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Series:   Elements in the Philosophy of Mathematics

Mathematical Rigour and Informal Proof

Published online by Cambridge University Press:  01 March 2024

Fenner Stanley Tanswell
Fenner Stanley Tanswell
Affiliation:
Loughborough University

Summary

This Element looks at the contemporary debate on the nature of mathematical rigour and informal proofs as found in mathematical practice. The central argument is for rigour pluralism: that multiple different models of informal proof are good at accounting for different features and functions of the concept of rigour. To illustrate this pluralism, the Element surveys some of the main options in the literature: the 'standard view' that rigour is just formal, logical rigour; the models of proofs as arguments and dialogues; the recipe model of proofs as guiding actions and activities; and the idea of mathematical rigour as an intellectual virtue. The strengths and weaknesses of each are assessed, thereby providing an accessible and empirically-informed introduction to the key issues and ideas found in the current discussion.
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Keywords

mathematical rigourinformal proofmathematical practiceformalisationepistemology of mathematics
Type
Element
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Series: Elements in the Philosophy of Mathematics
Online ISBN: 9781009325110
Publisher: Cambridge University Press
Print publication: 28 March 2024

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References

Aberdein, A. (2005). The uses of argument in mathematics. Argumentation, 19, 287–301.CrossRefGoogle Scholar
Aberdein, A. (2006). Managing informal mathematical knowledge: Techniques from informal logic. In Borwein, J. M., & Farmer, W. M. (eds.), MKM 2006, Lecture Notes in Computer Science 4108 (pp. 208–21). Berlin, Springer.Google Scholar
Aberdein, A. (2007). The informal logic of mathematical proof. In Van Kerkhove, B., & Van Bendegem, J. P. (eds.), Perspectives on Mathematical Practices (pp. 135–51). Dordrecht, Springer.Google Scholar
Aberdein, A. (2010). Observations on sick mathematics. In Van Kerkhove, B., Van Bendegem, J. P., & de Vuyst, J. (eds.), Philosophical Perspectives on Mathematical Practice (pp. 269–300). Suwanee, GA, College Publications.Google Scholar
Aberdein, A. (2013). The parallel structure of mathematical reasoning. In Aberdein, A., & Dove, I. (eds.), The Argument of Mathematics (pp. 361–80). Dordrecht, Springer.CrossRefGoogle Scholar
Aberdein, A. (2021). Dialogue types, argumentation schemes, and mathematical practice: Douglas Walton and mathematics. Journal of Applied Logics, 8(1), 159–82.Google Scholar
Aberdein, A. (2023). Deep disagreement in mathematics. Global Philosophy, 33(1), 17.Google Scholar
Aberdein, A., Rittberg, C. J., & Tanswell, F. S. (2021). Virtue theory of mathematical practices: An introduction. Synthese, 199, 10167–80.CrossRefGoogle Scholar
Alcolea Banegas, J. (1997). L’argumentació en matemàtiques. In Casaban i Moya, E. (ed.), XIIè Congrés Valencià de Filosofia, València (pp. 135–47).Google Scholar
English translation (2013). Argumentation in mathematics. In Aberdein, A., & Dove, I. (eds.), The Argument of Mathematics (pp. 47–60). Dordrecht, Springer.CrossRefGoogle Scholar
Andersen, L. E. (2017). On the nature and role of peer review in mathematics. Accountability in Research, 24(3), 177–92.CrossRefGoogle ScholarPubMed
Andreatta, M., Bezdek, A., & Boronski, J. P. (2011). The problem of Malfatti: Two centuries of debate. The Mathematical Intelligencer, 33(1), 72–6.CrossRefGoogle Scholar
Anscombe, G. E. M. (1958). Modern moral philosophy. Philosophy, 33(124), 1–19.CrossRefGoogle Scholar
Antonutti Marfori, M. (2010). Informal proofs and mathematical rigour. Studia Logica, 96(2), 261–72.CrossRefGoogle Scholar
Appel, K., & Haken, W. (1977). Every planar map is 4-colorable. Part I: Discharging. Illinois Journal of Mathematics, 21(3), 429–90.Google Scholar
Appel, K., Haken, W., & Koch, J. (1977). Every planar map is four colorable. Part II: Reducibility. Illinois Journal of Mathematics, 21(3), 491–567.Google Scholar
Aristotle. (2009). The Nicomachean Ethics. Ross, W. D. (trans.), 2nd edition revised and with notes by Brown, L. Oxford, Oxford University Press.Google Scholar
Atiyah, M., Borel, A., Chaitin, G. J. et al. (1994). Responses to: A. Jaffe and F. Quinn, ‘Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics’. Bulletin of the American Mathematical Society, 30(2), 178–207.CrossRefGoogle Scholar
