Abstract
‘Grounding and the indispensability argument’ presents a number of ways in which nominalists can use the notion of grounding to rebut the indispensability argument for the existence of mathematical objects. I will begin by considering the strategy that puts grounding to the service of easy-road nominalists (“Nominalistic content meets grounding” section). I will give some support to this strategy by addressing a worry some may have about it (“A misguided worry about the grounding strategy” section). I will then consider a problem for the fast-lane strategy (“Grounding and generality: a problem for the fast lane” section) and a problem for easy-road nominalists willing to accept Liggins’ grounding strategy (“More on the grounding strategy and easy-road nominalism” section). Both are related to the problem of formulating nominalistic explanations at the right level of generality. I will then consider a problem that Liggins only hints at (“Mathematics and covering generalizations” section). This problem has to do with mathematics’ function of providing the sort of covering generalizations we need in scientific explanations.
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Notes
I will follow Liggins [3] in considering grounding a relation between facts, so I should speak of what grounds \(\langle S\rangle \) (the fact that S) rather than speaking of what grounds S. Moreover, I will avail myself of the phrase “the ground of S” to convey what, according to the platonist, grounds \(\langle S\rangle \). According to the nominalist, nothing grounds \(\langle S\rangle \), given that it is not the case that S.
This is partly because I think some of the problems for the fast lane strategy mentioned by Liggins ([3], pp. 12–13) may not be easy to fix. In any case, the choice of this starting point is partly instrumental. As we shall see in Sect. 3, the issue I raise poses problems also for the fast lane strategy.
Liggins’ [3] ‘abstract expressionists’ are usually hermeneutic fictionalists: mathematics enhances our expressive powers precisely because sometimes the only way to express a proposition p is by uttering a mathematics-infused sentence S such that p = ||S||. If this happens, it means we cannot find a non-mathematics-infused sentence S* such that p =|S*|.
I am assuming that the existence of numbers and other mathematical objects is contingent relative to the conditions of the physical world. See Yablo ([7], p. 1013) for a defense of the legitimacy of this assumption. See also Liggins’ [3] footnote 3 for reasons to call into question the assumption that mathematical entities are modally extreme (i.e., necessary-or-impossible).
The question at stake here is why no one has ever managed to have a round tour of Königsberg crossing each bridge exactly one time.
As a referee noted, this is a simplification. In the Königsberg case, more than one disjunct holds, so there is more than one disjunct grounding the disjunction. But this does not affect the substance of the point. First, we can easily think of analogous cases where the disjunction is such that only one disjunct holds (see for instance the example in the next paragraph). Moreover, a nominalistic explanation mentioning all the (finitely many) holding disjuncts still seems to be run at the wrong level of generality.
Could fast-lane nominalists reply that both S* and S** ground S? I’m not sure that such a reply would help here: if S is grounded by S* and S**, but S** is more fundamental than S*, then in explaining why S obtains we should mention S**.
I consider this example rather than the one in the previous section because it facilitates the illustration of some points I am interested in.
In this case, only one disjunct holds, so it is the truth of that disjunct that grounds the truth of the disjunction. Note also that construing ||S|| as having quantificational structure wouldn’t change much: the truth of ||S|| would still be grounded in the truth of one of its instances.
The moral is not that these examples pose problems for a Field-style or an easy-road-style nominalist. The dialectic is different. Liggins [3] has put forward a new nominalistic strategy and claims that it overcomes some of the problems of the old ones: it is more generally applicable than Field’s strategy, and offers a generalized reply to the content challenge. However, if it turns out that Liggins’ strategy suffers from other problems (formulating explanations at the wrong level of generality, in this case), it will not be immediate that nominalists should prefer it over classic strategies. Nominalists could stick with the old strategies, or show that they can be harmonized with the grounding approach, or show that the grounding approach is indeed superior and can overcome the problem I am pressing here.
See Yablo ([7], pp. 1024–1025): What did Euler discover, then? He discovered logical truths to the effect that anything with the structural features postulated in the axioms (Königsberg, for example) has thus and such other features as well”.
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Acknowledgments
Many thanks to David Liggins.
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Plebani, M. Nominalistic content, grounding, and covering generalizations: Reply to ‘Grounding and the indispensability argument’. Synthese 193, 549–558 (2016). https://doi.org/10.1007/s11229-014-0428-z
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DOI: https://doi.org/10.1007/s11229-014-0428-z