[FOM] 461: Reflections on Vienna Meeting
friedman at math.ohio-state.edu
Wed May 11 21:24:47 EDT 2011
NEW TRENDS IN LOGIC meeting was held in Vienna, at the Celebration
Hall of the Austrian Academy of Sciences, April 29-30, 2011.
The meeting was sponsored by the John Templeton Foundation.
The meeting featured lectures by the 10 winners of the Kurt Goedel
Research Prize Fellowships (5 from 2008 and 5 from 2011), together
with lectures by members of the juries.
There was a lot of interesting mathematical logic presented.
Here I just want to focus on my and Angus MacIntyre's presentations,
as well as the interchange between us right after Angus' talk.
My talk, PAST, PRESENT, AND FUTURE DIRECTIONS IN THE FOUNDATIONS OF
MATHEMATICS, is available at http://www.math.osu.edu/~friedman/manuscripts.html
Lecture Notes, #55.
It was not meant to be comprehensive, which is not a reasonable goal
for an hour presentation.
As you can see, I began by emphasizing that there are distinct forms
of intellectual lives, where I briefly discuss
The Mathematical Life
The Philosophical Life
The Foundational Life
These have very different value systems and aims. The overlapping
concerns between the Mathematical Life and the Foundational Life, and
between the Philosophical Life and the Foundational Life, dwarf the
overlapping concerns between the Mathematical Life and the
Philosophical Life, at least at the present time and for considerable
I professed insufficient knowledge and experience to comment on
The Scientific Life
which again is quite different.
We were all looking forward to Anton Zeilinger's presentation, a very
well known quantum physicist. Unfortunately, he was unable to come at
the last moment.
I don't know if Anton would feel most appropriately associated with
the Scientific Life or with the Foundational Life - or perhaps both in
a significant way. In my own case, I find myself perfectly classified
in the Foundational Life, as many aspects of both the Mathematical
Life and the Philosophical Life are contrary to my ways of thinking.
Here is a guess: Anton would associate with the Scientific Life, with
a desire for greater Foundational Life - but the foundational side of
quantum mechanics is still in a very primitive stage.
I certainly don't want to speak for Anton - we all wanted to hear from
Anticipating that Anton and many of his associates and students might
be there, I rather carefully discussed the emergence of the
Foundations of Mathematics starting around 1800, culminating in 1922
with the formulation of the ZFC set theory axiom system, still the
most generally accepted general foundational scheme for the whole of
I then focused on three preeminent driving foundational themes -
consistency, completeness, and incompleteness - and discussed their
past, present, and future directions. I presented a number of
expectations for the future.
Angus talked a few hours later in the afternoon session. Angus had to
suffer several intense interruptions (mostly but not entirely by me),
but maintained his thread admirably. After his talk, there was an
extended action packed exchange.
Naturally, I don't remember the exact order of points, and many of the
minute details. But I remember enough about what transpired to make
1. I asked very specifically whether
The consistency of Peano Arithmetic
is a legitimate mathematical problem in present day mathematical
I asked this because the Fields Medalist Voevodsky has explicitly
considered this to be an open problem.
To my shock, Angus declared, unequivocally, yes.
The reason I am shocked by this, and also in the case of Voevodsky, is
not that I discard finitism, or even ultrafinitism. In fact, I neither
accept nor reject these.
I am shocked, because there is a seemingly perfectly understandable
and normal mathematical proof of the consistency of PA which is
entirely within the current fabric of mathematical practice.
With Angus readily accepting Grothendieck's work, or at least large
portions of Grothendieck's work, as real mathematics, I just don't get
A possibility is that Angus has some idea of something unusual about
the usual proof of the consistency of PA that he can articulate, that
distinguishes it in a serious way from "ordinary mathematics". Of
course, that would be interesting. I don't have a clue as to what
Angus has in mind.
Then I met Thierry Coquand for the first time, a quiet, unassuming
well known French mathematician, who also regarded the consistency of
PA as an open problem. This I was not surprised by, because Coquand is
obviously directly concerned with issues in the foundations of
mathematics. In our chat, it was obvious that he had a very different
conception of "foundations of mathematics" than is standard, with a
lot in common with some forms of the constructivist school.
2. Reflecting on item 1, I see the great relevance of the new Strict
Reverse Mathematics. Here I identified a set of instantly recognizable
essential mathematical facts which interpret EFA = exponential
function arithmetic - no base theory involved whatsoever. This is to
establish that mathematics at least has some logical strength. This
paper has appeared in ASL LC06.
