[FOM] 461: Reflections on Vienna Meeting

[Harvey Friedman]

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  
time past.

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  
him directly.

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  
mathematics.

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  
this posting.

1. I asked very specifically whether

The consistency of Peano Arithmetic

is a legitimate mathematical problem in present day mathematical  
culture.

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  
it.

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  
essential mathematics.

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  
instead.

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  
Matthias Baaz.

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  
assembled.

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  
theories.

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  
real numbers.

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  
accordingly.

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  
this down.

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  
works with.

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  
not?"

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  
Reverse Mathematics.

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  
f.o.m. development.

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  
Gelfand.

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  
mathematicians,

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?"  
and

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  
mathematics.

*****************************************

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

Harvey Friedman