FOM: Epochal Significance?

[Harvey Friedman]

Some history on the FOM. I posted 12:Finite trees/large cardinals on 3/11/98.

Simpson wrote 3/12/98

>FOM: Friedman's independence results, an epochal f.o.m. advance
>I would like to call attention to Harvey Friedman's posting
>       FOM: 12:Finite trees/large cardinals
>of 11 Mar 1998 11:36:36.  It represents tremendously important
>progress in f.o.m.  One of the key issues in f.o.m. is whether new
>axioms are needed.  Opinion is divided, as witness the earlier
>discussion of Sol Feferman's paper "Does mathematics need new axioms?"
>here on the FOM list.  The background here is G"odel's incompleteness
>theorem, which can be interpreted as showing that new axioms are
>always needed, simply in order to prove the consistency of the current
>axioms.  But a key question that remains is the impact on mathematical
>practice.  Harvey's work tends to bring the incompleteness phenomenon
>into the realm of core mathematics.  In the research cited above,
>Harvey shows that large cardinal axioms (subtle cardinals, etc) are
>needed in order to decide some very natural combinatorial propositions
>about inserting new nodes into finite labeled trees.  This represents
>a new level of achievement in this important direction of
>f.o.m. research.

This led to a spirited discussion, with attackers and defenders of
Simpson's position - especially his use of the phrase "epochal advance."

Some of the attackers had an obvious emotional reaction against Simpson's
posting, faking bewilderment over the complexity of it all - a backhanded
way of saying that the independent statements were not natural. But one
attacker, in particular, was not faking anything, and I want to address his
reservations. This is Joe Shoenfield, who I have known since 1967, and
always have known him to be quite sceptical - or at least indifferent and
unmoved - by what I consider to be basic foundational research. The views
he expressed were no surprise to me.

My own view is that the foundational program in question is of clear and
obvious epochal significance. The posting 12:Finite trees/large cardinals
on 3/11/98 represents the state of the art on 3/11/98. The manuscript,
which has been submitted for publication, can be found on my website.

The question of how striking, convincing, stunning, illuminating, etcetera,
the state of the art was on 3/11/98 is hopefully moot. This is because I am
checking over a new series of four postings which represent the state of
the art now and which are clearly more striking, more convincing, more
stunning, more illuminating, more etcetera, than before. It seems likely
that these new postings will finally put an end to any fake bewilderment.

It may take some time for me to do the appropriate checking before I post
these new results. But the FOM has been comparatively quiet for some time,
and I have been strongly urged to stir up the pot now.

Shoenfield wrote 3/26/98:

>  The recent debate over the epochal significance of Harvey's
>results seems to ignore the history of set thery.   These results show
>that large cardinals can be used to prove combinatorial principles
>not provable in ZFC.

to prove FINITE combinatorial principles not probable in ZFC.

>Such results were obtained by Erdos and his
>colloaborators in the sixties.

Not FINITE ones.

>The combinatorial principles were
>infinite generalizations of well-known results in finite combinatorics.

Thanks for saying INFINITE.

>These results were not of much interest to logicians until Rowbottom
>discovered that they could be use to prove significant facts about
>constructible sets.   Rowbottom's result were considerably extended
>by Silver in his thesis; these techniques have become a standard tool
>in the study of core models.   Harvey's results may be different in
>some respects from these results, but hardly of epochal significance.

There is the extreme diffrerence of being FINITE. This is why the project
is of epochal significance.

Shoenfield wrote 3/39/98:

>     Two replies to my recent message on the above subject state that I
>failed to distinguish finite combinatorics from infinite combinatorics.

Yes.

>I would be much more impressed with this reply if it included a
>reasonable syntactic definition of "finite combinatorial statement"
>which included Harvey's principles but not Erdos's.

First of all, Erdos's principles are not finite, so there is a trivial
answer to this. Any relevant finite combinatorial statement is in class
pi-0-infinity, but not Erdos's.

> I would be still
>more impressed if one could show that these finite combinatorial
>statements had some absoluteness properties not shared by Erdos's
>results.   (Please take this as a challenge, not a criticism.)

The obvious absoluteness property is that these finite combinatorial
statements have the same truth value in all models of set theory (say ZFC),
or even arithmetic (say PA) with standard integers only. This is not true
of Erdos's statements. In fact, the statements are trapped between
1-consistency and consistency (sometimes I know it's one of these two).

