{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,11]],"date-time":"2025-06-11T18:22:31Z","timestamp":1749666151922,"version":"3.40.5"},"reference-count":67,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2021,12,3]],"date-time":"2021-12-03T00:00:00Z","timestamp":1638489600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>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<jats:italic>purity<\/jats:italic>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, Szemer\u00e9di\u2019s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (ergodic) proof due to Furstenberg (Section 3). Furstenberg\u2019s 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\u2019s proof is explanatory in light of its clear expression of a crucial structural result, which provides the \u201creason why\u201d Szemer\u00e9di\u2019s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources \u201cget a grip on\u201d 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<jats:italic>structural content<\/jats:italic>(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).<\/jats:p>","DOI":"10.1017\/s1755020321000538","type":"journal-article","created":{"date-parts":[[2021,12,3]],"date-time":"2021-12-03T07:37:14Z","timestamp":1638517034000},"page":"700-739","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":4,"title":["SZEMER\u00c9DI\u2019S THEOREM: AN EXPLORATION OF IMPURITY, EXPLANATION, AND CONTENT"],"prefix":"10.1017","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2384-7207","authenticated-orcid":false,"given":"PATRICK J.","family":"RYAN","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,12,3]]},"reference":[{"doi-asserted-by":"publisher","key":"S1755020321000538_r28","DOI":"10.1090\/S0273-0979-1982-15052-2"},{"key":"S1755020321000538_r13","first-page":"281","article-title":"Explanation in metaphysics and Bolzano\u2019s theory of ground and consequence","volume":"56","author":"Betti","year":"2010","journal-title":"Logique et Analyse"},{"key":"S1755020321000538_r3","first-page":"207","volume-title":"Simplicity: Ideals of Practice in Mathematics and the Arts","author":"Arana","year":"2017"},{"key":"S1755020321000538_r17","first-page":"183","article-title":"Recherches analytiques Sur la th\u00e9orie des nombres premiers","volume":"20","author":"de\u00a0la Vall\u00e9e\u00a0Poussin","year":"1896","journal-title":"Annales de la Societe Scientifique de Bruxelles"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r67","DOI":"10.1090\/mbk\/059"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r1","DOI":"10.1007\/BF01594179"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r35","DOI":"10.1093\/acprof:oso\/9780199296453.003.0007"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r59","DOI":"10.1093\/actrade\/9780198145622.book.1"},{"unstructured":"[63] Towsner, H. (2008). Some Results in Logic and Ergodic Theory. Ph.D. Thesis, Carnegie Mellon University.","key":"S1755020321000538_r68"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r48","DOI":"10.1023\/A:1010621314372"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r16","DOI":"10.1093\/philmat\/nkl009"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r56","DOI":"10.1017\/S0963548307008565"},{"key":"S1755020321000538_r69","first-page":"212","article-title":"Beweis einer Baudetschen Vermutung","volume":"15","author":"van\u00a0der Waerden","year":"1928","journal-title":"Nieuw Archief voor Wiskunde"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r14","DOI":"10.1080\/00029890.1950.11999523"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r47","DOI":"10.3406\/rhs.1999.1364"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r58","DOI":"10.1002\/rsa.20108"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r5","DOI":"10.1017\/S1755020312000020"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r31","DOI":"10.1017\/S0963548305007236"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r50","DOI":"10.1093\/acprof:oso\/9780199296453.001.0001"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r15","DOI":"10.1017\/CBO9780511974069.008"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r41","DOI":"10.2307\/2025559"},{"key":"S1755020321000538_r20","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-85729-021-2","volume-title":"Ergodic Theory with a View towards Number Theory","author":"Einsiedler","year":"2011"},{"volume-title":"Aristotle: Posterior Analytics","year":"1993","author":"Barnes","key":"S1755020321000538_r12"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r54","DOI":"10.1002\/rsa.20117"},{"key":"S1755020321000538_r44","first-page":"410","volume-title":"Scientific Explanation","author":"Kitcher","year":"1989"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r6","DOI":"10.1093\/philmat\/11.3.257"},{"key":"S1755020321000538_r38","first-page":"203","volume-title":"The Philosophy of Mathematics","author":"Isaacson","year":"1996"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r66","DOI":"10.1090\/crmp\/043\/08"},{"volume-title":"Remarks on the Foundations of Mathematics","year":"1967","author":"Wittgenstein","key":"S1755020321000538_r71"},{"volume-title":"Because