Element contents
Elements of Purity
Published online by Cambridge University Press: 03 December 2024
Summary
A proof of a theorem can be said to be pure if it draws only on what is 'close' or 'intrinsic' to that theorem. In this Element we will investigate the apparent preference for pure proofs that has persisted in mathematics since antiquity, alongside a competing preference for impurity. In Section 1, we present two examples of purity, from geometry and number theory. In Section 2, we give a brief history of purity in mathematics. In Section 3, we discuss several different types of purity, based on different measures of distance between theorem and proof. In Section 4 we discuss reasons for preferring pure proofs, for the varieties of purity constraints presented in Section 3. In Section 5 we conclude by reflecting briefly on purity as a preference for the local and how issues of translation intersect with the considerations we have raised throughout this work.
Keywords
Information
- Type
- Element
- Information
- Online ISBN: 9781009052719Publisher: Cambridge University PressPrint publication: 16 January 2025
References
Aigner, M., & Ziegler, G. M. (2010). Proofs from THE BOOK (Fourth ed.). Berlin: Springer.CrossRefGoogle Scholar
Allen, J. (2001). Inference from signs: Ancient debates about the nature of evidence. Oxford: Oxford University Press.CrossRefGoogle Scholar
Apter, E. (2013). Against world literature: On the politics of untranslatability. London: Verso.Google Scholar
Arana, A. (2007). Review of Toward a philosophy of real mathematics, by David Corfield. Mathematical Intelligencer, 29(2), 80–83.CrossRefGoogle Scholar
Arana, A. (2008). Logical and semantic purity. Protosociology, 25, 36–48. (Reprinted in Philosophy of mathematics: Set theory, measuring theories, and nominalism, Preyer, G. and Peter, G. (Eds.), Offenbach: Ontos, 2008.)Google Scholar
Arana, A. (2009). On formally measuring and eliminating extraneous notions in proofs. Philosophia Mathematica, 17, 208–219.CrossRefGoogle Scholar
Arana, A. (2014). Purity in arithmetic: Some formal and informal issues. In Link, G. (Ed.), Formalism and beyond: On the nature of mathematical discourse (pp. 315–335). Boston: De Gruyter.CrossRefGoogle Scholar
Arana, A. (2015). On the depth of Szemerédi’s Theorem. Philosophia Mathematica, 23(2), 163–176.CrossRefGoogle Scholar
Arana, A. (2017). On the alleged simplicity of impure proof. In Kossak, R. & Ording, P. (Eds.), Simplicity: Ideals of practice in mathematics and the arts (pp. 207–226). Cham: Springer.Google Scholar
Arana, A. (2023). Purity and explanation: Essentially linked? In Posy, C. & Ben-Menahem, Y. (Eds.), Mathematical knowledge, objects and applications: Essays in memory of Mark Steiner (pp. 25–39). Cham: Springer.CrossRefGoogle Scholar
Arana, A., & Burnett, H. (2023). Mathematical hygiene. Synthese, 202(110), 1–28.CrossRefGoogle Scholar
Arana, A., & Mancosu, P. (2012, June). On the relationship between plane and solid geometry. Review of Symbolic Logic, 5(2), 294–353.CrossRefGoogle Scholar
Arana, A., & Stafford, W. (2023). On the difficulty of discovering mathematical proofs. Synthese, 202(38), 1–29.CrossRefGoogle Scholar
Avigad, J. (2003). Number theory and elementary arithmetic. Philosophia Mathematica, 11, 257–284.CrossRefGoogle Scholar
Babbitt, D., & Goodstein, J. (2011, February). Federigo Enriques’s quest to prove the “Completeness Theorem.” Notices of the American Mathematical Society, 58(2), 240–249.Google Scholar
Bachelard, G. (1966). Le rationalisme appliqué (Third ed.). Paris: Presses Universitaires de France.Google Scholar
Bachelard, S. (1967). La représentation géométrique des quantités imaginaires au début du xixe siècle. Paris: Conférences du Palais de la Découverte.Google Scholar
