Mathematical depth and Szemerédi's theorem · Arana (Illinois/Paris) Depth Helsinki LC 2015 18 /...

Mathematical depth and Szemeredi’s theorem

Andrew AranaPhilosophy, University of Illinois at Urbana-Champaign (until August 2015)

Philosophy, University of Paris 1 Pantheon-Sorbonne, IHPST (after August 2015)

Logic Colloquium 2015, Helsinki, August 7, 2015

Arana (Illinois/Paris) Depth Helsinki LC 2015 1 / 26

Introduction

Introduction

Mathematicians frequently cite depth as an important value for their research.

For instance, the Annals of Mathematics since the 1920s has more than a hundred articlesemploying the modifier “deep”, referring to deep results, theorems, conjectures, questions,consequences, methods, insights, connections, and analyses.

However, there is no single, widely-shared understanding of mathematical depth.

Today I will attempt to bring some coherence to mathematicians’ understandings of depth, byusing Szemeredi’s theorem as a case study.


Szemeredi’s theorem


A set A of positive integers has positive density if the proportion of elements of A among theintegers from 1 to n approaches a positive number in the limit as n goes to infinity.

Szemeredi’s theorem, infinite form (1975)Each subset of the natural numbers of positive density contains an arithmetic progression ofarbitrary length.

Szemeredi’s theorem, finite form (1975)Let k � 3 be an integer and let 0 < � 1. Then there is a positive integer N = N(k , �) suchthat any subset of {1, 2, . . . ,N} of at least �N elements contains an arithmetic progression oflength k .




In 1936 Erdos and Turan had conjectured that every sufficiently “dense” subset of N containsan arithmetic progression of length three, and more strongly, of arbitrary length.

In 1953, Roth resolved this conjecture for k = 3, and his 1958 Fields Medal citation listed thisresult among his significant achievements.

In 1975 Szemeredi resolved the conjecture for arbitrary k .

Erdos, “Some problems and results in number theory” (1984)I offered $1,000 for [my conjecture] and late in 1972 Szemeredi found a brilliant but verydifficult proof of [my conjecture]. I feel that never was a 1,000 dollars more deserved. In factseveral colleagues remarked that my offer violated the minimum wage act.



What Szemeredi’s theorem asserts

Gowers, Abel Prize announcement for Szemeredi, 2012One way of understanding Szemeredi’s theorem is to imagine the following one-player game.You are told a small number, such as 5, and a large number, such as 10,000. Your job is tochoose as many integers between 1 and 10,000 as you can, and the one rule that you mustobey is that from the integers you choose it should not be possible to create a five-termarithmetic progression.

For example, if you were accidentally to choose the numbers 101, 1103, 2105, 3107 and 4109(amongst others), then you would have lost, because these five numbers form a five-termprogression with step size 1002.




GowersObviously you are destined to lose this game eventually, since. . . if you keep going for longenough you will eventually have chosen all the numbers between 1 and 10,000, which willinclude many five-term arithmetic progressions.

But Szemeredi’s theorem tells us something far more interesting: even if you play with thebest possible progression-avoiding strategy, you will lose long before you get anywhere nearchoosing all the numbers.




Szemeredi for k = 23, � = 11000 : There exists a positive integer N such that every subset of

{1, 2, . . . ,N} of at least N1000 elements contains an arithmetic progression of length 23.

GowersIf, for instance, we are trying to avoid progressions of length 23, Szemeredi’s theorem tells usthat there is some N (which may be huge, but the point is that it exists) such that if we playthe game with N numbers, then we cannot choose more than N

1000 of those numbers–that is, amere 0.1% of them–before we lose. And the same is true for any other progression length andany other positive percentage.



Proofs of Szemeredi’s theorem

There are three main proofs of Szemeredi’s theorem.

• Szemeredi’s combinatorial proof (1975; Abel Prize 2012)• Furstenberg’s ergodic-theoretic proof (1977; Wolf Prize 2006/7)• Gowers’ Fourier-analytic proof (1998; Fields Medal 1998)

Recently these proofs have been mixed together to yield yet further proofs.

