Abstract
The philosophy of mathematical practice sometimes investigates the social constitution of mathematics but does not always make explicit the philosophical-normative framework that guides the discussion. This chapter investigates some recent proposals in the philosophy of mathematical practice that compare social facts and mathematical objects, discussing similarities and differences. An attempt will be made to identify, through a comparison with three different perspectives in social ontology, the kind of objectivity attributed to mathematical knowledge, the type of representational or non-representational semantics adopted, and the justificatory or coordinative role entrusted to axiomatics. After a brief introduction to key issues in social ontology, Sect. 3 of the chapter offers a survey of contributions by Feferman, Ferreirós, and Cole, highlighting a difference between approaches based on rules, practices, and intentional states, respectively. Section 4.1 also discusses results by Carter, Collin, Giardino, and Pantsar, focusing on differences between the objectivity of knowledge and objectivity of objects and showing that many authors combine a realist, structuralist, and pragmatist perspective, as well as the idea that objectivity comes in degrees. Section 4.2 focuses on the kind of semantics that is adopted in socially oriented approaches. A representational semantics is preferred by authors grounding their views on mental states, whereas a non-representational semantics best fits with views based on practices. Section 5 considers how axiomatics can be understood in a social perspective. Axiomatics generally plays a justificatory role also in theories that aim to explain the social constitution of mathematical knowledge. Yet, if one shifts from approaches based on mental states to approaches anchored on rules or habits, axiomatics can be viewed as an institution based on obligations, functions, coordination problems, and agent’s actions and roles, thus playing an organizational and coordination role.
This is a preview of subscription content, log in via an institution to check access.
Notes
- 1.
- 2.
See, for example, Gandon’s (2013) criticism.
- 3.
On the effects that this change of viewpoint has on mathematical objectivity, see the TOPOI special issue “Mathematical practice and social ontology,” where different degrees of objectivity are compared (Cantù and Testa 2023).
- 4.
- 5.
A notable exception to the investigation of hypothetico-deductive axiomatics as a mainly formal activity is Schlimm, who, quoting Bourbaki, claims that formalization is just one aspect and the least interesting of axiomatics (Schlimm 2013, 41).
- 6.
We will not discuss the work of Paul Ernest either, who, like Hersh, analyzed the contributions of Maverick authors, but was more interested in discursive practices and in characterizing mathematical knowledge. Adopting Bloor’s theory of objectivity, he distinguishes between explicit and implicit mathematical knowledge, claiming that both produce an objective knowledge, i.e., a knowledge that is shared and intersubjective, deployed and used in public, for persons to witness (Ernest 1998: 144ff). For recent developments of Ernest’s views, see chapter “The Ethics of Mathematical Practice,” this volume.
- 7.
See Ferreirós (2022) for an explicit discussion of Feferman’s conceptual structuralism, in which, however, the author explicitly distances himself from the idea that the kind of objectivity proper to mathematics, called robust, is akin to the objectivity of social facts.
- 8.
On collective agency we refer to Yacin Hamami’s article “Agency in Mathematical Practice,” this volume. On language, symbolization, and artifacts, see Valeria Giardino 2018b and 2023. For further directions of research see also M. Kremakova’s article “The Sociology of Mathematical Practice,” this volume and the paper by M. Inglis and K. Weber “Education and Mathematical Practice,” this volume.
- 9.
According to Searle’s correspondentist theory of truth, an utterance is true if and only if it corresponds to the fact that makes it true. However, with Strawson, Searle agrees that facts cannot be considered as groups or agglomerations of things and that a picture theory of meaning, according to which sentences have the meanings they do because they are conventionalized pictures of facts, does not work. Facts are conditions in the world that satisfy the truth conditions of the statements.
- 10.
For example, to discover the function of the heart is to discover causal processes but also to place them in a teleological framework, which distinguishes hearts that function well from hearts that function poorly, depending on how they contribute to the maintenance of what we assume to be the highest value: life (Searle 1995: 14–15).
- 11.
