## Abstract

Additional theorizing about mathematical practice is needed in order to ground appeals to truly useful notions of the virtues in mathematics. This paper aims to contribute to this theorizing, first, by characterizing mathematical practice as being epistemic and “objectual” in the sense of Knorr Cetina (in: Schatzki, Knorr Cetina, von Savigny (eds) The practice turn in contemporary theory, Routledge, London, 2001). Then, it elaborates a MacIntyrean framework for extracting conceptions of the virtues related to mathematical practice so understood. Finally, it makes the case that Wittgenstein’s methodology for examining mathematics and its practice is the most appropriate one to use for the actual investigation of mathematical practice within this MacIntyrean framework. At each stage of thinking through mathematical practice by these means, places where new virtue-theoretic questions are opened up for investigation are noted and briefly explored.

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## Notes

- 1.
MacIntyre (1981, p. 186).

- 2.
Note that my claim isn’t that pursuing, say, a virtue-based epistemology for mathematical knowledge as in Tanswell (2016) needs to wait until we have a perfectly adequate understanding of mathematical practice. However, the more realistic and detailed our picture of the kind of practical and intellectual virtues a mathematical knower can be expected to exhibit is, the more likely a view of this sort is to be compelling. My contention is simply that a clearer view of mathematical practice can help provide this more realistic view of the virtues surrounding the practice.

- 3.
This type of realistic approach—characterized by “the realistic spirit,” which looks to pay close attention to our ordinary, everyday practices—is largely inspired by the work of Wittgenstein (1953/2009, for example), who in many ways is rightly seen as a philosopher of mathematical practice.

*Cf.*, e.g., Shanker (1987), Mühlhölzer (2010), Floyd (2012), and Mühlhölzer (2014). See also Diamond (1996), Laugier (2013, xi–xii) and Methven (2015, Ch. 1) on “ordinary realism” and the realistic spirit. - 4.
Floyd (2015, p. 17).

- 5.
Ferreirós (2016, p. 28, emphasis in the original).

- 6.
Carter (2019, 24).

- 7.
It’s not my intention to single out any particular definition of practice as being especially bad of course. On its own, the fact that there are so many attempts to say what a practice is supposed to be in the philosophy of mathematics already suggests that there’s still work to be done.

- 8.
See Carter (2019, pp. 25–26).

- 9.
Kitcher (1984, pp. 163–165).

- 10.
“\(\mathbf {Math[imatical]Pract[ice]} = \langle M, P, F, PM, C, AM, PS, \ldots \rangle \) (as a reminder: \(M = \) community of mathematicians, \(P =\) research program, \(F =\) formal language, \(PM =\) proof methods, \(C =\) concepts, \(AM =\) argumentative methods, \(PS =\) proof strategies)” (Van Bendegem and Van Kerkhove 2004, p. 534).

- 11.
See, e.g., Rouse (2003, Ch. 5) for discussion.

- 12.
I am, therefore, basically in agreement with Colin Rittberg that “[t]he philosophy of mathematics needs a body of knowledge which critically assesses our philosophical methods (to engage with mathematical practices and otherwise)” (Rittberg 2019, p. 14).

- 13.
See Cellucci (2013) for a discussion of “top-down” and “bottom-up” philosophy of mathematics.

- 14.
This version of the dilemma is taken almost verbatim from Pitt (2001, p. 373).

- 15.
Burian (2001, p. 388).

- 16.
Schatzki (1996, p. 89).

- 17.
Internal goods are also characterized by being less likely to be limited in supply and less likely to be limited to being good just for me than external goods. So, if I get a raise, that means there’s less money in the company available for you, and you’re not particularly benefited by my improved financial standing. But if I invent a new technique in painting—a technique that can perhaps be seen to be the good that it is only by those within the practice—the practice is now no less likely to develop other new techniques and you can also benefit from my advance nearly as much as I can.

- 18.
- 19.
- 20.
Knorr Cetina (2001, p. 184).

- 21.
Heidegger (1927/1962, par. 15:68–70).

