Abstract
In this paper I shall adopt a possible reading of the notions of ‘explanatory indispensability’ and ‘genuine mathematical explanation in science’ on which the Enhanced Indispensability Argument (EIA) proposed by Alan Baker is based. Furthermore, I shall propose two examples of mathematical explanation in science and I shall show that, whether the EIA-partisans accept the reading I suggest, they are easily caught in a dilemma. To escape this dilemma they need to adopt some account of explanation and offer a plausible answer to the following ‘question of evidence’: What is a genuine mathematical explanation in empirical science and on what basis do we consider it as such? Finally, I shall suggest how a possible answer to the question of evidence might be given through a specific account of mathematical explanation in science. Nevertheless, the price of adopting this standpoint is that the genuineness of mathematical explanations of scientific facts turns out to be dependent on pragmatic constraints and therefore cannot be plugged in EIA and used to establish existential claims about mathematical objects.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Metaphysical realism about mathematical entities is the belief that mathematical entities exist, and that this existence is mind- and language-independent.
This argument appeals to the notions of indispensability and quantification. However, Mark Colyvan has observed that it makes an implicit use of the Quinean doctrines of confirmational holism and naturalism (Colyvan 2001, p. 11). Actually, there are various versions of the Quine-Putnam indispensability argument. The version that explicitly makes reference to quantification over mathematical entities is generally attributed to Quine and Putnam and is the version which is commonly discussed in the literature. For alternative formulations of the argument see also Colyvan (2001), and Panza and Sereni (2013).
For instance, Mary Leng has rejected the argument on the grounds that mathematical explanations need not have true explanans, and therefore the objects posited by such explanations do not necessarily exist (Leng 2005).
Of course, not all the criticisms to EIA adopt \((\alpha )\) or \((\beta )\) as basic strategy. For instance, the criticism put forward by Christopher Pincock does not appeal to nominalistic reconstructions and it supports the claim that mathematics plays an explanatory role in science.
Note that this does not amount to say that \(T\) must offer explanations without employing a mathematical vocabulary. This formulation is very general and allows not only for \(T\)* to be a theory in which other mathematical entities (i.e. entities belonging to a mathematical theory different from \(M\)) yield explanatory power but also for \(T\)* to be a nominalized version of \(T\).
Here the notion of ‘indispensability of entities’ essentially makes reference to the mathematical theory to which these entities belong. Otherwise, to speak of mathematical entities tout court would be meaningless. On a similar characterization of indispensability and on the necessity to speak of indispensability of entities belonging to a mathematical theory see Panza and Sereni (2013, Chapter 6).
It is easy to stretch (E-DISP) in order to make it as explicitly referring to some entities. Nevertheless, for simplicity of notation, in (E-DISP) I make reference to the dispensability of a single mathematical entity \(x\).
I am grateful to one of the anonymous referees for drawing my attention to the following point: It might be thought that there exist cases in which an entity \(x\) could be explanatorily dispensable to a scientific theory, however the mathematical theory \(M\) is not. This scenario, which is not explicitly covered by my (E-DISP) criterion, is prima facie plausible and corresponds to a situation in which different entities from the very same mathematical theory are used to explain a scientific phenomenon within a scientific theory. Although reasonable, as far as I know there is no evidence in the literature on mathematical explanation of these sort of cases. Nevertheless, my rough intuition is that different entities belonging to a same mathematical theory \(M\) can always be expressed in terms of the vocabulary of that theory and therefore this scenario collapses to a specific setting that (E-DISP) allows for.
Offering a formal definition of how explanatory dispensability of entities might not affect the whole scientific theory but only a fraction of the theory is a complex issue that I will not pursue here. Although I presume that this further distinction might be made more formal, I shall not integrate it in (E-DISP). Instead, for the sake of the discussion at least, I shall leave this idea on an intuitive level and in the next section I shall offer a concrete example in which explanatory dispensability affects only a fraction of a scientific theory.