One key aim of reproof, especially from Bourgain 1986 (Fields Medal 1994) onward, has beento find better bounds on N .


The depth of Szemeredi’s theorem

Szemeredi’s theorem has been judged deep

Tao and Vu, Additive Combinatorics (2006)Of course, a “typical” additive set will most likely behave like a random additive set, which oneexpects to have very little additive structure. Nevertheless, it is a deep and surprising fact thatas long as an additive set is dense enough in its ambient group, it will always have some levelof additive structure. The most famous example of this principle is Szemer´edi’s theorem. . . [a]beautiful and important theorem.

Announcement of the Rolf Schock prize for Szemeredi, 2008[The prize is awarded] for his deep and pioneering work from 1975 on arithmetic progressionsin subsets of the integers, which has led to great progress and discoveries in several branchesof mathematics.



Why is it deep?

I have considered four types of criteria of depth, what I call

• Genetic criteria• Evidentialist criteria• Consequentialist criteria• Cosmological criteria

Today I will concentrate on evidentialist and cosmological criteria of depth.



Objectivity as a desideratum for an analysis of depth

In the course of assaying a critierion of depth, one issue of interest to many philosophers ofmathematics will recur.

Can judgments of mathematical depth be objective? And if so, in what sense?

In a preliminary investigation of depth like this one, it is not my goal to decide definitively onthe larger questions of whether depth is objective, what objectivity of the relevant sort wouldconsist in, and whether it is a good/bad thing that depth is/is not objective.

But I ask of each account of depth whether it can deliver a notion of depth that is not vague,is temporally stable, and is not essentially dependent on our contingent interests or our merelyhuman limitations.


The depth of Szemeredi’s theorem Evidentialist depth

Evidentialist criteria

I call a criterion of depth evidentialist if it links the depth of a theorem with some quality ofits proof.

One evidentialist criterion was identified by Daniel Shanks.

Shanks, Solved and Unsolved Problems in Number Theory (1978)We confess that although this term “deep theorem” is much used in books on number theory,we have never seen an exact definition. In a qualitative way we think of a deep theorem as onewhose proof requires a great deal of work�it may be long, or complicated, or difficult, or itmay appear to involve branches of mathematics the relevance of which is not at all apparent.

Shanks thus locates the depth of a theorem in the laboriousness of its proof.



Laboriousness of Szemeredi’s proof



Laboriousness and impurity

But there is a problem: Shanks has given four characteristics of proofs: length, complexity,difficulty, and impurity. But these are different!

A proof can be difficult and not deep: consider the proof of the four color theorem.

A proof can also be pure and deep: consider Szemeredi’s proof!

Which of these characteristics is best suited for an evidentialist criterion of depth?



The problem of multiple proofs

This question is compounded by another problem.

There are multiple proofs of Szemeredi’s theorem, by Szemeredi, Furstenberg, Gowers, and byothers.

Which proof should determine the depth of Szemeredi’s theorem on an evidentialist criterionof proof?



The problem of multiple proofs: difficulty

Start with difficulty: one could say that a theorem is deep if all its proofs are difficult.

But this is difficult to predict or prove.

One could say instead that a theorem is deep if some of its proofs are difficult.

But a single theorem can have both difficult and easy proofs (e.g. infinitude of primes).

Why should its having a difficult proof take precedence over its having an easy proof as far asdepth is concerned?



The problem of multiple proofs: impurity

Continue with impurity: one could say that a theorem is deep if all its proofs are impure.

But this is also difficult to predict or prove.

One could say instead that a theorem is deep if some of its proofs are impure.

But one can inject logically “inessential” impurities into an otherwise pure proof (for instance,diagrammatic moves in an otherwise algebraic proof).

Objection: they’re inessential! But how can one determine which impurities are inessential?



The problem of multiple proofs

In fact the problem of multiple proofs is a problem for all evidentialist criteria of depth.Another evidentialist strategy: a theorem is deep if its

every proof of italmost every proof of it canonical proofshortest proofsimplest proofleast deep proof

9>>>>=

>>>>;

is deep

These are parasitic on a prior criterion of depth of proof, though.

