Abstract
In this chapter, we assess the latest developments of the debate around the “Enhanced Indispensability Argument” by Baker (Philos Math 25:194–209, 2017) and Knowles and Saatsi (Erkenntnis:1–19. https://doi.org/10.1007/s10670-019-00146-x, 2019) to conclude that neither part succeeds in their ambition to overtake the other.
We also aim to clarify what is at stake by looking at the metaphysical pictures that result from the criteria for ontological commitment following from the characterizations of substantial mathematical explanation provided by our authors.
We conclude that (i) the strategy of relying on independently established theories of explanation does not settle the issue. Rather, the choice of one or another theory of explanation, or the way in which the theory is interpreted, seems to be inspired by each author’s preferred metaphysical view on mathematical entities; (ii) The explanatory version of the indispensability argument will not take the realist further than its more traditional version since it fails to establish that mathematical entities make a difference. Nevertheless, our nominalists both fail to establish their view and to disallow a moderate version of realism like the one advanced by Baron (Br J Philos Sci 70(3):683–717. https://doi.org/10.1093/bjps/axx062, 2019). There seems to be no fact of the matter to decide between these last two views. Baker 2003, pace Philos Math 25:194–209, 2017, seems to be right, in that even if mathematics is indispensable, now explanatory indispensable, some version of “the status of No-Difference is unclear, perhaps irredeemably so”
This contribution has received financial support from FEDER/Spanish Ministry of Economy and Competitiveness under the project FFI2013-41415-P, and from FEDER/Spanish Ministry of Science, Innovation and Universities - State Research Agency under the project FFI2017-82534-P.
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Notes
- 1.
See the introduction to this volume for a brief but illuminating account; Colyvan 2001 is a classic source.
- 2.
As Baron 2016b (ft. 5) claims “the case for realism about mathematics is supposed to ‘piggy-back’ on the case for scientific realism. The case for scientific realism, however, is made through explanation. So, the case for mathematical realism should proceed via explanation as well.”
- 3.
See Mancosu 2018 for a general account of the debate.
- 4.
“To give the inevitable example: What is it about a square peg that allows it to slip into a round hole (Putnam 1995, Garfinkel 1981)? The peg’s microphysical make-up involves too much unneeded detail; it would still have fit, had it been made of copper. The peg fits if and because the sides are less than \( \sqrt{2} \) times as long as the radius of the hole.” Yablo 2012, 1020–21.
- 5.
This strategy is what we know as easy-road nominalism. Note that the easy-road is not a single account, but a series of them: Azzouni (1997, 2004, 2012), Balaguer (1996a, b, 1998a, b), Bueno (2012), Leng (2010, 2012), Liggins (2012), Maddy (1995, 1997), Melia (2000, 2002) and Yablo (2002, 2005, 2012), among others. What they all have in common is the rejection of mathematical Platonism with attempts to provide an easier path to nominalism than that of Field’s.
- 6.
Marcus noted in his Marcus 2018 the argument “leaves open the question of how one is supposed to determine the commitments of an explanation. EI[A] refers to the theoretical posits postulated by explanations but does not tell us how we are supposed to figure out what an explanation posits.” For a similar point see Knowles and Liggins 2015, 3406.
- 7.
Baker (2017, 2, 5, 6) formulates MES and NES starting with premise number (4) in order to compare these versions of the argument with the versions he calls ‘Cicada MESGEN’ and ‘Skid Patch MESGEN’ (respectively, pages 8–9 and 11).
- 8.
NES∗ provides a general argument pattern: “For any given range [T1, ..., T2], of cicada life cycles as determined by the local ecological constraints there is a corresponding instantiation of schema (5/6)∗∗. If this instantiation of the argument pattern is true of the given range […], then it can be used to explain the life-cycle length of the given cicada”(Baker 2017, 6).
- 9.
Baker 2017, 9–10.
- 10.
Baker 2017, 11.
- 11.
In their own words: “Ontic accounts take an explanation’s explanatory power to at least partly derive from its latching onto worldly things that bear an objective, explanatorily relevant relation to the explanandum.” (on line version previous to Knowles and Saatsi 2019)
- 12.