To compare social facts and mathematical facts, it is useful to have an idea of several phenomena peculiar to social facts that Searle intends to explain with his theory (although they do not apply indiscriminately to all social facts): (1) self-referentiality (for something to fall under the concept of money, it must be believed to be used or be regarded as money, which is also expressed by saying that being a type of money is a constitutive property of money); (2) the creation by explicit performative utterances as in the example “I appoint you chairman”; (3) the existence of social facts on top of brute facts, which are logically prior; (4) the existence only in a set of systematic relations to other facts (e.g., money cannot be conceived independently of some exchange system and property ownership); (5) social objects are just the continuous possibility of a social activity, because the process is prior to the product; and (6) language is a constitutive property of social facts (Searle 1995: 33–37).
- 12.
The token-type distinction could be applied to mathematics, asking whether and how the number type differs from tokens related to specific number systems. Applying this theory to mathematical knowledge, it would then be a matter of explaining that some extension of the number system is non-arbitrary, not so much because it is the result of constraints imposed by status functions but because it results from a strategic interaction between agents guided by motivations, incentives, and expectations.
- 13.
An interesting research direction would be to compare abstraction procedures in mathematics with the way in which constitutive rules are obtained from regulatory rules through the introduction of theoretical terms.
- 14.
On the relationship between structuralism and mathematical practice, see Reck and Schiemer (2020), presenting in the first part mathematical practices related to structuralism from Grassmann to McLane. See also Cantù and Patras (2023) and Carter (2008). Carter distinguishes different ways in which structures operate in mathematics: structures are not only objects of study but also ways of studying sets initially conceived as devoid of structure, of which one investigates whether they have or can be given one or more structures. Cantù and Patras claim that structures often play a dual role in axiomatic practice: e.g., in Bourbaki (1948) the notion of structure plays the role of an epistemic object (formalized in the theory through an axiomatic definition) as well as the role of an epistemic concept (used in the architectural construction of the axiomatic framework).
- 15.
We will return to this point in the next section, but it is important to note that axiomatization is conceived by Feferman as a formal activity that presupposes the choice of one language and one logic and whose goal is basically to transform undefined into defined statements.
- 16.
On Dingler see also Gödel (1932).
- 17.
See also Heinzmann and Wolters (2021).
- 18.
- 19.
See, for example, Veronese (1909) and Veronese and Gazzaniga (1895–97), discussed in Cantù (2000). Another example suggested to me by Dirk Schlimm, whom I thank for several precious comments to a preliminary version of this chapter, would be Moritz Pasch, who claimed that both arithmetic and geometry can be constructed in stages from some practical activity. See in particular Pasch 1923.
- 20.
The distinction between the two is taken to be fuzzy, but symbolic frameworks generally contain ideograms, technical expressions, and symbolic methods (symbolic frameworks need not be only formal nor only linguistic), while theoretical frameworks include statements or propositions, proofs, theoretic or proof methods, and open questions (Ferreirós 2016: 55).
- 21.
One of Ferreirós’s goals is precisely to avoid the identification between frameworks and paradigms or disciplinary matrices à la Kuhn: that is why he claims that different frameworks can coexist in the same historical period and even in the same agent (Ferreirós 2016: 37).
- 22.
- 23.
Indeed, the biological and cognitive component of human abilities is included – it was recalled as one way to explain the cultural invariance of mathematics in Ferreirós 2005 (p. 64) – but the historical dimension clearly prevails. For example, Ferreirós claims that the number concept is a cultural product, even if it is based on some specific cognitive abilities of the human species (Ferreirós 2016: 68). Besides, the second part of the volume on the interplay of practices is entirely dedicated to historical case studies ranging from ancient China to contemporary mathematics. Furthermore, the term cognitive is used in such a broad sense (basic abilities such as perception, memory, and language) that cultural components also enter into such abilities (Ferreirós 2016: 77).
- 24.
As in the pragmatist tradition, meaning is elucidated in terms of use, and mathematical understanding in terms of elementary practices and cognitive roots, but Ferreirós tries at all costs to avoid reductionism, suggesting that it is a form of substantiation (Ferreirós 2016: 5).