That demands separate analytic work.



The criterion of multiple proofs

We can try to transform the problem of multiple proofs into a virtue.

We can say that a theorem is deep if it has many different proofs.

But this demands an account of the individuation of proofs.

The problem is that two apparently different proofs could be in fact the same.

Therefore it is difficult to determine how many different proofs a theorem has.

The problem of individuation of proofs is itself deep.


The depth of Szemeredi’s theorem Consequentialist depth

Consequentialist criteria

A consequentialist criterion of depth, by contrast, measures the depth of a theorem by somequality of its consequences or the consequences of its proofs, such as fruitfulness.

Green and Tao (2004) used Szemeredi’s theorem to prove that there are arbitrarily longarithmetic progressions consisting only of prime numbers (the first item cited in Tao’s FieldsMedal announcement).

But consequentialist criteria are interest-dependent; a theorem is deep only so long as itsconsequences are of interest to us.

Consequentialist criteria can be objective only if other values that mathematicians hold, suchas “interestingness”, are objective.


The depth of Szemeredi’s theorem Cosmological depth

Cosmological criteria

The last type of criterion of depth I want to discuss is what I will call a cosmological criterion.

I’ve chosen this word because of the Greek word kosmos, which means “order”.

Godefroy, Les math´ematiques, mode d’emploi (2011)Szemeredi’s theorem. . . is a deep combinatorial result that establishes (like Ramsey’stheorems) the inescapable presence of certain structures, lumps of order in a formless dough[grumeaux d’ordre dans une pˆate informe], even though we have a great deal of freedom ofconstruction since the only constraint imposed on A is positive density.



Cosmological criteria

Szemeredi identifies this feature of his theorem as well.

Interview with Szemeredi after Abel Prize awardIn finite objects we look for patterns, different shapes, and try to understand what conditionsmake different patterns emerge, this is one of the most fundamental questions. A slightlypompous and philosophical way to put it is that we want to prove that there is order in anychaos. in other words, if you give me a hostile structure, I will still be able to find orderly partsin it.



Order in chaos

Szemeredi’s theorem tells us that for any arithmetic progression of length k and any desiredpositive density, there’s an N such that we’re guaranteed to find such an arithmeticprogression choosing only from a subset of {1, 2, . . . ,N} of our desired density (so its size canbe very small relative to N).

What Godefroy calls a “formless dough” and what Szemeredi calls “chaos” is the chosen densesubset of {1, 2, . . . ,N}.

We choose these subsets without any other constraints besides positive density.

Szemeredi’s theorem shows that these “randomly” chosen sets must contain arithmeticprogressions of the desired length.

This “unexpected” structure is what makes Szemeredi’s theorem deep, on what I’m calling acosmological view of depth.



A precisification of cosmological depth

In a more formal direction, one could think about cosmological depth as follows.

Let us suppose there is a predicate S measuring the orderliness of what is expressed by astatement.

Then a theorem is cosmologically deep if S(premises) << S(conclusion).



A problem for cosmological depth

The cosmological view of depth does not fully capture depth as used in practice.

For instance, Wiles’ result that all elliptic curves arise from modular forms is surely deep.

But the premise (that E is an elliptic curve) is at least as orderly as the conclusion (that Earises from a modular form).

Thus this result is not deep according to the cosmological criterion.

Note also that the theorem that every modular form has an elliptic curve attached to it�theEichler-Shimura theorem�is surely also deep.

We then have two deep theorems with the logical forms, respectively, of “every A is a B” and“every B is an A”.

But if the << relation is antisymmetric, as seems reasonable, it would follow that only one ofthe two theorems can be cosmologically deep.


Conclusions

Conclusions

I have put forward Szemeredi’s theorem as a case of a deep theorem.

I’ve articulated four different accounts of its depth, and indicated some pros and cons of each.

Arithmetic combinatorics is a rich source of cases for thinking about depth, because itstheorems tend to be simply stated and readily understandable, yet strong.

Furthermore it has been a hotbed of activity in the last couple of decades, with severalarticulate mathematicians at its center.


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