The intention is to bootstrap up to stronger systems, once I have so
firmly opened the door.
So the result will be as follows:
i. There is a small set of instantly recognizable essential*
mathematical facts with the property that they themselves form a
formal system in which PA can be interpreted.
ii. This should be viewed as establishing the consistency of PA in
3. I have put an asterisk on essential. These will have to involve
infinite objects. This is required under the standard interpretation
of Goedel's results. The first convincing way of doing this will
probably involve infinite sequences of integers (I have drafts of
this). But a highly attractive alternative is to use real numbers
4. But the fact that I have just outlined something very serious
concerning PA and essential mathematics, does not relieve the shock at
Angus' assertion in 1 above. Georg Kreisel, a close associate of
Angus' for decades, would have ridiculed Angus' assertion in 1. In
fact, my hypothesis can be tested in Salzburg with the help of
5. Angus stated that the first incompleteness theorem was not
surprising. At first I was horrified by this. But Angus did make an
interesting point, at least in the case of ZFC.
Angus actually said that since ZFC was put together in a clumsy way
over a period of time, it was not surprising that it remains
incomplete after being clearly incomplete in the various stages before
arriving at ZFC.
Interesting point. Of course, this remark applies only to ZFC and to
*general* incompleteness - the existence of *some sentence* in the
language that cannot be proved or refuted.
6. But Angus' remark does not apply to, e.g., the arithmetic
incompleteness of ZFC. Here there is no idea that one was getting more
and more arithmetical consequences as the pieces of ZFC were being
7. And, of course, Angus' remarks do not reflect the additional deep
fact that one can find incompleteness from T under only the minimal
assumption that T is consistent (assuming T interprets Robinson's Q
and is axiomatizable by finitely many schemes, or more technically,
recursively axiomatizable), and no matter how haphazard the
construction of T is or was.
8. I didn't hear any attack by Angus on Goedel's Second Incompleteness
Theorem, but perhaps he did not have enough time for that.
9. A persistent feature of Angus' talk is to misrepresent the
intention of the ZFC system. For example, ZFC can be attacked on the
grounds that actual mathematics is generally not about sets, with sets
playing only a minor role. Also, that the actual formalization of
mathematics in set theory is cumbersome, and often impossible at any
practical level. Angus made dramatic and to my mind, misguided and
superficial assertions to the effect that actual formalization of
mathematics in set theory is entirely pointless.
10. To begin with, ZFC forms an unexpectedly comprehensive and
definite model of mathematical practice, which is clearly sufficient
to draw a number of startling and deep conclusions about the nature of
mathematical practice. Our great physical theories also have this
property - they are clearly sufficient to draw a number of startling
and deep conclusions about the nature of physical phenomena - and
none of them come close to reflecting all physical phenomena that we
seek to analyze.
11. However, it must be said that there are all kinds of phenomena of
mathematical practice that remain untouched by the current f.o.m. This
will require more refined models of mathematical practice than is
provided by ZFC. It is the supreme goal of f.o.m. to incorporate more
and more of such phenomena. The analogous statements hold of physical
12. In fact, Angus made an attack on the idea that ZFC does for
mathematical practice what it is supposed to. Angus referred to some
work on topos theory that "some topos theorists don't know how to
formalize in ZFC".
13. A serious follow through on this attack of Angus will either lead
to some laughable exposure of the incompetence of a number of
mathematicians - or possibly something of fundamental importance.
14. I want to interject a discussion about 20 years ago with Charles
Fefferman when he was an Editor of the Annals of Mathematics. He said
that, as Editor, his gold standard for acceptability of a paper was
its ready formalization within the usual ZFC axioms (in addition to
originality, importance, etc.). Any assumption not included in ZFC
must be specifically highlighted as an hypothesis.
15. It has been well known for decades that some mathematicians are
completely allergic to the usual set theoretic foundation for
mathematics - so much that they become paralyzed when confronted with
routine freshman or pre-college level exercises in formalization.
Lawvere is someone who is rather proudly in this category, essentially
claiming a fundamental incoherence in the usual setup.
I have little doubt that Angus can perform routine exercises in
formalization. However, I have my real doubts whether Angus can
properly discern between a case of deliberate incompetence or stunning
epiphany on the part of fancy mathematicians that he admires.