Auslander, J. (2009). On the roles of proof in mathematics. In Gold, B., & Simons, R. A. (eds.), Proof and Other Dilemmas: Mathematics and Philosophy (pp. 61–78). Washington, DC, The Mathematical Association of America.CrossRefGoogle Scholar
Austin, J. L. (1962). How to Do Things with Words. Oxford, Oxford University Press.Google Scholar
Avigad, J. (2021). Reliability of mathematical inference. Synthese, 198(8), 7377–99.Google Scholar
Azzouni, J. (2004). The derivation-indicator view of mathematical practice. Philosophia Mathematica (III), 12(2), 81–106.CrossRefGoogle Scholar
Azzouni, J. (2005). Is there still a sense in which mathematics can have foundations. In Sica, G. (ed.), Essays on the Foundations of Mathematics and Logic (pp. 9–48). Monza, Polimetrica.Google Scholar
Azzouni, J. (2009). Why do informal proofs conform to formal norms? Foundations of Science, 14(1), 9–26.CrossRefGoogle Scholar
Azzouni, J. (2020). The algorithmic-device view of informal rigorous mathematical proof. In Sriraman, B. (ed.), Handbook of the History and Philosophy of Mathematical Practice (pp. 1–82). Cham, Springer. https://doi.org/10.1007/978-3-030-19071-2_4-1.Google Scholar
Barany, M. J. (2011). God, king, and geometry: Revisiting the introduction to Cauchy’s Cours d’analyse. Historia Mathematica, 38(3), 368–88.CrossRefGoogle Scholar
Barany, M. J. (2013). Stuck in the middle: Cauchy’s Intermediate Value Theorem and the history of analytic rigor. Notices of the AMS, 60(10), 1334–8.Google Scholar
Barany, M. J. (2020). Impersonation and personification in mid-twentieth century mathematics. History of Science, 58(4), 417–36.CrossRefGoogle ScholarPubMed
Barton, N. (2012). Structural relativity and informal rigour. In Oliveri, G., Ternullo, C., & Boscolo, S. (eds.), Objects, Structures, and Logics: FilMat Studies in the Philosophy of Mathematics (pp. 133–74). Cham, Springer.Google Scholar
Bass, H. (2003). The Carnegie Initiative on the Doctorate: The case of mathematics. Notices of the AMS, 50(7), 767–76.Google Scholar
Battaly, H. (2008). Virtue epistemology. Philosophy Compass, 3(4), 639–63.CrossRefGoogle Scholar
Beall, J. (1999). From full blooded Platonism to really full blooded Platonism. Philosophia Mathematica (III), 7(3), 322–5.CrossRefGoogle Scholar
Beall, J. C., & Restall, G. (2005). Logical Pluralism. Oxford, Oxford University Press.CrossRefGoogle Scholar
Blåsjö, V. (2022). Operationalism: An interpretation of the philosophy of ancient Greek geometry. Foundations of Science, 27(2), 587–708.CrossRefGoogle Scholar
Bourbaki, N. (1968). Elements of Mathematics, Theory of Sets. Reading, MA, Addison-Wesley.Google Scholar
Brown, J. (1999). Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures. London, Routledge.Google Scholar
Buldt, B., Löwe, B., & Müller, T. (2008). Towards a new epistemology of mathematics. Erkenntnis, 68(3), 309–29.CrossRefGoogle Scholar
Burgess, J. P. (2015). Rigor and Structure. Oxford, Oxford University Press.Google Scholar
Burgess, J., & De Toffoli, S. (2022). What is mathematical rigor? Aphex, 25, 1–17.Google Scholar
Cantù, P., & Luciano, E. (2021). Giuseppe Peano and his school: Axiomatics, symbolism and rigor. Philosophia Scientiæ. Travaux d’histoire et de philosophie des sciences, 25(1), 3–14.Google Scholar
Carter, J. (2019). Philosophy of mathematical practice: Motivations, themes and prospects. Philosophia Mathematica (III), 27(1), 1–32.CrossRefGoogle Scholar
Cellucci, C. (2018). Definition in mathematics. European Journal for Philosophy of Science, 8(3), 605–29.CrossRefGoogle Scholar
Code, L. (1987). Epistemic Responsibility. Hanover, NH, University Press of New England.Google Scholar
Cook, R. T. (2000). Logic-as-Modeling: A New Perspective on Formalization. Doctoral dissertation, The Ohio State University. https://etd.ohiolink.edu/acprod/odb_etd/etd/r/1501/10?clear=10&p10_accession_num=osu1260202088.Google Scholar
Cook, R. T. (2010). Let a thousand flowers bloom: A tour of logical pluralism. Philosophy Compass, 5(6), 492–504.CrossRefGoogle Scholar
Cotnoir, A. J. (2018). Logical nihilism. In Wyatt, J., Pedersen, N. J. L. L., & Kellen, N. (eds.), Pluralisms in Truth and Logic (pp. 301–29). Cham, Palgrave Macmillan.Google Scholar
Coumans, V. J. W. (2021). Definitions (and concepts) in mathematical practice. In Sriraman, B. (ed.), Handbook of the History and Philosophy of Mathematical Practice (n. pag.). Cham, Springer. https://doi.org/10.1007/978-3-030-19071-2_94-1.Google Scholar
Coumans, V. J. W., & Consoli, L. (2023). Definitions in practice: An interview study. Synthese, 202(23), 1–32.CrossRefGoogle Scholar