Baldwin, J. (2018). Model theory and the philosophy of mathematical practice. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Bolzano, B. (1999). Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation. In Ewald, W. (Ed.), From Kant to Hilbert (Vol. 1, pp. 227–248). Oxford: Oxford University Press. (Originally published in 1817; translated by Russ, S.. )Google Scholar
Briançon, J., & Skoda, H. (1974). Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de cn. Comptes Rendus de l’Académie des Sciences, Série A, 278, 949–951.Google Scholar
Brigaglia, A., & Cilberto, C. (1995). Italian algebraic geometry between the two world wars. Kingston: Queen’s University. (Translated from Italian by Jeanne Duflot.)Google Scholar
Brouwer, L. E. J. (1913). Intuitionism and formalism. Bulletin of the American Mathematical Society, 20(2), 81–96.CrossRefGoogle Scholar
Buss, S. R. (1991). The undecidability of k-provability. Annals of Pure and Applied Logic, 53(1), 75–102.CrossRefGoogle Scholar
Carbone, A. (2009). Logical structures and genus of proofs. Annals of Pure and Applied Logic, 161(2), 139–149.CrossRefGoogle Scholar
Carnap, R. (1937). Logical syntax of language. London: Kegan Paul, Trench, Trubner and Co., Ltd. (Translated by Amethe Smeaton (Countess von Zeppelin).).Google Scholar
Cassin, B. (2004). Vocabulaire européen des philosophies: dictionnaire des intraduisibles. Paris: Seuil.Google Scholar
Cassin, B. (2014). Dictionary of untranslatables. Princeton: Princeton University Press.CrossRefGoogle Scholar
Cegielski, P. (1984). La theorie élémentaire de la divisibilité est finiment axiomatisable. Comptes Rendus de l’Académie des Sciences – Series I – Mathematics, 299(9), 367–369.Google Scholar
Cegielski, P., Matijasevich, Y., & Richard, D. (1996, June). Definability and decidability issues in extensions of the integers with the divisibility predicate. Journal of Symbolic Logic, 61(2), 515–540.CrossRefGoogle Scholar
Chang, H. (2012). Is water H2O? Evidence, realism and pluralism. Dordrecht: Springer.CrossRefGoogle Scholar
Chemla, K. (1998). Lazare Carnot et la généralité en géométrie. Variations sur le théorème dit de Menelaus. Revue d’histoire des máthematiques, 4, 163–190.Google Scholar
Chemla, K., & Keller, E. F. (Eds.). (2017). Cultures without culturalism: The making of scientific knowledge. Durham: Duke University Press.Google Scholar
Collison, M. J. (1977). The origins of the cubic and biquadratic reciprocity laws. Archive for History of Exact Sciences, 17(1), 63–69.CrossRefGoogle Scholar
Correia, F., & Schnieder, B. (2012). Metaphysical grounding. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Coxeter, H. S. M. (1948, January). A problem of collinear points. American Mathematical Monthly, 55(1), 26–28.CrossRefGoogle Scholar
Crombie, A. (1994). Styles of scientific thinking in the European tradition. London: Duckworth.Google Scholar
D’Aquino, P. (1992). Local behaviour of the Chebyshev theorem in models of IΔ0. Journal of Symbolic Logic, 57(1), 12–27.Google Scholar
Davenport, H. (2008). The higher arithmetic (Eighth ed.). Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Dawson, J. W. (2006). Why do mathematicians re-prove theorems? Philosophia Mathematica, 14(3), 269–286.CrossRefGoogle Scholar
Dawson, J. W. (2015). Why prove it again? Alternative proofs in mathematical practice. Heidelberg: Birkhäuser.CrossRefGoogle Scholar
Dean, W. (2016). Computational complexity theory. In Zalta, E. N. (Ed.), The Stanford encyclopedia of philosophy (Winter 2016 ed.). Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/win2016/entries/computational-complexity/.Google Scholar
Dean, W., & Walsh, S. (2017). The prehistory of the subsystems of second-order arithmetic. Review of Symbolic Logic, 10(2), 357–396.CrossRefGoogle Scholar
Dedekind, R. (1872). Continuity and irrational numbers. In Ewald, W. (Ed.), From Kant to Hilbert (Vol. 2, pp. 766–779). Oxford: Oxford University Press. (Translated by W. Ewald.)Google Scholar
Dedekind, R. (1932). Gesammelte mathematische Werke (Vol. III; Fricke, R., Noether, E., & Ore, Ö, Eds.). Braunschweig: Vieweg.Google Scholar
