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Unification and mathematical explanation in science

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Abstract

Mathematics clearly plays an important role in scientific explanation. Debate continues, however, over the kind of role that mathematics plays. I argue that if pure mathematical explananda and physical explananda are unified under a common explanation within science, then we have good reason to believe that mathematics is explanatory in its own right. The argument motivates the search for a new kind of scientific case study, a case in which pure mathematical facts and physical facts are explanatorily unified. I argue that it is possible for there to be such cases, and provide some toy examples to demonstrate this. I then identify a potential source of scientific case studies as a guide for future work.

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Notes

  1. For discussion, see Andersen (2016), Baker (2005), Baker (2009), Baker (2017), Bangu (2020), Baron (2020), Baron et al. (2017), Chirimuuta (2018), Colyvan (2001), Colyvan (2010), Jansson and Saatsi (2019), Knowles and Saatsi (2019),Knowles (forthcoming);Lange (2016), Leng (2010), Lyon (2012), Pincock (2015), Pincock (2007), Povich (2019), Saatsi (2017), Saatsi (2016).

  2. This view is defended by Baker (2005), Baron (2020), Colyvan (2010), Lange (2016), Potochnik (2007), Pincock (2007) among others.

  3. See, for instance, Bueno (2012),Liggins (2012), Leng (2010), Melia (2000), Saatsi (2011), Yablo (2012).

  4. The approach is also reminiscent of Steiner’s (1978b) account, on which mathematics plays a genuine explanatory role in a scientific explanation if, when we strip out all of the physical detail of the explanation, an explanation of a pure mathematical fact is left over. See Baker (2012) and Lyon (2012) for recent discussion.

  5. Note that the distinction between the thick and thin explanatory roles is not supposed to be the same as the distinction between the ontic and epistemic accounts of explanation. Saatsi’s idea is that even if we assume that explanations are generally ontic and that the explanatory power of our scientific representations comes from representing objective relations of explanatory relevance in the world, we can still draw a distinction between two roles for mathematics. Mathematics might appear in our scientific representations merely as a way of representing physical facts that are doing the explanatory work; or mathematics might itself be doing the explanatory work. The thick/thin distinction should thus be read as internal to the ontic account of explanation.

  6. Similar appeals have been made by Baker (2017) and by Colyvan (2001, 2002).

  7. I am grateful to a referee for the following way of stating the motivation.

  8. Why believe that our best scientific theories are true? This is due, primarily, to the so-called ‘no-miracles’ argument. This argument proceeds, roughly as follows: our current best scientific theories are extremely well-confirmed and enjoy incredible predictive success. The best explanation of this is that they are true. So we should believe that they are true. The literature on this argument is extensive. For an excellent overview and defense of the no-miracles argument, see Psillos (1999). For a compelling case against the no-miracles argument see Frost-Arnold (2010). For a critical discussion of the use of realism to support indispensability arguments, see Saatsi (2007).

  9. Some scientific realists argue only for the approximate truth of our best scientific theories. In the present context, the notion of approximate truth is potentially troubling. For it is compatible with the approximate truth of our best theories that vertical unification does not always imply horizontal unification, so long as our theories get it right enough of the time. Approximate truth poses a rather general problem for indispensability-based arguments that focus on explanation. For if our best theories can make mistakes, then one of those mistakes can just be the mistake of incorrectly implying that mathematics is playing a thick explanatory role. I think this is a genuine concern but, due to its generality, not one that I have the space to deal with here.

  10. See Zelcer (2013) for scepticism about mathematical explanation.

  11. Skow (2015) considers similar reasons to the two cited here. To a certain extent he concedes the force of the first reason, though he still maintains that physical facts can be used to explain mathematical facts in at least one case (involving Pythagoras’s theorem). Though I can’t argue the point here, the example he uses strikes me as one in which physical facts appear in the R-explanation for a mathematical fact, but are not part of the reason why the fact holds.

  12. If one remains worried, however, then there is a weaker version of Premise (3) that will do just as well. Namely: the O-explanation for M is mathematical. We can thus restrict the argument to those mathematical facts that do not hold for some purely logical reason. This, I take it, is something we may be able to determine using the tools of mathematics and logic (assuming there’s a clean split between the two). What if there are ‘brute’ mathematical facts? Such cases don’t pose a problem for the argument. Premise (3) is a universal claim and so it can be read in a conditional sense: if a mathematical fact has an O-explanation, then it’s O-explanation is mathematical. The fact that a mathematical fact is the output of an R-explanation is a reason to believe it has an O-explanation and thus that the antecedent is satisfied.

  13. I am grateful to a referee for raising this objection.

  14. See Baker (2009); Baker and Colyvan (2011); Baker (2017) for defence of the cicada case.

  15. One might worry about the example: haven’t I just shown that there is a pure mathematical explanation of a mathematical fact embedded within the cicada and gear cases? And doesn’t this just make the case into a variant on one of Steiner’s, which I have already said provide no support for the argument in §2? Note, however, that there is no reason why the sequences in the mathematical case have to be the same sequences as in the cicada or gear cases. Assume that they are different sequences. Then the mathematical explanation is still an instance of the general pattern and it is not embedded in the cicada and gear cases, because those cases involve entirely different sequences.

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Acknowledgements

I would like to thank audience members at the University of Lisbon for helpful discussion of an earlier version of this paper, particularly Amanda Bryant and David Yates. Research on this paper was supported by a Discovery Early Career Researcher Award from the Australian Research Council, DE180100414.

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  1. Dianoia Institute of Philosophy, Australian Catholic University, Victoria, Australia

    Sam Baron

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Baron, S. Unification and mathematical explanation in science. Synthese 199, 7339–7363 (2021). https://doi.org/10.1007/s11229-021-03118-3

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