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Some Preliminary Notes on the Objectivity of Mathematics

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Abstract

I respond to a challenge by Dieterle (Philos Math 18:311–328, 2010) that requires mathematical social constructivists to complete two tasks: (i) counter the myth that socially constructed contents lack objectivity and (ii) provide a plausible social constructivist account of the objectivity of mathematical contents. I defend three theses: (a) the collective agreements responsible for there being socially constructed contents differ in ways that account for such contents possessing varying levels of objectivity, (b) to varying extents, the truth values of objective, socially constructed contents are constrained to be what they are, and (c) typically, socially constructed mathematical contents are objective and possess truth values that are highly constrained by the intended applications of the mathematical facets of reality that they represent.

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Notes

  1. Following Dieterle and Second Wave Feminists, throughout, when I refer to genders, I am referring to gender roles (e.g., man/boy and woman/girl) not gender identities.

  2. Henceforth, I abbreviate ‘propositional content’ by ‘content’.

  3. Versions of the ontological and semantic senses of the objective-subjective distinction are discussed by Searle (2010) under the labels ontological and epistemic. The epistemic sense of the objective-subjective distinction is discussed in certain Feminist critiques of science—see, e.g., Harding (1987).

  4. Throughout, I employ ‘grounded’ in a non-technical, everyday sense rather than with a technical meaning.

  5. I thank an anonymous referee for observing that, in the sense outlined in the next paragraph, mathematical facets of reality are also atypical with respect to social constructs for which collective agreements are partially responsible. Thus, it would be just as problematic to ascribe general features to social constructs for which collective agreements are partially responsible based on them being possessed by mathematical facets of reality as it would be to ascribe them based on those features being possessed by genders.

  6. Henceforth, I shorten ‘man/boy’ to ‘man’ and ‘woman/girl’ to ‘woman’.

  7. The documents in Yale’s Avalon Project (http://avalon.law.yale.edu/), which collects important historical and contemporary legal documents, demonstrate that laws against murder and theft are present in nearly all legal codes, including the earliest surviving such codes.

  8. For a deeper discussion of this broader context, the reader is referred to Cole (2015).

  9. This claim is famously defended by Searle (1995, 2010). The most general definition of a social construct is a facet of reality for which the collective imposition of function onto reality is responsible. We restrict our attention to social constructs whose core functions are, at least in part, not consequences of their physical characteristics.

  10. The core functions of a given social construct might be only a subset of the functions that it serves and can change over time.

  11. Dieterle (2010) fully recognized this and incorporated it into the challenge that she offered to social constructivists: “if [Cole’s] account of objectivity is weak enough to proclaim that ‘Women are more nurturing than men’ is ([semantically]) objective, then [semantic] objectivity does not give us the robust sense of objectivity we want for mathematical truths” (p. 326). The weak account of objectivity to which Dieterle refers is unconstrained semantic objectivity. She goes on to recognize that I need to argue that mathematical contents typically possess a more robust variety of objectivity, to wit, constrained semantic objectivity.

  12. It is worth noting that the account of the constraints on the truth values of mathematical contents that I provide in Sect. 9 explains why there is widespread agreement about which mathematical structures we should collectively agree exist. Thus, the discussion in Sect. 9 not only establishes that socially constructed mathematical contents typically possess strongly constrained semantic objectivity but explains why such contents typically possess such objectivity.

  13. Above, I have observed that there is an uncommonly high level of agreement among mathematicians about three things: which mathematical contents are consequences of which other mathematical contents, how best to formulate the axioms of a particular branch of mathematics, and how best to formulate key mathematical concepts. Each of these observations demands an explanation that falls beyond the scope of this paper.

  14. I thank Rytilä (2021) for clearly articulating these flaws.

  15. Thus, contra Cole (2015), facts about in which relations it is logically possible for facets of reality to stand are ontologically subjective rather than ontologically objective. Moreover, the model underwriting the constrained semantic objectivity of mathematical contents has only two levels rather than the three described in §5.2 of Rytilä (2021).

  16. Molinini (2020) recognizes this constraint on the truth values of mathematical contents when he argues that internal and external crosschecking contribute to the weak objectivity of mathematics.

  17. See, for instance, Kennedy (2011) and Feferman (2011).

References

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Acknowledgements

I thank Barbara Olsafsky for conversations that helped inform the content of this paper, two anonymous referees for helpful insights into how to improve this paper, and the numerous people with whom I have discussed mathematical social constructivism over the years for the assistance that they have provided in informing my thoughts on the topics addressed by this paper.

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Correspondence to Julian C. Cole.

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Cole, J.C. Some Preliminary Notes on the Objectivity of Mathematics. Topoi 42, 235–245 (2023). https://doi.org/10.1007/s11245-022-09855-5

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