FOM: determinate truth values, coherent pragmatism

Martin Davis martin at
Tue Sep 5 19:12:56 EDT 2000

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. 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.

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.

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. 
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.

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


                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at
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