- 22.
*Cf*. Knorr Cetina (2001, p. 188). - 23.
Knorr Cetina (2001, p. 190).

- 24.
Occasional difficulties in applying tools or understanding how to make use of equipment are significant for Heidegger’s overall story in

*Being and Time*as well, but for different reasons. These sorts of problems—a piece of equipment’s conspicuousness (*Auffälligkeit*), obtrusiveness (*Aufdringlichkeit*), or obstinacy (*Aufsässigkeit*)—can reveal the otherwise hidden “worldliness of the world” to us, but they aren’t themselves motivators of further investigations into particular objects of concern. See Heidegger (1927/1962, par. 16). - 25.
Grosholz (2007, p. 47).

- 26.
- 27.
Knorr Cetina (2001, p. 194).

- 28.
See Knorr Cetina (1999, p. 11) for an account of science in general along these lines.

- 29.
E.g., one of the motivations for active research into computer-verified proofs, say, using Coq, Mizar, or Isabelle, is both to check long, complicated proofs and to provide an easily accessible store of mathematical results.

- 30.
Thinking in terms of the virtues seems to be becoming more prevalent in these fields in recent years as well. For example, the most recent edition of

*Engineering Ethics: Concepts and Cases*Harris et al. (2019), one of the most widely-used textbooks on the subject, has now added sections incorporating virtue ethics into the set of tools it hopes to provide its readers. - 31.
- 32.
MacIntyre (1981, p. 191).

- 33.
*Cf*. MacIntyre (1981, p. 64). - 34.
MacIntyre (1988).

- 35.
Making this case is one of the main goals of MacIntyre (1988).

- 36.
Moral traditions are also supposed to do some of the work of justifying something that might seem like a virtue: the virtue of understanding yourself and your place within a tradition. See MacIntyre (1981, p. 223).

- 37.
MacIntyre (1981, p. 222).

- 38.
MacIntyre (1981, p. 222).

- 39.
MacIntyre (1981, p. 275, emphasis in the original).

- 40.
MacIntyre (1988, 402).

- 41.
- 42.
Albers (1994, p. 4).

- 43.
See, again, Jones (2006) for historical discussion of the question of how mathematics can develop a person’s individual virtues.

- 44.
See Grayson (2018) for an introduction.

- 45.
MacIntyre (1988, p. 402).

- 46.
- 47.
See National Research Council (2013) for more along these lines.

- 48.
- 49.
MacIntyre (2016, p. 206).

- 50.
Rittberg (2019, p. 13) provides a long list of possible methodologies for pursuing the study of mathematical practice. The approach advocated here is closest to the one mentioned from Larvor (2010), but it’s not my intention to rule out any of the alternatives. Rather, the methodology to be considered simply suggests ways of thinking about the various objects of study focused on by these other approaches. I should note also that the approach doesn’t fit very naturally into the catalogue of Van Bendegem (2014, p. 221).

- 51.
- 52.
The American legal realists can be seen as being realistic in a similar fashion. See, for example, Leiter (2005, pp. 50–53).

- 53.
- 54.
See, e.g., Dummett (1959, p. 348) for the classic interpretation of this kind.

- 55.
See in particular the work of Juliet Floyd and Felix Mühlhölzer in the bibliography..

- 56.
See Wittgenstein (1939/1989, p. 39). It’s interesting to note that Wittgenstein immediately qualifies this claim, calling it an exaggeration and saying that it’s partly true and partly false.

- 57.
- 58.
- 59.
- 60.
The attempt to minimize philosophical background assumptions also helps to make room for the “pluralism in perspectives” suggested by Michelle Friend, another author that can be seen as attempting to find the best way to be realistic when philosophizing about mathematics. See Friend (2014, p. 25).

- 61.
This is Felix Mühlhölzer’s translation of the passage more familiarly rendered as “Mathematics is a motley” Wittgenstein (1956/1983, III §46). Mühlhölzer argues that the term ‘motley’ has negative connotations that don’t fit well with the general thrust of Wittgenstein’s remarks about the mixture of proof methods found in mathematics. I use his translation of this remark to signal my agreement on this point. See, however, Hacking (2014, p. 57) for a contrary view.

