Andrew Granville: Accepted Proofs: Objective Truth, or Culturally Robust?

Andrew Granville (home page; the comics book: “The Prime Suspects“)

Andrew Granville, a famous number theorist, wrote a wonderful paper about proofs in mathematics published at  the Annals of Mathematics and Philosophy. 

M×Φ

Accepted Proofs: Objective Truth, or Culturally Robust?

I will give some quotes from the paper and some remarks (that follow an email correspondence with Andrew). The paper touches on subjects of many mathematical discussions over the last 150 years (and more). Several of the issues in the paper were discussed here over the blog; let me especially mention my essay About Mathematics. You can learn from Andrew’s paper not only about mathematics and proofs but also about Andrew himself and his views on a variety of issues. There is a very nice Quanta Magazine’s article written by Jordana Cepelewicz about Andrew’s views: Why mathematical proof is a social compact.

This post has four parts: 1. On the Nature of mathematical proofs, and the relevance of Gödel’s theorem, 2. Footnote (15) – are simple number-theoretic conjectures independent from ZFC,  3. On the nature of mathematical explanation, 4. Computers and proofs. We conclude with a few selected footnotes. But first (sound of trumpets),

King Fredrick II’s opinions about mathematicians

The starting view of the paper is:

Vanity of vanities! Vanity of mathematics!– Frederick II of Prussia (1778)

See, the 0th footnote of the paper for the identity of the person King Fredrick was complaining about and to whom the complaint was addressed.

And here is Oscar Wilde’s view of mathematicians (from the  Happy Prince .)

[T]he Mathematical Master frowned and looked very severe, for he did not approve of children dreaming.

1. The Nature of mathematical proofs, and the relevance of Gödel’s theorem, 

The first sections in Granville’s paper discuss what is a mathematical proof, and Gödel’s theorems.

§ 1. — Proof– why and how.

Aristotle wrote

If … understanding is as we posited, it is necessary for demonstrative understanding … to depend on things which are true and primitive and immediate and more familiar than and prior to and explanatory of the conclusion.

and Dan Rockmore wrote

The proofs actually conjure mathematics into existence.

And here is a quote from the paper itself

Prularity The foundations of mathematics starts with a set of axioms, but which set? Hilbert’s proposals leave open the possibility of working within different axiomatic systems, and perhaps those different axiomatic systems will lead to contradictory conclusions to the same simple questions. Then how do we decide which axiomatic system is the correct one to use?

§ 2. — Living with, and ignoring, the Gödel crisis.

The general mathematical culture is to not worry about these things too much. If one works on a question close to where some problem has been found that is undecidable within the “standard” axiomatic framework then one needs to act carefully,(14) but for the most part it seems like a distant problem, and not one that we meet in our day-to-day work.(15)

Footnote (14) explains the notion of “undecidable”

(14) By “undecidable” we mean that if it is true it is not provable using the axioms
in the theory, and if it is false it is not refutable using those same axioms.

Footnote (15) expressed an interesting opinion about  suggestions that certain mathematical conjectures are undecidable. (In fact discussing this issue with Andrew was how I learned about the paper.) We will come back to it below, but let us first discuss other inherent obstacles for proving mathematical statements.

Other inherent obstacles for finding a proof

The end of Section 2 raises the following question:

Can there be a mathematical statement that is provable within our standard axiomatic framework, for which every proof is too long for humans’ timespan?

There is a brief discussion on the relation with the P ≠ NP problem, and a quote of Avi Wigderson

problems in NP are really all the problems we … mathematicians, can ever hope to solve, because [we need to] know if we have solved them.

In the  post Why is mathematics possible?  we talked about  three (potential) inherent obstacles for proving a (simple to state) mathematical assertion: 1) Independence from ZFC, 2) The proof is too long, 3) It is computationally intractable to find the proof. The post raised the question of why we can usually safely ignore these obstacles. Here is a link to a 1999 paper of Kazhdan, and to Tim Gowers’s view.

2. Are simple number-theoretic conjectures independent from ZFC?

Footnote (15) of Andrew’s paper expressed an interesting opinion about  suggestions that certain mathematical conjectures are undecidable. In fact, discussing this issue with Andrew was how I learned about the paper.