16. In particular, this matter is well worth investigating. I suspect
that it is simply some fancy mathematicians (purposefully) unfamiliar
with routine formalization. On the other hand, perhaps there is some
new way of going beyond ZFC in some interesting sense, not known to
people like me?
17. Of course, in a sense, this has already happened with
Grothendieck's universes. These are the very strong kinds of topoi he
introduced for maximum generality and power. But in all my discussions
with the relevant knowledgeable people, these are viewed as overkill,
with entirely straightforward eliminations for real purposes.
18. As a caricature, suppose you want to study algebraic number
fields. First you can study fields generally, and even form the
category of all fields, various subcategories, and various natural
transformations between them. If you push this far enough, you would
use class theory, and perhaps in a demonstrably necessary way.
Necessary for perhaps the higher order results, but not necessary for
the original purpose of studying algebraic number fields.
19. But would it be essential in any serious piece of algebraic number
field theory? Of course not, and everybody would know this.
20. Of course, my own work is aimed at discovering contexts where "of
course not" must be replaced with "of course". But I have yet to see
what of significance is to be learned from those not schooled in the
usual foundations of mathematics, making uninformed superficial
remarks, as I continue this work.
21. Let me come back to Angus' remark that formalization in ZFC,
particularly through proof assistant technology, is utterly pointless.
Let me use Angus' own position against this. Consider 12 above. This
indicates Angus' skepticism, following many other mathematicians, that
ZFC is indeed comprehensive for standard mathematical activity.
22. In fact, I believe that conventional wisdom among the very large
number of important mathematicians who are not skilled in routine
formalization, is that complete formalization is impossible in
principle. That there are simply too many hidden moves made by a
mathematician in a rich context to be broken down, even in principle.
A modified view which probably has considerably more following among
these people is that although this may be possible in theory, there is
a kind of exponential blowup that renders it impossible to construct
completely formal proofs because the physical universe is too small. A
weaker position, gaining even more following, is that although the
proofs may fit into the physical universe, it would take thousands of
years to construct them.
23. All such positions seem to be explicitly refuted by the work on
actual formalization. So, Angus, isn't that a very strong point in
favor of this research? That hardened and uninformed "intuitions" of
major figures in mathematics, perhaps including yourself, are
explicitly refuted? One can do worse.
24. A persistent theme throughout Angus' talk was the utter
irrelevance of the set theoretic approach to geometry. Of course, the
usual set theoretic foundations simply treats geometry like any other
kind of mathematics. Indeed, mathematicians, in some sense, also do
this at least at the publication level, by using formal set theoretic
structures such as topological spaces, manifold charts, etcetera.
25. I saw this as an opportunity to make it clear that foundations of
mathematics is, at least in principle, far broader than the already
huge and massively open ended topic of using ZFC as a model of
mathematical practice. So in an interchange with Angus the next day, I
told him about the work I did regarding geometric properties of a
complete linear ordering with endpoints, [0,1].
26. In particular, I explicitly formulated the program of determining
which fundamental geometric properties of [0,1] force [0,1] to be
separable. I.e., force [0,1] to be isomorphic to the interval [0,1] of
27. A very basic case of this general program uses the condition that
there exists a continuous f:[0,1] x [0,1] into [0,1] such that x < y
implies x < f(x,y) < y. I.e., a continuous betweenness operator. See http://www.cs.nyu.edu/pipermail/fom/2005-February/008773.html
which also includes a proof.
28. I didn't pursue this further. But high on the list would be, for
example, a continuous bijection from the right triangle with legs 1,1,
and the unit square.
29. With little expectations, I mentioned this program and roughly
this result to Angus, and to my surprise, he regarded this as
interesting, and fitting into what he had in mind.
30. I have always regarded the development of the concept of o-
minimality and the fundamental results as a piece of f.o.m.
Furthermore, I have done some basic work in this area. "What is 0-
minimality?", APAL, November 2008. Also see 11,12,14 of http://www.math.osu.edu/~friedman/publications.html
for work connected with the tame/wild dichotomy. This is clearly
part of f.o.m. as I practice it, and Angus's remarks that current
f.o.m. does not deal with geometric ideas needs to be adjusted
31. Angus maintains throughout his talk that there is a very
substantial difference between the so called "concrete mathematically
natural independence results" of mine, and what goes on in
mathematical practice. His comments on this in his talk were based on
the examples given in my book, Boolean Relation Theory and
Incompleteness (available on my website). They were not based on the
new work he heard about only a few hours earlier, but clearly did not
regard this as changing his picture of the situation.