Shoenfield wrote 4/14/98

>     I think my differences with Steve on the above topic are due to a
>different notion of what fom is about.   I was rather shocked to see
>him write some time ago that he prefers fom to mathematics.

F.o.m. is part of a general subject called foundations - not just
foundations of mathematics. It is by far the most stunning of the
foundations yet. As such, it is of the highest general intellectual
interest, and has a special interest quite independent of mathematics.
Viewing it as a branch of mathematics is a gross mischaracterization that
grossly diminishes its importance and depth. However, there is no question
that f.o.m. is a mathematical subject whose methodology has very
substantial mathematical components.

> In my
>view, fom is a branch of mathematics whose subject is the structure
>of mathematics.

I do not think that the study of the structure of mathematics is a branch
of mathematics. It is a mathematical subject. So is computer science.
Computer science is not mathematics - it is a mathematical subject.
Statistics is not mathematics. It is a mathematical subject.

>Its object is to replace our intuitive notions
>about this structure with precisely defined objects and then formulate
>and prove statements about those objects.   (I think this is the moti-
>vating idea behind Harvey's series of communications.)

This is too narrow a characterization of the point of f.o.m. for me.

>From this
>point of view, finite combinatorics becomes interesting only when it
>has been precisely defined.

I know of no finite combinatorician in the world who has a *precise*
definition of finite combinatorics.

>I suggested a syntactical definition
>because it seemed to me that we should be able to identify a finite
>combinatorial statement by looking at the sentence which expresses it.
>     The fact that Harvey's combinatorial statement is pi-0-2 is
>certainly significant.   It suggest to me that I should have com-
>pared his result not to Erdos-Rowbottom but to Godel's result:
>ConZFC is not provable in ZFC, but is provable if one assumes an
>inaccessible cardinal.   ConZFC is pi-0-1, and I think it is clearly
>a finite combinatorial statement.

No. ConZFC is not a finite combinatorial statement, since in order to
understand it, one must be thinking of sets of enormous size. If one is not
thinking of sets of enormous size, then it is an incomprehensible pile of
ad hoc unintelligle nonsense. (I leave open the possibility that one might
find an intelligible interpretation of ZFC that does not involve sets of
enormous size, and perhaps thereby Con(ZFC) wouldn't require sets of
enormous size for its understandability. But that has certainly not been
done in a suitable way). And finite combinatorists do not think of sets of
enormous size when doing finite combinatorics.

>Harvey's result is a natural
>extension.

Natural extension of ZFC? Certainly not. It is about the successive
insertion of new nodes in finite trees.

>Perhaps his methods could lead to a series of results
>on the relationship between combinatorial statements having certain
>forms in the arithmetical hierarchy and large cardinals.

I think that the statements involved are sufficienctly simple - especially
some new ones - that this project is feasible and worth doing. But it
certainly isn't required in order to draw a clear distinction them and
consistency statements.

>Possibly
>one could find surprising and interesting connections similar to
>those in the Martin-Steel Theorem.   This would be fine mathematics,
>but perhaps not all that Steve claims for Harvey's result.   (Perhaps
>my objection is due to my reluctance to use words like "epochal"
>unless they are clearly called for.)

The word "epochal" is clearly called for for the project that is being
contributed to. The history of advances in it indicate that progress will
likely continue to be a matter of degree. The result that

"e + pi is irrational" is equivalent to the 1-consistency of ZFC

would be a matter of extremely high degree. And here "epochal" would be a
gross understatement. If you proved that tomorrow, it would rightly be
regarded as the most stunning mathematical advance ever made by one person
in the history of the human race. And you should be extremely insulted if
such absurdly minimal evaluations as "epochal advance" were used. Of
course, it is almost certainly false, and you are not working on it.

Shoenfield wrote 4/20/98:

>   Steve's recent communication states our differences clearly.   He
>slightly misrepresents me on one point.   I don't object to discussion
>of informal notions; but I think that if the discussion is to serve
>the purposes of fom, it must aim at replacing these notions by formal
>ones.   Steve shows no interest in replaing the notion of "finite
>combinatorial statement" by a formal notion.

