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The Epistemological Subject(s) of Mathematics

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Handbook of the History and Philosophy of Mathematical Practice

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Abstract

Paying attention to the inner workings of mathematicians has led to a proliferation of new themes in the philosophy of mathematics. Several of these have to do with epistemology. Philosophers of mathematical practice, however, have not (yet) systematically engaged with general (analytic) epistemology. To be sure, there are some exceptions, but they are few and far between. In this chapter, I offer an explanation of why this might be the case and show how the situation could be remedied. I contend that it is only by conceiving the knowing subject(s) as embodied, fallible, and embedded in a specific context (along the lines of what has been done within social and feminist epistemology) that we can pursue an epistemology of mathematics sensitive to actual mathematical practice. I further suggest that this reconception of the knowing subject(s) does not force us to abandon the traditional framework of epistemology in which knowledge requires justified true belief. It does, however, lead to a fallible conception of mathematical justification that, among other things, makes Gettier cases possible. This shows that topics considered to be far removed from the interests of philosophers of mathematical practice might reveal to be relevant to them.

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Notes

  1. 1.

    Throughout the chapter, I will use epistemology to mean analytic epistemology. This clarification is needed in a context like the one of the present volume, which assembles heterogenous contributions belonging to different traditions. As I use the term, analytic epistemology is the theory of knowledge (and justification) in the analytic philosophical tradition of the Anglo-American world – a very different topic compared, for example, with épistémologie in the French tradition, which has to do with the critical and historical study of science and scientific knowledge.

  2. 2.

    To be sure, there are significant exchanges already, albeit not systematic ones – I will turn to some of those in the conclusion. See, for example, Azzouni (2006), Easwaran (2015), and Tanswell and Kidd (2020).

  3. 3.

    Lakatos (1976), Kitcher (1984), and Ferreirós (2016).

  4. 4.

    See, for example, the discussion in Kitcher (1992).

  5. 5.

    This view is explicitly endorsed in Ferreirós (2016) and implicitly assumed in a plethora of works, such as the ones discussing (like Netz’s) the roles of diagrams and notations in mathematics; see, for example (Chemla 2018; Manders 2008).

  6. 6.

    Thanks to Gisele Secco whose criticism helped me realize the importance of this issue.

  7. 7.

    See, for example J.A. Carter and Pritchard (2015), Bengson and Moffett (2011), and Habgood-Coote (2019).

  8. 8.

    Note that this “or” is inclusive. It is the standard disjunctive connective of classical logic: “A or B” is true unless both A and B are false.

  9. 9.

    This lead to knowledge first approaches in epistemology; see Williamson (2000).

  10. 10.

    It is therefore somewhat bizarre that the famous problem of access to abstract objects originally formulated in Benacerraf (1973) involved Goldman’s causal theory of knowledge (more on this later).

  11. 11.

    There are ways to apply them to mathematics, however. See, for example, Clarke-Doane (2020).

  12. 12.

    This has not always been the case. For instance, Descartes’s most radical form of skepticism involved mathematical beliefs, but these do not feature in most contemporary discussions.

  13. 13.

    Frege’s third realm, however, differently from Popper’s third world, only accommodates necessary immutable truths, leaving out the laws of science.

  14. 14.

    As Susan Haack (1979) explains, it is not trivial to pin down the exact locus of the negative feature of the type of subjectivism Popper had in mind. First, the objective/subjective dichotomy can be conceived in several different ways. Second, the notion of intersubjectivity can be placed in between, thus leaving the dichotomy to give space to something more nuanced.

  15. 15.

    For an analysis of the reasons (some of which are institutional in nature) why psychologism started to be conceived as a major sin in the German philosophy departments of the beginning of the twentieth century, see Kusch (1995).

  16. 16.

    To be sure, this is a simplification. Frege critique of psychologism is sophisticated and developed across multiple works.

  17. 17.

    See Sinaceur (2019).

  18. 18.

    And in fact, the most virulent form of psychologism, against which Frege’s criticism were levelled, arises from the application of psychology to logic and mathematics.

  19. 19.

    Think, for example, about conventionalism and Quine’s holistic empiricism. For an elaboration on these ideas, see De Toffoli (2021a).

  20. 20.

    See Burgess (2015, 145) illuminating discussion of different types of indifference in mathematical practice.

  21. 21.

    For a recent endeavor in that direction, see Pettigrew (2021).

  22. 22.

    See De Toffoli (2021b).

  23. 23.

    See Easwaran (2009) and De Toffoli (2021a).

  24. 24.

    In my view, transferability can also be understood in terms of the a priori – given a minimal, fallible conception of a priori justification. In this volume, Danielle Macbeth (2021, 13) also connects transferability to the a priori: “The proof is, as it is sometimes put, “transferable” (Easwaran 2009). Indeed, it is in just this sense that mathematical knowledge is a priori: It is a priori not because it is infallible but because and insofar as it does not rely on empirical evidence, whether that of one’s own senses or that given on the testimony of another.”

  25. 25.

