CT vs ST thread

CT vs ST thread

[Todd Wilson]

I'd like to thank the contributors to what has turned out to be
another Category Theory vs Set Theory thread -- Steve Vickers, Michael
Barr, Colin McLarty, Charles Wells, Zinovy Diskin, and John Duskin --
especially Steve and Zinovy for their lengthy replies.  My original
post was, in a way, meant to be a step in the direction of ending such
CT vs ST debates, but there still seems to be some interest in
establishing "oppositions" between the two theories.  Here is my
attempt to frame the debate and to respond to some comments of the
contributors in the process.  I apologize for the length of this post.

Zinovy Diskin observes that the debate involves "methodologically
quite different problems whose merging under the same title CTvsST may
be misleading", echoing the earlier statement by Bill Lawvere that
"[s]o much confusion has been accumulated that an opposition of the
form `set-theoretical versus non-set-theoretical' has at least seven
wholly distinct meanings."  I quite agree, and this was exactly the
point of the end of my original post:

    And, finally, shouldn't "better" really be "better for what"?  In
    other words, aren't the two communities really just arguing past
    one another, like people arguing over types of automobile?  What
    really is the issue here?

Colin McLarty has asked for clarification of these questions.  My
point was simply that it is pointless to claim that one theory or
subject or approach is better than another without specifying what it
is supposed to be better *for*.  (Is a pick-up truck better than a
2-seater sports-car?  Is a Mercedes Benz better than a BMW?)  Here is
a list of eight possible uses to which CT and ST may be put:

1.  Offering an axiomatic foundation upon which all of mathematics may
    be developed, with a view towards

    a.  establishing or making manifest its consistency
    b.  providing a standard of rigor
    c.  providing a common framework for the cooperation between
        different mathematical domains

2.  Acting as a language in which certain mathematical ideas can be
    expressed, so that

    a.  better use can be made of them (in applying them to specific
        instances, seeing connections between them, highlighting their 
        more important features, etc.)
    b.  they can be communicated more effectively to other
        mathematicians
    c.  they better please our aesthetic sense

3.  Providing models for specific phenomena (physical or
    computational), with a view towards

    a.  informally illuminating their properties and connections
    b.  predicting the outcomes of experiments involving them

All of these uses are more or less concrete enough that comparisons
between CT and ST could be undertaken empirically using these criteria
(although some of them, for example 2a and 2c, are clearly more
subjective than the others).  My expectation is that CT and ST would
each be "winners" on some non-trivial subset of the criteria.

In addition to these eight uses for CT and ST, there is another
important role that these theories play, namely, as formal theories of
pre-mathematical concepts -- "collection" and "membership" in the case
of ST, and "transformation" and "comparison" in the case of CT.  From
this point of view, I am left somewhat mystified by Lawvere's
reference to the "totally arbitrary `singleton' operation of Peano
with the resulting chains of mathematically spurious rigidified
membership".  Whatever its other "faults", ST seems to me to be a
pretty accurate theory of "recursive collections" (or "classes" or
"containers"), that is, collections of collections of ....  Indeed, we
can easily play with real containers of various sizes, and put them
inside each other in various ways, and ST can be seen to be an
accurate description of these configurations.  The part of our
conceptual apparatus that deals with container relations such as these
is real and deserves to be modeled by a formal theory, and I find it
hard to criticize ST on its appropriateness for this role.  (Of
course, ST is also about infinite collections, and its role there is
more open to debate.)

Continuing from this point of view, one feature of ST that I find
especially interesting is that, although it is ostensibly a theory of
"recursive collections", it nevertheless is quite successful, without
any substantial additions, in many other roles -- see the list of
eight uses above.  It is an interesting philosophical and practical
question to ask why this might be so, but one can hardly dispute this
success.  Thus, I also find it difficult to appreciate claims (such as
Lawvere's) that set theory is "ill-suited for mathematics"; doesn't a
simple comparison of the mathematical achievements before and after
its "arithmetization" by set theory suggest otherwise?

