hoped from Friedman is
not one more,  even if sweeter,  axiomatic
presentations of ZFC,  but a far superior rival
to topos thoery -- a theory that manages to naturally
develop within itself something like a full range of
classes of set-like objects,  including ZFC-sets as
one of the classes,  soemthing like Brouwerian sets
(non-well-pointed) for another. . .  A theory
which furthermore would enable us to understand
the mathematical _VIRTU_ of each of the classes is
disclosed.

                robert tragesser