32. Of course, Angus is fully aware that this can be taken to
absurdity: that the statements I prove are independent are not the
same as the statements mathematicians are proving! But Angus wants to
claim that there is such a difference in kind that my results have no
significance for mathematics.
33. First of all, the intention is not specifically to affect
mathematical practice. E.g., a theory of the big bang is not meant to
affect the big bang, or even the cosmological future. Yes, I fully
expect the work on concrete mathematical incompleteness to eventually
profoundly affect mathematical practice, but that is still premature.
34. Rather, the issue is whether or not it reveals profound
revelations concerning mathematical practice. In my view, Goedel did
this, and the inevitable sets in. Namely, over the years, one focuses
on major features of mathematical practice that are not taken into
account in Goedel, calling for sharpened forms of Goedel's results.
This is probably a never ending process, resulting ultimately in a
major overhaul in mathematical practice of the kind Goedel envisioned.
35. Angus closed his talk by focusing on the case of the late great
broad Russian mathematician I.M. Gelfand. Gelfand ran a very famous
seminar in mathematics in Moscow, which was highly influential in
several dimensions. He brought his operation, to the extent possible,
to Rutgers after he emigrated to the USA.
36. Angus pointed out that Gelfand never had any connection with
logic, despite his legendary breadth. Angus related his experience in
being invited to present at the Gelfand Seminar at Rutgers.
37. Since I was also invited, through the Rutgers Professor Cherlin,
to present at Gelfand's seminar, and remember the experience vividly,
including references by Gelfand to Angus' presentation, my mind was
racing to collect my thoughts for the question period for Angus' talk.
So I don't remember too well what point Angus was making - other than
Gelfand is this great broad mathematician who never saw any reason to
get involved in foundations of mathematics or mathematical logic.
However, Angus probably said more than that, and I will try to pin
38. In any case, when the question period began, I was handed the
microphone. I related, in much detail, my experience at the Gelfand
Seminar. I prefaced it by saying that all of it can be confirmed by
Rutgers Professor Cherlin, who was present during my entire
interaction with the Gelfand Seminar.
39. I started my presentation by writing down the Thin Set Theorem,
with the idea that this leads quickly to Boolean Relation Theory and
Incompleteness (also the title of my book on my website).
40. This already invoked an extremely interesting reaction from
Gelfand. He immediately wanted to know (in Russian, through Rutgers
Professor Retakh and also Kontsevich (Fields medal winning student of
Gelfand - although his Ph.D. advisor was Zagier according to http://en.wikipedia.org/wiki/Maxim_Kontsevich)
, what the hypotheses are. See, the Thin Set Theorem, states that for
every f:Z^k into Z, there exists infinite A contained in Z, such that
f[A^k] is not Z.
41. So Gelfand wants to know what the hypothesis is on f! I think he
found it hard to imagine that something new to him of this level of
immediacy, about an arbitrary f:Z^k into Z.
42. So, eerily, Gelfand said that he did not understand what on earth
the Thin Set Theorem said!
43. This went around and around through translations between Russian
and English, till finally Gelfand fully realized that there were no
hypotheses, and that this statement was new to him!
44. As an indication of the impression the Thin Set Theorem made, he
looked at Kontsevich, who was sitting in the front row, and asked him
if he could prove the Thin Set Theorem, and getting no indication of
this, he asked him to prove this overnight and report back to him! The
next day, I saw Kontsevich and asked him if he saw how to do this, and
he indicated that he did not.
45. Actually, for core mathematicians, the proof of the Thin Set
Theorem is not at all apparent or naturally obtainable. It is easy for
those schooled in the infinite Ramsey Theorem - as a large number of
mathematical logicians are. I have had similar experiences of great
interest from but unsuccessful attempts at proof by such
mathematicians as Kazhdan, Lovasz, and Zagier.
46. I then proceeded to present the idea of Boolean Relation Theory,
moving to 2 functions and three sets, where the functions are from ELG
= expansive linear growth (on N), and the three sets are from INF =
the infinite subsets of N. I stated that there was such a statement
that was independent of ZFC, and in fact can be proved using certain
well studied large cardinals but not without.