Without Cause","year":"2017","author":"Lange","key":"S1755020321000538_r45"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r2","DOI":"10.1093\/philmat\/nku036"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r10","DOI":"10.4064\/fm210-3-2"},{"volume-title":"Analytic Philosophy and the Foundations of Mathematics","year":"2019","author":"Arana","key":"S1755020321000538_r4"},{"volume-title":"Science without Numbers","year":"1980","author":"Field","key":"S1755020321000538_r22"},{"key":"S1755020321000538_r63","first-page":"107","article-title":"Aristotle on kind-crossing","volume":"54","author":"Steinkrueger","year":"2018","journal-title":"Oxford Studies in Ancient Philosophy"},{"unstructured":"[67] Zhao, Y. Szemer\u00e9di\u2019s Theorem via Ergodic Theory. Completed by the author in partial fulfillment of the requirements of the Part III Tripos in Mathematics at Cambridge University.","key":"S1755020321000538_r72"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r26","DOI":"10.1090\/S0273-0979-1981-14932-6"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r70","DOI":"10.1007\/s000170050045"},{"volume-title":"Klassiker der Exacten Wissenschaften","year":"1905","author":"Ostwald","key":"S1755020321000538_r55"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r52","DOI":"10.2178\/bsl\/1286284558"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r25","DOI":"10.1007\/BF02813304"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r42","DOI":"10.1086\/289019"},{"volume-title":"The Nature of Mathematical Knowledge","year":"1984","author":"Kitcher","key":"S1755020321000538_r43"},{"key":"S1755020321000538_r65","first-page":"581","article-title":"The dichotomy between structure and randomness, arithmetic progressions, and the primes","volume":"I","author":"Tao","year":"2006a","journal-title":"Proceedings of the International Congress of Mathematicians"},{"volume-title":"Realism, Mathematics and Modality","year":"1989","author":"Field","key":"S1755020321000538_r23"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r30","DOI":"10.1007\/s00039-001-0332-9"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r64","DOI":"10.4064\/aa-27-1-199-245"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r29","DOI":"10.1007\/BF02790008"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r39","DOI":"10.1305\/ndjfl\/1193667707"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r57","DOI":"10.1002\/rsa.20017"},{"unstructured":"[46] Mancosu, P. (2018). Explanation in mathematics. Stanford Internet Encyclopedia of Philosophy. Available from: https:\/\/plato.stanford.edu\/ entries\/mathematics-explanation.","key":"S1755020321000538_r51"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r18","DOI":"10.1093\/acprof:oso\/9780199296453.003.0008"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r53","DOI":"10.1017\/CBO9780511527333"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r60","DOI":"10.1007\/978-0-8176-4781-0"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r33","DOI":"10.4007\/annals.2008.167.481"},{"key":"S1755020321000538_r19","first-page":"1","article-title":"Purity of methods","volume":"11","author":"Detlefsen","year":"2011","journal-title":"Philosophers\u2019 Imprint"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r7","DOI":"10.1016\/j.apal.2008.09.001"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r61","DOI":"10.1016\/0315-0860(80)90036-1"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r40","DOI":"10.1016\/0039-3681(75)90024-2"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r49","DOI":"10.1093\/acprof:oso\/9780199296453.003.0006"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r24","DOI":"10.2307\/2024924"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r34","DOI":"10.24033\/bsmf.545"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r36","DOI":"10.1093\/acprof:oso\/9780199296453.003.0009"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r32","DOI":"10.4249\/scholarpedia.3446"},{"key":"S1755020321000538_r37","first-page":"651","article-title":"Review 10, 595c (mr0029411)","volume":"45","author":"Ingham","year":"2008","journal-title":"Bulletin of the American Mathematical Society"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r62","DOI":"10.1007\/978-3-642-59971-2"},{"doi-asserted-by":"publisher","key":"S1755020321000538_r21","DOI":"10.2307\/2269764"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020321000538","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T15:14:34Z","timestamp":1699888474000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020321000538\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,3]]},"references-count":67,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["S1755020321000538"],"URL":"https:\/\/doi.org\/10.1017\/s1755020321000538","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"type":"print","value":"1755-0203"},{"type":"electronic","value":"1755-0211"}],"subject":[],"published":{"date-parts":[[2021,12,3]]},"assertion":[{"value":"\u00a9 The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}