Davies, B., Alcock, L., & Jones, I. (2020). Comparative judgement, proof summaries and proof comprehension. Educational Studies in Mathematics, 105(2), 181–97.CrossRefGoogle Scholar
Davies, B., Alcock, L., & Jones, I. (2021). What do mathematicians mean by proof? A comparative-judgement study of students’ and mathematicians’ views. The Journal of Mathematical Behavior, 61, 100824.CrossRefGoogle Scholar
Davies, B., Miller, D., & Infante, N. (2021). The role of authorial context in mathematicians’ evaluations of proof. International Journal of Mathematical Education in Science and Technology, 54(5), 725–39.Google Scholar
De Morgan, A. (1838). Mathematical induction. The Penny Cyclopedia, 12, 465–6.Google Scholar
De Toffoli, S. (2021a). Groundwork for a fallibilist account of mathematics. The Philosophical Quarterly, 71(4), 1–22.CrossRefGoogle Scholar
De Toffoli, S. (2021b). Reconciling rigour and intuition. Erkenntnis, 86, 1783–802.CrossRefGoogle Scholar
De Toffoli, S. (2023). Who’s afraid of mathematical diagrams? Philosophers’ Imprint, 23, 9. https://doi.org/10.3998/phimp.1348.CrossRefGoogle Scholar
De Toffoli, S., & Fontanari, C. (2022). Objectivity and rigor in classical Italian algebraic geometry. Noesis: Objectivity in Mathematics, 38, 195–212.Google Scholar
De Toffoli, S., & Fontanari, C. (2023). Recalcitrant disagreement in mathematics: An ‘endless and depressing controversy’ in the history of Italian algebraic geometry. Global Philosophy, 33(4), 1–29.Google Scholar
De Toffoli, S., & Giardino, V. (2015). An inquiry into the practice of proving in low-dimensional topology. In Lolli, G., Panza, M., & Venturi, G. (eds.), From Logic to Practice (pp. 315–36). Cham, Springer.Google Scholar
Dean, W., & Kurokawa, H. (in press). On the methodology of informal rigour: Set theory, semantics, and intuitionism. In Antonutti Marfori, M., & Petrolo, M. (eds.), Intuitionism, Computation, and Proof: Selected Themes from the Research of G. Kreisel, Springer.Google Scholar
Delarivière, S., Frans, J., & Van Kerkhove, B. (2017). Mathematical explanation: A contextual approach. Journal of Indian Council of Philosophical Research, 34(2), 309–29.CrossRefGoogle Scholar
Detlefsen, M. (2009). Proof: Its nature and significance. In Gold, B., & Simons, R. A. (eds.), Proof and Other Dilemmas: Mathematics and Philosophy (pp. 3–32). Washington, DC, The Mathematical Association of America.Google Scholar
Dove, I. J. (2013). Towards a theory of mathematical argument. In Aberdein, A., & Dove, I. (eds.), The Argument of Mathematics (pp. 291–308). Springer, Dordrecht.CrossRefGoogle Scholar
Dutilh Novaes, C. (2011). The different ways in which logic is (said to be) formal. History and Philosophy of Logic, 32(4), 303–32.CrossRefGoogle Scholar
Dutilh Novaes, C. (2021). The Dialogical Roots of Deduction: Historical, Cognitive, and Philosophical Perspectives on Reasoning. Cambridge, Cambridge University Press.Google Scholar
Ernest, P. (1998). Social Constructivism as a Philosophy of Mathematics. Albany, NY, State University of New York Press.Google Scholar
Ernest, P. (2021). Mathematics, ethics and purism: An application of MacIntyre’s virtue theory. Synthese, 199(1), 3137–67.CrossRefGoogle Scholar
Eves, H. (1965). A Survey of Geometry, Volume 2. Boston, MA, Allyn and Bacon.Google Scholar
Ferreirós, J. (2008). The crisis in the foundations of mathematics. In Gowers, T., Barrow-Green, , & Leader, I. (eds.), The Princeton Companion to Mathematics (pp. 142–56). Princeton, NJ, Princeton University Press.Google Scholar
Fine, K. (2005). Our knowledge of mathematical objects. In Gendler, T. S., & Hawthorne, J. (eds.), Oxford Studies in Epistemology (pp. 89–109). Oxford, Oxford Academic.CrossRefGoogle Scholar
Foot, P. (1978,) Virtues and Vices and Other Essays in Moral Philosophy. Oxford: Blackwell.Google Scholar
Franks, Curtis (2015). Logical nihilism. In Hirvonen, Å., Kontinen, J., Kossak, R., & Villaveces, A. (eds.), Logic without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, pp. 147–66. Berlin, De Gruyter.Google Scholar
Frege, G. (1884) The Foundations of Arithmetic. Austin, J. L. (trans.), 1953. Oxford, Blackwell.Google Scholar
Fricker, M. (2007). Epistemic Injustice: Power and the Ethics of Knowing. Oxford, Oxford University Press.CrossRefGoogle Scholar
Geist, C., Löwe, B., & Kerkhove, B. V. (2010). Peer review and knowledge by testimony in mathematics. In Löwe, B., & Müller, T. (eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. Research Results of the Scientific Network PhiMSAMP (pp. 1–24). London, College Publications.Google Scholar