    See Habgood-Coote and Tanswell (2021).

  26. 26.

    See Garavaso (2018).

  27. 27.

    See Longino (2002).

  28. 28.

    See, for example, Soler et al. (2014).

  29. 29.

    Its beginning could be pinned to Kuhn’s (1962) influential work on scientific revolutions; later on, emphasis on practice was prominent among historians and sociologists of science.

  30. 30.

    Although there are important precursor, like (Lakatos 1976), the philosophy of mathematical practice took off much later. See, for example, Mancosu (2008).

  31. 31.

    For instance, this would be in line with Kornblith (2012).

  32. 32.

    See, for example, Hieronymi (2006).

  33. 33.

    Note that such a belief might not be an unqualified belief in the conclusion of the proof but can be the conditional belief that the conclusion is implied by the (explicit or implicit) premises.

  34. 34.

    The example is adapted from the classic Singer (1972).

  35. 35.

    See Chignell (2018).

  36. 36.

    This is also related to John Dewey’s critique to traditional epistemology according to which the subject of knowledge, like the classical subject of perception, is passive.

  37. 37.

    Some steps in this directions have been taken in (Clarke-Doane 2020).

  38. 38.

    See Tanswell (2016).

  39. 39.

    See, for example, Silva (2022).

  40. 40.

    Actually, the Goldbach conjecture says that every even number larger than 2 is the sum of two primes.

  41. 41.

    This is done in Pettigrew (2021).

  42. 42.

    See, for example, Berto and Nolan (2021).

  43. 43.

    A similar approach can be (and has been) adopted with respect to science. See for example the discussion in Bird (2010).

  44. 44.

    Reported in Dummett (1978, xxviii).

  45. 45.

    See, for example, Burgess (1983).

  46. 46.

    There are exceptions, however. Sometimes the truth of certain axioms (such as the axiom of choice) is debated by practicing mathematicians.

  47. 47.

    This is in line with Peirce’s view of mathematics. According to him, in deductive reasoning we start from an “hypothetical state of things and are led to conclude that, however it may be with the universe in other respects, wherever and whenever the hypothesis may be realized, something else not explicitly supposed in that hypothesis will be true invariably” (1998, 212).

  48. 48.

    Even though this story is not universally accepted. For example, it is criticized by constructivist mathematicians. This unveils other instances (in this case linked to intuitionism) in which metaphysical positions can indeed lead one to endorse particular epistemological views.

  49. 49.

    See Avigad (2021).

  50. 50.

    To be sure, this holds nowadays, not of the historical practice.

  51. 51.

    It is worth noting that he proposes a different account for elementary mathematics.

  52. 52.

    Interesting similarities could perhaps be drawn from Putnam’s internal realism, but Ferreirós does not discuss them.

  53. 53.

    Ironically, Justin Clarke-Done (2020) uses examples similar to the ones by Ferreirós to argue for mathematical realism and against mathematical objectivity. In his view, an objective question admits a single answer. He uses the case of non-Euclidean geometry precisely to highlight the non-objective character of mathematics. Although there is much more to be said about this, for reasons of space I have to postpone the discussion to another time.

  54. 54.

    To be sure, this claim can be interpreted in many ways – see Pedersen and Wright (2018).

  55. 55.

    A coherence notion of truth for mathematics is developed by Crispin Wright (1992, 1999). For a detailed discussion on how to apply Wright’s ideas to mathematics, see (S. Shapiro 2007).

  56. 56.

    This example is actually due to Carl Ginet.

  57. 57.

    I changed the notation for the subject and the proposition to make it coherent with the one adopted in the rest of the chapter.

  58. 58.

    See the discussion in De Toffoli (2022).

  59. 59.

    Other relevant approaches in epistemology are ones inspired by the embodied cognition research program, broadly understood to include embedded, enactive, and extended cognition (Shapiro and Spaulding 2021).

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Acknowledgments

I am indebted to the following people for helpful feedback: José Ferrirós, Pieranna Garavaso, Valeria Giardino, Andrea Sereni, and Guido Tana.

This research was financed by MUR – Ministero dell’Università e della Ricerca tramite PRIN PNRR Missione 4 “Istruzione e Ricerca” – Componente C2 Investimento 1.1, “Fondo per il Programma Nazionale di Ricerca e Progetti di Rilevante Interesse Nazionale (PRIN)” Finanziato dall'Unione Europea Next GenerationEU (Progetto: “Understanding Scientific Disagreement and its Impact on Society,” n. P2022A8F82) – CUP I53D23006880001.

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  1. Scuola Universitaria Superiore IUSS Pavia, Pavia, Italy

    Silvia De Toffoli

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  1. Silvia De Toffoli
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Correspondence to Silvia De Toffoli .

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  1. Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

    Bharath Sriraman

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  1. CNRS, Nancy, France

    Valeria Giardino

  2. University of Lorraine, Nancy, France

    Valeria Giardino

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De Toffoli, S. (2024). The Epistemological Subject(s) of Mathematics. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-031-40846-5_51

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