FOM: the Urbana meeting

[Harvey Friedman]

Reply to Davis, 10:58AM 6/27/00 and 11:24AM 6/27/00:

>> > Some time ago in a telephone conversation, Harvey told
>> >me that I am an "extreme Platonist". Being a great fan of Harvey's
>> >work on the necessary use of large cardinals, I took his comment
>> >quite seriously and began to wonder. Is that really me?
>>
>>Yes, because I under the impression that you think that any
>>intelligible set theoretic question, quantifying even over all sets -
>>regardless of where they lie in the cumulative hierarchy - is a well
>>defined mathematical problem in the same sense as, say, the Riemann
>>hypothesis or the twin prime conjecture.
>
>I do believe that every sentence in the language of set theory has a
>definite truth value (whether human beings will ever be able to determine
>it or not). If this makes me an "extreme platonist" - so be it. My own
>understanding is that the term usually is taken to involve ontological
>commitments that I'm not prepared to make.

I stand corrected on terminology. I just talked to Neil Tennant, who is
fully versed in the terminology in this slippery area. A suggested name for
your stated position about truth values is "truth value determinism."

I do not see how come to this startling conclusion without believing in
some underlying objective reality of a set theoretic nature. Perhaps, the
underlying objective reality that you base this on is some definite
structure. I.e., somehow the cumulative hierarchy of sets under membership
is uniquely determined as a structure up to isomorphism? That this is "out
there" in the sense that the physical universe is "out there"? If so, this
is a variant of extreme Platonism that is obviously closely related.
Tennant has mentioned the term "structuralist."

By the way, my own view is that "truth value determinism" for GENERAL SET
THEORY is an extreme view that is rational to hold only if it becomes
supported by appropriate scientific developments. At the moment such a view
is nowhere near being sufficiently supported by current scientific
developments.

Instead of taking extreme views with nonexistent or minimal supporting
evidence as you are doing, I work on trying to find appropriate scientific
developments that help us carry on intelligible discussions of such
matters. In my view, it is possible to carry on only very superficial
discussions of such matters without the requisite new and deep scientific
developments.

What scientific developments could make a more than superficial discussion
of your extreme view possible? That will be a subject of a series of
postings titled:

What I Think.

>I'm certainly aware of the
>difference in kind between such problems as RH and twin primes on the one
>hand and CH or the existence of inaccessibles on the other. I don't know
>how Harvey intends "in the same sense as" to be taken.

Now we are getting somewhere. What is this difference in kind? After all,
in both cases, from your point of view, one is imply trying to determine
what an already determinate truth value is.

>>That there is an absolute
>>right and an absolute wrong answer. And that it is part of normal
>>mathematical activity to work on such questions just as it is
>>to work on RH or TP, at least in the sense that there is no
>>special difference in kind between the two activities that justifies
>>calling one "normal mathematical activity" and the other "not normal
>>mathematical activity."
>
>Here is Harvey's tendency to turn all such questions into questions of
>mathematical sociology as mathematics is practiced today.

Here is Martin's tendency to ignore the important lessons to be learned by
the existing general mathematical community.

Obviously I don't believe that mathematicians are good logicians or
philosophers, but doing foundations of mathematics without regard to the
way mathematicians think makes no sense to me.

As I continue to emphasize, mathematicians do not view set theory as part
of normal mathematics, but only an interpretation of normal mathematics. To
ignore this is to ignore one of the great lessons of mathematics.

An issue of transcendental importance is: to what extent and in what sense
is set theory interpretable into normal mathematics? Even though set theory
is not normal mathematics, it may well be interpretable into normal
mathematics, and this would be an incredible surprise to the general
mathematical community, with all sorts of profound consequences in theory
and in practice.

>But - if we must,
>of course there is such a difference today (though I would not choose a
>pejorative like "not normal" which only obscures the real issues).

Not using "not normal" only obscures the real issues.

>The
>paradox is that it is Harvey's own work which is doing so much to erase
>this distinction!

I am trying to prove that set theory (including large cardinals) is
interpretable into normal mathematics. That does not make set theory normal
mathematics, and does not erase that distinction. It may, however,
eventually make set theory an essential tool in normal mathematics.

An analogy is this. Writing compiliers and computer software is not normal
mathematics. Yet it is being increasingly used in normal mathematical
investigations, even in proofs of normal mathematical results.

OK, I will go this far: I will agree that looking out very very far into
the future, large cardinals may be a required or semirequired part of the
normal graduate mathematics curriculum.

>> >He also implied that the traditional set theory community is on the
>> >wrong track.

>>Yes, of course I believe that the results that I get are TRUE, but what
>>results? The results that it is necessary and sufficient to use large
>>cardinals to get such and such, or such and such can only be done with
>>large cardinals, or such and such is outright equivalent to the
>>1-consistency or consistency of large cardinals, etcetera.
>>
>>That is where my role as f.o.m. expert ends, and where, if I wish to
>>continue, my role as ph.o.m. begins.
>>
>>Namely, my f.o.m. expert role is to show that basic natural elementary
>>universally accessible concrete mathematics - part of the unremoveable
>>furniture of mathematics as we know it - is inexorably tied up with large
>>cardinals, through their 1-consistency or consistency.

>It is precisely when foundational issues come to the fore that
>philosophical questions become inescapable.

My business is to bring us to the point where philosophical questions
become inescapable for the general mathematical community - not to resolve
these philosophical questions.

I do not believe that Godel resolved any of the philosophical questions
raised by, suggested by, or addressed by, his 2nd incompleteness theorem.
How do you think Godel's 2nd incompleteness theorem compares as an
intellectual achievment to his and others writings about related
philosophical questions?

>Harvey has written on "The
>Necessary Use of  Large Cardinals" not on "Equivalences with 1-Consistency
>of Large Cardinals" presumably because he really believes (though he's
>reluctant to admit it) that the combinatorial consequences he drew were
>TRUE. Otherwise, why bother?

Why bother? Because the equivalence of Boolean relation theory with the
status of large cardinals joins the issues in a serious way for
mathematicians for the first time. No position on the status of large
cardinals is needed to see that the status of large cardinals is now
seriously joined.

The title means: large cardinals are necessary (and sufficient) in order to
prove certain kinds of statements. Also it is necessary and sufficient to
settle the status (as to consistency or 1-consistency) of large cardinals
in order to settle the status (true or false) of certain kinds of
statements.

As I said in the Urbana meeting, I am willing only to go this far: that
this work will justify that some large cardinals  be elevated to the status
of working hypotheses for the general mathematical community. And before
this kind of work is completed, the general mathematical community will,
quite appropriately for their purposes, continue to totally ignore large
cardinals and remain totally uninterested in their status.