Gelfert, A. (2022). Thinking with notations: Epistemic actions and epistemic activities in mathematical practice. In Friedman, M., & Krauthausen, K. (eds.), Model and Mathematics: From the 19th to the 21st Century: Trends in the History of Science (pp. 333–62). Cham, Birkhäuser.Google Scholar
Goethe, N. B., & Friend, M. (2010). Confronting ideals of proof with the ways of proving of the research mathematician. Studia Logica, 96(2), 273–88.CrossRefGoogle Scholar
Gonthier, G. (2008). Formal proof – The four-color theorem. Notices of the American Mathematical Society, 55(11), 1382–93.Google Scholar
Greco, J. (2010). Achieving Knowledge. Cambridge, Cambridge University Press.CrossRefGoogle Scholar
Greiffenhagen, C., & Sharrock, W. (2005). Gestures in the blackboard work of mathematics instruction. Paper presented at Interacting Bodies: Proceedings of 2nd Conference of the International Society for Gesture Studies (Lyon, 15–18 June 2005), pp. 1–24. http://gesture-lyon2005.ens-lyon.fr/IMG/pdf/Greiffenhagen-Gesture.pdf.Google Scholar
Habgood-Coote, J., & Tanswell, F. S. (2023). Group knowledge and mathematical collaboration: A philosophical examination of the classification of finite simple groups. Episteme, 20(2), 281–307.CrossRefGoogle Scholar
Haffner, E. (2021). The shaping of Dedekind’s rigorous mathematics: What do Dedekind’s drafts tell us about his ideal of rigor? Notre Dame Journal of Formal Logic, 62(1), 5–31.CrossRefGoogle Scholar
Hales, T. C. (2008). Formal proof. Notices of the AMS, 55(11), 1370–80.Google Scholar
Hamami, Y. (2019). Mathematical rigor and proof. Review of Symbolic Logic, 15(2), 409–49.Google Scholar
Hamami, Y., & Morris, R. L. (2020). Philosophy of mathematical practice: A primer for mathematics educators. ZDM, 52(6), 1113–26.CrossRefGoogle Scholar
Hanna, G., & Larvor, B. (2020). As Thurston says? On using quotations from famous mathematicians to make points about philosophy and education. ZDM: Mathematics Education, 52(6), 1137–47.CrossRefGoogle Scholar
Hardy, G. H. (1929). Mathematical proof. Mind, 38(149), 1–25.CrossRefGoogle Scholar
Hegel, G. W. F. (1807). Phenomenology of Spirit. Miller, A. V. (trans.), 1977. Oxford, Clarendon Press.Google Scholar
Heinze, A. (2010). Mathematicians’ individual criteria for accepting theorems and proofs: An empirical approach. In Hanna, G., Jahnke, H. N., & Pulte, H. (eds.), Explanation and Proof in Mathematics (pp. 101–11). Boston, MA, Springer.Google Scholar
Hengel, E. (2022). Publishing while female: Are women held to higher standards? Evidence from peer review. The Economic Journal, 132(648), 2951–91. https://doi.org/10.1093/ej/ueac032.CrossRefGoogle Scholar
Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–99.CrossRefGoogle Scholar
Hersh, R. (1997). Prove – Once more and again. Philosophia Mathematica (III), 5(2), 153–65.CrossRefGoogle Scholar
Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig, Teubner.Google Scholar
Hookway, C. (2010). Some varieties of epistemic injustice: Reflections on Fricker. Episteme, 7(2), 151–63.CrossRefGoogle Scholar
Hornsby, J. (2012). Ryle’s knowing-how, and knowing how to act. In Bengson, J., & Moffett, M. A. (eds.), Knowing How: Essays on Knowledge, Mind and Action (pp. 80–98). Oxford, Oxford University Press.CrossRefGoogle Scholar
Hunsicker, E., & Rittberg, C. J. (2022). On the epistemological relevance of social power and justice in mathematics. Axiomathes, 32, 1147–68.CrossRefGoogle Scholar
Hursthouse, R., & Pettigrove, G. (2022). Virtue ethics. In Zalta, E. N., & Nodelman, U. (eds.), Stanford Encyclopedia of Philosophy (Winter 2022 Edition). https://plato.stanford.edu/archives/win2022/entries/ethics-virtue/.Google Scholar
Inglis, M., & Aberdein, A. (2015). Beauty is not simplicity: An analysis of mathematicians’ proof appraisals. Philosophia Mathematica (III), 23(1), 87–109.CrossRefGoogle Scholar
Inglis, M., & Aberdein, A. (2016). Diversity in proof appraisal. In Larvor, B. (ed.), Mathematical Cultures (pp. 163–79. Cham, Birkhäuser.Google Scholar
Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43(4), 358–90.CrossRefGoogle Scholar
Inglis, M., Mejía‐Ramos, J. P., Weber, K., & Alcock, L. (2013). On mathematicians’ different standards when evaluating elementary proofs. Topics in Cognitive Science, 5(2), 270–82.CrossRefGoogle ScholarPubMed
Isaacson, D. (2011). The reality of mathematics and the case of set theory. In Noviak, Z. & Simonyi, A. (eds.), Truth, Reference, and Realism (pp. 1–75). Budapest, Central European University Press.Google Scholar