Turning to more specific topics, several respondents to my original
post realized that my discussion of Cartesian products was a bit
muddled, and that my suggestion of an "unordered product" was
incoherent.  Colin McLarty answered my question about the technical
work involved in dealing with the a/the distinction by suggesting that
"nothing is involved if we introduce a product operator".  However, if
we have a category with many non-trivial isomorphisms, then the task
of introducing a product operator does involve something: making some
arbitrary choices.  Without getting into the details, I just wanted to
ask whether such choices themselves added a kind of "spurious element"
to the construction at least qualitatively similar to those in ST
referred to by Lawvere.

Steven Vickers nicely spelled out in more detail some of the ideas
behind Lawvere's "cohesive and variable sets".  However, I didn't mean
to suggest in my post that I didn't appreciate the difference in
approach (for example, I did my PhD thesis on frame theory, an
algebraic underpinning for point-free topology).  Rather, I fail to
understand the point of criticizing ST for presenting a *particular*
view of cohesion.  Isn't it interesting that a surprisingly general and
mathematically fruitful definition of cohesion/nearness among a set of
points can be given in terms of just a single element of the double
powerset of those points?  The many achievements of "point-set" or
"general" topology would suggest so.  Now, I certainly agree that this
view of cohesion may not be the best one for some, or even many,
applications (which is what initially led to my interest in frame
theory), but why does that have to turn into a criticism of the
set-theoretic framework?  We don't criticize the rectangular
representation of complex numbers because we also have a polar
representation.

Vickers also makes the point that set-theoretic topology "does not
generalize well to situations where you want to vary the set theory,"
and also mentions situations where it is fruitful to vary the
underlying logic as well.  These topics are among my favorites in all
of mathematics, and I have been a proselytizer for these ideas in
other forums, but the reaction of set theorists is understandable:
varying the ST and logic makes it harder to relate what you are doing
in the different "universes".  A mathematician wanting to take
advantage of a shift in ST and logic is faced with the problem of
(re-)interpreting the results in the original framework.  The extra
overhead involved in moving between universes (not to mention the
large "start-up costs" involved in learning the framework in the first
place) is seldom ever justified by the advantages that ones gains,
however real they can be.  This is a very practical matter, involving
mathematicians' choice of where to invest their time, and, again, I
don't see what is to be gained by criticizing them for their choices.

CONCLUSION:  PLURALISM AND A CHALLENGE

To sum up, I think that any debate on CT vs ST should take place in
the context of a concrete and particular use of the two theories,
where it is possible to investigate, more or less empirically, the
advantages and disadvantages of each.  In any other context, the
debate reduces to a battle over personal preference, artistic sense,
working habits, and other such subjective issues, and is unlikely to
get anywhere.

Second, I think we ought to foster a more pluralistic viewpoint.  Each
theory has its strengths and weaknesses, and we should choose the most
appropriate tool for whatever job it at hand.  If someone contends
that there is a significant difference in appropriateness between two
approaches, then, for this contention to be taken seriously, the
difference has to be made clear and concrete for the "worker in the
field".  I would make the same point to computer scientists who are
involved in the endless "Language Wars" over which programming
language is the "best".

And finally, I would like to offer a challenge (or challenges).  For
those mathematicians and computer scientists enamored with the vistas
opened up by category theory, and topos theory in particular (and I
count myself as among these),

- Can we build computer-implemented formal systems that make it easier
  to navigate through several universes, work simultaneously with
  several logics, and help with re-interpretation when necessary?

- Can we write books that help reduce the start-up costs involved in
  "outsiders" learning and using the framework?

- Can we discuss in public places and in detail the importance of the
  topos-theoretic or category-theoretic outlook in obtaining our
  mathematical results?

- And can we all the while hold off on our criticism of other
  approaches and instead let the results speak for themselves?

-- 
Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh

Re: CT vs ST thread

[Colin McLarty]

	Todd Wilson suggested there are many different issues in
contrasting CT to ST. He admits he was partly "mystified" by Lawvere's
earlier post, which said the same thing and actually advanced the
discussion by specifically laying out several different threads. And when
I asked Wilson which issues he wanted to discuss he mistook my meaning,
and thought I was asking him to explain how there are different issues.
This kind of repeatedly starting from scratch is not helpful, though it is
regrettably common in the "debate" over ST and CT.

