[FOM] replies to Friedman's points
Nik Weaver
nweaver at math.wustl.edu
Sun Feb 26 19:23:37 EST 2006
In his latest post in the "BETA(N)" thread Harvey Friedman
posed several questions to me. It seems that I have answered
similar questions already and they are posed here mainly for
rhetorical effect. Will it help to explain my position one
more time?
Okay, here goes. There is a basic distinction between
predicative and impredicative mathematics. The latter by
definition involves objects which in principle cannot be
defined in a non-circular way. This only makes sense if
one believes in the objective existence of some kind of
metaphysical world of abstract entities. Unfortunately,
this view is literally nonsensical and arises from a series
of grammatical confusions, as Hartley Slater has convincingly
shown. Moreover, it immediately leads to contradictions.
These can apparently be blocked by artificially weakening
the formal system in which one reasons, but there is no clear
philosophical basis for doing so. As von Neumann said,
"There is, to be sure, a certain justification for the axioms
in the fact that they go into evident propositions of naive
set theory if in them we take the word `set', which has no
meaning in the axiomatization, in the sense of Cantor. But
what is omitted from naive set theory --- and to circumvent
the antinomies some omission is essential --- is absolutely
arbitrary." (J. von Neumann, "An axiomatization of set theory")
So: predicative mathematics has a clear philosophical basis
in terms of manipulation of entities that are conceptually
unproblematic (marks on paper, or whatever); impredicative
mathematics does not.
But what does predicative mathematics cover? Rough answer:
all normal, "core" mathematics and very little else. This
is necessarily a loose assertion since what constitutes
"normal" mathematics is obviously debatable. However,
striking examples can be given, for example in functional
analysis, where predicative reasoning cuts off at precisely
the point where mainstream usage cuts off. (The sequence
space c_0, its first dual l^1, and its second dual l^infinity,
are all everyday objects in functional analysis, and all three
are predicative. The third dual of c_0, the space of regular
Borel measures on beta N, is impredicative and is almost never
encountered in mainstream mathematics. There are a host of
examples of this type.)
So, should we forbid/condemn impredicative mathematics? Of
course not. A more subtle question: would mathematicians
be better off if they were to explicitly use predicative
reasoning? I don't have any special insight into this issue.
It's a bit like asking whether physicists ought to be more
rigorous. You can argue that too much rigor stifles creativity,
and that might be true, though Faraday used this argument to
disparage Maxwell's equations (I wish I could remember the source
for this), so the argument can be pushed too far. An analogy
could also be drawn with the naive use of infinitesimals. Are
we better off using epsilon-delta proofs? Most mathematicians
would say yes, most physicists would probably say no. I see
it as basically a matter of taste. But in any case, it's not
my place to make any recommendations of this sort. It *is* my
place, minimally, to encourage interest among the foundations
community in the unjustly neglected predicativist foundational
stance.
So here are my answers to Friedman's points.
1, 2, 7. Friedman: Alleged examples of impredicative reasoning
in algebraic geometry, countable algebra, and general topology.
He challenges me to "address the issue". Well, some math is
predicative and some isn't. What am I supposed to say? I'd be
surprised if anything important in algebraic geometry really
requires impredicative reasoning. I'm sure there are plenty of
things in general topology that are genuinely impredicative.
As for the unspecified example in countable algebra, I am
withholding judgement.
What's the question here? If it's "how dare you condemn
such-and-such a theorem" the answer is I don't. If it's "how
dare you say that such-and-such a theorem is proven on the
basis of axioms that have no clear philosophical basis" the
answer is, because it's true.
3. Friedman: ACA_0 matches core mathematics just as well as
predicativism. This is basically true, and indeed ACA_0 (with
relatively minor modifications) is a reasonable system for
predicativists to use. However, there are a few things that
can be proven predicatively but not in ACA_0, such as Kruskal's
theorem. Friedman denies this and wants to say only that there are
different versions of predicativism, some of which prove Kruskal's
theorem and some of which don't. He hasn't read my Gamma_0 paper.
It isn't a case of there being different versions. The conventional
view that predicativism stops at Gamma_0 is simply not correct.
Friedman: "predicativity is not any particularly distinguished
match for core or normal mathematical practice." It is a much
better match, a strikingly better match, than Cantorian set
theory, and that is a point in its favor. If there are other
compelling foundational stances that are similarly good matches,
fine. I haven't seen any yet.
4. Friedman: Why has ZFC gained "overwhelming acceptance" in the
mathematics community? Well, I don't think it has. That's a very
misguided impression on Friedman's part. Most core mathematicians
don't know or care much about foundations and just don't want to worry
about it. Among those who do think about it, there is substantial
skepticism. Support for ZFC among ordinary mathematicians is not
as monolithic as Friedman makes out.
5. "You have not addressed the issue that I raised that referring
to impredicativity as `having no clear philosophical basis' is a
misuse of the word `philosophy'."
I think it's a silly argument. If anyone finds it convincing,
that's their business.
6. "You have not addressed the issue of the nature of your
condemnation of impredicative mathematical activity ..."
Simply a misrepresentation of my views, and one that I have
repeatedly corrected.
----------
A couple of other points from Friedman's post.
> I wrote this ["You have already hinted at an expansion of
> predicativity via the **Pi11 comprehension axiom scheme**
> which has been roundly rejected as predicative"] because you
> mentioned that it was open whether or not the graph minor
> theorem was predicatively provable, in an earlier posting
> of yours. That's how I got the impression.
>
> GMT cannot be proved in Pi11-CA0 and furthermore it proves the
> consistency of Pi11-CA0. Prizes have been awarded by the AMS
> and other organizations largely based on GMT. Do you wish to
> dismiss GMT as not "core" or not "normal" mathematics? GMT is
> certainly yet another major challenge to predicativists - one
> that you have not acknowledged as far as I recall.
One that I have not acknowledged?? If you look a few sentences
back, you see yourself referring to my response about GMT.
Since your memory seems to be playing tricks, I'll repeat my
answer: it is open whether one can predicatively prove a type of
well-ordering statement that is sufficiently strong to imply
GMT. If we could, that would also imply consistency of the
system Pi-1-1-CA_0, which is of course totally different from
saying that we should expand predicativism by adopting Pi-1-1
comprehension.
Then on the question about whether there predicatively exist
nontrivial ultrafilters over N:
> How are you going to state in the system that all sets are
> constructible?
V = L?
> First of all, it is mathematically obnoxious,
Whatever. It's an infinitary analog of the Church-Turing thesis.
> Secondly, the only way to state it, generally, is to use the
> concept of *well ordering* which is rather atrocious predicatively.
You don't understand predicative well-ordering. Read my Gamma_0
paper, particularly sections 2.4 and 2.5.
Nik
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