47. It was apparent that Gelfand immediately grasped the concept of
BRT, that there are only finitely many (2^512) instances, and that
some of them have exotic logical properties. I believe I did write
down an example, but not as nice as the EXOTIC CASE in my book.
48. The Seminar ended with Gelfand making a rather long summary
statement in English. He made the following points.
a. He has seen that mathematical logic can be useful and clarifying in
certain mathematical contexts, for example from Angus MacIntyre at the
Seminar a little while ago. This he acknowledged was interesting - but
in a limited way.
b. He also said that he was aware of set theoretic problems like the
continuum hypothesis which were independent of ZFC. He acknowledged
strong work in this vein, but said that he had no interest in such
results, as it was so different in character from the mathematics he
c. "But Friedman here has presented readily understandable concrete
mathematical contexts in which strong incompleteness enters, and this
is entirely different than a,b."
d. Gelfand then directly spoke to every person in the audience (except
me), and said to each "this is different and this is important, is it
e. As I expected, this was sufficiently unusual, that nobody there had
the internal fortitude to engage Gelfand's question. But he certainly
made his point in a dramatic way.
49. The question period was ended after I finished this long counter
to Angus' talk.
50. I had one other meeting with Gelfand, and this time I was alone
with him, with Retakh present to translate. This time the topic was
51. I explained the essence of Reverse Mathematics, taken well into
account that Gelfand was pretty well known to have not studied
mathematical logic - as opposed to some other Russian luminaries,
particularly Yuri Manin (others would include Markov, Kolmogorov).
52. Gelfand asked the right questions, many surrounding crucial
robustness issues, and was satisfied by my explanations. He made it
clear that he regarded RM = Reverse Mathematics as obviously
interesting and important. That encounter was also deeply satisfying.
53. The whole experience with Gelfand concerning f.o.m. leads me to
the unmistakable conclusion that had I been at Rutgers with him, I
would have deeply interested him in f.o.m. and he likely would have
made enough perceptive remarks about how it could attack various
mathematical issues I have not thought about, to have influenced
54. Furthermore, my idea that foundations of mathematics was of the
greatest general intellectual interest, at the top of mathematical
activity, was, in my opinion, fully supported by my experience with
55. The next day, I came up with a reasonably clear test which Angus
confirmed would be convincing to Angus.
56. Angus and I assemble a list of prominent mathematicians outside
logic, known for power and depth and judgment in the wider
mathematical community. We even mentioned some names, and we have many
names in common for this purpose. I would present a concrete
mathematically natural theorem which I know can be proved with small
large cardinals but not in ZFC.
57. Instead of asking these mathematicians, "were you working on this
proposition or closely related propositions?", we instead ask these
58. "now that you know this information (relative to the consistency
or 1-consistency of or existence of large cardinals), is this, on
reflection, something that you are happy to know and keep in mind as
part of your toolbox of possibly relevant mathematical facts that
could be useful at any time? and
59. "is this information something whose intrinsic mathematical
interest is comparable to important theorems in existing mathematics?"
60. "do you want to hear about future developments of this kind that
go more deeply into such contexts and explain what is going on in a
more powerful way?" and
61. "is such and such a variant of this statement something you would
personally work on, or at least recommend to graduate students in your
Math Dept who may not be in your field?"
62. Angus agreed that positive answers to questions like these from a
majority of such mathematicians would be entirely convincing to him
regarding the clear importance of the foundations of mathematics for
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 461st in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-449 can be found
in the FOM archives at http://www.cs.nyu.edu/pipermail/fom/2010-December/015186.html
450: Maximal Sets and Large Cardinals II 12/6/10 12:48PM
451: Rational Graphs and Large Cardinals I 12/18/10 10:56PM
452: Rational Graphs and Large Cardinals II 1/9/11 1:36AM
453: Rational Graphs and Large Cardinals III 1/20/11 2:33AM
454: Three Milestones in Incompleteness 2/7/11 12:05AM
455: The Quantifier "most" 2/22/11 4:47PM
456: The Quantifiers "majority/minority" 2/23/11 9:51AM
457: Maximal Cliques and Large Cardinals 5/3/11 3:40AM
458: Sequential Constructions for Large Cardinals 5/5/11 10:37AM
459: Greedy CLique Constructions in the Integers 5/8/11 1:18PM
460: Greedy Clique Constructions Simplified 5/8/11 7:39PM
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