Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-18T00:01:31.738Z Has data issue: false hasContentIssue false

SZEMERÉDI’S THEOREM: AN EXPLORATION OF IMPURITY, EXPLANATION, AND CONTENT

Published online by Cambridge University Press:  03 December 2021

PATRICK J. RYAN*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA, USA

Abstract

In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I call structural content (Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5).

Type
Research Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Ackermann, W. (1937). Die Widerspruchsfreiheit der allgemeinen Mengenlehre. Mathematische Annalen, 114, 305315.Google Scholar
Arana, A. (2015). On the depth of Szemerédi’s theorem. Philosophia Mathematica, 23(2), 163176.Google Scholar
Arana, A. (2017). On the alleged simplicity of pure proof. In Kossak, R., and Ording, P., editors. Simplicity: Ideals of Practice in Mathematics and the Arts. Berlin: Springer, pp. 207229.Google Scholar
Arana, A. (2019). Elementarity and purity. In Alvarez, C., and Arana, A., editors. Analytic Philosophy and the Foundations of Mathematics. London: Palgrave-Macmillan.Google Scholar
Arana, A., & Mancosu, P. (2012). On the relationship between plane and solid geometry. The Review of Symbolic Logic, 5(2), 294353.Google Scholar
Avigad, J. (2003). Number theory and elementary arithmetic. Philosophia Mathematica, 11(3), 257284.Google Scholar
Avigad, J. (2009). The metamathematics of ergodic theory. Annals of Pure and Applied Logic, 157, 6476.Google Scholar
Avigad, J., & Towsner, H. (2010). Metastability in the Furstenberg-Zimmer tower. Fundamenta Mathematicae, 210, 243268.Google Scholar
Barnes, J. (1993). Aristotle: Posterior Analytics, second edition. Clarendon Aristotle Series. Oxford: Oxford University Press.Google Scholar
Betti, A. (2010). Explanation in metaphysics and Bolzano’s theory of ground and consequence. Logique et Analyse, 56(211), 281316.Google Scholar
Bourbaki, N. (1950). The architecture of mathematics. The American Mathematical Monthly, 57(4), 221232.Google Scholar
Burnyeat, M. (2012). Aristotle on understanding knowledge. In Explorations in Ancient and Modern Philosophy. Cambridge: Cambridge University Press, pp. 115144.Google Scholar
Dawson, J. W. (2006). Why do mathematicians re-prove theorems? Philosophia Mathematica, 3(14), 269286.Google Scholar
de la Vallée Poussin, C. (1896). Recherches analytiques Sur la théorie des nombres premiers. Annales de la Societe Scientifique de Bruxelles, 20, 183256.Google Scholar
Detlefsen, M. (2008). Purity as an ideal of proof. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 179198.Google Scholar
Detlefsen, M., & Arana, A. (2011). Purity of methods. Philosophers’ Imprint, 11(2), 120.Google Scholar
Einsiedler, M., & Ward, T. (2011). Ergodic Theory with a View towards Number Theory. Graduate Texts in Mathematics, Vol. 259. Berlin: Springer-Verlag.Google Scholar
Feferman, S. (1964). Systems of Predicative Analysis, I. The Journal of Symbolic Logic, 29, 130.Google Scholar
Field, H. (1980). Science without Numbers. Princeton: Princeton University Press.Google Scholar
Field, H. (1989). Realism, Mathematics and Modality. Oxford: Basil Blackwell.Google Scholar
Friedman, M. (1974). Explanation and scientific understanding. Journal of Philosophy, 71(1), 119.Google Scholar
Furstenberg, H. (1977). Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. Journal d’ Analyse Mathématique, 34, 204256.Google Scholar
Furstenberg, H. (1981). Poincaré recurrence and number theory. Bulletin of the American Mathematical Society, 5(3), 211234.Google Scholar
Furstenberg, H., Katznelson, Y., & Ornstein, D. (1982). The ergodic theoretical proof of Szemerédi’s theorem. Bulletin of the American Mathematical Society, 7(3).Google Scholar
Furstenberg, H., & Weiss, B. (1978). Topological dynamics and combinatorial number theory. Journal d’ Analyse Mathématique, 34, 6185.Google Scholar
Gowers, T. (2001). A new proof of Szemerédi’s theorem. Geometric and Functional Analysis, 11, 465588.Google Scholar
Gowers, T. (2006). Quasirandomness, counting and regularity for 3-uniform hypergraphs. Combinatorics, Probability and Computing, 15(1-2), 143184.Google Scholar
Green, B., & Tao, T. (2007). Szemerédi’s theorem. Scholarpedia, 2(7), 3446.Google Scholar
Green, B., & Tao, T. (2008). The primes contain arbitrarily long arithmetic progressions. Annals of Mathematics, 167, 481547.Google Scholar
Hadamard, J. (1896). Sur la distribution des zéros de la fonction $\zeta (s)$ et ses conséquences arithmétiques. Bulletin de la Société Mathématique de France, 24,199220.Google Scholar
Hafner, J., & Mancosu, P. (2008). Beyond unification. In Mancosu, P., editor. The Philosophy of Mathematical Practice. New York: Oxford University Press, pp. 151178.Google Scholar
Hallett, M. (2008). Reflections on the purity of method in Hilbert’s Grundlagen der Geometrie. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 198255.Google Scholar