- 25.
See Ferreirós (2016, p. 14). In other words, Ferreirós assumes that there is a difference between mathematical and physical experiments and retains a version of the indispensability thesis, even if the emphasis on the interplay between mathematical and non-mathematical practices apparently dissolves the problem of applicability.
- 26.
Ferreirós, unlike the enculturation tradition (see Pantsar 2023 for a review of enculturation studies), does not provide an explanation of this difference based on cognitive science. He does not attempt to explain but merely notes this difference, suggesting that historical evidence supports a cross-cultural universality of mathematics (Ferreirós 2023).
- 27.
Epstein distinguishes three main kinds of grounding: (a) cognitive states, as in the theories of Searle and Levi-Strauss; (b) practices, considered as embodied routines in the social world combined with physical actions and symbolic systems (Bourdieu) or in the interaction with social environment (Giddens and Sewell); and (c) functions, as in Millikan and Boyd’s theory (Epstein 2016: 245–246).
- 28.
“Mathematics is not given a priori, but is the ‘science that is sought’ (to abuse words of Aristotle), constantly being reshaped and reconstructed.” (Ferreirós 2016: 314).
- 29.
On the analysis of the relationship between the practical turn in philosophy of mathematics and developments in pragmatism as a philosophical method, a topic we cannot develop here, see Heinzmann (2015).
- 30.
Of particular interest for a comparison with mathematics is Searle’s claim that is possible to reiterate the imposition of status functions, affirming formulas in which X occurs as Y in some preceding formula: for example, when I affirm that “only a citizen of the United States can become President.” Another point he makes – and that might be relevant in a perspective, like that of Ferreirós or Carter, based on an interplay of practices – is that the persistence of a social fact depends on the possibility of continuously reactivating not only a single social activity but also its interactions with other social activities (Searle 1995: 80–81).
- 31.
First, one might ask why one should use the tools of social ontology to investigate axiomatics in logic and mathematics. The chosen ontology might have consequences on the kind of problems and puzzles one must solve in the development of a given science. Taking axiomatics to be a method or a system is not the same as taking it to be a mode of discourse or symbolic order; an informal institution based on norms, conventions, customs, laws, organizations, groups, and roles; or an entity sustained in existence by collective recognition and serving status functions. The explanation of axiomatics’ persistence in time has been partly investigated by an analysis of the function of the axiomatic method, and eventually of its changes in time, but has not been related to an analysis of its specific constituents.
- 32.
Even if Lorenzen’s operative approach is non-axiomatic, it is “compatible with the axiomatic approach in its non-fundamentalist interpretation” (Heinzmann 2021: 15).
- 33.
A good example is Walton’s New Dialectic (1998).
- 34.
Similarly, Otto Hölder (1924) considers that mathematical logic should be a normative logical theory that is derived from the investigation of mathematics itself, as in Lambert’s unachieved anatomy of mathematics.
- 35.
- 36.
For an epistemological approach to mathematics, see, e.g., De Toffoli (chapter “The Epistemological Subject(s) of Mathematics,” this volume).
- 37.
See, e.g., Hamami (chapter “Agency in Mathematical Practice,” this volume) and Hamami and Morris (2020a).
- 38.
- 39.
See, e.g., Tanswell and Inglis (chapter “The Language of Proofs: A Philosophical Corpus Linguistics Study of Instructions and Imperatives in Mathematical Texts,” this volume), Schlimm and González (chapter “Mathematical Experiments on Paper and Computer,” this volume), van Bendegem JP (chapter “Experiments in Mathematics: Fact, Fiction, or the Future?,” this volume), and more generally the section on experimental mathematics in this volume.
- 40.
