Abstract
Easy-road mathematical fictionalists grant for the sake of argument that quantification over mathematical entities is indispensable to some of our best scientific theories and explanations. Even so they maintain we can accept those theories and explanations, without believing their mathematical components, provided we believe the concrete world is intrinsically as it needs to be for those components to be true. Those I refer to as “mathematical surrealists” by contrast appeal to facts about the intrinsic character of the concrete world, not to explain why our best mathematically imbued scientific theories and explanations are acceptable in spite of having false components, but in order to replace those theories and explanations with parasitic, nominalistically acceptable alternatives. I argue that easy-road fictionalism is viable only if mathematical surrealism is and that the latter constitutes a superior nominalist strategy. Two advantages of mathematical surrealism are that it neither begs the question concerning the explanatory role of mathematics in science nor requires rejecting the cogency of inference to the best explanation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For an overview of such arguments see (Colyvan 2019).
This is the program laid out in (Field 1980).
See Urquhart (1990) for reasons to doubt that Field’s (1980) hard-road nominalization strategy can be successfully applied to the theory of general relativity. See (Malament 1982) for reasons to doubt that it can be successfully applied to quantum mechanics, (Balaguer 1998, ch. 6) for a response, and (Bueno 2002) for a counter response.
The terminological distinction between “hard-road” and “easy-road” nominalist strategies is introduced by Colyvan (2010). While Colyvan’s distinction applies to a wider class of views, this paper is concerned with the contrast insofar as it applies to versions of mathematical fictionalism.
While Balaguer and Yablo both defend the feasibility of what might be classified as versions of easy-road mathematical fictionalism, they do so not in service of nominalism, but rather the view that there is no fact of the matter concerning whether mathematical entities exist.
The label is adapted from Leplin (1987), who coined the term ‘surrealism’ as a label for scientific anti-realist views that attempt to explain the success of science by positing that the non-observable portion of the world is constituted so as to make the observable portion behave as if the theories in question were true.
Modal structuralist views such as those advocated by Hellman (1989) and Horgan (1984, 1987), as well as the constructibilism of Chihara (2004), might also be construed along surrealist lines. Although Hellman regards his view as a realist one, and Chihara takes a neutral stance regarding the content of mathematical theories as they are actually used.
The mathematical surrealists’ suggestions are “paraphrase strategies” not in the sense that they supply us with semantically or logically equivalent formulations of the original theories, but in the sense that they offer a way of taking mathematically imbued theories and systematically generating nominalistically acceptable alternatives.
Leng (2010: 10–12) characterizes her position as a version of scientific anti-realism. But she is a nominalistic scientific realist in Balaguer’s (1998: 131) sense (i.e. she believes that the concrete world is intrinsically as it needs to be in order for our best scientific theories and explanations to be at least approximately correct, in spite of the fact that the abstract entities to which they are committed do not exist).
In the original quotation this claim is not asserted but embedded in a question. It is clear from the ensuing discussion however that Leng endorses it.
This is how Baker (2009: 628) reads her 2005 account.
Yablo (2005) also appeals to Walton’s account.
Of course, mathematical fictionalists will not want simply to identify the relevant notion of consistency with the model-theoretic notion, since that would commit them to the existence of sets. Here Leng (2010: 56–57, 97–98) follows Field (1989: 30–38) in taking the relevant notion of consistency to be a primitive one, albeit one that extensionally coincides with the model-theoretic notion (at least for standard first-order logic) on the hypothesis that there are the sort of set-theoretic entities invoked in standard treatments of metalogic.
Colyvan (2010: 297) offers a similar criticism of Melia’s view.
Or rather, they are nominalistically acceptable on the assumption that there are nominalistically kosher ways of stating metalogical claims pertaining to logical consequence. See notes 16, 29 and Sect. 5 for further discussion.
Since I think we have a sufficiently intuitive grasp of what it is for a claim to be “solely about the concrete world” in this sense (and since, in any case, this is a notion that easy-road fictionalists require), I will have little by way of explicit things to say concerning which claims belong in this class. One thing that I will be assuming in the arguments that follow however is that this class is closed under possible negation. That is, I will be assuming that for any claim P*, if P* is solely about the concrete world, and it is possible that ~ P* is true, then ~ P* is also solely about the concrete world. I believe this assumption to be evident upon reflection. If whether P is true is determined solely by the intrinsic character of the concrete world, but it is possible for P to be false, then whether ~ P is true must also be so determined.
I will be assuming that the uninterpreted logical language with which we are working is of some standard variety (suited for mathematical applications within science) except for being supplemented by the following two additional features: First, the language somehow marks syntactically those sentences that are to be regarded as expressing propositions solely about the concrete world (with permissible interpretations constrained accordingly). Second, the ‘CAST(…)’ operator is to be regarded as part of the language’s logical vocabulary with permissible interpretations of sentences involving it constrained by the truth condition for that operator (as stated in the next paragraph of the main text).