Ingham, A.E. (2008). Review 10, 595c (mr0029411). Bulletin of the American Mathematical Society, 45(4), 651654.Google Scholar
Isaacson, D. (1996). Arithmetical truth and hidden higher-order concepts. In Hart, W., editor. The Philosophy of Mathematics. New York: Oxford University Press, pp. 203224.Google Scholar
Kaye, R., & Wong, T. L. (2007). On interpretations of arithmetic and set theory. Notre Dame Journal of Formal Logic, 48(4), 497510.Google Scholar
Kitcher, P. (1975). Bolzano’s ideal of algebraic analysis. Studies in History and Philosophy of Science, 6, 229269.Google Scholar
Kitcher, P. (1976). Explanation, conjunction, and unification. Journal of Philosophy, 73(8), 207212.Google Scholar
Kitcher, P. (1981). Explanatory unification. Philosophy of Science, 48, 507531.Google Scholar
Kitcher, P. (1984). The Nature of Mathematical Knowledge. Oxford: Oxford University Press.Google Scholar
Kitcher, P. (1989). Explanatory unification and the causal structure of the world (sections 1-4.5). In Kitcher, P., and Salmon, W., editors, Scientific Explanation. Minnesota Studies in the Philosophy of Science, Vol. XIII. Minneapolis: University of Minnesota Press, pp. 410437.Google Scholar
Lange, M. (2017). Because Without Cause. Oxford: Oxford University Press.Google Scholar
Mancosu, P. (1999). Bolzano and Cournot on mathematical explanation. Revue d’histoire des sciences, 52(3/4), 429455.Google Scholar
Mancosu, P. (2001). Mathematical explanation: Problems and prospects. Topoi, 20, 97117.Google Scholar
Mancosu, P. (2008a). Mathematical explanation: Why it matters. In Mancosu, P., editor. The Philosophy of Mathematical Practice. New York: Oxford University Press, pp. 134150.Google Scholar
Mancosu, P. editor (2008b). The Philosophy of Mathematical Practice. Oxford: Oxford University Press.Google Scholar
Mancosu, P. (2018). Explanation in mathematics. Stanford Internet Encyclopedia of Philosophy. Available from: https://plato.stanford.edu/ entries/mathematics-explanation.Google Scholar
McLarty, C. (2010). What does it take to prove Fermat’s last theorem? Grothendieck and the logic of number theory. The Bulletin of Symbolic Logic, 16(3), 359377.Google Scholar
Morrison, M. (2000). Unifying Scientific Theories. Cambridge: Cambridge University Press.Google Scholar
Nagle, B., Rödl, V., & Schacht, M. (2006). The counting lemma for regular $k$ -uniform hypergraphs. Random Structures and Algorithms, 28(22), 113179.Google Scholar
Ostwald, W. (1905). Klassiker der Exacten Wissenschaften, Vol. 153. Leipzig: Wilhelm, Engelmann.Google Scholar
Rödl, V., & Schacht, M. (2007). Regular partitions on hypergraphs: Counting lemmas. Combinatorics, Probability and Computing, 16(6), 887901.Google Scholar
Rödl, V., & Skokan, J. (2004). Regularity lemma for uniform hypergraphs. Random Structures and Algorithms, 25(1), 142.Google Scholar
Rödl, V., & Skokan, J. (2006). Applications of the regularity lemma for uniform hypergraphs. Random Structures and Algorithms, 28(2), 180194.Google Scholar
Ross, W., & Minio-Paluello, L. (1964). Aristotelis Analytica Priora et Posteriora. Oxford: Oxford University Press.Google Scholar
Rota, G.-C. (1997). Indiscrete Thoughts. Boston: Birkhäuser.Google Scholar
Russ, S. (1980). A translation of Bolzano’s paper on the intermediate value theorem. Historia Mathematica, 7, 156185.Google Scholar
Simpson, S. (1999). Subsystems of Second Order Mathematics. Berlin: Springer.Google Scholar
Steinkrueger, P. (2018). Aristotle on kind-crossing. Oxford Studies in Ancient Philosophy, 54, 107158.Google Scholar
Szemerédi, E. (1975). On sets of integers containing no k elements in arithmetic progression. Acta Arithmetica, XXVII, 199245.Google Scholar
Tao, T. (2006a). The dichotomy between structure and randomness, arithmetic progressions, and the primes. Proceedings of the International Congress of Mathematicians, I, 581608.Google Scholar
Tao, T. (2007). The ergodic and combinatorial approaches to Szemerédi’s theorem. In Granville, A., Nathanson, M., and Solymosi, J., editors. Additive Combinatorics. Providence, RI: American Mathematical Society, pp. 145193.Google Scholar
Tao, T. (2008). Structure and Randomness: Pages from Year One of a Mathematical Blog. Providence, RI: American Mathematical Society.Google Scholar
Towsner, H. (2008). Some Results in Logic and Ergodic Theory. Ph.D. Thesis, Carnegie Mellon University.Google Scholar
van der Waerden, B. (1928). Beweis einer Baudetschen Vermutung. Nieuw Archief voor Wiskunde, 15, 212216.Google Scholar
van der Waerden, B. (1998). Wie der Beweis der Vermutung von Baudet gefunden wurde. Elemente der Mathematik, 53, 139148.Google Scholar
Wittgenstein, L. (1967). Remarks on the Foundations of Mathematics. Cambridge, MA: M.I.T. Press.Google Scholar
Zhao, Y. Szemerédi’s Theorem via Ergodic Theory. Completed by the author in partial fulfillment of the requirements of the Part III Tripos in Mathematics at Cambridge University.Google Scholar