- 62.
*Cf*. Ferreirós (2016, p. 37). - 63.
Burgess (2015, p. 60).

- 64.
Serre et al. (1999, p. 35).

- 65.
- 66.
- 67.
Wittgenstein (1935/1958, p. 4, emphasis in the original).

- 68.
See Frege (1903/1960, §88). This is Frege’s way of restating the views of E. Heine and J. Thomae.

- 69.
Wittgenstein (1935/1958, p. 4).

- 70.
Wittgenstein (1953/2009, §432, emphasis in the original).

- 71.
Wittgenstein (1953/2009, §38).

- 72.
Wittgenstein (1953/2009, §116).

- 73.
What exactly ‘metaphysical’ is supposed to mean in this statement is the matter of a debate that needn’t be settled here. For the record, I’m roughly in agreement with Gordon Baker, who suggests that metaphysical uses try to express essences or to pass themselves off as being scientific but are not.

*Cf*. Baker (2009, pp. 96–100). - 74.
Thurston (2006, p. 167).

- 75.
- 76.
- 77.
Inglis and Aberdein (2016, p. 168).

- 78.
See Baz (2012).

- 79.
Baz (2012, p. 105).

- 80.
The methodological principles advocated in this section are similar to those accepted in ethnomethodology and the sociology of scientific knowledge. (See, e.g., Livingston 1986, p. 1; Lynch 1993, pp. 14–15), and more recently François and Van Kerkhove (2010) for ethnomethodology. Barnes et al. (1996) is a good example of the sociology of knowledge that deals with mathematics in its final chapter.) Many of the authors within these fields also take inspiration from Wittgenstein, so the resemblance isn’t coincidental. The goal of “pure description” for which ethnomethodologists put this kind of methodology to use is, however, likely to be different from the goals of philosophers of mathematics who make use of the methodology outlined here. Being a methodology though, the realistic view on offer doesn’t seek to dictate the uses to which it’s put.

- 81.
*Cf*. Toulmin (1972, pp. 505–506). - 82.

## References

Aigner, M., & Ziegler, G. (2000).

*Proofs from THE BOOK*. Berlin: Springer.Albers, D., & Dyson, F. (1994). Freeman dyson: Mathematician, physicist, and writer.

*The College Mathematics Journal*,*25*(1), 2–21.Avigad, J. (2008). Understanding proofs. In P. Mancosu (Ed.),

*The philosophy of mathematical practice*. Oxford: Oxford University Press.Baker, A. (2009). Mathematical accidents and the end of explanation. In Ø. Bueno & O. Linnebo (Eds.),

*New waves in philosophy of mathematics*. London: Palgrave Macmillan.Barnes, B., Bloor, D., & Henry, J. (1996).

*Scientific knowledge: A sociological analysis*. Chicago: University of Chicago Press.Baz, A. (2012).

*When words are called for: A defense of ordinary language philosophy*. Cambridge: Harvard University Press.Bourdieu, P. (1977).

*Outline of a theory of practice*. Cambridge: Cambridge University Press.Burgess, J. (2015).

*Rigor and structure*. Oxford: Oxford University Press.Burian, R. M. (2001). The dilemma of case studies resolved: The virtues of using case studies in the history and philosophy of science.

*Perspectives on Science*,*9*(4), 383–404.Carter, J. (2019). Philosophy of mathematical practice—Motivations, themes and prospects.

*Philosophia Mathematica (III)*,*27*(1), 1–32.Cellucci, C. (2013). Top-down and bottom-up philosophy of mathematics.

*Foundations of Science*,*18*(1), 93–106.Corfield, D. (2012). Narrative and the rationality of mathematical practice. In A. Doxiadis & B. Mazur (Eds.),

*Circles disturbed: The interplay of mathematics and narrative*. Princeton: Princeton University Press.D’Alessandro, W. (2018). Mathematical explanation beyond explanatory proof.

*The British Journal for the Philosophy of Science*,*71*(2), 581–603.Davis, P., & Hersh, R. (1981).