Jaffe, A., & Quinn, F. (1993). ’Theoretical mathematics’: Toward a cultural synthesis of mathematics and theoretical physics. Bulletin of the American Mathematical Society, 29(1), 1–13.CrossRefGoogle Scholar
Kirsh, D., & Maglio, P. (1994). On distinguishing epistemic from pragmatic action. Cognitive Science, 18(4), 513–49.CrossRefGoogle Scholar
Kitcher, P. (1984). The Nature of Mathematical Knowledge. Oxford, Oxford University Press.Google Scholar
Kneebone, G. T. (1957). The philosophical basis of mathematical rigour. Philosophical Quarterly, 7(28), 204–23.CrossRefGoogle Scholar
Knipping, C., & Reid, D. A. (2019). Argumentation analysis for early career researchers. In Kaiser, G. & Presmeg, N. (eds.), Compendium for Early Career Researchers in Mathematics Education (pp. 3–31). Cham, Springer.Google Scholar
Kreisel, G. (1967). Informal rigour and completeness proofs. In Lakatos, I. (ed.), Studies in Logic and the Foundations of Mathematics, Vol. 47 (pp. 138–86). Amsterdam, Elsevier.Google Scholar
Kunen, K. (1980). Set Theory: An Introduction to Independence Proofs. Amsterdam, North-Holland.Google Scholar
Kurji, A. H. (2021). What the Heck Is Logic? Logics-as-Formalizations, a Nihilistic Approach. Doctoral dissertation, University of Bristol. https://research-information.bris.ac.uk/en/studentTheses/what-the-heck-is-logic.Google Scholar
Lakatos, I. (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge, Cambridge University Press.CrossRefGoogle Scholar
Lane, L., Martin, U., Murray-Rust, D., Pease, A., & Tanswell, F. (2019). Journeys in mathematical landscapes: Genius or craft? In Hanna, G., Reid, D. A., & de Villiers, M. (eds.), Proof Technology in Mathematics Research and Teaching (pp. 197–212). Cham, Springer.CrossRefGoogle Scholar
Larvor, B. (2001). What is dialectical philosophy of mathematics? Philosophia Mathematica (III), 9, 212–29.CrossRefGoogle Scholar
Larvor, B. (2012). How to think about informal proofs. Synthese, 187(2), 715–30.CrossRefGoogle Scholar
Larvor, B. (2016). Why the naïve derivation recipe model cannot explain how mathematicians’ proofs secure mathematical knowledge. Philosophia Mathematica (III), 24(3), 401–4.CrossRefGoogle Scholar
Leitgeb, H. (2009). On formal and informal provability. In Bueno, O., & Linnebo, Ø. (eds.), New Waves in Philosophy of Mathematics (pp. 263–99). London, Palgrave Macmillan.Google Scholar
Lob, H., & Richmond, H. W. (1930). On the solutions of Malfatti’s problem for a triangle. Proceedings of the London Mathematical Society, 2(1), 287–304.CrossRefGoogle Scholar
Lombardi, G. (2022a). Proving the solution of Malfatti’s marble problem. Rendiconti del Circolo Matematico di Palermo, Series 72, 1751–82. https://doi.org/10.1007/s12215-022-00759-2.Google Scholar
Lombardi, G. (2022b). Demistifying Malfatti’s marble problem. Medium, 27 June. https://medium.com/@giancarlolombardi_25894/demistifying-malfattis-marble-problem-fcb0a4b98b36.Google Scholar
Löwe, B. (2016). Philosophy or not? The study of cultures and practices of mathematics. In Ju, S., Löwe, B., Müller, T., & Xie, Y. (eds.), Cultures of Mathematics and Logic (pp. 23–42). Cham, Birkhäuser.Google Scholar
Löwe, B., & Müller, T. (2008). Mathematical knowledge is context-dependent. Grazer Philosophische Studien, 76, 91–107.CrossRefGoogle Scholar
Löwe, B., & Müller, T. (2010). Skills and mathematical knowledge. In Löwe, B., & Müller, T. (eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice (pp. 265–80). London, College Publications.Google Scholar
Löwe, B., & Müller, T. (2011). Data and phenomena in conceptual modelling. Synthese, 182(1), 131–48.CrossRefGoogle Scholar
Löwe, B., & Van Kerkhove, B. (2019). Methodological triangulation in empirical philosophy (of mathematics). In Aberdein, A., & Inglis, M. (eds.), Advances in Experimental Philosophy of Logic and Mathematics (pp. 15–37). New York, Bloomsbury Academic Publishers.Google Scholar
Mac Lane, S. (1986). Mathematics: Form and Function. New York, Springer-Verlag.CrossRefGoogle Scholar
MacIntyre, A. (1985). After Virtue, 2nd Edition. London, Duckworth.Google Scholar
MacKenzie, D. (2004). Mechanizing Proof: Computing, Risk, and Trust. London, MIT Press.Google Scholar
Maddy, P. (2017). Set-theoretic foundations. In Caicedo, A., Cummings, J., Koellner, P., & Larson, P. B. (eds.), Contemporary Mathematics 690: Foundations of Mathematics (pp. 289–322). Providence, RI, American Mathematical Society.Google Scholar
Maddy, P. (2019). What do we want a foundation to do? Comparing set-theoretic, category-theoretic, and univalent approaches. In Centrone, S., Kant, D., & Sarikaya, D. (eds), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts (pp. 293–311). Cham, Springer.CrossRefGoogle Scholar