An orthogonal matrix \(\mathbf {A}\) is a real square matrix with the following property: the transpose matrix of \(\mathbf {A}\) coincides with the inverse of \(\mathbf {A}\). The class of \(n \times n\) orthogonal matrices is a group under matrix multiplication. The group of real orthogonal \(n \times n\) matrices is called the orthogonal group, and it is denoted by \(O(n)\). Every orthogonal matrix has determinant either \(+1\) or \(-1\), and the special orthogonal group, denoted by \(SO(n)\), is the subgroup of \(O(n)\) formed by the orthogonal \(n \times n\) matrices with determinant +1.
At least, this is the feeling that I had reading his papers explicitly devoted to the subject, where there is no claim that Steiner’s case study concerning the displacement of a rigid body around a fixed axis is not a genuine mathematical explanation of a physical phenomenon (cf. Baker 2005, 2009, 2012).
Although I am following Colyvan here, this is a delicate point. Indeed, there is no consensus among physicists and philosophers on the fact that the Minkowski metric is explanatory. For instance, it might be thought that the basic explanation of relativistic effects such as the length contraction is that rods and clocks are embedded in Minkowski spacetime, with its flat pseudo-Euclidean metric of Lorentzian signature. And therefore the metric would turn out to be a mere heuristic device that encodes the facts about the behaviour of rods and clocks (cf. Brown 2007; Smart 1990). I thank an anonymous referee for having brought my attention to this important point.
Note that here I am following Colyvan in considering a geometrical explanation as genuinely mathematical (cf. also Lyon and Colyvan 2008). And this intuition seems to reflect the opinion of various physicists and philosophers. For instance, Clifford Truesdell observed that “rational mechanics is mathematics, just as geometry is mathematics” (Truesdell 1967, p. 47). Pure Euclidean geometry is mathematics, and it is when it is interpreted so that the geometric points are taken to be representations of positions in actual space that the (interpreted) theory becomes a physical theory of space (Einstein 1923; Carnap 1966, p. 183).
Although SpecRel consists of only four axioms, the details of how the axiomatixation is put in place are fairly technical and they are not essential to the point I want to make here. This is the reason why I am skipping here the technicalities and I am reporting only a general account of how SpecRel is built. The interested reader will find a comprehensive treament of SpecRel in Andréka et al. (2002) and in chapter 3 of Székely (2009). Andréka et al. (2002) only develops special relativity. However, in Andréka et al. (2012) the 2002 theory is extended to general relativity.
In passing, let me note that the example of the length contraction explained by using ZF represents an exemplar of a case in which a mathematical explanation can be rephrased in first-order logic. Alan Baker claims that no first-order paraphrase of the number-theoretic apparatus invoked in the explanation of cicada life cycles is possibile (Baker 2009, pp. 618–619). On the other hand, in the length contraction case this paraphrase can be offered.
One of these examples, the explanation of the length contraction through StandSpecRel, has been explicitly considered by Mark Colyvan in connection with the notion of explanatory indispensability (of mathematical entities) that appears in EIA.
The real orthogonal and real special orthogonal groups have, of course, geometric interpretations. However they are defined in different mathematical theories and are therefore different mathematical entities.
There might be, of course, other cases of mathematical explanations in science for which the validity of IBE and EIA could still be maintained. And my argument here is not a general one and does not rule out these situations. By taking into consideration Baker’s example, however, what I want to suggest here is that as far as the genuine character of these explanations will be grounded in (EVID), the examples will face the same problem that affects Baker’s case, namely that (EVID) seems to be a rather weak criterion in order to apply IBE and EIA. Again, whether the EIA-supporter has (EVID) alone in her hands, the burden of the proof is still on her side.
Obviously, it would take more space than I have to illustrate here how the account proposed in Molinini (2011) fits the case of the logical relativity theory project. Nevertheless, such an analysis is provided in Friend (2012), so I shall not reproduce it here but I shall offer only the general strategy.