(15) One does hear people suggest that popular unsolved problems might be “undecidable” within our axiomatic framework, which I regard as hubris; just because we have not yet found a good understanding of something does not make it an eternal mystery. Knuth [37] even makes the tenuous argument “the Goldbach conjecture … [is] a problem that’s never going to be solved. I think it might not even have a proof. It might be one of the unprovable theorems that Gödel showed exist … we now know that in some sense almost all correct statements about mathematics are unprovable,” and goes on to claim that Goldbach must be “true because it can’t be false” for which he then gives a standard heuristic. The only salvageable truth from this is that Goldbach might not be provable in Peano arithmetic since there might be a different model of integers that satisfy Peano arithmetic yet for which Goldbach fails; however if so then we tinker with our axioms and add one to ensure we remain in the usual integers and then Goldbach should be provable. People have made analogous fatuous claims about the Riemann Hypothesis, the twin prime conjecture, etc. with no real substance to back their claims. There is a good discussion of all this at https://mathoverflow.net/questions/27755/

The very interesting MathOverflow question referred to in Footnote (15) deals with Knuth’s intuition that Goldbach’s conjecture is independent from ZFC. 

Here is something I learned from the comments there: referring to the existence of odd perfect numbers, the iteration of numerical functions, the existence of infinitely many Fermat primes, etc.,  Enrico Bombieri opined (1976):

Some of these questions may well be undecidable in arithmetic; the construction of arithmetical models in which questions of this type have different answers would be of great importance.

Gowers’s answer to the MO question mentions Don Zagier who talked of a general intuition “that the quality that may make some natural statements unprovable is the quality of telling you just what you would expect to happen anyway.”

As seen from footnote (15), Andrew is skeptical of intuitions that various simple number theoretical conjectures are independent from ZFC. Andrew’s interpretation of progress he himself has witnessed in his life time is strongly against such ZFC-independence speculations.

My view is that it may be important (as it was important in the past) to find mathematical frameworks for understanding inherent obstacles for proving number theoretical conjectures, whether obstacles of specific techniques or obstacles of the entire ZFC. I find the following idea quite appealing 

From ZFC’s perspective,  primes are “psedorandom” entities and the set of primes looks like a “sample” of integers based on some “approximated Cramer heuristics”.

I completely agree that there is no evidence for this idea. 

A diversion: Cramer heuristics

Talking about Cramer heuristic, Andrew himself is perhaps the world expert.

Indeed in  a 2007 Annals paper An uncertainty principle for arithmetic sequences, by Andrew Granville Kannan Soundararajan, Andrew and Sound established an uncertainty principle which more-or-less shows that there is no formulation of Cramer type heuristics that can give completely correct predictions! They use among other things results and ideas from the theory of irregularity of distributions and especially results of Jiri Matousek and Joel Spencer. 

 

My impression is that the coarse insights from Cramer’s theory (e.g. “there is always a prime between and ,”)  are not harmed by such deviations and perhaps my “large deviation heuristics”  can give some (conjectural) insights about the worse case behavior of possible deviations. Regarding systematic deviations from Cramer’s heuristics let me mention an older 1995 paper by Andrew (published in the Scandinavian Actuarial Journal) entitled Harald Cramér and the distribution of prime numbers, and a related paper by William Banks, Kevin Ford, and Terence Tao.

 

The question about the plurality of the natural numbers

The following question is very interesting:

Is there some reasons to think that for the natural numbers there is a (platonic) notion of “true” natural numbers along with a variety of pseudo-natural numbers that satisfies the same axioms, or a better intuition is that of geometries (without the parallel axiom), namely, that there are genuinely different models with different properties, with no “priority” of one of the models?

(See Joel David Hamkins answer to the MO question mentioned above, and comments to Joel’s answer.) Of course, the same question is important when we refer to group theory and (mostly) to set theory rather than to number theory. 

3. The nature of mathematical explanation

Sections 3-6 and 13 in Andrew’s paper discuss the nature of mathematical explanations, acceptance of proofs as correct, famous cases of mathematical mistakes, and objectivity in mathematics. 

§ 3. — Formal proof vs culturally appropriate, intuitive explanation.

Section 3 discusses formal proofs, “good proofs,” and beauty. Here are two very nice quotes:

There is no … mathematician so expert in his science as to
place entire confidence in his proof immediately on his
discovery of it…Every time he runs over his proofs, his
confidence increases; but still more by the approbation
of his friends; and is rais’d to its utmost perfection by the
universal assent and applauses of the learned world.
. — David Hume (1739)

Mathematics …[does not] reward passive consumption.
Understanding a mathematical paper is like visualizing
a building based on the architect’s drawings: the text and
formulas are only a blueprint that the reader must use to
reconstruct the author’s imaginary world in her mind. If
she does that, however, then the best mathematical theories
have the same breathtaking quality as the image
of Paris folded on itself. The experience can be both
exhilarating and addictive. — Izabella Laba

§ 4. — What is an accepted proof in pure mathematics?