Malfatti, G. (1803). Memoria sopra un problema sterotomico. Memorie di matematica e di fisica della Societá Italiana delle Scienze, 10(1), 235–44.Google Scholar
Martin, J. V. (2021). Prolegomena to virtue-theoretic studies in the philosophy of mathematics. Synthese, 199(1), 1409–34.CrossRefGoogle Scholar
Maxwell, E. A. (1959). Fallacies in Mathematics. Cambridge, Cambridge University Press.CrossRefGoogle Scholar
Mejía-Ramos, J. P., & Weber, K. (2014). Why and how mathematicians read proofs: Further evidence from a survey study. Educational Studies in Mathematics, 85(2), 161–73.CrossRefGoogle Scholar
Mihaljević, H., & Santamaría, L. (2022). Mathematics publications and authors’ gender: Learning from the Gender Gap in Science project. European Mathematical Society Magazine, 123, 34–8.Google Scholar
Mihaljević, H., Santamaría, L., & Tullney, M. (2016). The effect of gender in the publication patterns in mathematics. PLoS One, 11(10), e0165367. https://doi.org/10.1371/journal.pone.0165367.CrossRefGoogle Scholar
Montmarquet, J. (1993). Epistemic Virtue and Doxastic Responsibility. Lanham, MD, Rowman & Littlefield.Google Scholar
Moore, R. C. (2016). Mathematics professors’ evaluation of students’ proofs: A complex teaching practice. International Journal of Research in Undergraduate Mathematics Education, 2(2), 246–78.CrossRefGoogle Scholar
Morris, R. L. (2021). Intellectual generosity and the reward structure of mathematics. Synthese, 199(1), 345–67.CrossRefGoogle Scholar
Müller-Hill, E. (2009). Formalizability and knowledge ascriptions in mathematical practice. Philosophia Scientiæ: Travaux d’histoire et de philosophie des sciences, 13(2), 21–43.CrossRefGoogle Scholar
Müller-Hill, E. (2011). Die epistemische Rolle formalisierbarer mathematischer Beweise. Doctoral dissertation, University of Bonn. https://bonndoc.ulb.uni-bonn.de/xmlui/handle/20.500.11811/4850.Google Scholar
Mumma, J. (2010). Proofs, pictures, and Euclid. Synthese, 175(2), 255–87.CrossRefGoogle Scholar
Nelsen, R. B. (1993). Proofs without Words: Exercises in Visual Thinking. Washington, DC, Mathematical Association of America.Google Scholar
Nelson, R. B. (2000). Proofs without Words II: More Exercises in Visual Thinking. Washington, DC, Mathematical Association of America.Google Scholar
Nelsen, R. B. (2008). Visual gems of number theory. Math Horizons, 15(3), 7–31.CrossRefGoogle Scholar
Nelsen, R. B. (2015). Proofs without Words III: Further Exercises in Visual Thinking. Washington, DC, Mathematical Association of America.CrossRefGoogle Scholar
Ohlhorst, J. (2022). Dual processes, dual virtues. Philosophical Studies, 179(7), 2237–57.CrossRefGoogle ScholarPubMed
Ording, P. (2019). 99 Variations on a Proof. Princeton, NJ, Princeton University Press.Google Scholar
Panse, A., Alcock, L., & Inglis, M. (2018). Reading proofs for validation and comprehension: An expert–novice eye-movement study. International Journal of Research in Undergraduate Mathematics Education, 4(3), 357–75.CrossRefGoogle Scholar
Pelc, A. (2009). Why do we believe theorems? Philosophia Mathematica (III), 17(1), 84–94.CrossRefGoogle Scholar
Pettigrew, R. (2016). Review of John P. Burgess’s Rigor and Structure. Philosophia Mathematica (III), 24, 129–46.Google Scholar
Popper, K. (1959) The Logic of Scientific Discovery. London, Hutchinson. (First published in German as Logik der Forschung, 1934.)Google Scholar
Popper, K. (1963) Conjectures and Refutations: The Growth of Scientific Knowledge. London, Routledge & Kegan Paul.Google Scholar
Priest, G. (1987). In Contradiction: A Study of the Transconsistent. Oxford, Oxford University Press.CrossRefGoogle Scholar
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica (III), 7(1), 5–41.CrossRefGoogle Scholar
Rittberg, C. J. (2021). Intellectual humility in mathematics. Synthese, 199(3), 5571–601.CrossRefGoogle Scholar
Rittberg, C. J. (2023). Justified epistemic exclusion in mathematics. Philosophia Mathematica (III), 31(3), 330–59,CrossRefGoogle Scholar
Rittberg, C. J., Tanswell, F. S., & Van Bendegem, J. P. (2020). Epistemic injustice in mathematics. Synthese, 197(9), 3875–3904.CrossRefGoogle Scholar
Rodin, A. (2014). On constructive axiomatic method. arXiv preprint, arXiv:1408.3591. https://arxiv.org/abs/1408.3591.Google Scholar
Rolfsen, D. (1976). Knots and Links. Berkeley, CA: Publish or Perish.Google Scholar
Rotman, B. (1988). Towards a semiotics of mathematics. Semiotica, 72, 1–35.CrossRefGoogle Scholar
Ruffino, M., San Mauro, L., & Venturi, G. (2021). Speech acts in mathematics. Synthese, 198(10), 10063–87.CrossRefGoogle Scholar
