FOM: HF's attacks superficial?
Harvey Friedman
friedman at math.ohio-state.edu
Fri Jan 23 19:08:59 EST 1998
This is a reply to Tragesser 7:36PM 1/23/98.
>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.
How do you square that with the quote I often make from Sol: 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.
Tragesser writes:
>[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.
My emphasis on "superficial." Thanks for the compliment. My attacks are the
standard reasons why they are almost universally rejected as f.o.m. I
always agreed that they form some sort of organizational structure for some
math. And there are other organizational structures for some other parts of
math, too. For instance, graph theory. A huge amount of applied math is now
couched from the beginning in terms of graph theory - especially in
computer science. Do I go around calling graph theory "foundations of
computer science?" Hardly. I might model circuits as certain graph
theoretic structures, and call that modelling (part of) "foundations of
computer science." But call graph theory "foundations of computer science?"
No!!
Another one: general topology. Is general topology f.o.m.?? Give me a break.
Your calling my attack "superficial" is itself superficial.
> 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?
This is not the issue I was addressing. Similarly, graph theory is more to
the point in talking about circuits than is set theory. So what??
> Does SET have this clarity? I think: nothing like.
Compared to what? And SET has many very very very important restrictions
which are much much much more than enough to give a suitable foundation for
mathematics.
>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.
Before you lump together Lavine and my work in this direction, why don't
you explain both of them to the fom? Warning: you may have to avoid being,
in your terms, superficial, in order to provide us with this service.
Heart of darkness? Compared to what? Even all of ZFC makes incomparably
more sense, say, than string theory, or even elementary quantum mechanics.
>For another, the conception of the
>cumulative hierarchy is shot through with problems.
What problems? Depends on what standards you want to apply. It has no
problems at all compared to black holes.
> 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).
Have to? Elaborate. Axiomatic set theory stands alone as the single most
successful, problem free, impressive foundation of anything of all time.
>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.
By what standards? For the practicing mathematician, the paradoxes are
completely solved. They trivially avoid them with no work. I am interested
in them because I think they are a source of inspiration for new, important
things. And I am interested in setting in even higher standards for set
theory. But it is already so far beyond anything else.
> I don't see CAT has many advantages.
It is simply a technically convenient way of looking at a lot of
mathematics, and is best thought of as gounded in ordinary set theory.
>[3] I favor the view that in SET and CAT we have two
>powerful methods for clarifying concepts (in loosely
>analogous ways:
CAT doesn't clarify any well defined concept. Zealots are proud of this
fact, because it creates diversity. You know, maybe like multiculturalism.
>we have local foundational problems with both,
By normal standards, there are no problems with set theory. The conception
is clear, and we need only very very small doses of it to found all of
normal mathematics. To be sure, I am about to announce some concrete
combinatorial investigations that require use of far reaching additional
set theoretic axioms. But that's besides the point. Set theory is totally
obviously natural, conceptually coherent, and fully adequate as global
foundations in the normal accepted sense of the word. And, by the way,
nothing else comes close in this regard.
>[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".
This is, as you put it, superficial. It is one of many ways to do useful
semantics of intuitionistic systems. I have even used it in some of my own
work, to some extent. Start with a complete Heyting algebra, and build a
Heyting valued set theoretic universe in order to get a model of
intuitionistic systems. Wrote papers using it. So what? Am I a zerlot for
topos theory as f.o.m.? No. But I also use a lot of technical constructions
for this purpose - recursive realizability, cut elimination, Kleene's
slash, Kripke models, the whole nine yards. Gee, maybe Kripke models is
genuine f.o.m.!! Don't make me ill.
> 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:
Long live the classical real number continuum!!!
> 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
I have it. ZFC.
>-- 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,
You mean, you want a new solution to the Paradoxes? Well, I'll tell you
later about some stuff that I did, but only when we've figured out how, in
your words, "superficial" I have been.
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