Abstract
In the past 10 years, contemporary philosophy of mathematics has seen the development of a trend that conceives mathematics as first and foremost a human activity and in particular as a kind of practice. However, only recently the need for a general framework to account for the target of the so-called philosophy of mathematical practice has emerged. The purpose of the present article is to make progress towards the definition of a more precise general framework for the philosophy of mathematical practice by exploring two strategies. A first strategy is to turn to philosophy of mind and Edwin Hutchins' view of distributed cognition in order to better understand the cognitive issues at play when considering a community of mathematicians; a second strategy is to refer to philosophy of language and focus on Robert Brandom's inferentialism and mathematical conceptual content. A possible combination of these two views, called enhanced material inferentialism, is then put forward as a promising framework to account for the philosophy of mathematical practice.
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Notes
Kitcher (1984) defined a mathematical practice as the quintuple <L, M, S, R, Q> composed by Language, Metamathematical views, accepted Statements, Reasoning methods and Questions (chapters 7 and 8). van Kerkhove and van Bendegem (2004) extended Kitcher’s proposal so as to include mathematical communities, thus obtaining a seventuple < M, P, F, PM, C, AM, PS > composed by a mathematical community M of individual mathematicians, a research program P, a formal language F, a set PM of proof methods, a set C of concepts, a set AM of argumentative methods and a set PS of proof strategies.
van Bendegem (1993) claims that, differently from philosophy of science, the philosophy of mathematics of the 1990s was dominated by a “received view”, according to which mathematics is considered—wrongly—as “the exact science”. More recently, van Bendegem (2014) identified eight perspectives that look at mathematics from the point of view of its practice: (1) the Lakatosian approach; (2) the descriptive analytical naturalizing approach; (3) the normative analytical naturalizing approach; (4) the sociology of mathematics approach; (5) the mathematics educationalist approach; (6) the ethno-mathematical approach; (7) the evolutionary biology of mathematics; and (8) the cognitive psychology of mathematics. For him, only the first three have a distinct philosophical nature, which shows to what extent the study of mathematics may involve several disciplines.
This is clear also from the profiles of the members of the Association for the Philosophy of Mathematical Practice (APMP) created in 2009. For reference, see the APMP website: http://www.philmathpractice.org.
An alternative—and not incompatible with my own—way of parsing the literature is provided in Carter (2019) where three different, in some cases overlapping, “strands” in the philosophy of mathematical practice are identified: (1) the agent based strand, (2) the historical strand, and (3) the epistemological strand.
Ferreirós (2016, chapter 3) points out that Kitcher’s quintuple is misleading because it is still based on an analysis of the production of scientific knowledge as depending mainly on linguistic knowledge; as a consequence, he rather thinks in terms of the couple Framework plus Agent, where frameworks are of two kinds: theoretical and symbolic. Moreover, the Framework-Agent couple is not identified with mathematical practice but is at the core of the practice and of the production and reproduction of knowledge.
This view resonates with Michael Detlefsen’s interpretation of Poincaré, according to which rigor is conceptualized in terms of understanding (see Detlefsen 1992).
Ferreirós suggests that there might also be elementary or “practical” parts of mathematics that connect to the physical world. I agree with this possibility, but if this is the case, then it is necessary to allow for some kind of mathematical content that does not only depend only on inferential connections to other concepts.
The case of material anchors is more difficult to discuss, and I will go back to this issue in the next section.
In the same footnote, Brandom claims that McDowell (1994)’s conceptualism about perceptual experience presents the strongest argument against his view.
It has to be noted here that this is another point where Brandom and the view I propose here move away from Hutchins’ framework, which is representational.
Note that here I deliberately use “materially” and not “visually”, because I want to focus on the dynamical features of the use of these tools and the invariances that emerge and not so much on their visual properties, that are in general subject to irrelevant variations (at the exception of tools that are designed with visual features conveying relevant meaning, as for example in Byrne 1847).
See Derry (2017) for a discussion of inferentialism in mathematics education.
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Acknowledgements
Previous versions of this paper were presented in Aix-en Provence at the International Workshop on Mathematical Practice, in Parma at a conference on Mathematical Practice and Social Ontology and in Toulouse at the Annual Conference of the European Network for the Philosophy of the Social Sciences. I thank all the participants to these events for their questions. Many thanks to two anonymous reviewers and to Paola Cantù whose precious comments helped me improve the paper, and to Pietro Salis who many years ago suggested to me to have a look at Robert Brandom's philosophy of language in relation to my views on diagrammatic reasoning. This work was supported by the Agence Nationale de la Recherche: Grant Number EUR FrontCog grant ANR-17-EURE-0017.
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Giardino, V. The Practice of Mathematics: Cognitive Resources and Conceptual Content. Topoi 42, 259–270 (2023). https://doi.org/10.1007/s11245-022-09861-7
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DOI: https://doi.org/10.1007/s11245-022-09861-7