Similarly, the perspective on mathematical explanation in science offered in Molinini (2011) can be used to account for the two mathematical explanations of the existence of an instantaneous axis of rotation. For instance, in Euler’s explanation geometrical reasoning is the intellectual tool used by Euler. For Euler geometrical reasoning defines a natural, ‘pure’ conceptual path which leads us from the content of the theorem to the result. This intellectual tool is used on a reconceptualization permitted by the use of the concepts of Euclidean geometry.
In passing, note that this account is compatible with the pluralist positions put forward by others philosophers engaged in the debate on mathematical explanation in science, as for instance those proposed by Christopher Pincock and Robert Batterman (Molinini 2011, Chapter 8).
References
Andréka, H., Madarász, J. X., & Németi, I. (2002). On the logical structure of relativity theories. Alfréd Rényi Institute of Mathematics, Budapest. With contributions from A. Andai, G. Sági, I. Sain, Cs. Tőke. http://www.math-inst.hu/pub/algebraic-logic/olsort.html. (E-book).
Andréka, H., Madarász, J. X., & Németi, I. (2007). Logic of spacetime and relativity. In M. Aiello, I. Pratt-Hartmann, & J. van Benthem (Eds.), Handbook of spatial logics (pp. 607–711). New York: Springer.
Andréka, H., Madarász, J., Németi, I., & Székely, G. (2012). A logic road from special relativity to general relativity. Synthese, 186, 633–649.
Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114, 223–238.
Baker, A. (2009). Mathematical explanation in science. British Journal of Philosophy of Science, 60, 611–633.
Baker, A. (2012). Science-driven mathematical explanation. Mind, 121(482), 243–267.
Baker, A., & Colyvan, M. (2011). Indexing and mathematical explanation. Philosophia Mathematica, 19(3), 323–334.
Bangu, S. (2008). Inference to the best explanation and mathematical realism. Synthese, 160, 13–20.
Batterman, R. (2010). On the explanatory role of mathematics in empirical science. British Journal for the Philosophy of Science, 61(1), 1–25.
Brown, H. (2007). Physical relativity: Space-time structure from a dynamical perspective. Oxford Scholarship Online. Philosophy module. Oxford: Oxford University Press.
Bueno, O., & Colyvan, M. (2011). An inferential conception of the application of mathematics. Noûs, 45(2), 345–374.
Carnap, R. (1966). Philosophical foundations of physics. New York: Basic Books.
Colyvan, M. (2001). The Indispensability of Mathematics. New York: Oxford University Press.
Colyvan, M. (2002). Mathematics and aesthetic considerations in science. Mind, 11, 69–78.
Daly, C., & Langford, S. (2009). Mathematical explanation and indispensability arguments. Philosophical Quarterly, 59, 641–658.
Detlefsen, M. (2008). Purity as an ideal of proof. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 179–197). Oxford: Oxford University Press.
Detlefsen, M., & Arana, A. (2011). Purity of methods. Philosophers’ Imprint, 11(12), 1–20.
Einstein, A. (1923). Sidelights on relativity. New York: Dutton.
Euler, L. (1750). Découverte d’un nouveau principe de mécanique [e177]. In Opera Omnia, volume 5 of II, pages 81–108. Originally published in Mémoires de l’académie des sciences de Berlin 6, 1752, pp. 185–217.
Friend, M. (2012). The epistemological significance of giving axioms for the relativity theories in the language of first-order Zermelo-Fraenkel set theory. Unpublished paper presented at the First International Conference on Logic and Relativity: honoring István Németi’s 70th birthday, Budapest, Hungary, September 8–12, 2012.
Friend, M., & Molinini, D. (2014). When mathematics explains a scientific theory. unpublished typescript.
Goldstein, H. (1957). Classical mechanics (1st ed.). Reading, MA: Addison Wesley.
Koukkari, W. L., & Sothern, R. (2006). Introducing biological rhythms. New York: Springer.
Leng, M. (2005). Mathematical explanation. In C. Cellucci & D. Gillies (Eds.), Mathematical reasoning, heuristics and the development of mathematics (pp. 167–189). London: King’s College Publications.