Russell, G. (2018). Logical nihilism: Could there be no logic? Philosophical Issues, 28(1), 308–24.CrossRefGoogle Scholar
Russell, G. (2019). Logical pluralism. In Zalta, E. N. (ed.), Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/logical-pluralism/.Google Scholar
Ryle, G. (1946). Knowing how and knowing that: The presidential address. Proceedings of the Aristotelian Society, 46, 1–16.CrossRefGoogle Scholar
Ryle, G. (1971). Thinking and self-teaching. Journal of Philosophy of Education, 5, 216–28.CrossRefGoogle Scholar
Sangwin, C. (2023). Sums of the first n odd integers. Mathematical Gazette, 107(568), 10–24.CrossRefGoogle Scholar
Sangwin, C. J., & Kinnear, G. (2021). Investigating insight and rigour as separate constructs in mathematical proof. EdarXiv preprint. https://doi.org/10.35542/osf.io/egks4.CrossRefGoogle Scholar
Sangwin, C., & Tanswell, F. S. (2023). Developing new picture proofs that the sums of the first n odd integers are squares. Mathematical Gazette, 107(569), 249–62.CrossRefGoogle Scholar
Schlimm, D. 2012. Mathematical concepts and investigative practice. In Feest, U. & Steinle, F. (eds.), Scientific Concepts and Investigative Practice (pp. 127–47). Berlin, de Gruyter GmbH.Google Scholar
Secco, G. D., & Pereira, L. C. (2017). Proofs versus experiments: Wittgensteinian themes surrounding the four-color theorem. In Silva, M. (ed.), How Colours Matter to Philosophy (pp. 289–307). Cham, Springer.CrossRefGoogle Scholar
Shapiro, S. (2014). Varieties of Logic. Oxford, Oxford University Press.CrossRefGoogle Scholar
Shapiro, S., & Roberts, C. (2021). Open texture and mathematics. Notre Dame Journal of Formal Logic, 62(1), 173–91.CrossRefGoogle Scholar
Shin, S. J. (1994). The Logical Status of Diagrams. Cambridge, Cambridge University Press.Google Scholar
Sosa, E. (2009). Knowing full well: The normativity of beliefs as performances. Philosophical Studies, 142(1), 5–15.CrossRefGoogle Scholar
Steiner, J. (1826). Einige geometrische Betrachtungen. Journal für die reine und angewandte Mathematik, 1826 (1), 161–84. https://doi.org/10.1515/crll.1826.1.161. Reprinted as Steiner, J. (1901) in Stern, R. (ed.), Einige geometrische Betrachtungen (section 14, pp. 25–7). Leipzig: Verlag von Wilhelm Engelmann. https://archive.org/details/einigegeometris01steigoog/page/n29/mode/2up?view=theater.Google Scholar
Steiner, M. (1975). Mathematical Knowledge. Ithaca, NY, Cornell University Press.Google Scholar
Su, F. (2017). Mathematics for human flourishing. American Mathematical Monthly, 124(6), 483–93.CrossRefGoogle Scholar
Su, F. (2020). Mathematics for Human Flourishing. New Haven, CT, Yale University Press.Google Scholar
Tanswell, F. (2015). A problem with the dependence of informal proofs on formal proofs. Philosophia Mathematica (III), 23(3), 295–310.CrossRefGoogle Scholar
Tanswell, F. S. (2016a). Saving proof from paradox: Gödel’s paradox and the inconsistency of informal mathematics. In Andreas, H., & Verdée, P. (eds.), Logical Studies of Paraconsistent Reasoning in Science and Mathematics (pp. 159–73). Cham, Springer.Google Scholar
Tanswell, F. S. (2016b). Proof, Rigour and Informality: A Virtue Account of Mathematical Knowledge. Doctoral dissertation, University of St Andrews. https://research-repository.st-andrews.ac.uk/handle/10023/10249.Google Scholar
Tanswell, F. (2017). Playing with LEGO® and proving theorems. In Cook, R. T., & Bacharach, S. (eds.), LEGO® and Philosophy: Constructing Reality Brick by Brick (pp. 217–26). Hoboken, NJ, Wiley Blackwell.Google Scholar
Tanswell, F. S. (2018). Conceptual engineering for mathematical concepts. Inquiry, 61(8), 881–913.CrossRefGoogle Scholar
Tanswell, F. S. (in press). Go forth and multiply! On actions, instructions and imperatives in mathematical proofs. In Bueno, O., & Brown, J. (eds.), Essays on the Philosophy of Jody Azzouni. Cham, Springer.Google Scholar
Tanswell, F. S., & Inglis, M. (2023) The language of proofs: A philosophical corpus linguistics study of instructions and imperatives in mathematical texts. In Sriraman, B. (ed.), Handbook of the History and Philosophy of Mathematical Practice (pp. 1–30). Cham, Springer. https://link.springer.com/referenceworkentry/10.1007/978-3-030-19071-2_50-1.Google Scholar
Tanswell, F. S., & Kidd, I. J. (2021). Mathematical practice and epistemic virtue and vice. Synthese, 199(1), 407–26.CrossRefGoogle Scholar
Tanswell, F. S., & Rittberg, C. J. (2020). Epistemic injustice in mathematics education. ZDM: Mathematics Education, 52(6), 1199–1210.CrossRefGoogle Scholar
