Mathematical depth and Szemerédi’s theorem Andrew Arana Philosophy, University of Illinois at Urbana-Champaign (until August 2015) Philosophy, University of Paris 1 Panthéon-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 articles employing 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, by using Szemerédi’s theorem as a case study. Arana (Illinois/Paris) Depth Helsinki LC 2015 2 / 26 Szemerédi’s theorem Szemerédi’s theorem A set A of positive integers has positive density if the proportion of elements of A among the integers from 1 to n approaches a positive number in the limit as n goes to infinity. Szemerédi’s theorem, infinite form (1975) Each subset of the natural numbers of positive density contains an arithmetic progression of arbitrary length. Szemerédi’s theorem, finite form (1975) Let k 3 be an integer and let 0 <  1. Then there is a positive integer N = N(k, ) such that any subset of {1, 2, . . . , N} of at least N elements contains an arithmetic progression of length k. Arana (Illinois/Paris) Depth Helsinki LC 2015 3 / 26 Szemerédi’s theorem Szemerédi’s theorem In 1936 Erdős and Turán had conjectured that every sufficiently “dense” subset of N contains an 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 this result among his significant achievements. In 1975 Szemerédi resolved the conjecture for arbitrary k. Erdős, “Some problems and results in number theory” (1984) I offered $1,000 for [my conjecture] and late in 1972 Szemerédi found a brilliant but very difficult proof of [my conjecture]. I feel that never was a 1,000 dollars more deserved. In fact several colleagues remarked that my offer violated the minimum wage act. Arana (Illinois/Paris) Depth Helsinki LC 2015 4 / 26 Szemerédi’s theorem What Szemerédi’s theorem asserts Gowers, Abel Prize announcement for Szemerédi, 2012 One way of understanding Szemerédi’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 to choose as many integers between 1 and 10,000 as you can, and the one rule that you must obey is that from the integers you choose it should not be possible to create a five-term arithmetic 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-term progression with step size 1002. Arana (Illinois/Paris) Depth Helsinki LC 2015 5 / 26 Szemerédi’s theorem What Szemerédi’s theorem asserts Gowers Obviously you are destined to lose this game eventually, since. . . if you keep going for long enough you will eventually have chosen all the numbers between 1 and 10,000, which will include many five-term arithmetic progressions. But Szemerédi’s theorem tells us something far more interesting: even if you play with the best possible progression-avoiding strategy, you will lose long before you get anywhere near choosing all the numbers. Arana (Illinois/Paris) Depth Helsinki LC 2015 6 / 26 Szemerédi’s theorem What Szemerédi’s theorem asserts 1 Szemerédi for k = 23, = 1000 : There exists a positive integer N such that every subset of N {1, 2, . . . , N} of at least 1000 elements contains an arithmetic progression of length 23. Gowers If, for instance, we are trying to avoid progressions of length 23, Szemerédi’s theorem tells us that there is some N (which may be huge, but the point is that it exists) such that if we play N the game with N numbers, then we cannot choose more than 1000 of those numbers–that is, a mere 0.1% of them–before we lose. And the same is true for any other progression length and any other positive percentage. Arana (Illinois/Paris) Depth Helsinki LC 2015 7 / 26 Szemerédi’s theorem Proofs of Szemerédi’s theorem There are three main proofs of Szemerédi’s theorem. Szemerédi’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 been to find better bounds on N. Arana (Illinois/Paris) Depth Helsinki LC 2015 8 / 26 The depth of Szemerédi’s theorem Szemerédi’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 one expects to have very little additive structure. Nevertheless, it is a deep and surprising fact that as long as an additive set is dense enough in its ambient group, it will always have some level of additive structure. The most famous example of this principle is Szemerédi’s theorem. . . [a] beautiful and important theorem. Announcement of the Rolf Schock prize for Szemerédi, 2008 [The prize is awarded] for his deep and pioneering work from 1975 on arithmetic progressions in subsets of the integers, which has led to great progress and discoveries in several branches of mathematics. Arana (Illinois/Paris) Depth Helsinki LC 2015 9 / 26 The depth of Szemerédi’s theorem 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. Arana (Illinois/Paris) Depth Helsinki LC 2015 10 / 26 The depth of Szemerédi’s theorem 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 of mathematics 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 on the larger questions of whether depth is objective, what objectivity of the relevant sort would consist 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 merely human limitations. Arana (Illinois/Paris) Depth Helsinki LC 2015 11 / 26 The depth of Szemerédi’s theorem Evidentialist depth Evidentialist criteria I call a criterion of depth evidentialist if it links the depth of a theorem with some quality of its 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 one whose proof requires a great deal of work�it may be long, or complicated, or difficult, or it may 