References
Aberdein A, Rittberg CJ, Tanswell FS (2021) Virtue theory of mathematical practices: an introduction. Synthese 199(3–4):10167–10180
Arrighi C, Cantù P, de Zan M, Suppes P (eds) (2010) Logic and pragmatism: selected essays by Giovanni Vailati. Center for the Study of Language and Information, Stanford
Aspray W, Kitcher P (1988) History and philosophy of modern mathematics. University of Minnesota Press, Minneapolis
Benacerraf P (1965) What numbers could not be. Philos Rev 74:47–73
Bourbaki N (1948) L’architecture des mathématiques. In: Le Lionnais F (ed) Les grands courants de la pensée mathématique. Cahiers du Sud, Paris
Brandom RB (1998) Making it explicit reasoning, representing, and discursive commitment. Harvard University Press, Cambridge, MA
Bratman ME (2013) Shared agency: a planning theory of acting together. Oxford University Press, New York
Cantù P (2000) L’insegnamento della geometria nelle scuole medie inferiori. Una lettera inedita di Giuseppe Veronese a Giovanni Vailati. Il Voltaire 5:109–118
Cantù P (2021) Peano’s philosophical views between structuralism and logicism, Chapter 9. In: Boccuni F, Sereni A (eds) Origins and varieties of logicism. A foundational journey in the philosophy of mathematics. Routledge, New York
Cantù P (2022) What is axiomatics? Ann Math Philos 1(1):1–24
Cantù P, Patras F (2023) Les structures bourbakistes: objets ou concepts épistémiques ? Philos Sci 27(2):3–29
Cantù P, Testa I (2006) Teorie dell’argomentazione. Un’introduzione storico- filosofica alle logiche del dialogo. Testi e Pretesti. Bruno Mondadori, Milano
Cantù P, Testa I (2023) Introduction: from social ontology to mathematical practice, and back again. Topoi 42(1):187–198
Carter J (2005) Motivations for realism in the light of mathematical practice. Croat J Philos 5(1):17–29
Carter J (2008) Structuralism as a philosophy of mathematical practice. Synthese 163(2):119–131
Carter J (2019) Philosophy of mathematical practice – motivations. Themes Prospects Philos Math 27(1):1–32
Carter J (2023) Mathematical practice, fictionalism and social ontology. Topoi 42(1):211–220
Cole JC (2008) Mathematical domains: social constructs? In: Gold B, Simons R (eds) Proof and other dilemmas: mathematics and philosophy. Mathematics Association of America, Washington, DC, pp 109–128
Cole JC (2010) Mathematical structuralism today. Philos Compass 5(8):689–699
Cole JC (2013) Towards an institutional account of the objectivity, necessity, and atemporality of mathematics. Philos Math 21(1):9–36
Cole JC (2015) Social construction, mathematics, and the collective imposition of function onto reality. Erkenntnis 80(6):1101–1124
Cole J (2017) Humanism about abstract objects. In: Sriraman B (ed) Humanizing mathematics and its philosophy: essays celebrating the 90th birthday of Reuben Hersh. Birkhäuser, Cham, pp 151–165
Cole JC (2023) Some preliminary notes on the objectivity of mathematics. Topoi 42(1):235–245
Collin JH (2023) Of marriage and mathematics: Inferentialism and social ontology. Topoi 42(1):247–257
Curry H (1951) Outlines of a formalist philosophy of mathematic. North-Holland Publishing Company, Amsterdam
De Toffoli S, Giardino V (2014) Forms and roles of diagrams in knot theory. Erkenntnis 79(4):829–842
Dingler H (1931) Philosophie der Logik und Arithmetik. Reinhardt, München
Dutilh Novaes C (2020) The dialogical roots of deduction: historical, cognitive, and philosophical perspectives on reasoning. Cambridge University Press, Cambridge
Dutz J, Schlimm D (2021) Babbage’s guidelines for the design of mathematical notations. Stud Hist Phil Sci 88:92–101
Epstein B (2014) How many kinds of glue hold the social world together? In: Galloti M, Michael J (eds) Social ontology and social cognition. Springer, Dordrecht
Epstein B (2016) Social ontology. In: McIntyre AR (ed) The Routledge companion to philosophy of social science. Routledge philosophy companions. Routledge, New York, pp 240–253
