FOM: determinate truth values, coherent pragmatism

[Harvey Friedman]

Reply to Davis 4:12PM 9/5/00:

>At 03:44 PM 9/5/00 -0400, Harvey Friedman wrote:
>>I have given no point of view directly, but simply gave an analysis of how
>>the general mathematical community thinks and acts with regard to axioms
>>-or at least will think and will act with regard to axioms.
>>
>>It appears to me to be very fruitful and interesting to analyze this and to
>>obtain results that are directly motivated by such an analysis. One then
>>has the expectation of having a substantial impact on the general
>>mathematical community.
>
>I believe this is correct, but I believe that it results in some of our
>discussion being at cross purposes. Informed speculation about how
>mathematicians today can be expected to react to certain developments is
>certainly a reasonable topic for discussion.

Furthermore, a great deal can be learned from how mathematicians today do
react and can be expected to react. After all, they are the ones who
develop mathematics, and have developed a keen sense of what they view
mathematics to be concerned with, what is important, etcetera. Certainly,
they are not in the business of articulating this with scientific clarity,
so one has to interact with them and listen to them particularly carefully,
and help them formulate such things in clearer terms. And they do not
always agree with each other. But there is still a tremendous amount of
commonality to work with and learn from.

>And I agree with Harvey
>Friedman that his work is more likely to have an immediate impact on
>mathematicians today than the work of the set-theory community. It is
>certainly striking how well received some of his most recent work has been.

I attribute this to my listening a lot to them for over 30 years, and
getting a sense of how they view foundational matters.The subtleties
involved are not apparent if one thinks only in inflexible categories
mapped out in mathematical logic and the philosophy of mathematics. My main
research programs can be viewed as directly responsive to their concerns.

>But what seems to happen is that when I talk about things from what I hope
>and believe is a general methodological/philosophical point of view about
>ultimate developments, possibly decades away, Harvey replies with what the
>mathematical community today will be willing to accept. Of course he is
>free to think and say that my speculations are pie-in-the-sky nonsense, but
>telling me what the folks in the Princeton or Harvard math departments
>today will think of them is not really relevant to what I'm talking about.

Yes, I do think that your speculations are pie-in-the-sky (although they
are not nonsense). Remember, your speculations involve statements about the
prospective views of the general mathematical community. And what Princeton
and Harvard and the like think about such things now clearly is relevant to
how they might be expected to think about things "decades away." This is
especially true if you recognize a coherence in their attitudes. Coherent
attitudes held deeply and for a long time don't just go away. There
normally has to be major scientific developments of just the right kind
that address these attitudes head on. What kind? That's what I am in the
business of trying to analyze and trying to act on.

>To return to my point that acceptance of the use of large cardinal axioms
>would require some reason to believe that their consequences are true:
>Harvey responds that assessment of consistency is all that we can expect.

I said assessment of consistency and coherent pragmatism. I discussed the
difficulties in getting any other means to assess truth values of
propositions, particularly the most crucial of all - arithmetic
propositions.

>This puzzles me, since of course we know that ZFC can not prove the
>consistency of these large cardinal axioms. We can know only that up to
>now, we haven't run into a contradiction.

You are talking about assessment of consistency.

>Far more important is the kind of cohesive theory that emerges when the
>consequences of these axioms are drawn.

This is part of coherent pragmatism.

>And now I bring out the war-horse
>of PD and the theory it has enabled for the projective hierarchy. Harvey
>will likely respond that mathematicians will prefer the theory coming from
>V=L, and I will respond that I wasn't talking about what mathematicians,
>especially those who haven't really looked into the matter, think.

As I said earlier, higher descriptive set theory don't speak to them
because it is, fundamentally, too far removed in context from normal
mathematics. Since the kinds of sets of reals that they need/want have no
logical difficulties, why overhaul the axioms of mathematics based on this?

Do you really think that if Princeton/Harvard were to really look into this
matter, including what I have been saying, that they would come out with
any different view than they have now?