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. Arana (Illinois/Paris) Depth Helsinki LC 2015 12 / 26 The depth of Szemerédi’s theorem Evidentialist depth Laboriousness of Szemerédi’s proof Arana (Illinois/Paris) Depth Helsinki LC 2015 13 / 26 The depth of Szemerédi’s theorem Evidentialist depth 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 Szemerédi’s proof! Which of these characteristics is best suited for an evidentialist criterion of depth? Arana (Illinois/Paris) Depth Helsinki LC 2015 14 / 26 The depth of Szemerédi’s theorem Evidentialist depth The problem of multiple proofs This question is compounded by another problem. There are multiple proofs of Szemerédi’s theorem, by Szemerédi, Furstenberg, Gowers, and by others. Which proof should determine the depth of Szemerédi’s theorem on an evidentialist criterion of proof? Arana (Illinois/Paris) Depth Helsinki LC 2015 15 / 26 The depth of Szemerédi’s theorem Evidentialist depth 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 as depth is concerned? Arana (Illinois/Paris) Depth Helsinki LC 2015 16 / 26 The depth of Szemerédi’s theorem Evidentialist depth 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? Arana (Illinois/Paris) Depth Helsinki LC 2015 17 / 26 The depth of Szemerédi’s theorem Evidentialist depth 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 9 every proof of it > > > almost every proof of it canonical proof > = shortest proof is deep > > simplest proof > > ; least deep proof These are parasitic on a prior criterion of depth of proof, though. That demands separate analytic work. Arana (Illinois/Paris) Depth Helsinki LC 2015 18 / 26 The depth of Szemerédi’s theorem Evidentialist depth 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. Arana (Illinois/Paris) Depth Helsinki LC 2015 19 / 26 The depth of Szemerédi’s theorem Consequentialist depth Consequentialist criteria A consequentialist criterion of depth, by contrast, measures the depth of a theorem by some quality of its consequences or the consequences of its proofs, such as fruitfulness. Green and Tao (2004) used Szemerédi’s theorem to prove that there are arbitrarily long arithmetic progressions consisting only of prime numbers (the first item cited in Tao’s Fields Medal announcement). But consequentialist criteria are interest-dependent; a theorem is deep only so long as its consequences are of interest to us. Consequentialist criteria can be objective only if other values that mathematicians hold, such as “interestingness”, are objective. Arana (Illinois/Paris) Depth Helsinki LC 2015 20 / 26 The depth of Szemerédi’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 mathématiques, mode d’emploi (2011) Szemerédi’s theorem. . . is a deep combinatorial result that establishes (like Ramsey’s theorems) the inescapable presence of certain structures, lumps of order in a formless dough [grumeaux d’ordre dans une pâte informe], even though we have a great deal of freedom of construction since the only constraint imposed on A is positive density. Arana (Illinois/Paris) Depth Helsinki LC 2015 21 / 26 The depth of Szemerédi’s theorem Cosmological depth Cosmological criteria Szemerédi identifies this feature of his theorem as well. Interview with Szemerédi after Abel Prize award In finite objects we look for patterns, different shapes, and try to understand what conditions make different patterns emerge, this is one of the most fundamental questions. A slightly pompous and philosophical way to put it is that we want to prove that there is order in any chaos. in other words, if you give me a hostile structure, I will still be able to find orderly parts in it. Arana (Illinois/Paris) Depth Helsinki LC 2015 22 / 26 The depth of Szemerédi’s theorem Cosmological depth Order in chaos Szemerédi’s theorem tells us that for any arithmetic progression of length k and any desired positive density, there’s an N such that we’re guaranteed to find such an arithmetic progression choosing only from a subset of {1, 2, . . . , N} of our desired density (so its size can be very small relative to N). What Godefroy calls a “formless dough” and what Szemerédi calls “chaos” is the chosen dense subset of {1, 2, . . . , N}. We choose these subsets without any other constraints besides positive density. Szemerédi’s theorem shows that these “randomly” chosen sets must contain arithmetic progressions of the desired length. This “unexpected” structure is what makes Szemerédi’s theorem deep, on what I’m calling a cosmological view of depth. Arana (Illinois/Paris) Depth Helsinki LC 2015 23 / 26 The depth of Szemerédi’s theorem Cosmological 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 a statement. Then a theorem is cosmologically deep if S(premises) << S(conclusion). Arana (Illinois/Paris) Depth Helsinki LC 2015 24 / 26 The depth of Szemerédi’s theorem Cosmological depth 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 E arises 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�the Eichler-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 of the two theorems can be cosmologically deep. Arana (Illinois/Paris) Depth Helsinki LC 2015 25 / 26 Conclusions Conclusions I have put forward Szemerédi’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 its theorems 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 several articulate mathematicians at its center. Arana (Illinois/Paris) Depth Helsinki LC 2015 26 / 26