Abstract
Indispensability arguments are used as a way of working out what there is: our best science tells us what things there are. Some philosophers think that indispensability arguments can be used to show that we should be committed to the existence of mathematical objects (numbers, functions, sets). Do indispensability arguments also deliver conclusions about the modal properties of these mathematical entities? Colyvan (in Leng, Paseau, Potter (eds) Mathematical knowledge, OUP, Oxford, 109-122, 2007) and Hartry Field (Realism, mathematics and modality, Blackwell, Oxford, 1989) each suggest that a consequence of the empirical methodology of indispensability arguments is that the resulting mathematical objects can only be said to exist (or not exist) contingently. Kristie Miller has argued that this line of thought doesn’t work (Miller in Erkenntnis, 77 (3), 335-359, 2012). Miller argues that indispensability arguments are in direct tension with contingentism about mathematical objects, and that they cannot tell us about the modal status of mathematical objects. I argue that Miller’s argument is crucially imprecise, and that the best way of making it clearer no longer shows that the indispensability strategy collapses or is unstable if it delivers contingentist conclusions about what there is.
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Notes
An earlier version of this paper explored the relevant sense of modality involved here: should we understand necessitism / contingentism as concerning metaphysical, logical, or epistemic modality? I argued that the only interpretation on which Miller’s epistemological objection (the main focus of this paper) seems to have some scope for working is if these positions are concerned with metaphysical modality, so this is how I will proceed from hereon. Reviewers for this journal sagely encouraged me to omit the bulk of that discussion, but to include a note pointing out that much of the debate between Field, Colyvan, Hale and Wright has concerned ‘conceptual necessity’, whereby a statement is conceptually possible if its negation is not true in virtue of its meaning (see Hale & Wright, 1992; Field, 1993: 285; Colyvan, 2000: 88).
Here’s a statement of such an indispensability argument given (but not endorsed) by Harty Field:
‘[I]f our belief in electrons and neutrinos is justified by something like inference to the best explanation, isn’t our belief in numbers and functions and other mathematical entities equally justified by the same methodology?’ (Field, 1989: 16).
In contrast, the so-called ‘easy road’ to nominalism is to accept that our best scientific theories imply that there are mathematical objects, but to resist platonism by finding ways to show that the particular role that mathematical objects play in science is not ontologically committing. Joseph Melia’s (2000) proposals for ‘weaselling out of ontological commitments’ is a key example of such an ‘easy road’ approach. See Knowles & Liggins (2015) for recent discussion of Melia’s strategy.
Colyvan has discussed, defended, but not quite endorsed mathematical contingentism on the basis of such an indispensability argument–in (2001) he says “Although I’m inclined to think that mathematics is contingent, it may be that indispensabilists can go either way on this issue”. See Miller (2012) footnote 14.
In what follows I won’t examine why indispensabilists accept the possibility of Newtonian worlds, or whether they’re correct in doing so. It suffices for our argument that some indispensabilists do in fact accept this possibility, in some sense.
This is not Field’s argument for contingentist nominalism (see Field, 1993 for a defence of his view).
As I suggested in the introduction, Miller’s paper isn't solely directed at contingentism (in either its platonic or nominalist forms). She has a much wider objective: to show that indispensabilism turns out to be no guide to the modal status of what exists, and so PQO should be abandoned. I won’t discuss the wider objective here, instead I focus on just this component argument.
Loosely speaking, indispensabilism is scientific ‘inference to the best explanation’ when it is adopted as a general methodology for doing ontology / metaphysics.
The corollary case is w2; if the best theory of the world can dispense with mathematics then Field’s line of argument (in §2 above) might be employed to infer that mathematical objects do not exist. But if the matching claim is correct then there is another world w2* which is physically indistinguishable from w2 in which mathematical objects do exist.
In an earlier version of this paper I argued that alternative ways of understanding the modality implicit in the matching claim (in terms of epistemic possibility and logical possibility) do not generate the epistemic objection. A reviewer for this journal noted that as a consequence, neither of them are sensible interpretations of Miller’s view, so I’ve omitted these arguments. I leave it as an exercise for the interested reader to reproduce these arguments.
This is not a totally precise statement as other differences in their positions persist. For instance, nominalists are under no obligation to endorse the view that ‘in possible worlds where the best scientific theories are committed to numbers, numbers exist and numbers are abstract objects’.
In saying that MM makes a stronger claim than MMC, I just mean that MM entails MMC, but MMC does not entail MM.
In §2 above, this claim was presented as Premise 5: ‘Newtonian worlds are possible, i ipso nominalism is possible’.
The epistemology of self-warranting beliefs might count as a kind of exception which proves the rule.
Miller spells out the ‘Epistemic Objection’ as an extra step which I have skipped over here.
It is perhaps useful to clearly distinguish the two claims here in a footnote. Fallibilism asserts that it is possible that the evidence E supports p and yet not-p. Contingentism says that it is possible that p and it is possible that not-p. Fallibilism does not entail contingentism.
This kind of fallibilism is not the same as (academic) skepticism, which proceeds by undermining the available evidence for p and thereby undermines any knowledge that p. Rather, it concedes that not-p is consistent with the best evidence adduced in support of p.
It is no coincidence that Miller’s epistemic objection has this family resemblance to Laudan’s pessimistic meta-induction argument against IBE; both involve the same kind of ‘self-defeat’, and indispensabilism is a variety of IBE.
I note Alexander Bird argues that a key component of scientific method involves a form of inference to the best explanation that he calls ‘Holmesian eliminative abduction’, or ‘inference to the ‘only’ explanation.’ Bird argues that Holmesian inferences are non-ampliative and infallibilist. But he does not think that this is the only form of reasoning employed in science. See Bird (2005) and (2010).
This paper has been under development for a long time. I orginally started working on it after a long and pleasant weekend of conversation with Richard Woodward in 2014, and I'm massively grateful for his continued support and input. An early version was presented to the Department of Philosophy at the University of Glasgow in 2014, and much a later one to the TilPS seminar at the University of Tilburg in 2019, and I'd like to thank the participants in those sessions for their valuable contributions, and the organisers for inviting me. Extra special super gold plated acknowledgement and utmost thanks go to Nathan Wildman, Naomi Thompson, Darragh Byrne and Adrian Downey for being the excellent readers and commenters that they are. I've been encouraged by discussions with Roger Clarke and Tom Walker, thanks to them both. And tremendous thanks to the two anonymous reviewers for this journal, who have been diligent and generous with critical, thoughtful comments.
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Morrison, J. What Can Our Best Scientific Theories Tell Us About The Modal Status of Mathematical Objects?. Erkenn 88, 1391–1408 (2023). https://doi.org/10.1007/s10670-021-00407-8
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DOI: https://doi.org/10.1007/s10670-021-00407-8