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Importance and Explanatory Relevance: The Case of Mathematical Explanations

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Abstract

A way to argue that something (e.g. mathematics, idealizations, moral properties, etc.) plays an explanatory role in science is by linking explanatory relevance with importance in the context of an explanation. The idea is deceptively simple: a part of an explanation is an explanatorily relevant part of that explanation if removing it affects the explanation either by destroying it or by diminishing its explanatory power, i.e. an important part (one that if removed affects the explanation) is an explanatorily relevant part. This can be very useful in many ontological debates. My aim in this paper is twofold. First of all, I will try to assess how this view on explanatory relevance can affect the recent ontological debate in the philosophy of mathematics—as I will argue, contrary to how it may appear at first glance, it does not help very much the mathematical realists. Second of all, I will show that there are big problems with it.

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Notes

  1. This kind of strategy is explicitly suggested in Baker (2009, 625).

  2. I will use “mathematical explanations” throughout this paper to refer to mathematical explanations of physical phenomena.

  3. This is only a presentation of what I take to be the idea behind this indispensability argument for mathematical realism and it should not be confused with the actual argument.

  4. For example Achinstein (2001), Azzouni (2004), Psillos (2009), Maddy (1992) and Sober (1993).

  5. I use “explanatorily active” for those entities (or their properties) which are mentioned in an explanatorily relevant part of an explanation.

  6. This is not, of course, Baker’s argument, but what I take to be his argumentative strategy.

  7. Although see the arguments in Morrison (2012) and Marcus (2014) for a different opinion.

  8. Let us call this the IA (indispensability requirement).

  9. See for example Lyon’s (2012) program explanation account.

  10. Besides Steiner (1978), we can find this approach in Colyvan (2003) and Lyon and Colyvan (2008).

  11. Cf. Busch and Morrison (2016).

  12. A similar formulation can be found in Cartwright (2011, 15).

  13. ER is not meant as a definition of explanatory relevance, so what comes after ‘if’ should not be taken as sufficient conditions for this thing but as a good sign that we are dealing with such a relation.

  14. We do speak of degrees of explanatory power, but that is a different matter.

  15. For this to make sense, one needs to assume that an explanation can have parts that do not have explanatory value; otherwise we would not be talking about a part of an explanation in this situation, but about an explanation. I do not consider this to be a controversial assumption, but for those unconvinced, Sect. 4. of this paper can be taken as an argument supporting it.

  16. P stands for the pressure of an amount of gas, n for the moles of molecules, V for volume and T for absolute temperature.

  17. This is one of the postulates of the kinetic theory.

  18. We can dub an explanation composed only of such a part a minimal explanation.

  19. A similar kind of reasoning can be easily developed for the honeycomb example: without the mathematical theorem there is no way to make sense of the evolutionary advantage that using such shapes to build the honeycombs is supposed to have, so the honeycomb theorem is essential for the explanation and that makes it genuinely explanatory.

  20. See Baker (2009, 617) and Baker and Colyvan (2011, 330–331).

  21. See for example Saatsi (2011, 153).

  22. Mathematics is a weakly explanatorily relevant part of an explanation if that explanation has, due to its mathematical component, a greater explanatory power than any nominalistic alternatives.

  23. For more about this requirement see the discussion in Sect. 2.

  24. Mancosu’s version of EIA runs as follows:

    1. (a)

      There are genuinely mathematical explanations of empirical phenomena;

    2. (b)

      We ought to be committed to the theoretical posits postulated by such explanations; thus,

    3. (c)

      We ought to be committed to the entities postulated by the mathematics in question. (Mancosu 2008, 137).

  25. We are so used to applying mathematics that we sometimes forget that it is not about the physical world.

  26. A similar point is made in Saatsi (2011). Unlike Saatsi, who says that “there is no ‘mathematical part’ to this evolutionary explanation” (Saatsi 2011, 146), I take the mathematical theorem in (3) to be a genuine and very important part of the honeycomb explanation.

  27. There is a Steiner inspired possible objection that can be raised at this point: the fact that the mathematical theorem represents a physical fact does not make it less explanatory. Actually, if Steiner (1978) is right, this is a sort of prerequisite for mathematics to be explanatory in such a context: if the mathematical theorem represents a physical fact, then the mathematical explanation of the mathematical theorem can be transferred to the physical fact, hence we will have a mathematical explanation for it. We do not need to reject Steiner’s view in order to dismiss this objection. It is sufficient to point to the fact that the explanandum in our case is the hexagonal structure of the honeycombs. But, even if the proof of the mathematical theorem formulated in (3) can be taken as an explanation for some physical fact, it will be for the fact that a hexagonal cell structure is the most economical, in terms of the wall material used, partition of some portion of the physical space. So, its role in the honeycomb structure explanation is at most that of enhancing our justification for one of the explanatory relevant facts.

  28. See the Boyle’s law example discussed above.

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Acknowledgements

This paper was written while I held a Visiting Fellowship offered by The European Philosophy of Science Association (EPSA) at the Centre for Logic and Philosophy of Science at Ghent University. I wish to express my gratitude to EPSA and to the Centre for the opportunity to develop my research, as well as for hospitality and support during my visit. I owe very special thanks to Professor Erik Weber for many helpful discussions and for providing invaluable comments on earlier drafts of this paper.

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  1. Centre for Logic and Philosophy of Science, Ghent University (UGent), Ghent, Belgium

    Gabriel Târziu

  2. Institute for Research in the Humanities, University of Bucharest, 1 Dimitrie Brandza, 060102, Bucharest, Romania

    Gabriel Târziu

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Târziu, G. Importance and Explanatory Relevance: The Case of Mathematical Explanations. J Gen Philos Sci 49, 393–412 (2018). https://doi.org/10.1007/s10838-018-9424-1

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