Tappenden, J. (2008). Mathematical concepts and definitions. In Mancosu, P. (ed.), The Philosophy of Mathematical Practice (pp. 256–75). Oxford, Oxford University Press.Google Scholar
Tatton-Brown, O. (2021). Rigour and intuition. Erkenntnis, 86, 1757–81.CrossRefGoogle Scholar
Tatton-Brown, O. (2023). Rigour and proof. Review of Symbolic Logic, 16(2), 480–508.CrossRefGoogle Scholar
Termini, M. (2019). Proving the point: Connections between legal and mathematical reasoning. Suffolk University Law Review, 52, 5–35.Google Scholar
Thomas, R. S. D. (2007). The comparison of mathematics with narrative. In Van Kerkhove, B., & Van Bendegem, J. P. (eds.), Perspectives on Mathematical Practices (pp. 43–59). Dordrecht, Springer.Google Scholar
Thomas, R. S. D. (2015). The judicial analogy for mathematical publication. In Zack, M., & Landry, E. (eds.), Research in History and Philosophy of Mathematics (pp. 161–70. Cham, Birkhäuser.Google Scholar
Thomas, R. S. D. (2017). Beauty is not all there is to aesthetics in mathematics. Philosophia Mathematica (III), 25(1), 116–27.Google Scholar
Thompson, C. J. (1986). The contributions of Mark Kac to mathematical physics. Annals of Probability, 14(4), 1129–38.CrossRefGoogle Scholar
Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–77.CrossRefGoogle Scholar
Toulmin, S. E. (1958). The Uses of Argument. Cambridge, Cambridge University Press.Google Scholar
Toulmin, S., Rieke, R., & Janik, A. (1979). An Introduction to Reasoning. London, Macmillan.Google Scholar
Van Bendegem, J. P. (2014). The impact of the philosophy of mathematical practice on the philosophy of mathematics. In Soler, L., Zwart, S., Lynch, M., & Israel-Jost, V. (eds.), Science After the Practice Turn in the Philosophy, History, and Social Studies of Science (pp. 215–26). Abingdon, UK, Taylor & Francis.Google Scholar
Vecht, J. J. (2023). Open texture clarified. Inquiry, 66(6), 1120–40.CrossRefGoogle Scholar
Vučković, A., & Sikimić, V. (2023). How to fight linguistic injustice in science: Equity measures and mitigating agents. Social Epistemology, 37(1), 80–96.CrossRefGoogle Scholar
Waismann, F. (1968). Verifiability. In Flew, A. (ed.), Logic and Language (pp. 118–44). Oxford, Basil Blackwell.Google Scholar
Walton, D. N. (1998). The New Dialectic: Conversational Contexts of Argument. Toronto, University of Toronto Press.CrossRefGoogle Scholar
Walton, D., & Krabbe, E. C. (1995). Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning. Albany, NY, State University of New York Press.Google Scholar
Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 39(4), 431–59.CrossRefGoogle Scholar
Weber, K. (2023). Instructions and constructions in set theory proofs. Synthese, 202(2), 1–17.CrossRefGoogle Scholar
Weber, K., & Czocher, J. (2019). On mathematicians’ disagreements on what constitutes a proof. Research in Mathematics Education, 21(3), 251–70.CrossRefGoogle Scholar
Weber, K., & Tanswell, F. S. (2022). Instructions and recipes in mathematical proofs. Educational Studies in Mathematics, 11(1), 73–87.CrossRefGoogle Scholar
Weber, K., Mejía-Ramos, J. P., & Volpe, T. (2022). The relationship between proof and certainty in mathematical practice. Journal for Research in Mathematics Education, 53(1), 65–84.Google Scholar
Weir, A. (2016). Informal proof, formal proof, formalism. Review of Symbolic Logic, 9(1), 23–43.CrossRefGoogle Scholar
Weisgerber, S. (2022). Visual proofs as counterexamples to the standard view of informal mathematical proofs? In Giardino, V., Linker, S., Burns, R., et al. (eds.), Diagrammatic Representation and Inference. 13th International Conference, Diagrams 2022, Rome, 14–16 September. Lecture Notes in Computer Science, vol. 13462. Cham, Springer. https://doi.org/10.1007/978-3-031-15146-0_3.Google Scholar
Whitehead, A. N., & Russell, B. (1910). Principia Mathematica, Volume I. Cambridge, Cambridge University Press.Google Scholar
Wiedijk, F. (2008). Formal proof – Getting started. Notices of the American Mathematical Society, 55(11), 1408–14.Google Scholar
Zagzebski, L. T. (1996). Virtues of the Mind. Cambridge, Cambridge University Press.CrossRefGoogle Scholar
Zalgaller, V. A., & Los’, G. A. (1994). The solution of Malfatti’s problem. Journal of Mathematical Sciences, 72(4), 3163–77.CrossRefGoogle Scholar
Zayton, B. (2022). Open texture, rigor, and proof. Synthese, 200(4), 1–20.CrossRefGoogle Scholar
Zeilberger, D. (1993). Theorems for a price: Tomorrow’s semi-rigorous mathematical culture. Notices of the American Mathematical Society, 40, 978–81.Google Scholar

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