Abstract
The author of “Parsimony and inference to the best mathematical explanation” argues for platonism by way of an enhanced indispensability argument based on an inference to yet better mathematical optimization explanations in the natural sciences. Since such explanations yield beneficial trade-offs between stronger mathematical existential claims and fewer concrete ontological commitments than those involved in (as it were) merely good mathematical explanations, one must countenance the mathematical objects that play a theoretical role in them via an application of the relevant mathematical results. The nominalist’s challenge is thus to undermine the platonistic force of such explanations by way of alternative nominalistic ones. The author’s contention is that such nominalistic explanations should provide a paraphrase of the proofs of the mathematical results being applied. There are reasons to doubt that proofs, construed here as formal derivations, actually contribute to the platonistc force to be undermined and, by parity, that nominalized proofs should bear responsability for the corresponding undermining. A discussion of two examples (Baker’s magicicadas, the hexagonal shape of honeycombs) and of associated arguments by Lange, Pincock, Steiner and Tallant, point to a a wealth of worries concerning the construal of this explanatory role. Among those figure the distinction between the weak and strong role of proofs, the distinction between causal or “ordinary” explanations and genuine mathematical ones, and the unifying role of optimization explanations. More generally, the very idea that the explanatory advantages yielded by applied mathematical claims may be construed as gradual or progressive and the associated notion that the feasibility of their nominalistic paraphrases decreases as the generality and force of these claim increases, deserves a closer attention.
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Notes
For the original distinction between the Easy Road and the Hard Road paths to nominalism, see Colyvan (1998).
The author doesn’t mention this particular line of argument. This might be because of the strength of Shapiro’s objections (Shapiro 1983) to Field’s programme (Field 1980). I suspect there is a better reason, from the author’s point of view, than Shapiro’s objection and what the author might consider weak in Field’s rejoinder to it in Field (1989). The programme does not consist, as Field himself insists, in a “reinterpretation” of applied mathematics but in an explanation of applicability that, on the contrary, takes the mathematics needed in a wide range of different kinds of applications at face-value. The author also favors the face-value approach, as opposed to an approach that would not take mathematics’ subject-matter to be inherently and constitutively abstract. My point here is that Field’s approach to the question of applicability counts as what the author judges to be holistic (Sect. 4, 2nd para) in the sense that, if it works, it does for all of applied mathematics by way of “nominalistic counterparts” (although not by way of counterparts obtained by means of a translation or by means of a paraphrase).
As far as holism is concerned in this context, I am thankful to an anonymous referee for pointing out the distinction between two components of Field’s programme: a global component and a local component. The global component is supplied by the conservativeness thesis proper defended in chapter 1 of Field (1980), along with the mathematical set-theoretic and proof-theoretic arguments offered in its support in the appendix of that chapter. The local component is supplied by the representation theorems needed for distinct nominalization cases, e.g for the case of geometry and distance (Field op. cit., ch. 3) and that of Newtonian space-time (Field op. cit., ch. 6), each theorem requiring different axioms for different theories. Insofar as they show that statements about space alone, free of reference to numbers, are equivalent to their abstract counterparts, which do talk about numbers (first case), or that the arbitrary coordinate system and distance function on which spatio-temporal relations depend are useful devices that help us derive “conclusions about spatio-temporal betweenness, simultaneity, and spatial congruence [...] which could be derived without ever bringing in numbers at all” (second case), the Hilbertian representation theorems show how the conclusion of the global component (the thesis that mathematics is conservative) might as it were be implemented or fleshed out locally. In Field’s view, all of mathematics required for application is indeed conservative; in that respect, his approach to nominalization could hardly be regarded as piecemeal. Nevertheless, as the referee correctly points out, the definitions, in empirically meaningful terms, of the differential equations of each of the theories that are relevant to the nominalization also belong to the local component of Field’s nominalistic programme. This gives us a cogent reason for not qualifying Field’s programme as genuinely holistic.
They may also surface at the level of the local implications of the results on which the explanation rests (although, obviously, not at the level of the purely mathematical implications of anything also purely mathematical that would be used to prove the needed results themselves), but I will not take into consideration the Piecemeal Hard Road nominalism agenda that strives to reproduce such implications of the mathematical results needed for the best mathematical explanation on a nominalistic basis. As the author points out (Sect. 4, 4th para), not only is the proposed nominalization weak and generality lost in such reproductions, but, along with them, the very strength of the original platonistic explanation. By contrast, generality is indeed taken into account by the two other kinds of piecemeal nominalistic approaches the author considers and that I therefore discuss critically infra in Sects. 2 and 3.
Strictly speaking, Lemma 1 makes no existential claims. Given Quine’s criterion, in order to qualify as a genuine assertion of the existence of mathematical objects one may furthermore decide to take at face-value, the lemma would have to be rephrased so that natural numbers, prime numbers, coprime numbers and lowest common multiples appear as values of variables bound by the objectual existential quantifier.
They do, e.g., play such a role in Field’s discussion of the usefulness of arithmetic but, in that particular case, to the practical disadvantage of the nominalistic counterparts (see Field 1980, ch. 2).
Notice in this respect that the questions the author considers in Sect. 5(a) in reference to the Honeycomb Theorem are all why questions. See Sect. 3 of my Comments for a discussion.
I am grateful to an anonymous referee for pointing out the relevance of Lange’s paper to the discussion. The more recent Lange (2014), where it is argued that there are differences in depth between distinct proofs of the same theorems, is also relevant.
See Bernays op.cit., p. 267.
References
Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114, 223–238.
Bernays, P. ([1935]1983). On platonism in mathematics. In P. Benacerraf & H. Putnam, Ed., Philosophy of mathematics—Selected readings, 2nd ed. (pp. 258–271) (C. D. Parsons, Eng. Trans.). Cambridge, MA: Cambridge University Press.
Colyvan, M. (1998). There is no easy road to nominalism. Mind, 119(474), 285–306.
Field, H. H. (1980). Science without numbers: A defence of nominalism. Princeton, NJ: Princeton University Press.
Field, H. H. (1989). Realism, mathematics and modality. Oxford: Basil Blackwell.
Lange, M. (2013). What makes a scientific explanation distinctively mathematical? British Journal for the Philosophy of Science, 64(3), 485–511.
Lange, M. (2014). Depth and explanation in mathematics. Philosophia Mathematica. doi:10.1093/philmat/nku022.
Pincock, C. (2012). Mathematics and scientific representation. Oxford: Oxford University Press.
Rizza, D. (2011). Magicicada, mathematical explanation and mathematical realism. Erkenntnis, 74, 101–114.
Shapiro, S. (1983). Conservativeness and incompleteness. Journal of Philosophy, 80(9), 521–531.
Steiner, M. (1978). Mathematics, explanation, and scientific knowledge. Nous, 12, 17–28.
Tallant, J. (2013). Optimus prime: Paraphrasing prime number talk. Synthese, 190, 2065–2083.
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Pataut, F. Comments on “Parsimony and inference to the best mathematical explanation”. Synthese 193, 351–363 (2016). https://doi.org/10.1007/s11229-015-0706-4
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DOI: https://doi.org/10.1007/s11229-015-0706-4