FOM: the Urbana meeting

[Martin Davis]

I was on the Program Committee for the Urbana ASL meeting, and the 
committee was enthusiastic about the proposed panel on the need for "new 
axioms". 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?

In this posting, I will comment on Harvey's presentation in Urbana and on 
what my own views really are. In his talk, Harvey said that he was bringing 
some very bad news, but also some good news. The bad news is that set 
theory in particular and foundational studies in general are on hard times 
and that unchecked, things would only get worse. The good news is that 
Harvey's new Boolean Relation Theory will save the day: Because it is a 
single appealing theme, whose necessary methods range from what 
mathematicians are used to, all the way through Mahlo cardinals, and 
eventually all the way up the large cardinal hierarchy, mathematicians will 
be led to accept these methods because they will see that they are needed 
to solve problems that interest them. He also implied that the traditional 
set theory community is on the wrong track. I certainly applaud Harvey's 
program, but I assume that since Harvey is devoting himself to this 
program, he believes that the results he gets are TRUE. What I don't 
understand is what he'll tell mathematicians who want to know why they 
should believe this.

The question of whether people could hope to use rational methods to 
investigate completed infinitudes is an old one, and I know only a little 
of the history. Gauss famously insisted that infinity in mathematics is 
always just a manner of speaking, referring to a limiting process. Much 
earlier Leibniz had embraced completed infinities while insisting there 
could be no infinite cardinal numbers (because of the "paradox" of a whole 
having the same cardinality as one of its parts). Cantor boldly ventured 
forth and developed a coherent mathematics of the infinite. Now, I have no 
belief that there are a priori methods by which we can guarantee the 
validity of this undertaking. Goedel spoke of an "organ" by which we intuit 
abstract concepts; I haven't the least faith in the existence of such an 
organ. What I believe to be true (and it is hard for me to see how it can 
be doubted) is that our reasoning faculty that presumably evolved to help 
our hunter/gatherer ancestors with mundane tasks is effective over a far 
greater range than this purpose would suggest - after all this is what 
makes possible such things as science,  mathematics, and fiction. I think 
the question of to what extent this faculty can be effective in exploring 
the infinite is an empirical question that can only be decided by trying 
and analyzing the results. Do we obtain a coherent picture? Or does it all 
dissolve in vagueness and contradictions? (I think this view is close to 
Maddy's "naturalism".)

 From this point of view, the work of set-theorists has been crucial in 
suggesting that it is the former that is the case. The use of PD in 
providing an elegant theory of the projective hierarchy and the discovery 
that PD is implied by large cardinal axioms encourages the view that one is 
dealing with a situation where there is an objective fact-of-the-matter 
with respect to the propositions being studied. The more recent work 
showing that consistency strength alone of certain of these axioms suffices 
to determine the truth values of sentences of given complexity, further 
enhances this perception. I am at a loss to understand why Harvey thinks 
that this work and his are at cross-purposes; it is clear to me that each 
needs the other: Harvey to show concretely that the higher infinities have 
specific interesting consequences way down, the set-theorists to map out 
the infinite terrain and provide a convincing case for a coherent robust 
state of affairs.

Sol Feferman is a brilliant learned expositor, his original article for the 
MONTHLY on the need for new axioms is very beautifully written, and his 
talk in Urbana maintained the same high standard. Sol maintains that CH is 
inherently vague and for that reason it is pointless to expect that the 
question will ever be resolved. This conclusion is not surprising since he 
finds that he does not believe that the concept of the continuum (or 
equivalently, the power set of omega)  is well-defined. Since CH is about 
the cardinality of this very set, naturally, for someone with Sol's 
beliefs, CH can have no determinate truth value. Sol's belief about the 
continuum has been held by such great mathematicians as Brouwer and Weyl, 
and so he is certainly in excellent company. However what bothers me is how 
his conclusion will be received by readers of his MONTHLY article with 
little training in foundations. In my experience, such mathematicians 
presume that the Goedel-Cohen independence results have settled the matter 
about CH, imagining that it is quite like the situation with the parallel 
postulate, and there is nothing more to be said. Such folk hearing Sol's 
conclusion about CH will likely nod their heads. But typically, they work 
with the continuum every day, and by no means are likely to share Sol's 
belief that it is a questionable concept, the belief on which his 
conclusion is based.

My own presentation in Urbana was part of a panel on the history of logic 
in the 20th century. I chose to talk about what I like to call Goedel's 
Legacy. Goedel emphasized that the right way to think about the 
incompleteness phenomenon for Pi-0-1 sentences, is that such sentences are 
seen to be true by proceeding to a higher type, that is, by climbing up a 
level in the cumulative hierarchy. From this point of view the explorations 
of the higher transfinite serve to provide possible strong axioms for 
realizing Goedel's idea. Of course the question remains: are any problems 
of genuine mathematical interest likely to be examples of the 
incompleteness phenomenon, even such problems of central importance as the 
Riemann Hypothesis (as Goedel ventured to suggest). In my talk, I suggested 
that on this question, interested people could be divided into three 
classes: optimists (people who think that such interesting undecidable 
propositions will be found - or even, are already being found), skeptics 
(people who think that Gödel incompleteness will not affect propositions of 
real interest to mathematicians), and pessimists(thinks that even if there 
are such propositions, it will be hopeless to prove them). In replying to a 
question from Dana Scott, I admitted that I am an optimist.

Martin




                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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                        http://www.eipye.com