	Finally though, he gets to some specifics.

>  Here is a list of eight possible uses to which CT and ST may be put:
>
>1.  Offering an axiomatic foundation upon which all of mathematics may
>    be developed, with a view towards
>
>    a.  establishing or making manifest its consistency
>    b.  providing a standard of rigor
>    c.  providing a common framework for the cooperation between
>        different mathematical domains

	Work going back to Osius and Mitchell and others in the 1970s shows that
CT can accomplish 1a and 1b in pretty much the same way that ST does, if
you want to do it that way. Perhaps the best comprehensive citation now is
to Johnstone TOPOS THEORY and its chapter on topos theory and set theory.
See also Mac Lane and Moerdijk SHEAVES IN GEOMETRY AND LOGIC and their
chapter on topoi and logic. Simpson on the FOM list likes to call this
"slavish imitation" of set theory, thus agreeing that CT can do what ST
does. It can also accomplish 1a and 1b in its own terms, if you like that.  

	1c is definitively answered in mathematical practice. Category
theory has been the leading common framework for linking domains for half
a century. ST is not a candidate on any practical level.

>2.  Acting as a language in which certain mathematical ideas can be
>    expressed, so that
>
>    a.  better use can be made of them (in applying them to specific
>        instances, seeing connections between them, highlighting their 
>        more important features, etc.)
>    b.  they can be communicated more effectively to other
>        mathematicians
>    c.  they better please our aesthetic sense


	Again, in practice, category theory is the leading framework for 2a and
2b. No one proposes using ZF on that level. 

	As to 2c, anyone may vote as they please. 

	I make no citations on these because they are perfectly obvious. If any
result can "speak for itself" surely the categorical methods in topology,
geometry, analysis, and arithmetic, can for themselves.


>3.  Providing models for specific phenomena (physical or
>    computational), with a view towards
>
>    a.  informally illuminating their properties and connections
>    b.  predicting the outcomes of experiments involving them

	As to physical phenomena, the principled foundations of math don't
seem to bear very directly on them as practiced today. Who could seriously
argue that Hawking's or Witten's work is "actually" founded on ZF versus
the category of sets? On the level of methods, people do propose
categorical methods for quantum gravity, quantum groups, and so on. I
don't believe anyone is seriously exploring ZF for the same role.

	People who believe practice would advance better, if foundations were
brought closer to practice so that the whole structure of math was more
harmonious, will probably favor category theory. 

	Computational phenomena bring us back to the issue that started the
thread. Here I think serious discussion is warrented because there is a lot
to know. I'm no computer expert but I will reply to:

	
>Turning to more specific topics, several respondents to my original
>post realized that my discussion of Cartesian products was a bit
>muddled, and that my suggestion of an "unordered product" was
>incoherent.  Colin McLarty answered my question about the technical
>work involved in dealing with the a/the distinction by suggesting that
>"nothing is involved if we introduce a product operator".  However, if
>we have a category with many non-trivial isomorphisms, then the task
>of introducing a product operator does involve something: making some
>arbitrary choices.

	No. Look at it this way: If I say in a computer program "x=0" am I
making an "arbitrary choice" of how to represent 0 by a set? We know that
in CT or ST there are alternative representations of 0. But the computer
does not occupy itself with those. It has its internal representation of 0
as a data value (perhaps several) and uses that (or, one of them).

	Similarly, suppose I have a product operator. That is actually,
one binary operator on objects _x_ with object values, and two binary
operators on objects p0_,_ and p1_,_ with arrow values. When implementing
these I give the machine a way of internally representing these as data
values, and the machine uses those representations.

	Of course, in implementing the categorical product, as in implementing
integer arithmetic or anything else, the programmer makes many more or less
"arbitrary" choices of details of how the machine will handle them. That
has nothing to do with non-identity isomorphisms, and nothing to do with ST
versus CT.

	In principle, or for foundations, the issue here is the difference
between choices and operators. There is a huge logical literature on it
and it is uncontroversial.

best, Colin