*The mathematical experience*. Boston: Houghton Mifflin.Diamond, C. (1996). Wittgenstein, mathematics, and ethics: Resisting the attractions of realism. In

*The Cambridge companion to Wittgenstein*. Cambridge University Press.Dummett, M. (1959). Wittgenstein’s philosophy of mathematics.

*Philosophical Review*,*68*(3), 324–348.Dunne, J. (2005). An intricate fabric: Understanding the rationality of practice.

*Pedagogy, Culture and Society*,*13*(3), 367–389.Ernst, M., Heis, J., Maddy, P., McNulty, M., & Weatherall, J. (2015). Forward to special issue on mathematical depth.

*Philosophia Mathematica (III)*,*23*(2), 155–162.Ferreirós, J. (2016).

*Mathematical knowledge and the interplay of practices*. Princeton: Princeton University Press.Floyd, J. (2001). Prose versus proof: Wittgenstein on gödel, tarski and truth.

*Philosophia Mathematica (III)*,*9*(3), 280–307.Floyd, J. (2011). On being surprised: Wittgenstein on aspect-perception, logic, and mathematics. In W. Day & V. Krebs (Eds.),

*Seeing Wittgenstein Anew: New essays on aspect seeing*. Cambridge: Cambridge University Press.Floyd, J. (2012). Das Überraschende: Wittgenstein on the surprising in mathematics. In J. Ellis & D. Guevara (Eds.),

*Wittgenstein and the philosophy of mind*. Oxford: Oxford University Press.Floyd, J. (2015). Depth and clarity.

*Philosophia Mathematica (III)*,*23*(2), 1–22.Font, V., Godino, J., & Gallardo, J. (2013). The emergence of objects from mathematical practices.

*Educational Studies in Mathematics*,*82*(1), 97–124.François, K., & Van Kerkhove, B. (2010). Ethnomathematics and the philosophy of mathematics (education). In B. Löwe & T. Müller (Eds.),

*PhiMSAMP. Philosophy of mathematics, sociological aspects and mathematical practice*. London: College Publications.Frege, G. (1903/1960). Grundgesetze der Arithmetik. In P. Geach & M. Black (Eds.),

*Translations from the philosophical writings of Gottlob Frege*(Vol. II). Oxford: Basil Blackwell.Friederich, S. (2011). Motivating Wittgenstein’s perspective on mathematical sentences as norms.

*Philosophia Mathematica (III)*,*19*(1), 1–19.Friedman, H. (1975). Some systems of second order arithmetic and their use. In

*Proceedings of the international congress of mathematicians (Vancouver, 1974)*(Vol. 1). Canadian Mathematical Congress.Friend, M. (2014).

*Pluralism in mathematics: A new position in philosophy of mathematics*. Berlin: Springer.Geuss, R. (2008).

*Philosophy and real politics*. Princeton: Princeton University Press.Grayson, D. (2018). An introduction to univalent foundations for mathematics.

*Bulletin of the American Mathematical Society*,*55*(4), 427–450.Grosholz, E. (2007).

*Representation and productive ambiguity in mathematics and the sciences*. Oxford: Oxford University Press.Hacking, I. (2014).

*Why is there philosophy of mathematics at all?*. Cambridge: Cambridge University Press.Harris, C., Pritchard, M., James, R., Englehardt, E., & Rabins, M. (2019).

*Engineering ethics: Concepts and cases*. Boston: Cengage.Heidegger, M. (1927/1962).

*Being and time*. Oxford: Basil Blackwell.Henriksen, M. (1993). There are too many bad mathematicians.

*The Mathematical Intelligencer*,*15*(1), 6–9.Hicks, D., & Stapleford, T. (2016). The virtues of scientific practice: Macintyre, virtue ethics, and the historiography of science.

*Isis*,*107*(3), 449–472.Inglis, M., & Aberdein, A. (2015). Beauty is not simplicity: An analysis of mathematicians’ proof appraisals.