This section discusses the refereeing process.

Famously, Littlewood would ask  — Is it new? Is it correct? Is it surprising?

An interesting side issue is the question of blind refereeing. I remember discussing this matter with Andrew over Facebook a couple years ago.

Famous mistakes in mathematics in the last 50 years

Sections § 5. and §6. are about Mistakes and Rethinking axioms and language. Section 5 deals with famous cases of mathematical mistakes. In some cases the mistakes were corrected and in some cases they were not. Section 6 deals with a couple of cases where mistakes led to rethinking various aspects of mathematics.

§ 13. — Myths of objectivity.

Andrew opens this section with the following question:

In confirming that a proof is correct we believe that we can recognize and establish an objective truth. But can we? It is easy to believe in one’s own objectivity, or that of an “unbiased machine”, but are such beliefs valid, or are they self-serving?

He then referred to Turing who remarks that in the time of Galileo, the quotations “The sun stood still … and delayed going down about a whole day” —Joshua 10:13 “He laid the foundations of the earth, that it should not move at any time” —Psalm104:556 were considered by many to be an objective refutation of the Copernican theory. 

Here is a great blog post by Thomas Vidick: What it is that we do about objectivity of proofs, with some discussion between Thomas and me.  

My opinion is that objectivity of mathematical proofs is not a myth and the objectivity of mathematical truths is an important pillar of human culture. 

4. Computers and proofs

The second half of Granville’s paper deals with computers.

§ 7. – § 12.— Computers and proofs.

Section 7 starts with three skeptical views about computer-generated proofs

I don’t believe in a proof done by a computer … I believe
in a proof if I understand it. — Pierre Deligne
I’m not interested in a proof by computer … I prefer to
think. — John H. Conway [53]
Computer-generated proofs…teach us very little(55).
— Vladimir Voevodskĭ [53]

Section 8 lucidly describes some famous cases of using computers in major theorems. For more examples see this MathOverflow question and this one. Let me also mention Doron Zeilberger’s important role and notable opinions regarding computers and mathematics. 

The following sections in Andrew’s paper discuss some aspects of computers and proofs.

Andrew raises the question

Can we design a computer verifier to learn and think like a human?

What do you think?

Section 12 is devoted to the Lean Theorem prover (see this post and this one). Here is a quote of Kevin Buzzard

In the near future I believe that maybe computers will be able to help humans like myself (an arithmetic geometer) to do mathematics research, by filling in proofs of lemmas, and offering powerful search tools for theorems … but there is still a huge amount of work to do before this happens.

§ 14. — Will machines change accepted proof?

The concluding Section 14 starts with the assertion

In this article I have asserted that proof verification does little to change the central tenets of proof as a social construction.

And there is a short reference to quantum computing

For us the question is whether quantum computing could be adapted to the task of finding proofs (for example, if one uses a ridiculously large search tree).

I agree with the spirit of Andew’s  concluding words

Indeed it is only a matter of time before we learn how to uncover tremendous possibilities for mathematics and for proofs revealed by computing power, software, and brilliant
programming ideas.

 A few noteworthy footnotes

Double-blind refereeing: Footnote (40)

Andrew opined that “…the expert can truly make the difference in helping an author who has good ideas but perhaps has not yet developed the technical skills to take their ideas all the way. In my experience as an editor many referees are encouraging and helpful in these circumstances, particularly if the author is an “unknown”, explaining how they might modify what they have done to make the argument correct, or the theorem stronger or more general.(40)

(40) This is why I am against “double blind-reviewing” in which the authors’ name is concealed from the referee, as it discourages the referee’s generosity: It seems referees tend to assume an anonymous author “should know better” than to make that mistake, and so give a terse explanation about a mistake, rather than a helpful one.

What is a leaner? 

(74) A leaner is someone who implements a proof in Lean.

Ofer Gabber

(86) A mathematician who is known to insist on the right details. (Referring to Ofer Gabber, a great mathematician whom I first met in 1971 when we were both teenagers.)

Footnote (96): Examples

(96) In pure mathematics, finding extensive examples is often a precursor to better understanding, and thus truly new Theorems.

In my view, examples are of central importance in mathematics just like “theorems” and “proofs”. For many examples see the MO question fundamental examples. (While the nature of scientific proofs is very different for different sciences, the central role of examples is common to many academic areas.)