It is worth noting that this formulation does not require that we have in our grasp some single proposition that corresponds to set theory, which may not be the case. (There may not be such a proposition because set theory is not finitely axiomatizable, or because not all relevant instances of the axiom schemas are stateable in our vocabulary, or perhaps because the non-existence of sets renders some of the vocabulary of set theory semantically defective). Nevertheless, there will be a class of sentences in the uninterpreted logical language in which we are working that constitute the theorems of set theory. And we may say that some claims P1, P2, … jointly logically implyP the falsity of set theory when there are some sentences SP1*, SP2*, … in the language that jointly logically implyS the falsity of some of the theorems of set theory and which under some permissible interpretation express P1*, P2*, … (respectively). We may also say that some claims P1*, P2*, … Q* together with set theory jointly logically implyP some claim Q* when there are some sentences in the language SP1*, SP2*, … SQ* which under some permissible interpretation express P1*, P2*, … Q* (respectively) and SP1*, SP2*, … together with the theorems of set theory logically implyS SQ*.
This follows, at any rate, given the assumption I said I would be making in note 21, namely that the class of claims solely about the concrete world is closed under possible negation (i.e. for any claim P*, if P* is solely about the concrete world and it is possible that ~ P* is true, then ~ P* is also solely about the concrete world). Proof: Suppose that P is solely about the concrete world and that CAST(P) is true. Assume for reduction that ~ P is true. Since ~ P is true, it is possible that ~ P is true. So since P is solely about the concrete world and the class of such claims is closed under possible negation, it follows that ~ P is solely about the concrete world. Since it is true that CAST(P), it follows from TC that there are no true claims solely about the concrete world that jointly logically imply the falsity of set theory. It also follows, however, that there are some true claims solely about the concrete world that together with set theory jointly logically imply P. Suppose that Q1, Q2,… are such claims. Note that since Q1, Q2,… together with set theory jointly logically imply P, it follows that Q1, Q2,… together with ~ P jointly logically imply the falsity of set theory. But since Q1, Q2, … and ~ P are each true claims that are solely about the concrete world, it follows from this that there are in fact some true claims solely about the concrete world that jointly logically imply the falsity of set theory, contrary to what was previously established. So by reductio ad absurdum we may conclude that P is true. By conditionalization, generalization from the arbitrary case, and necessitation, we may further conclude that necessarily, for any claim P* that is solely about the concrete world, if CAST(P*) is true then P* is true.
Bangu (2008) criticizes Baker’s cicada example for having an explanandum that does not pertain solely to physical(/biological) phenomena but also makes reference to mathematical entities and properties.
It is not (non-vacuously) true in general, however, that for any given mathematical claim, M*, if M* is true then so is CAST(M*). That will not be (non-vacuously) true, for example, for cases in which the mathematical claim in question is one concerning which set theory is silent (such as the continuum hypothesis).
Q may not be not logically equivalent to CAST(Q) since there may be possible worlds in which Q is true but the intrinsic character of the concrete world precludes the truth of set theory. But since we are taking it for granted as part of the assumed background information that set theory is adequate for the purposes of science, we are also taking it to be part of the assumed background information that the truth of set theory is not actually so precluded.
Field (1989: 30–31, 1991, 1994) addresses a similar worry by employing a primitive logical implication operator in place of quantifying over model-theoretic entities and by taking a deflationary stance toward quantification over proposition-like entities. These strategies are in turn endorsed by Leng (2010: 51–57).
It should be noted that Leng herself does not believe that an easy-road fictionalist strategy is required to handle the sort of apparent quantification over abstract entities that occurs in the context of metalogic (see note 29).
References
Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114, 223–238.
Baker, A. (2009). Mathematical explanation in science. British Journal for the Philosophy of Science, 60, 611–633.
Baker, A. (2016). Parsimony and inference to the best mathematical explanation. Synthese, 193, 333–350.
Baker, A. (2017a). Mathematics and explanatory generality. Philosophia Mathematica, 25, 194–209.
Baker, A. (2017b). Mathematical spandrels. Australasian Journal of Philosophy, 95, 779–793.
Balaguer, M. (1998). Platonism and anti-platonism in mathematics. New York: Oxford University Press.
Balaguer, M. (2009). Fictionalism, theft, and the story of mathematics. Philosophia Mathematica, 17, 131–162.
Bangu, S. I. (2008). Inference to the best explanation and mathematical realism. Synthese, 160, 13–20.
Berenstain, N. (2017). The applicability of mathematics to physical modality. Synthese, 194, 3361–3377.
Bueno, O. (2002). Is it possible to nominalize quantum mechanics? Philosophy of Science, 70, 1424–1436.
Chihara, C. S. (2004). A structural account of mathematics. New York: Oxford University Press.
Colyvan, M. (1999). Confirmation and indispensability. Philosophical Studies, 96, 1–19.
Colyvan, M. (2001). The indispensability of mathematics. New York: Oxford University Press.
Colyvan, M. (2002). Mathematics and aesthetic considerations in science. Mind, 111, 69–74.
Colyvan, M. (2006). Scientific realism and mathematical nominalism: A marriage made in hell. In C. Cheyne & J. Worrall (Eds.), Rationality and reality: Conversations with Alan Musgrave (pp. 225–237). Dordrecht: Springer.
Colyvan, M. (2010). There is no easy road to nominalism. Mind, 119, 285–306.
Colyvan, M. (2019). Indispensability arguments in the philosophy of mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved from September 27, 2019, (https://plato.stanford.edu/entries/mathphil-indis/).
Dorr, C. (2008). There are no abstract objects. In T. Sider, J. Hawthorne, & D. W. Zimmerman (Eds.), Contemporary debates in metaphysics (pp. 32–63). Malden: Blackwell Publishing.
Dorr, C. (2010). Of numbers and electrons. Proceedings of the Aristotelian Society, 110, 133–181.
Field, H. (1980). Science without numbers: A defense of nominalism. Princeton: Princeton University Press.
Field, H. (1989). Realism, mathematics, and modality. New York: Basil Blackwell.
Field, H. (1991). Metalogic and modality. Philosophical Studies, 62, 1–22.
Field, H. (1994). Deflationist views of meaning and content. Mind, 103, 249–285.
Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. New York: Oxford University Press.
Horgan, T. (1984). Science nominalized. Philosophy of Science, 51, 529–549.
Horgan, T. (1987). Science nominalized properly. Philosophy of Science, 54, 281–282.
Leng, M. (2005). Mathematical explanation. In C. Cellucci & D. Gillies (Eds.), Mathematical reasoning and heuristics (pp. 167–189). London: King’s College Publications.
Leng, M. (2010). Mathematics and reality. New York: Oxford University Press.
Leng, M. (2012). Taking it easy: A response to Colyvan. Mind, 121, 983–995.
Leng, M. (2017). Reasoning under a presupposition and the export problem: The case of applied mathematics. Australasian Philosophical Review, 1, 133–142.
Leplin, J. (1987). Surrealism. Mind, 96, 519–524.
Lyon, A. (2012). Mathematical explanations of empirical facts, and mathematical realism. Australasian Journal of Philosophy, 90, 559–578.
Lyon, A., & Colyvan, M. (2008). The explanatory power of phase spaces. Philosophia Mathematica, 16, 227–243.
Maddy, P. (1997). Naturalism in mathematics. New York: Oxford University Press.
Malament, D. (1982). Review of field. Journal of Philosophy, 79, 523–534.
Melia, J. (1995). On what there’s not. Analysis, 55, 223–229.
Melia, J. (2000). Weaseling away the indispensability argument. Mind, 109, 455–479.
Melia, J. (2002). Response to Colyvan. Mind, 111, 75–79.
Melia, J. (2008). A world of concrete particulars. In D. W. Zimmerman (Ed.), Oxford studies in metaphysics (Vol. 4, pp. 99–124). New York: Oxford University Press.
Rosen, G. (2001). Nominalism, naturalism, epistemic relativism. Nous, 35(s15), 69–91.
Urquhart, A. (1990). The logic of physical theory. In A. D. Irvine (Ed.), Physicalism in mathematics (pp. 145–154). Norwell: Kulwer Academic Publisher.
Walton, K. L. (1990). Mimesis as make-believe: On the foundations of the representational arts. Cambridge: Harvard University Press.
Walton, K. L. (1993). Metaphor and prop oriented make-believe. European Journal of Philosophy, 1(1), 39–56.
Woodward, R. (2010). Fictionalism and inferential safety. Analysis, 70, 409–417.
Wright, C., & Hale, B. (1992). Nominalism and the contingency of abstract objects. The Journal of Philosophy, 89, 111–135.
Yablo, S. (2001). Go figure: A path through fictionalism. Midwest Studies in Philosophy, 25, 72–102.
Yablo, S. (2002). Abstract objects: A case study. Philosophical Issues, 12, 220–240.
Yablo, S. (2005). The myth of seven. In M. E. Kalderon (Ed.), Fictionalism in metaphysics (pp. 88–115). New York: Oxford University Press.
Yablo, S. (2012). Explanation, extrapolation, and existence. Mind, 121, 1007–1029.
Acknowledgements
Special thanks are owed to Sarah Boyce and to an anonymous referee for helpful comments on previous drafts.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Boyce, K. Mathematical surrealism as an alternative to easy-road fictionalism. Philos Stud 177, 2815–2835 (2020). https://doi.org/10.1007/s11098-019-01340-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11098-019-01340-x