*Philosophia Mathematica (III)*,*23*(1), 87–109.Inglis, M., & Aberdein, A. (2016). Diversity in proof appraisal. In B. Larvor (Ed.),

*Mathematical cultures: The London meetings 2012–2014*. Boston: Birkhäuser.Jones, M. (2006).

*The good life in the scientific revolution: Descartes, Pascal, Leibniz, and the cultivation of virtue*. Chicago: University of Chicago Press.Kienzler, W., & Grève, S. (2016). Wittgenstein on gödelian ‘incompleteness’, proofs and mathematical practice: Reading remarks on the foundations of mathematics, part i, appendix iii, carefully. In S. Grève & M. Jakub (Eds.),

*Wittgenstein and the creativity of language*. London: Palgrave Macmillan.Kitcher, P. (1984).

*The nature of mathematical knowledge*. Oxford: Oxford University Press.Knorr-Cetina, K. (1981).

*The manufacture of knowledge: An essay on the constructivist and contextual nature of science*. Oxford: Pergamon Press.Knorr Cetina, K. (1999).

*Epistemic cultures: How the sciences make knowledge*. Cambridge: Harvard University Press.Knorr Cetina, K. (2001). Objectual practice. In T. Schatzki, K. Knorr Cetina, & E. von Savigny (Eds.),

*The practice turn in contemporary theory*. London: Routledge.Lange, M. (2010). What are mathematical coincidences (and why does it matter)?

*Mind*,*119*(474), 307–340.Lange, M. (2017). Mathematical explanations that are not proofs.

*Erkenntnis*,*83*(8), 1–18.Larvor, B. (2001). What is dialectical philosophy of mathematics?

*Philosophia Mathematica (III)*,*9*(2), 212–229.Larvor, B. (2010). Syntactic analogies and impossible extensions. In B. Löwe & T. Müller (Eds.),

*PhiMSAMP—Philosophy of mathematics: Sociological aspects and mathematical practice*. London: College Publications.Laugier, S. (2013).

*Why we need ordinary language philosophy*. Chicago: University of Chicago Press.Leiter, B. (2005). American legal realism. In M. Golding & W. Edmundson (Eds.),

*The Blackwell guide to the philosophy of law and legal theory*. Hoboken: Blackwell Publishing.Livingston, E. (1986).

*The ethnomethodological foundations of mathematics*. London: Routledge & Kegan Paul.Lynch, M. (1993).

*scientific practice and ordinary action: Ethnomethodology and social studies of science*. Cambridge: Cambridge University Press.MacIntyre, A. (1981).

*After virtue: A study in moral theory*. Notre Dame: University of Notre Dame Press.MacIntyre, A. (1988).

*Whose justice? Which rationality?*. Notre Dame: University of Notre Dame Press.MacIntyre, A. (1990).

*Three rival versions of moral enquiry: Encyclopaedia, genealogy, and tradition*. Notre Dame: University of Notre Dame Press.MacIntyre, A. (2006). Epistemological crises, dramatic narrative, and the philosophy of science. In

*The tasks of philosophy: Selected essays*(Vol. 1). Cambridge University Press.MacIntyre, A. (2016).

*Ethics and the conflicts of modernity: An essay on desire, practical reasoning, and narrative*. Cambridge: Cambridge University Press.Mazur, B. (1997). Conjecture.

*Synthese*,*111*(2), 197–210.Methven, S. (2015).

*Frank Ramsey and the realistic spirit*. London: Palgrave Macmillan.Mühlhölzer, F. (2002). Wittgenstein and surprises in mathematics. In R. Haller & K. Puhl K (Eds.),

*Wittgenstein and the future of philosophy: A reassessment after 50 years*. öbt & hpt.Mühlhölzer, F. (2010).

*Braucht die Mathematik eine Grundlegung? Eine Kommentar des Teils III von Wittgenstein Bermerkungen über die Grundlagen der Mathematik*. Vittorio Klostermann.Mühlhölzer, F. (2014). On live and dead signs in mathematics. In G. Link (Ed.),

*Formalism and beyond: On the nature of mathematical discourse*. Berlini: De Gruyter.National Research Council. (2013). Important trends in the mathematical sciences. In

*The mathematical sciences in 2025*. The National Academies Press.Pitt, J. C. (2001). The dilemma of case studies: Toward a heraclitian philosophy of science.

