FOM: the Urbana meeting

[Harvey Friedman]

Reply to Martin Davis 6/20/00 1:35PM:

>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?

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. 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."

>Harvey said .. 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.

This is more or less accurate, but I said it slightly differently,
and let me say it here even slightly differently than I said it in
Urbana.

Mathematical logic is in very bad shape sociologically and
politically, and that part of the difficulties come from the focus of
contemporary research.

Virtually all major scientific areas are born with the discovery and
development of striking new models of some phenomenon that exists
independently of that new scientific area. In most cases, the subject
really attracts attraction and attains an identity through striking
findings that are based on the striking new models.

Mathematical logic fits very much into this framework. We all know
the great models of the phenomenon known as logical reasoning (through
predicate calculus),
mathematical reasoning (through set theory), and algorithmic
procedures, and others. And we all know the striking findings in the
early part of the 20th century that really gave the subject its
identity.

In order to maximize the impact of a subject on the wider
intellectual community, one must periodically - better yet,
continuously - strive for the renewal and fresh perspective one gains
by revisiting its origins in order to reflect more subtle features of
the seminal phenomena. This is normal and standard.

E.g., in partial differential equations, one continually strives to
get more subtle information about more equations that model more
closely more subtle physical phenomena. The same is true of
mathematical economics, etcetera.

However, it is extremely important not to try to force the pace of
this natural evolution beyond what can be productively accommodated
at any given stage in the development of the subject. This merely
leads to a counterproductive negativity that is unwarranted. Every
subject, including the most successful and revered areas of pure
mathematics, look like dismal failures when looked at with such
unrealistic expectations. There is a natural evolution of subjects.

Nevertheless, I have insisted that the commonly referred to four main
areas of mathematical logic, set theory, model theory, recursion
theory, proof theory, are in dire need of such renewal. Actually,
some appropriate renewal is already taking place in some parts of some of
the areas, but not in others. I like to think that I am always striving to
point the way towards renewal.

>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.

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.

This is in a context in which it is conventional wisdom among
mathematicians that basic natural elementary universally accessible
concrete mathematics - part of the unremoveable furniture of mathematics as
we know it - is in no way tied up with large cardinals, or their
1-consistency or consistency.

More explicitly, this is in a context in which set theory is not viewed as
part of mathematics, but rather as a scheme for establishing rigor in
mathematics. Set theory is regarded as a framework for interpreting
mathematics, and not a part of mathematics itself. This is the modern view,
hardened by the brief and fleeting experience with experimenting with set
theoretic questions taken literally. Nowadays, set theoretic formulations
are used only when they simplify the underlying mathematics. When they
create their own peculiar problems - arising out of pathological cases
which have nothing to do with the underlying mathematics - then they
discard them as utterly irrelevant and useless.

I think that I have some singular contributions to make to this situation
as an f.o.m. expert, but I am less sure that I have, at this time, some
singular contributions to make to this situation as a ph.o.m. expert. (Of
course, there is the question of just who does have singular contributions
to make to such issues as ph.o.m. experts).

>... 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 will take your use of the word "infinite" to mean "the absolutely
unrestricted infinite" - not just the natural numbers, or even sets of
natural numbers. I.e., full blown set theory.

It is obvious that we get a coherent picture that does not appear to
dissolve into contradictions. But does it dissolve into vagueness if we
push it too hard? I certainly think that the experience with the continuum
hypothesis and related questions definitely makes it at least *appear* that
it all dissolves into vagueness when things are pushed too literally.

I know that some set theorists are hopeful that the continuum hypothesis
and related questions will not continue to make it appear that it all
dissolves into vagueness when pushed literally. However, at this point, the
set theorists have pretty much abandoned the idea that there will be a fix
(i.e., resolution of such problems as the continuum hypothesis) that can be
readily understood and accepted by people who are not experts in set
theory. I.e., in the same sense that the usual axioms of ZFC can be readily
understood and accepted, even with such additional axioms as strongly
inaccessible cardinals, or even such additional axioms as the existence of
a probability measure on all subsets of [0,1].

I am doubtful that the *appearance* that it all dissolves into vagueness
when pushed literally will be erased by any esoteric "fix" understandable
only by experts in set theory.

Undoubtedly there will be a great effort ultimately made to reduce any such
esoteric "fix" to commonly understandable - and commonly convincing -
terms.

Equally surely, there will be a great effort ultimately made to show that
there is no "simple" fix that is as "simple" as the usual axioms of set
theory. I have a plan for this.

Of the prospects for the last two paragraphs - I'll put my money on the
second paragraph.