De Morgan, A. (1849). Trigonometry and double algebra. London: Taylor, Walton, and Maberly.Google Scholar
Descartes, R. (1898). Oeuvres (Vol. II; Adam, C. & Tannery, P., Eds.). Paris: Léopold Cerf.Google Scholar
Descartes, R. (1902). Oeuvres (Vol. VI; Adam, C. & Tannery, P., Eds.). Paris: Léopold Cerf.Google Scholar
Detlefsen, M. (1988). Fregean hierarchies and mathematical explanation. International Studies in the Philosophy of Science 3(1), 97–116.CrossRefGoogle Scholar
Detlefsen, M. (1990b, November). On an alleged refutation of Hilbert’s program using Gödel’s first incompleteness theorem. Journal of Philosophical Logic, 19(4), 343–377.CrossRefGoogle Scholar
Detlefsen, M. (1996). Philosophy of mathematics in the twentieth century. In Philosophy of Science, Logic, and Mathematics (Vol. 9, pp. 50–123). London and New York: Routledge. (Edited by Shanker, Stuart G. )Google Scholar
Detlefsen, M. (2008). Purity as an ideal of proof. In Mancosu, P. (Ed.), The Philosophy of Mathematical Practice (pp. 179–197). Oxford: Oxford University Press.CrossRefGoogle Scholar
Detlefsen, M. (2010). Rigor, Re-proof and Bolzano’s Critical Program. In Bour, P. E., Rebuschi, M., & Rollet, L. (Eds.), Construction. Festschrift for Gerhard Heinzmann (pp. 171–184). London: King’s College Publications.Google Scholar
Detlefsen, M., & Arana, A. (2011, January). Purity of methods. Philosophers’ Imprint, 11(2), 1–20.Google Scholar
Diamond, H. G. (1982). Elementary methods in the study of the distribution of prime numbers. Bulletin of the American Mathematical Society, 7(3), 553–589.CrossRefGoogle Scholar
Dieudonné, J. (1969). Linear algebra and geometry. Boston: Houghton Mifflin Co. (Translation of J. Dieudonné, Algèbre linéaire et géométrie élémentaire, Hermann, Paris, third edition, 1964.)Google Scholar
, Diophantus. (1621). Diophanti alexandrini arithmeticorum libri sex, et de numeris multangulis liber uns. Paris: Sebastiani Cramoisy. (Latin edition translated from the Greek with commentaries by Claude Gaspar Bachet Sieur de Méziriac)Google Scholar
Dirichlet, G. L. (1856). Sur l’équation t2 + u2 + v2 + w2 = 4m [Extrait d’une Lettre de M. Lejeune-Dirichlet à M. Liouville]. Journal de mathématiques pures et appliquées, 1, 210–214.Google Scholar
Eastaugh, B. (2019). Set existence principles and closure conditions: Unravelling the standard view of reverse mathematics. Philosophia Mathematica, 27(2), 153–176.CrossRefGoogle Scholar
Eilenberg, S., & Steenrod, N. E. (1945, April). Axiomatic approach to homology theory. Proceedings of the National Academy of Sciences of the United States of America, 31(4), 117–120.CrossRefGoogle ScholarPubMed
Eilenberg, S., & Steenrod, N. (1952). Foundations of algebraic topology. Princeton: Princeton University Press.CrossRefGoogle Scholar
Eisenstein, G. (1847). Neue Theoreme der höheren Arithmetik. Journal für die reine und angewandte Mathematik, 35, 117–136.Google Scholar
Erdős, P. (1949). On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proceedings of the National Academy of Sciences, USA, 35, 374–384.CrossRefGoogle Scholar
Euler, L. (1758). De numeris, qui sunt aggregata duorum quadratorum. Novi Commentarii academiae scientiarum Petropolitanae, 4, 3–40.Google Scholar
Ewald, W. (Ed.). (1999). From Kant to Hilbert (Vol. II). Oxford: Oxford University Press.Google Scholar
Farrell, J., Farrell, K., & Rodgers, T. (2016). Martin Gardner and Marilyn vos Savant: A not always easy collaboration. Word Ways, Vol. 49(4).Google Scholar
Feferman, S. (1998). In the light of logic. New York: Oxford University Press.CrossRefGoogle Scholar
Feferman, S. (2005). Predicativity. In Shapiro, S. (Ed.), Handbook of the philosophy of mathematics and logic. Oxford: Oxford University Press.Google Scholar
Fermat, P. d. (1894). Oeuvres de Fermat. Tome 2 (Tannery, P. & Henry, C., Eds.). Paris: Gauthier-Villars.Google Scholar