We follow Reutlinger’s description in his Reutlinger 2016 who mentions, “Nagelian Bridge laws, symmetry assumptions, limit theorems, and other modeling assumptions” (Idem, 738) as examples of auxiliary assumptions.”
- 13.
“or a conditional probability P(E|S1, …, Sn) – where the conditional probability need not be ‘high’ in contrast to Hempel’s covering-law account.” (Idem ft.14)
- 14.
See Ylikoski and Kuorikoski 2010 p. 215.
- 15.
The train case goes as follows: “… suppose we want to explain why a train T arrives at a station, S, at 3:00 pm. The explanation is as follows: T left another station, S∗, 10 kilometers away at 2:00 pm and headed towards S at 10Kph. Obviously, this explanation exploits some basic mathematics. Numbers are used to state the distance between stations as well as the speed of the train and a very basic mathematical calculation is deployed, namely 10/10 = 1. However, the mathematics itself does not do any explanatory work. Rather, it is facts about the speed of the train and the distance between stations that fully explain why the train arrived when it did. The mathematics just helps us to express the explanatory facts at issue.” (Baron 2016a, b, 460)
- 16.
- 17.
- 18.
See Woodward and Hitchock 2003.
- 19.
He chooses relevant logic as background logic. The reason for that is that classical logic is monotonic, and since mathematical theorems are true in every model, then, it is possible to add any mathematical theorem to our set of premises and we still get a sound explanation, but one in which irrelevant information can appear. Anyway, he acknowledges that it is possible to adopt classical logic as a background logic by adding the following proviso:
Containment: The premises of an extra-mathematical explanation must contain all of the information contained within the conclusion and each premise must contribute some part of that information. (Baron 2019, 704)
- 20.
“A non-mathematical claim P is essentially deducible from a premise set S that includes at least one mathematical sentence M just when for an appropriate choice of expressive resources there is a sound derivation of P from S and either for the same choice of expressive resources there is no sound derivation of P from a premise set S ∗ that includes only physical sentences or all sound derivations of P from premise sets S1...Sn each of which includes only physical sentences are worse than the mathematical derivation or for all appropriate choices of expressive resources the best derivations use M.” (Baron 2019)
- 21.
Following Lewis (1983), he also refers to ‘unity’ as strength.
- 22.
See Baron 2019 §7 for the problems his proposal faces in relation to unity and how they are to be solved.
- 23.
It is important to note, that Baron thinks that different explanations are to be compared in terms of the consequences they have for physical claims only. This last restriction has to do with the fact that he intends to characterize extra-mathematical explanation, he is not interested in mathematical explanations of mathematical facts.
- 24.
“The empirical setup is the relevant bits of the empirical world, not a mathematics-free description of it.” (Bueno and Colyvan 2011, 354) “The results are read back down into the physical system via some structural mapping relation between the mathematical and physical structures. The structural mapping used in the interpretation step need not be the inversion of the mapping used at the immersion step.” Baron (2019, 33) refers to Batterman 2010 for an account of cases in which no immersion is possible.
- 25.
He claims that this view “dovetails nicely with Rizza’s (2013) account of applied mathematics.” Rizza’s account, as described by Baron, contends that some cases of applied mathematics operate by first identifying a formal property of a physical system; second reasoning mathematically about the physical system; and third using those mathematical results to get further detail about the particular physical system.
- 26.
We follow Rayo 2007.
- 27.
- 28.
Moderate versions of platonism related to the original indispensability argument, for instance the one advocated by Hellman and Putnam, demand that the mathematical and the concrete are orthogonal, therefore, in that case, the mathematical does not make a difference. See Martínez-Vidal 2018.
- 29.
- 30.
Idem.
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Acknowledgments
We would like to thank an anonymous referee for her very helpful comments and the audience of the Workshop “Abstract Objects?” celebrated in Santiago de Compostela in 2018 for their patience and feed-back.
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Martínez-Vidal, C., Rivas-de-Castro, N. (2020). Description, Explanation and Ontological Committment. In: Falguera, J.L., Martínez-Vidal, C. (eds) Abstract Objects. Synthese Library, vol 422. Springer, Cham. https://doi.org/10.1007/978-3-030-38242-1_3
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