>From this point of view, the work of set-theorists has been crucial
>in suggesting that it is the former that is the case.

And the work of set-theorists has been crucial in suggesting that this is
not the case, because the set theorists have shown that so many set
theoretic problems like the continuum hypothesis are independent of ZFC
together with so many additional axioms.

>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 fact that other axioms solve the same problems differently cuts in the
other direction. A perfectly legitimate conclusion from all this is that
there is no "objective fact-of-the-matter" since it is not an "objective
fact-of-the-matter" whether V = L is true or whether large cardinals are
true. It is just that both of these hypotheses are sufficient to settle
these particular questions.

As I have said in the Urbana meeting, the set theorist wants to accept
large cardinals because of the extra delicate and interesting set theoretic
structure that entails, and is missing under V = L. But the mathematician
doesn't welcome such extra set theoretic structure - as it is irrelevant
and sharply different in flavor to underlying mathematical issues. So if
forced to make a choice, mathematicians would greatly prefer V = L.

>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.

This sentence is mathematically false on its face. An accurate statement is
far more technical, making any "perception" less convincing.

> 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.

Cross-purposes is not the way I would put it. It's more like this. There is
a poker game going on in a barn. The players are in a long and heated
dispute as to whether or not it is legal for one of them to raise the pot
for the third time. This argument is going on unabated while the barn is
being consumed in a devastating fire.

I'm sitting here working my god damn xxx off to put out the fire while set
theorists are arguing about their rules of poker, calmly sitting in the
middle of that fire.

To put out the fire, one needs only to use some cardinal that is at least
beyond ZFC. To the general mathematical community, anything that large is
already grotesquely large - it takes experts in set theory to discern one
from the other.

The cardinals involved in what I am doing to put the fire out were already
set up in the years 1911-1913.

And while the fire is raging, nobody cares about some esoteric explanation
of just how "coherently robust the state of affairs is" to experts in set
theory. After all, these experts in set theory talking about "coherent
robust states of affairs" are starting to be being consumed in the fire!!

>Sol maintains that CH is inherently vague and for that reason it is
>pointless to expect that the question will ever be resolved. ... he does
>not believe
>that the concept of the continuum (or equivalently, the power set of
>omega)  is well-defined.

The equivalence between the continuum and the power set of omega is not
generally accepted by some leading core mathematicians because the
correspondence is not a natural mathematical object. In fact, many do not
believe that our usual model of the continuum in terms of Dedekind cuts and
the like fairly represents the continuum.

This is in consonance with my earlier statement about set theory being an
interpretation of mathematics rather than a part of mathematics.

>...for someone with Sol's beliefs, CH can have no
>determinate truth value. ...has been
>held by such great mathematicians as Brouwer and Weyl... what bothers me
>is how his
>conclusion will be received by readers of his MONTHLY article with
>little training in foundations.

I am uncomfortable with other aspects of Sol's article, also. I plan to
write something for the Monthly in due course about BRT. Why don't you
write something for the Monthly about your extreme Platonism?

>...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.

There is a lot to be said for this preumption, given what has actually been
accomplished. However, what they don't generally realize is

i) how demonstrably irrelevant the CH is for any questions they really care
about, even if they suspect it;
ii) that there is an axiom of restriction, V = L, that dispenses with the
CH and all related questions.

>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.

But the way they work with the continuum is not as a set theoretic object.
So the relationship between what mathematicians are doing and what Sol's
views are is, to my mind, quite unclear. Perhaps you ought to ask Sol about
what he thinks of this.

> I chose to talk about Goedel's Legacy. ... 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). ...I suggested
>that ... 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.

My own view is this:

i) it is likely that every non set theoretic statement in the normal
mathematical literature is decided by ZFC; (this can be made very precise,
but that's for another time and place).

ii) BRT will be accepted as having results "of genuine mathematical
interest", and is chock full of incompleteness phenomena. The most
immediately convincing ones will be in terms of classifications and
specific theorems about classifications.

iii) additional classification problems throughout mathematics - even more
mathematically friendly than BRT - will surface following the lead of BRT,
and the incompleteness phenomena will also routinely appear there. E.g.,
instead of looking at Boolean relations, one looks at solutions to
algebraic equations.

iv) the hallmark of these new kinds of classification problems is that
there are always just a finite number of cases. The number of cases is
generally quite large, like 2^512. The idea is that large cardinals are
supposed to settle all of the instances, but some instances cannot be
settled without large cardinals. Also, there are general features of the
classification that are stated as single theorems which can only be proved
using large cardinals.