Ferraro, G., & Panza, M. (2012). Lagrange’s theory of analytical functions and his ideal of purity of method. Archive for History of Exact Sciences, 66(2), 95–197.CrossRefGoogle Scholar
Ferreirós, J., & Gray, J. J. (Eds.). (2006). The architecture of modern mathematics. New York: Oxford University Press.CrossRefGoogle Scholar
Fisch, M. (1999). The making of Peacock’s Treatise on Algebra: A case of creative indecision. Archive for History of Exact Sciences, 54, 137–179.CrossRefGoogle Scholar
Floyd, J. (2021). Wittgenstein’s philosophy of mathematics. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Frege, G. (1980). The Foundations of Arithmetic. Evanston, IL: Northwestern University Press. (Translated by Austin, J. L. )Google Scholar
Frege, G. (1984). Formal theories of arithmetic. In McGuinness, B. (Ed.), Collected papers on mathematics, logic, and philosophy (pp. 112–121). Oxford: Blackwell. (Translated by E.-H. W. Kluge)Google Scholar
Friedman, H. M. (1975). Some systems of second order arithmetic and their use. In James, R. D. (Ed.), Proceedings of the 1974 International Congress of Mathematicians (Vol. 1, pp. 235–242). Montreal: Canadian Mathematical Congress.Google Scholar
Friedman, H. M. (1976, June). Systems of second order arithmetic with restricted induction. I. Journal of Symbolic Logic, 41(2), 557–558.Google Scholar
Fukaya, K., & Ono, K. (1999). Arnold conjecture and Gromov–Witten invariant. Topology, 38(5), 933–1048.CrossRefGoogle Scholar
Gentzen, G. (1934–1935). Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39, 405–431. (Translated as “Investigations into logical deduction” in The collected papers of Gerhard Gentzen, Szabo, M. E. (Ed.), North-Holland, 1969.)Google Scholar
Gilmore, C., Göbel, S. M., & Inglis, M. (2018). An introduction to mathematical cognition. London: Routledge.CrossRefGoogle Scholar
Girard, J.- Y. (1987). Proof theory and logical complexity (Vol. 1). Naples: Bibliopolis.Google Scholar
Gowers, T. (Ed.). (2008a). The Princeton companion to mathematics. Princeton, NJ: Princeton University Press.Google Scholar
Gowers, T. (2008b). What is mathematics about? In Gowers, T. (Ed.), The Princeton companion to mathematics (pp. 1–7). Princeton, NJ: Princeton University Press.Google Scholar
Gray, J. (2012). Henri Poincare: A Scientific Biography. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Gray, J. (2015). The real and the complex: A history of analysis in the 19th century. Heidelberg: Springer.Google Scholar
Hacking, I. (1992). ‘Style’ for historians and philosophers. Studies in History and Philosophy of Science, 23, 1–20.CrossRefGoogle Scholar
Hadamard, J. (1945). The psychology of invention in the mathematical field. Princeton: Princeton University Press.Google Scholar
Hájek, P., & Pudlák, P. (1998). Metamathematics of first-order arithmetic. Berlin: Springer. (Second printing)Google Scholar
Hallett, M. (2008). Reflections on the purity of method in Hilbert’s Grundlagen der Geometrie. In Mancosu, P. (Ed.), The philosophy of mathematical practice (pp. 198–255). Oxford University Press.CrossRefGoogle Scholar
Hamami, Y. (2014). Mathematical rigor, proof gap and the validity of mathematical inference. Philosophia Scientiæ, 18(1), 7–26.CrossRefGoogle Scholar
Hardy, G. H., & Littlewood, J. (1919). On a Tauberian theorem for Lambert’s series, and some fundamental theorems in the analytic theory of numbers. Proceedings of the London Mathematical Society, 19(1), 21–29.Google Scholar
Hardy, G. H., & Wright, E. M. (1979). An introduction to the theory of numbers (Fifth ed.). New York: Oxford University Press.Google Scholar
Harris, M. (2019, June). Why the proof of Fermat’s Last Theorem doesn’t need to be enhanced. Quanta Magazine. (https://www.quantamagazine.org/why-the-proof-of-fermats-last-theorem-doesnt-need-to-be-enhanced-20190603)Google Scholar
