FOM: Categorical "foundations"?

[Harvey Friedman]

This is a brief reply to Pratt 4:12PM 1/22/98.

One basic issue is raised by the question of Silver:

>From: Silver
>What I'm looking for is a development of category theory, or
>perhaps topos theory, that is based on some underlying *conception*.

And I answered:

>Hopeless. As foundations of mathematics, it is conceptually completely
>incoherent. Proponents are zealots who want to hide this from you in the
>worst way.

Another basic issue is raised by the statement of Feferman 7:15PM 1/16/98:

>... 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. ...
>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). '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.

These points, which are related, are among many objections that can be made
to categorical "foundations". They are completely fatal.

Now Pratt has suggested that categorical "foundations" can be ressurrected,
or reworked in terms of an ostensibly sensible idea of taking the continuum
as primitive, or a jumping off point, rather than collections. But I do not
see how to do this so that is has what Silver is looking for. It could
conceivably meet Sol's objection if it is done just right - but that is far
from clear.

What I don't see is this. If you start with the notion of collection that
little children see - such as three cards on a table grouped together and
referred to as one - then it is easy to jump off from there to have not
only more elements, but also have elements which are themselves not
physical, and in fact elements which are themselves collections also. This
quickly generates a lot of axioms of set theory. Perhaps not the full
cumulative hierarchy, which involves additional considerations. But already
so many powerful axioms that one quickly has a totally standard kind of set
theoretic foundations, and at a level vastly superior and simpler to any
known alternative scheme.

Let us say that we instead start with the continuum as Pratt suggests.
Where do I go from here in a philosophically careful way? Don't get me
wrong. I think this may be fruitful, but I don't see topoi in it.

It is obvious that Silver is not going to find what he wants, and the
sooner people pushing these alternative foundational schemes fess up to
this the better off we all are. I then want to see what concept of
"foundation" is served when one is not able to answer Silver. This applies
to all of your algebraic postings which you call "foundations" as well.

I should remind everybody that there are lots of philosophically
sophisticated people on this e-mail list who are not going to be razzle
dazzled by a lot of mathematical notation regardess of how pretty it can be
made to look. They are going to look for things that make philosophical
sense - e.g., like propositional calculus, predicate calculus, and Peano
arithmetic. They are not easily fooled. E.g., they know the difference
between an existential quantifer and a map.

>What Harvey and I think many others on this particular list see in
>categorical foundations is an incoherent mess being promoted by either
>misguided zealots or charlatans.

How could you tell?

>What the category theorists see is a
>simple and elegant yet rich and powerful basis for mathematical thought.

Mathematical thought? Whose mathematical thought? Philosophers are going to
want to understand and examine very carefully some of the relevant
mathematical thought to see if it is conceptually coherent. Remember,
physics, chemistry, biology, medicine, etcetera seem to be about something
in the way of fixed objective reality. So is "practical" set theory. They
will want to know what topoi are *about.*

>Both sides are well populated with technically brilliant mathematicians.

The other "side" is now weak because the idea of categorical foundations
largely ran its course in the 70's. People still use the language for
trivialities, but also the trend in math is to get away from "abstract
nonsense" and get more concrete - so you can compute. Even MacLane has
abandoned talk of autonomous categorical foundations, even when I last saw
him in the 80's (or early 90's). I wouldn't know who, *today*, is the most
technically brilliant mathematician to seriously endorse topoi as an
autonomous foundation for mathematics on a par with the usual set theoretic
foundations. Do you?

>Both sides are notorious for their inability to convince each other
>of their respective viewpoints.  I'd say we have here a breakdown in
>communication approaching that in Ireland or the Middle East.

I am obviously convincing McLarty and others. It takes a little time. If
more of the sensible silent majority like Silver would comment, then the
conversion process would be sped up. How about it, silent majority?