Epstein B (2018) Social ontology. In Zalta EN (ed) The stanford encyclopedia of philosophy. Metaphysics Research Lab, Stanford University
Ernest P (1998) Social constructivism as a philosophy of mathematics. State University of New York Press, New York
Feferman S (2011) Is the continuum hypothesis a definite mathematical problem? Manuscript available at http://logic.harvard.edu/EFI_Feferman_IsCHdefinite.pdf
Feferman S (2014) Logic, mathematics, and conceptual structuralism. In: Rush P (ed) The metaphysics of logic. Cambridge University Press, Cambridge, UK, pp 72–92
Ferreirós J (2005) Peano and Dedekind. In Grattan-Guinness I (ed) Landmark writings in western mathematics 1640–1940. Elsevier, Amsterdam, San Diego, Oxford and London, pp 613–626
Ferreirós J (2016) Mathematical knowledge and the interplay of practices. Princeton University Press, Princeton
Ferreirós J (2022) Conceptual structuralism. J Gen Philos Sci 54:125
Ferreirós J (2022b) The two sides of modern axiomatics: Dedekind and Peano, Hilbert and Bourbaki. In: Ferreira F, Kahle R, Sommaruga G (eds) Axiomatic thinking I. Springer, Cham, pp 83–103
Ferreirós J (2023) Degrees of objectivity? Mathemata and social objects. Topoi 42(1):199–209
Gandon S (2013) Quelle philosophie pour quelle mathématique ? Arch Philos 76(2):197–216
Giardino V (2017) The practical turn in philosophy of mathematics: a portrait of a young discipline. Philos Mind 12:18–28
Giardino V (2018a) Tools for thought: the case of mathematics. Endeavour 2(42):172–179
Giardino V (2018b) Manipulative imagination: how to move things around in mathematics. Theoria 33(2):345–360
Giardino V (2023) The practice of mathematics: cognitive resources and conceptual content. Topoi 42(1):259–270
Giardino V, Greenberg G (2015) Introduction: varieties of iconicity. Rev Philos Psychol 6(1):1–25
Gödel K (1932) Review of Dingler 1931. Monatshefte Math Phys 89:3
Guala F (2016a) Understanding institutions. Princeton University Press, Princeton
Guala F (2016b) Naturalism and anti-naturalism in the philosophy of social science. In: Humphreys P (ed) The Oxford handbook of philosophy of science. Oxford University Press, New York, pp 43–61
Hamami Y, Morris R (2020a) Plans and planning in mathematical proofs. Rev Symb Log 14(4):1030–1065
Hamami Y, Morris RL (2020b) Philosophy of mathematical practice: a primer for mathematics educators. ZDM 52:1113–1112. https://doi.org/10.1007/s11858-020-01159-5
Hartimo M, Rytilä J (2023) No magic: from phenomenology of practice to social ontology of mathematics. Topoi 42(1):283–295
Haslanger S (1995) Ontology and social construction. Philos Top 23(2):95–125
Heinzmann G (2015) Pragmatism and the practical turn in philosophy of mathematics: explanatory proofs. In: Agazzi E, Heinzmann G (eds) The practical turn in philosophy of science. Franco Angeli, Milan, pp 113–129
Heinzmann G (2021) Operation and predicativity: Lorenzen’s approach to arithmetic. In Heinzmann G, Wolters G (eds) Paul Lorenzen – mathematician and logician, logic, epistemology, and the unity of science. Springer International Publishing, pp 11–22
Heinzmann G, Wolters G (eds) (2021) Paul Lorenzen – mathematician and logician. Springer International Publishing, Cham
Hersh R (1997) What is mathematics, really? Oxford University Press, New York
Hindriks F, Guala F (2015) Institutions, rules, and equilibria: a unified theory. J Institutional Econ 11(3):459–480
Hölder O (1924) Die mathematische Methode. Springer, Berlin
Inglis M, Aberdein A (2015) Beauty is not simplicity: an analysis of mathematicians’ proof appraisals. Philos Math 23(1):87–109
Kamlah W, Lorenzen P (1967) Logische Propädeutik: Vorschule des vernünftigen Redens, 3rd 1996 ed. Springer. Stuttgart-Weimar
Kitcher P (1984) The nature of mathematical knowledge. Oxford University Press, New York
Lakoff G, Núñez RE (2000) Where mathematics comes from how the embodied mind brings mathematics into being. Basic Books, New York