Hartnett, K. (2017, February). A fight to fix geometry’s foundations: When two mathematicians raised pointed questions about a classic proof that no one really understood, they ignited a years-long debate about how much could be trusted in a new kind of geometry. Quanta Magazine.Google Scholar
Hasse, H. (1930). Die moderne algebraische Methode. Jahresbericht der Deutschen Mathematiker-Vereinigung, 39, 22–33.Google Scholar
Hilbert, D. (1901). Mathematische Probleme. Archiv der Mathematik und Physik (3rd series), 1, 44–63, 213–237. (English translation by M. W. Newson in the Bulletin of the American Mathematical Society 8: 437–479, 253–297, 1902.)Google Scholar
Hilbert, D. (1931). Die Grundlegung der elementaren Zahlenlehre. Mathematische Annalen, 104, 485–494.CrossRefGoogle Scholar
Hilbert, D. (1971). Foundations of Geometry. La Salle, IL: Open Court. (English translation of Grundlagen der Geometrie (B. G. Teubner, Leipzig, 1899), by L. Unger)Google Scholar
Hilbert, D. (2004). David Hilbert’s lectures on the foundations of geometry, 1891–1902 (Hallett, M. & Majer, U., Eds.). Berlin: Springer.Google Scholar
Hirschfeldt, D. (2014). Slicing the truth: On the computable and reverse mathematics of combinatorial principles. Singapore: World Scientific.CrossRefGoogle Scholar
Ingham, A. (1932). The distribution of prime numbers. Cambridge: Cambridge University Press.Google Scholar
Isaacson, D. (1996). Arithmetical truth and hidden higher-order concepts. In Hart, W. (Ed.), The philosophy of mathematics (pp. 203–224). New York: Oxford University Press. ( First published in Logic Colloquium ’85, the Paris Logic Group (Eds.), Amsterdam, North-Holland, 1987, pp. 147–169.)Google Scholar
Ito, K. (2017). Cultural difference and sameness: Historiographic reflections on histories of physics in modern Japan. In Chemla, K. & Keller, E. F. (Eds.), Cultures without culturalism: The making of scientific knowledge. Durham: Duke University Press.Google Scholar
Jacobi, C. G. J. (1828). Note sur la décomposition d’un nombre donné en quatre quarrés. Journal für die reine und angewandte Mathematik, 1828(3), 191.Google Scholar
Jacobi, C. G. J. (1829). Fundamenta nova theoriae functionum ellipticarum. Regiomonti: Sumtibus Fratrum Borntraeger.Google Scholar
Jacobi, C. G. J. (1834). De compositione numerorum e quatuor quadratis. Journal für die reine und angewandte Mathematik, 3, 167–172.CrossRefGoogle Scholar
Jacobi, C. G. J. (1848). Über unendliche Reihen, deren Exponenten zugleich in zwei verschiedenen quadratischen Formen enthalten sind. Journal für die reine und angewandte Mathematik, 37, 61–94, 221–254.Google Scholar
Kahle, R., & Pulcini, G. (2018). Towards an operational view of purity. In Arazim, P. & Lávička, T. (Eds.), The Logica yearbook 2017 (pp. 125–138). College Publications.Google Scholar
Kitcher, P. (1981, December). Explanatory unification. Philosophy of Science, 48(4), 507–531.CrossRefGoogle Scholar
Kitcher, P. (1989). Explanatory unification and the causal structure of the world. In Kitcher, P. & Salmon, W. (Eds.), Scientific explanation (pp. 410–505). Minneapolis: University of Minnesota Press.Google Scholar
Klein, J. (1992). Greek mathematical thought and the origin of algebra. New York: Dover Publications. (Translated by E. Brann)Google Scholar
Kleiner, I. (2012). Excursions in the history of mathematics. New York: Birkhäuser.CrossRefGoogle Scholar
Knorr, W. R. (1978). Archimedes and the spirals. Historia Mathematica, 5, 43–75.CrossRefGoogle Scholar
Knorr Cetina, K. (1999). Epistemic cultures: How the sciences make knowledge. Cambridge, MA: Harvard University Press.CrossRefGoogle Scholar
Kreisel, G. (1969). Luitzen Egbertus Jan Brouwer: 1881–1966. Biographical Memoirs of Fellows of the Royal Society, 18, 39–68.Google Scholar
Kreisel, G. (1980). Kurt Gödel. Biographical Memoirs of Fellows of the Royal Society, 26, 149–224.Google Scholar
Lacroix, S.- F. (1797). Traité du calcul différentiel et du calcul intégral (Vol. 1). Paris: J.-B.-M. Duprat.Google Scholar