Steve is just saying, and I agree with him, that such a replacement is not
even remotely needed in order to clearly see the gross difference between
consistency stsatements and natural statements about successively inserting
nodes in finite trees. Having said that, I will very likely try my hand at
coming up with little  languages for extremely basic discrete mathematics,
and at least observe that my independent statements are very very short.
And conjecturing that if they are much shorter, than they are not
independent. Also, there always seem to be candidates for decision
procedures when one tries this sort of thing, where the decision procedure
can only be verified to work with large cardinals.

>If I correctly inter-
>pret him, we have an unresolvable difference over the purpose of fom.
>     I did not defend my statement that ConZFC is a combinatorial
>statement because I didn't think it would be controversial.   Steve
>says that a combinatorial statement is a statement about the arrange-
>ment of objects in patterns.   ConZFC ia about the arrangement of
>sentences of set theory in the patterns of proofs.

Proofs in set theory (ZFC) involving sets of enormous size. As indicated
above, that is not finite combinatorics.

>Steve's objection
>seems to be that ConZFC is not interesting to certain people: people
>with little knowledge of logic and specialists in finite combinatorics.
>It seems he is replacing "finite combinatorial statement" by "finite
>combinatorial statement of interest to certain people".   I do not
>regard this informal notion as of much interest, especially with the
>peculiar (to me) choice of these people.

There is no problem giving a combinatorics seminar or computer science
seminar on the successive insertion of nodes in finite trees, and related
matters. I know from perpsonal experience. I am always testing my
statements out in terms of naturalness with people who actually do finite
combinatorics and combinatorial computer science. This is a way of testing
out my ideas. They do not react the same way to a discussion of the
consistency of ZFC or of the consistency of ZFC + large cardinals.

SHoenfield wrote 4/23/98:

>     What does matter?   There has been much discussion of the state-
>ment: ZFC is a foundation of mathematics.   To most of us, this means:
>all of accepted mathematics can be translated into ZFC in a fairly
>straightforward way.   I think most of us agree that this statement
>is true.   Why does it matter?   There has been some discussion of
>this, sometimes not well-considered.   For example, it has been
>stated that it gives a precise definiton of acceptable mathematics.
>Perhaps it does, but not in a way that matters.   The mathematicians
>who verified the correctness of Wiles theorem did not concern them-
>selves with ZFC.   I think the statement matters because it has prac-
>tical applications of the following kind.   Cohen proved, by an
>ingenious study of certain models, that CH is not provable in ZFC;
>he then used the statement to conclude the much more interseting
>result that CH is not provable by accepted mathematics.   This
>application alone wouls seem yo me to justify the hard work which
>has been put into verifying the statement.

I like this and I agree with this. But I want to go further. For the vast
majority of mathematicians, a lot of set theoretic notions "don't matter."
For them, arbitrary sets of reals "don't matter." For the large majority of
them, even arbitrary Borel sets of reals "don't matter." For very many of
them, arbitrary continuous functions on the reals "don't matter." As the
regularity conditions get stronger, and as you move into the finite, it
"matters" for more and more of them. Putting aside the question for the
moment of whether or not they are right, they are looking at these things
very much like you were looking at much discussion on the FOM when you
wrote: (also in your 4/23/98 posting)

>    When I was a youth, I was somewhat interested in the mind-matter
>problem; but it was finally settled for me by the immortal answer:
>never mind, it doesn't matter.    (Does anyone know the author of
>this remark?)   I think some of the longest, dullest, and most
>abrasive discussions on fom are about things that don't matter, at
>least to people whose primary interest is the nature and structure
>of mathematics.   Let me give a couple of examlples (without, I
>hopre, insulting anyone).
>     (1) Was Lagrange's theorem true of pebbles on a Jurassic beach?
>Mathematicians understand this theorem, know that it is true, and
>know how to apply it (eg, to undecidablity of certain theories of
>the integers).   Even if we could agree whether or not it was true
>in Jurassic times, it would not affect the activity of mathemati-
>cians or their understanding of the theorem.
>     (2) Are Boolean algebras the same as Boolean rings?   The
>exact send in which this two notions are equivalent is well known
>and explained in many texts.   Again mathematicians know how to use
>the equivalence, eg, by using theorems on rings to prove facts about
>Boolean algebras.    Even if we could decide whether they are "really"
>the same, it wouldn't matter.

I.e., the mathematicians are looking at the kind of advances in foundations
that you stress like you look at these things. The various projects that I
stress are of a very different character.