Linnebo Ø (2018) Thin objects: an abstractionist account. Oxford University Press, Oxford
Lorenzen P (1955) Einführung in die operative Logik und Mathematik. Springer, Berlin
Mancosu P (ed) (2008) The philosophy of mathematical practice. Oxford University Press, Oxford
Pantsar M (2023) From maximal intersubjectivity to objectivity: an argument from the development of arithmetical cognition. Topoi 42(1):271–281
Peano G (1892) Sopra la raccolta di formule di matematica. Rivista di Matematica 2:76–77
Reck EH, Schiemer G (eds) (2020) The prehistory of mathematical structuralism. Oxford University Press, New York
Schlimm D (2013) Axioms in mathematical practice. Philos Math 21(1):37–92
Schlimm D (2018) On Frege’s Begriffsschrift notation for propositional logic: design principles and trade-offs. Hist Philos Logic 39(1):53–79
Schlimm D (2021a) Peano’s dot notation and design principles for notation. In The Peano school: epistemology: logic, epistemology and didactics, Special Issue ed. by P. Cantù and E. Luciano, Philosophiae Scientiae 25(1):95–126
Schlimm D (2021b) How can numerals be iconic? more varieties of iconicity. In Basu A, Stapleton G, Linker S, Legg C, Manalo E, Viana P (eds) Diagrammatic representation and inference. 12th International Conference, Diagrams 2021, Virtual, Proceedings. Springer, pp 520–528
Searle JR (1995) The construction of social reality. Free Press, New York
Searle JR (2005) What is an institution? J Inst Econ 1(1):1–22
Sieg W (2014) The ways of Hilbert’s axiomatics: structural and formal. Perspect Sci 22(1):133–157
Sundholm G (2002) What is an expression. In: LOGICA yearbook 2001. Filosofia Publishers, Czech Academy of Science, Prague, pp 181–194
Testa I (2017) Dewey’s social ontology. A pragmatist alternative to Searle’s approach to social reality. Int J Philos Stud 25(1):40–62
Testa I (2020) A habit ontology for cognitive and social sciences methodological individualism, pragmatist interactionism, and 4e cognition. In: Caruana F, Testa I (eds) Habits: pragmatist approaches from cognitive science, neuroscience, and social theory. Cambridge University Press, Cambridge, pp 269–295
van Bendegem JP (2016) Philosophy of mathematical practice: what is it all about? In: Paul E (ed) The philosophy of mathematics education. Springer, New York, pp 13–18
van Bendegem JP (2023) Experiments in mathematics: fact, fiction or the future? In Sriraman B (ed) Handbook of the History and Philosophy of Mathematical Practice, Springer
Veronese G (1909) Elementi di geometria intuitiva ad uso delle scuole tecniche. Drücker, Padova
Veronese G, Gazzaniga P (1895–97) Elementi di Geometria, ad uso dei licei e degli istituti tecnici (primo biennio). Drucker, Verona/Padova
Walton DN (1998) The new dialectic: conversational contexts of argument. University of Toronto Press, Toronto
Acknowledgments
I would like to thank Italo Testa, because without our discussions the project of comparing social ontology and mathematical practice would never have come into being, and also the institutions that made it possible to organize two workshops on the topic (the Faculty of Arts at the University of Aix-Marseille and the Department of Humanities at the University of Parma), the participants in these workshops, two referees very generous with valuable advice, and numerous colleagues (especially Dirk Schlimm and Jessica Carter) who carefully read a preliminary version of this chapter.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2024 Springer Nature Switzerland AG
About this entry
Cite this entry
Cantù, P. (2024). The Social Constitution of Mathematical Knowledge: Objectivity, Semantics, and Axiomatics. 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_57
Download citation
DOI: https://doi.org/10.1007/978-3-031-40846-5_57
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-40845-8
Online ISBN: 978-3-031-40846-5
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering