Abstract
Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely mathematical, according to Baker (Mind 114(454):223–238, 2005; Br J Philos Sci 60(3):611–633, 2009). Furthermore, the result is then also used to strengthen the platonist position (e.g. Baker in Aust J Philos 95(4):779–793, 2017a). We pick up the circularity problem brought up by Leng (in: Cellucci, Gillies (eds) Mathematical reasoning, heuristics and the development of mathematics, King’s College Publications, London, pp 167–189, 2005) and Bangu (Synthese 160(1):13–20, 2008). We will argue that Baker’s attempt to solve this problem fails, if Hume’s Principle is analytic. We will also provide the opponent of the Enhanced Indispensability Argument with the so-called ‘interpretability strategy’, which can be used to come up with alternative explanations in case Hume’s Principle is non-analytic.
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Notes
In (2017a, pp. 783–784), Baker gives an alternative version of the argument. But the first three steps are the same as in the initial formulation. Baker points out that the length of the cicada life cycles can be expressed as sums of perfect squares, which introduces more sophisticated mathematics.
For completeness, both the 13- as well as the 17-year life cycles were mentioned in (D). However, to prevent too much repetition, in what follows only the first datum will be used.
We follow the notation of (Boolos 1998, p. 139)
In fact, any metaphysical theory that claims to be true should be compatible with a theory of truth of sentences, which depends on a theory of syntax, which in turn is hardly conceivable without some kind of theory of concatenation. See Tarski (1936).
Goodman and Quine (1947, p. 106) decline to assume that there are infinitely many objects. As it will turn out, QT has no finite models. So, QT goes beyond what they are willing to assume. However, other nominalists, notably Field, have no scruples in assuming that there are infinitely many objects.
Interestingly, Q is not interpretable in mereology, another favourite of nominalists. This is a consequence of the following theorem listed by Niebergall (2011): there is no consistent mereological theory in which AST is relatively interpretable. This is certainly relevant for the discussion about the evaluation of the nominalistic paraphrase of statements about primeness given by Tallant (2013), because he uses mereological fusions. Taking it further, he hopes that mereology will be enough to paraphrase all “scientifically explanatory mathematics”. The latter cannot include Robinson Arithmetic, if his project is to succeed.
Our strategy is in this sense similar to a strategy suggested by Bueno (2001, p. 119). His idea was for a nominalist to capitalize on the so-called reverse mathematics programme, initiated by Harvey Friedman, and aiming at discovering what the weakest axioms are to prove a theorem. There are a couple of differences though. One is that Bueno is aiming at a nominalistic reconstruction of as much of mathematics as possible, whereas we are only talking about what Tallant (2013, p. 2074) calls “scientifically explanatory mathematics”. Another difference is that Bueno focuses on abstraction principles, which he wants to paraphrase using a combination of plural logic and mereology.
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Acknowledgements
We would like to thank two anonymous reviewers for their very helpful comments. Earlier versions of this paper have been presented at the Paris-Leuven Workshop (Paris, 11 October 2018), the IX Conference of the Spanish Society for Logic, Methodology and Philosophy of Science (Madrid, 13 Nov 2018–16 Nov 2018), the seminar of the Philosophy department in Stellenbosch (Stellenbosch, 7 December 2019) and the CLPS Seminar (Leuven, 1 March 2019). We would like to thank all the audiences for their useful feedback. Finally, we would also like to thank Leon Horsten for discussing the topic of this paper with one of the authors.
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Heylen, J., Tump, L.A. The Enhanced Indispensability Argument, the circularity problem, and the interpretability strategy. Synthese 198, 3033–3045 (2021). https://doi.org/10.1007/s11229-019-02263-0
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DOI: https://doi.org/10.1007/s11229-019-02263-0