Lagrange, J.- L. (1772). Démonstration d’un théorème d’arithmétique. Nouveaux Mémoires de l’Académie royale des Sciences et Belles-Lettres de Berlin, année 1770, 123–133.Google Scholar
Lagrange, J.- L. (1799). Discours sur l’objet de la théorie des fonctions analytiques. Journal de l’École Polytechnique, 2(6), 232–235.Google Scholar
Lagrange, J.- L. (1876). Leçons sur mathematiques elementaires. In Oeuvres de lagrange (Vol. VII). Paris: Gauthier-Villars. (Edited by Joseph-Alfred Serret)Google Scholar
Landau, E. (1909). Handbuch der Lehre von der Verteilung der Primzahlen. Leipzig: B.G. Teubner.Google Scholar
Lange, M. (2019). Ground and explanation in mathematics. Philosophers’ Imprint, 19(33), 1–18.Google Scholar
Lehet, E. (2021, 1). Impurity in contemporary mathematics. Notre Dame Journal of Formal Logic, 62.CrossRefGoogle Scholar
Leibniz, G. W. (1996). New essays on human understanding. Cambridge: Cambridge University Press. (Edited and translated by Remnant, Peter and Bennett, Jonathan )CrossRefGoogle Scholar
Lipman, J., & Teissier, B. (1981). Pseudo-rational local rings and a theorem of Briançon-Skoda about integral closures of ideals. Michigan Mathematical Journal, 28, 97–116.CrossRefGoogle Scholar
Lorenat, J. (2015). Figures real, imagined, and missing in Poncelet, Plücker, and Gergonne. Historia Mathematica, 42(2), 155–192.CrossRefGoogle Scholar
Lorenat, J. (2016). Synthetic and analytic geometries in the publications of Jakob Steiner and Julius Plücker (1827–1829). Archive for History of Exact Sciences, 70, 413–462.CrossRefGoogle Scholar
Luchins, A., & Luchins, E. (1990). The Einstein–Wertheimer correspondence on geometric proofs and mathematical puzzles. Mathematical Intelligencer, 12(2), 35–43.CrossRefGoogle Scholar
Maddy, P. (2000). Does mathematics need new axioms? Bulletin of Symbolic Logic, 6(4), 413–422.Google Scholar
Maddy, P. (2001). Some naturalistic reflections on set theoretic method. Topoi, 20, 17–27.CrossRefGoogle Scholar
Mancosu, P. (2021). Mathematical style. In Zalta, E. N. (Ed.), The Stanford encyclopedia of philosophy (Winter 2021 ed.). Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/win2021/entries/mathematical-style/.Google Scholar
Marshall, D. B. (2023, 10). Internal applications and puzzles of the applicability of mathematics. Philosophia Mathematica, 32(1), 1–20.CrossRefGoogle Scholar
Martinot, R. (2023). Ontological purity for formal proofs. The Review of Symbolic Logic, 1–40.Google Scholar
Mathias, A. (1992). The ignorance of Bourbaki. Mathematical Intelligencer, 14(3), 4–13.CrossRefGoogle Scholar
Mazur, B. (1991). Number theory as gadfly. American Mathematical Monthly, 98(7), 593–610.CrossRefGoogle Scholar
Mazzotti, M. (2023). Reactionary mathematics: A genealogy of purity. Chicago: University of Chicago Press.CrossRefGoogle Scholar
McCarthy, T. (2021, 1). Induction, constructivity, and grounding. Notre Dame Journal of Formal Logic, 62.CrossRefGoogle Scholar
McDuff, D., & Wehrheim, K. (2015). Kuranishi atlases with trivial isotropy: The 2013 state of affairs.Google Scholar
McLarty, C. (2020). The large structures of Grothendieck founded on finite-order arithmetic. Review of Symbolic Logic, 13(2), 296–325.CrossRefGoogle Scholar
Mercer, I. (2009, April). On Furstenberg’s proof of the infinitude of primes. American Mathematical Monthly, 116, 355–356.CrossRefGoogle Scholar
Merz, J. T. (1903). On the development of mathematical thought during the nineteenth century. In European thought in the nineteenth century (Vol. II, pp. 627–740). Edinburgh: William Blackwood and Sons.Google Scholar
Michel, N. (2020). The values of simplicity and generality in Chasles’s geometrical theory of attraction. Journal for General Philosophy of Science, 51, 115–146.CrossRefGoogle Scholar
Negri, S., & von Plato, J. (2001). Structural proof theory. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Newton, I. (1972). The mathematical papers of Isaac Newton. Vol. V: 1683–1684 (Whiteside, D., Ed.). Cambridge, UK: Cambridge University Press.Google Scholar