*Perspectives on Science*,*9*(4), 373–382.Putnam, H. (1980). Models and reality.

*Journal of Symbolic Logic*,*45*(3), 464–482.Radford, L. (2008). The ethics of being and knowing: Towards a cultural theory of learning. In L. Radford, G. Schubring, & F. Seeger (Eds.),

*Semiotics in mathematics education: Epistemology, history, classroom, and culture*. Rotterdam: Sense Publishers.Rittberg, C. J. (2019). On the contemporary practice of philosophy of mathematics.

*Acta Baltica Historiae et Philosophiae Scientiarum*,*7*(1), 5–26.Rouse, J. (2003).

*How scientific practices matter: Reclaiming philosophical naturalism*. Chicago: University of Chicago Press.Sartre, J. P. (1943/1993).

*Being and nothingness: An essay on phenomenological ontology*. Washington Square Press.Schatzki, T. (1996).

*Social practices: A Wittgensteinian approach to human activity and the social*. Cambridge: Cambridge University Press.Serre, J. P., Chong, C. T., & Leong, Y. (1999). An interview with Jean-Pierre Serre. In R. Wilson & J. Gray (Eds.),

*Mathematical conversations: Selections from the mathematical intelligencer*. Berlin: Springer.Shanker, S. (1987).

*Wittgenstein and the turning-point in the philosophy of mathematics*. London: Routledge.Shapiro, S. (1991).

*Foundations without foundationalism: A case for second-order logic*. Oxford: Oxford University Press.Simpson, S. (1999).

*Subsystems of second order arithmetic*. Berlin: Springer.Soler, L. (2012). Préface to “from practice to results in logic and mathematics” issue.

*Philosophia Scientiae*,*16*(1), 1–3.Su, F. (2020).

*Mathematics for human flourishing*. New Haven: Yale University Press.Tanswell, F. S. (2016).

*Proof, rigour and informality: A virtue account of mathematical knowledge*. Ph.D. Dissertation, University of St. Andrews.Thurston, W. (2006). On proof and progress in mathematics. In R. Hersh (Ed.),

*18 Unconventional essays on the nature of mathematics*. Berlin: Springer.Toulmin, S. (1972).

*Human understanding: General introduction and part I*(Vol. I). London: Clarendon Press.Van Bendegem, J. (2014). The impact of the philosophy of mathematical practice on the philosophy of mathematics. In L. Soler, S. Zwart, M. Lynch, & V. Israel-Jost (Eds.),

*Science after the practice turn in the philosophy, history, and social studies of science*. London: Routledge.Van Bendegem, J. P., & Van Kerkhove, B. (2004). The unreasonable richness of mathematics.

*Journal of Cognition and Culture*,*4*(3), 525–549.Voevodsky, V. (n.d.).

*Unimath. HFL 2015*, 25 August 2015, Heidelberg. https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2015_08_25_HLF_Heidelberg.pdf. Accessed 23 July 2020.Weber, M. (1968).

*Economy and society: An outline of interpretive sociology*. Berkeley: University of California Press.Wittgenstein, L. (1930/1975).

*Philosophical remarks*. Basil Blackwell.Wittgenstein, L. (1935/1958).

*The blue and brown books: Preliminary studies for the philosophical investigations*. Blackwell.Wittgenstein, L. (1939/1989). Wittgenstein’s lectures on the foundations of mathematics, Cora Diamond, (Ed.). University of Chicago Press.

Wittgenstein, L. (1953/2009).

*Philosophical investigations*. Wiley-Blackwell.Wittgenstein, L. (1956/1983).

*Remarks on the foundations of mathematics*. The MIT Press.

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Martin, J.V. Prolegomena to virtue-theoretic studies in the philosophy of mathematics.
*Synthese* (2020). https://doi.org/10.1007/s11229-020-02802-0

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### Keywords

- Virtue
- Mathematical practice
- MacIntyre
- Wittgenstein