Lyon, A., & Colyvan, M. (2008). The explanatory power of phase spaces. Philosophia Mathematica, 16(2), 227–243.
Madarász, J., Németi, I., & Székely, G. (2006). Twin paradox and the logical foundation of relativity theory. Foundations of Physics, 36, 681–714.
Madarász, J., Németi, I., & Székely, G. (2007). First-order logic foundation of relativity theories. In D. Gabbay, M. Zakharyaschev, & S. Goncharov (Eds.), Mathematical problems from applied logic II. volume 5 of International mathematical series (Vol. 5, pp. 217–252). New York: Springer.
Mancosu, P. (2011). Explanation in mathematics. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. Summer 2011 edition. http://plato.stanford.edu/archives/sum2011/entries/mathematics-explanation/.
Melia, J. (2000). Weaseling away the indispensability argument. Mind, 109(435), 455–480.
Molinini, D. (2011). Toward a pluralist approach to mathematical explanation of physical phenomena. Lille: ANRT.
Molinini, D. (2012). Learning from Euler. From mathematical practice to mathematical explanation. Philosophia Scientiae, 16(1), 105–127.
Panza, M., & Sereni, A. (2013). Plato’s problem. An introduction to mathematical platonism. Basingstoke, UK: Palgrave MacMillan.
Pincock, C. (2004). A revealing flaw in Colyvan’s indispensability argument. Philosophy of Science, 71(1), 61–79.
Pincock, C. (2007). A role for mathematics in the physical sciences. Noûs, 41(2), 253–275.
Putnam, H. (1971). Philosophy of logic. New York: Harper & Row.
Railton, P. (1989). Explanation and metaphysical controversy. In P. Kitcher & W. Salmon (Eds.), Scientific explanation. Minnesota studies in the philosophy of science (Vol. XIII, pp. 220–252). Minneapolis: University of Minnesota Press.
Saatsi, J. (2011). The enhanced indispensability argument: Representational versus explanatory role of mathematics in science. The British Journal for the Philosophy of Science, 62(1), 143–154.
Sereni, A. (2014). Equivalent explanations and mathematical realism. Reply to “Evidence, explanation and enhanced indispensability”. In D. Molinini, F. Pataut, & A. Sereni (Eds.), Synthese, Special Issue on Indispensability and Explanation. doi:10.1007/s11229-014-0491-5.
Smart, J. J. C. (1990). Explanation: Opening address. In D. Knowls (Ed.), Explanation and its limits (pp. 1–19). Cambridge: Cambridge University Press.
Steiner, M. (1978a). Mathematical explanation. Philosophical Studies, 34, 135–151.
Steiner, M. (1978b). Mathematics, explanation and scientific knowledge. Noûs, 12, 17–28.
Székely, G. (2009). First-Order Logic Investigation of Relativity Theory with an Emphasis on Accelerated Observers. PhD Thesis, Eötvös Loránd University.
Targ, S. M. (1968). Theoretical Mechanics: a short course (V. Talmy, Trans. from the Russian). Moscow: Mir.
Truesdell, C. A. (1967). Foundations of continuum mechanics. In M. Bunge (Ed.), Delaware seminar in the foundations of physics. volume 1 of Studies in the Foundations Methodology and Philosophy of Science (pp. 35–48). Berlin: Springer.
Conflict of interest
The author reports no real or perceived vested interests that relate to this article that could be construed as a conflict of interest.
Author information
Authors and Affiliations
Corresponding author
Additional information
I am indebted to the organizers of the workshop “Indispensability and Explanation” (Paris, IHPST, November 19–20, 2012) for their kind invitation and to all the participants for insights and ideas shared at the meeting. I also want to thank Andrea Sereni for fruitful discussions that have helped me sharpen my presentation of the issues considered in this paper and two anonymous referees for their valuable comments and suggestions.
Rights and permissions
About this article
Cite this article
Molinini, D. Evidence, explanation and enhanced indispensability. Synthese 193, 403–422 (2016). https://doi.org/10.1007/s11229-014-0494-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-014-0494-2