Newton, I. (2022). Unarranged fragments, mostly relating to the dispute with Leibniz. Cambridge, UK: Cambridge University Library. (MS Add. 3968, ff. 594r-619v. https://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00385)Google Scholar
O’Hara, C., & Ward, D. (1937). An introduction to projective geometry. Oxford: Clarendon Press.Google Scholar
Ording, P. (2019). 99 variations on a proof. Princeton, NJ: Princeton University Press.Google Scholar
Pambuccian, V. (2001, March). Fragments of Euclidean and hyperbolic geometry. Scientiae Mathematicae Japonicae, 53(2), 361–400.Google Scholar
Pambuccian, V. (2005). Euclidean geometry problems rephrased in terms of midpoints and point–reflections. Elemente der Mathematik, 60, 19–24.CrossRefGoogle Scholar
Pambuccian, V. (2009). A reverse analysis of the Sylvester–Gallai theorem. Notre Dame Journal of Formal Logic, 50(3), 245–260.CrossRefGoogle Scholar
Pambuccian, V. (2011). The axiomatics of ordered geometry I. Ordered incidence spaces. Expositiones Mathematicae, 29, 24–66.CrossRefGoogle Scholar
Panza, M. (1997). Classical sources for the concept of analysis and synthesis. In Otte, M. & Panza, M. (Eds.), Analysis and synthesis in mathematics (pp. 365–414). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
Paris, J., & Harrington, L. (1977). A mathematical incompleteness in Peano Arithmetic. In Barwise, J. (Ed.), Handbook of mathematical logic (pp. 1133–1142). Amsterdam: North-Holland.CrossRefGoogle Scholar
Paseau, A. (2010). Proofs of the compactness theorem. History and Philosophy of Logic, 31(1), 73–98.CrossRefGoogle Scholar
Pel, B. (2023). ‘A remarkable artifice’: Laplace, Poisson and mathematical purity. The Review of Symbolic Logic, 1–37.CrossRefGoogle Scholar
Picard, E., & Simart, G. (1906). Théorie des fonctions algébriques de deux variables indépendantes (Vol. 2). Paris: Gauthier-Villars.Google Scholar
Pillay, A. (2021, 1). Remarks on purity of methods. Notre Dame Journal of Formal Logic, 62.CrossRefGoogle Scholar
Pincock, C. (2023). Mathematics and explanation. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
, Plato. (1997a). Collected works. Indianapolis: Hackett. (Edited by John M., Cooper )Google Scholar
, Plato. (1997b). Meno. In Plato (1997a) (pp. 870–897). (Translated by Grube, G. M. A. )Google Scholar
Poggiolesi, F., & Genco, F. (2023). Conceptual (and hence mathematical) explanation, conceptual grounding and proof. Erkenntnis, 88(4), 1481–1507. doi: 10.1007/s10670-021-00412-xCrossRefGoogle Scholar
Poincaré, H. (1891). Les géométries non euclidiennes. Revue générale des sciences pures et appliqueés, 2, 769–774. (Reprinted as chapter 3 of Science and Hypothesis, Walter Scott Publishing, London, 1905, pp. 35–50.)Google Scholar
Poincaré, H. (1902). Sur la nature du raisonnement mathématique. In La science et l’hypothèse. Paris: Flammarion.Google Scholar
, Proclus. (1992). A commentary on the first book of Euclid’s Elements. Princeton, NJ: Princeton University Press. (Translated from Greek by Glenn R. Morrow)Google Scholar
Rabouin, D. (2009). Mathesis universalis – l’idée de « mathématique universelle » d’aristote à descartes. Paris: PUF.CrossRefGoogle Scholar
Rabouin, D. (2017). Styles in mathematical practice. In Chemla, K. & Keller, E. F. (Eds.), Cultures without culturalism: The making of scientific knowledge. Durham: Duke University Press.Google Scholar
Robinson, J. (1949). Definability and decision problems in arithmetic. Journal of Symbolic Logic, 14, 98–114.CrossRefGoogle Scholar
Rourke, C. P., & Sullivan, D. P. (1971). On the Kervaire obstruction. Annals of Mathematics, 94(3), 397–413.CrossRefGoogle Scholar
Rusnock, P. (2022, 02). Grounding in Practice: Bolzano’s Purely Analytic Proof in Light of the Contributions. In Bolzano’s Philosophy of Grounding: Translations and Studies. Oxford: Oxford University Press.Google Scholar
