FOM: Toposy-turvey

[Harvey Friedman]

Feferman on 1/16/98 put his finger on one of the principal issues in this
debate about whether category theory and its elaborations have a
significant role to play in f.o.m. I was waiting for a reply from Mclarty
to my posting of 8:57AM 1/14/98. However, Mclarty's reply to Feferman dated
3:18PM 1/17/98 is so revealing that I couldn't wait any longer. (By the
way: how about a reply to my 8:57AM 1/14/98?)

Feferman writes:

>>I didn't think I would get drawn again into a discussion of category
>>theory vs. set theory as a foundation of mathematics, but Aczel's FOM
>>drugs are at work.  First, I appreciate Peter's effort to clarify what the
>>conflict is about.  But I think he has left out something essential, and
>>that was what I emphasized in an earlier related posting of 19 Nov 23:46.
>>Namely, the notion of topos is a relatively sophisticated mathematical
>>notion which assumes understanding of the notion of category and that in
>>turn assumes understanding of notions of collection and function.

Mclarty responds with the astonishing and revealing:

>	To me, understanding the notion of category is the best way
>known today of understanding the notions of "collection and function".

I can begin to see how one can, quite falsely, begin to think that the
notion of category helps in understanding the notion of function. But I
have been completely unable to see how one can begin to think that the
notion of category helps in understanding the notion of collection.
Obviously, the way that the concept of collection is assimilated in little
children is by placing two or three actual physical objects into a bunch or
group and talking about the bunch or group instead of just talking about
the actual physical objects. Of course, later there is the leap that the
items in a collection can themselves be of a rather arbitrary character -
e.g., a collection itself. Of course, this leads to all kinds of
interesting things including the paradoxes, but that is another story for
another time.

>	To be precise: Pace Feferman's theory, the first thing you
>need to know about collections and functions is that a pair of functions
>may have a composite, and this composition is associative when it is
>defined.

Again, I can begin to see how one can, again quite falsely, begin to think
that the first thing you need to know about functions is that "a pair of
functions may have a composite, and this composition is associative when it
is defined." But again I have been completely unable to see how one can
begin to think that the first thing you need to know about collections is
that "a pair of functions may have a composite, and this composition is
associative when it is defined."

Probably the first thing you need to know about collections is that any two
collections are identical if and only if they have the same members. At
least, that's the first thing that's normally taught after the concept is
"explained."

And probably the first thing you need to know about functions is whether or
not one is talking about functions as rules, or functions as black boxes.
The latter obeys extensionality, whereas the former does not. In other
words, do you only care about the input/output relationship, or do you care
about the process? The history of mathematics shows that it is more
productive to concentrate on the extensional concept, and analyze the
intensional concepts in terms of extensional concepts. E.g., analyze the
notion of algorithm in terms of ordinary extensional concepts in the theory
of partial recursive functions, programming languages, and models of
computation, etcetera.

A red flag for you here should have been that "the first thing you need to
know" and "may have.. and is ... when it is defined" don't go together very
well.

>And now I have completely stated the category axioms.

You haven't stated the axioms completely since you have hidden some
complications. E.g., that it is a many sorted theory, with partially
defined operations. Also, you only have an axiomatic system, and that
doesn't explain the concepts of collection and functions at all. You see,
an axiomatic theory of something doesn't do much to explain the underlying
concepts unless a little bit goes a long way; i.e., a few simple basic
axioms allow one to derive an unexpectedly massive amount of diverse
important facts. Without that, it goes almost nowhere in terms of an
explanation.

>This
>seems to me not very sophisticated mathematics.

Its even less sophisticated f.o.m.

>The topos axioms are
>more sophisticated, but no more so than ZF or Feferman's theory of
>collections and functions.

The ZF axioms can be viewed as an extrapolation of the axioms of finite set
theory. The systematic carrying out of this point of view is a major
research project of mine, and I have a number of major findings in this
connection in a somewhat different context which I hope I can convert to
this context. In any case, the axioms of finite set theory are extremely
simple - much simpler than ZF - and incomparably simpler than topos axioms.
Also, the comparison of topos with ZF is unfair; one should compare topos
with Z = Zermelo set theory, which is significantly simpler than Z.

>>  I do
>>not mean these in the sense identified in the cumulative hierarchy or any
>>supposed standard model for that, but rather in the informal everyday
>>mathematical sense.  Thus there is both a logical and psychological
>>priority for the latter notions to the former.  'Logical' because what a
>>topos is requires a definition in order to work with it and prove theorems
>>about it, and this definition ultimately returns to the notions of
>>collection (class, set, or whatever word you prefer) and function
>>(or operation).
>
>	Actually, when you pose a topos as a foundation you do not
>use a *definition* of a topos, you use *axioms* on objects and arrows.
>
>	This is exactly the same point as saying that Feferman's
>foundation begins not by *defining* a universe of collections and
>operations, but by giving *axioms* on collections and functions.

The big difference is that in the set theoretic axioms, a little bit goes a
very long way. If topos were presented in a way where a little bit goes a
very long way, it would simply look like a crude restatement of the set
theoretic axioms - and incomparably more complicated and cumbersome to use.

One major reason for this gross difference is the following. The usual set
theoretic foundations leverages over the universal language for all of
logical thought - for all, there exists, and, or, not, if then, iff,
equals. This counts for zero complexity since it underlies all systematic
thought. (Of course, there is much much much more to thought than just
this!). Categorical foundations seeks to replace or explain these notions
in terms of other "notions." This may be a nice idea for certain technical
constructions, which may be of some use in limited contexts in mathematics
- e.g., as some sort of "generalized set theory" or "generalized algebra"
or the like. In fact, as Freyd, Scott, Solovay (in no order) have
indicated, such ideas can be used to give an exposition of the independence
results of Cohen. I should note that there are other, more "foundational",
ways of reexpositing forcing in terms of partial information or randomness
- but this is besides the point. Thus category theory may illuminate some
situations in mathematics from a technical point of view. But this hardly
counts as foundational in any of the senses I discussed in my posting of
11:55PM 1/14/98, which I suggest that you look at again.

Some category theorists have suggested that category theoretic foundations
may become competitive because categorical foundations doesn't need: for
all, there exists, and, or, not, if then, iff, equals, whereas set
theoretic foundations does. But the actual development of actual
mathematics done this way would be incomprehensible.

>>'Psychological' because you can't understand what a topos
>>is unless you have some understanding of those notions.  Just writing down
>>the "axioms" for a topos does not provide that understanding.  If that is
>>granted, and I can't continue further in this discussion if it is not,
>>then what work in f.o.m. has to accomplish is a clarification of the
>>notions of collection (or class) and function (or operation).

>...I even grant that you need informal
>understanding of a lot of things: arithmetic, functions, explicit
>deductive reasoning, and more, before any kind of foundational axioms
>will make sense to you.

Explicit deductive reasoning is a universal given, across the intellectual
landscape. You should make the most effective use of it, since it is
inevitable. Don't hide it inside some jargon. Of course, I admit that
hiding it in some jargon creates a technical generalization - which may
well have some technical uses in limited contexts.

>I do not believe that discrete structureless
>collections stand out among mathematical objects as THE ONES you have
>to know about to understand a foundation.

The finite structureless collections seem to play a special role in thought
and intellectual development. Bow to the inevitable!

>	I wonder if that much disagreement really does itself preclude
>meaningful discussion. It might but I hope not.

You can always admit that you are grossly mistaken. Then we can have a
meaningful discussion on how you became grossly mistaken. How about it?