Ryan, P. J. (2021). Szemerédi’s theorem: An exploration of impurity, explanation, and content. The Review of Symbolic Logic, 1–40.Google Scholar
Sabourau, S. (2010). Local extremality of the Calabi–Croke sphere for the length of the shortest closed geodesic. Journal of the London Mathematical Society, 82(3), 549–562.CrossRefGoogle Scholar
Saint-Gervais, H. P. d. (2010). Uniformisation des surfaces riemann. Lyon: ENS Éditions. (Translated from the French by Robert G. Burns 2016)Google Scholar
Saint-Gervais, H. P. d. (2016). Uniformization of Riemann surfaces. Zurich: European Mathematical Society. (Translated from the French by Robert G. Burns)CrossRefGoogle Scholar
Segal, S. (2003). Mathematicians under the Nazis. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Segre, C. (1891). Su alcuni indirizzi nelle investigazioni geometriche. Rivista di Matematica, 1, 42–66.Google Scholar
Selberg, A. (1949). An elementary proof of the prime-number theorem. Annals of Mathematics, 50, 305–313.CrossRefGoogle Scholar
Simpson, S. G. (2009). Subsystems of second order arithmetic (Second ed.). Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Smith, K. E., Kahanpää, L., Kekäläinen, P., & Traves, W. (2000). An invitation to algebraic geometry. New York: Springer.CrossRefGoogle Scholar
Stanley Tanswell, F. (2024). Mathematical rigour and informal proof. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Statman, R. (1974). Structural complexity of proofs (Unpublished doctoral dissertation). Stanford University.Google Scholar
Steiner, M. (1978). Mathematics, explanation, and scientific knowledge. Nous, 12(1), 17–28.CrossRefGoogle Scholar
Steinkrüger, P. (2018). Aristotle on kind-crossing. In Caston, V. (Ed.), Oxford studies in ancient philosophy (pp. 107–158). Oxford: Oxford University Press.Google Scholar
Suppes, P. (1978). The plurality of science. In Psa: Proceedings of the biennial meeting of The Philosophy of Science Association (Vol. 1978, pp. 3–16).Google Scholar
Tappenden, J. (2005). Proof style and understanding in mathematics. I. Visualization, unification and axiom choice. In Mancosu, P., Jørgensen, K. F., & Pedersen, S. A. (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 147–214). Dordrecht: Springer.CrossRefGoogle Scholar
Tappenden, J. (2008). Mathematical concepts and definitions. In Mancosu, P. (Ed.), The philosophy of mathematical practice (pp. 256–275). Oxford University Press.CrossRefGoogle Scholar
Toffoli, S. D., & Fontanari, C. (2023). Recalcitrant disagreement in mathematics: An “endless and depressing controversy” in the history of Italian algebraic geometry. Global Philosophy, 33(38), 1–29.Google Scholar
University, C. W. R. (2013, March 4). Professor shows Fermat’s Last Theorem can be proved more simply. College of Arts and Sciences News. (https://artsci.case.edu/news/philosophy-professor-shows-fermats-last-theorem-can-be-proved-more-simply/)Google Scholar
van Dalen, D. (1995). Hermann Weyl’s intuitionistic mathematics. Bulletin of Symbolic Logic, 1(2), 145–169.CrossRefGoogle Scholar
vos Savant, M. (1993). The world’s most famous math problem. New York: St. Martin’s Press.Google Scholar
Weil, A. (2005). A 1940 letter of André Weil on analogy in mathematics. Notices of the American Mathematical Society, 52(3), 334–341. (Translated into English by Martin H. Krieger)Google Scholar
Weyl, H. (1918). Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis. Leipzig: Veit.CrossRefGoogle Scholar
Wussing, H. (2007). The genesis of the abstract group concept: A contribution to the history of the origin of abstract group theory. Mineola, NY: Dover.Google Scholar
Accessibility standard: Unknown
Why this information is here
This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.Accessibility Information
Accessibility compliance for the HTML of this element is currently unknown
and may be updated in the future.
- 9
- Cited by