Mathematical Explanations and the Piecemeal Approach to Thinking About Explanation

Gabriel Tȃrziu


A new trend in the philosophical literature on scientific explanation is that of starting from a case that has been somehow identified as an explanation and then proceed to bringing to light its characteristic features and to constructing an account for the type of explanation it exemplifies. A type of this approach to thinking about explanation – the piecemeal approach, as I will call it – is used, among others, by Lange (2013) and Pincock (2015) in the context of their treatment of the problem of mathematical explanations of physical phenomena. This problem is of central importance in two different recent philosophical disputes: the dispute about the existence on non-causal scientific explanations and the dispute between realists and antirealists in the philosophy of mathematics. My aim in this paper is twofold. I will first argue that Lange (2013) and Pincock (2015) fail to make a significant contribution to these disputes. They fail to contribute to the dispute in the philosophy of mathematics because, in this context, their approach can be seen as question begging. They also fail to contribute to the dispute in the general philosophy of science because, as I will argue, there are important problems with the cases discussed by Lange and Pincock. I will then argue that the source of the problems with these two papers has to do with the fact that the piecemeal approach used to account for mathematical explanation is problematic.


Almgren, F. & Taylor, J. (1976). The Geometry of Soap Films and Soap Bubbles. Scientific American, 235(1): 82-93.

Baker, A. (2005). Are There Genuine Mathematical Explanations of Physical Phenomena? Mind, 114(454): 223-238.

Baker, A. (2009). Mathematical Explanation in Science. British Journal for the Philosophy of Science, 60(3): 611-633.

Baker, A. (2011). Explaining The Applicability Of Mathematics In Science. Interdisciplinary Science Reviews, 36(3): 255-267.

Baker, A. (2012). Science-Driven Mathematical Explanation. Mind, 121(482): 243-267.

Batterman, R. (2002). The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence. Oxford/New York: Oxford University Press.

Batterman, R. (2009). Idealization and Modeling. Synthese, 169(3): 427 - 446.

Batterman, R. (2010). On the Explanatory Role of Mathematics in Empirical Science. British Journal for the Philosophy of Science, 61(1): 1-25.

Batterman, R. & Rice, C. (2014). Minimal Model Explanations. Philosophy of Science, 81(3): 349-376.

Berkovski, S. (2002). Surprising User-Friendliness. Logique Et Analyse, 45(179-180): 283-297.

Bokulich, A. (2008). Can Classical Structures Explain Quantum Phenomena? British Journal for the Philosophy of Science, 59(2): 217-235.

Carnap, R. (1950) Logical foundation of probability. London: Routledge and Keegan Paul.

Cartwright, N. (1983). How the Laws of Physics Lie. Oxford, UK: Clarendon Press.

Colyvan, M. (2001). The Indispensability of Mathematics. Oxford/New York: Oxford University Press.

Craver, C. (2006). When Mechanistic Models Explain. Synthese, 153(3): 355-376.

Craver, C. (2007). Explaining the Brain: Mechanisms and the Mosaic Unity of Neuroscience. Oxford: Clarendon Press.

Daly, C. & Langford, S. (2009). Mathematical Explanation and Indispensability Arguments. Philosophical Quarterly, 59(237): 641-658.

Friedman, M. (1974). Explanation and Scientific Understanding. Journal of Philosophy, 71(1): 5-19.

Grimm, S. (2008). Explanatory Inquiry and the Need for Explanation. British Journal for the Philosophy of Science, 59(3): 481-497.

Irvine, E. (2015). Models, Robustness, and Non-Causal Explanation: A Foray Into Cognitive Science and Biology. Synthese, 192(12): 3943-3959.

Hempel, C. & Oppenheim, P. (1948). Studies in the Logic of Explanation. Philosophy of Science, 15(2): 135-175.

Hempel, C. (1965). Aspects of Scientific Explanation and Other Essays in the Philosophy of Science. New York: The Free Press.

Huneman, P. (2010). Topological Explanations and Robustness in Biological Sciences. Synthese, 177(2): 213-245.

Kaplan, D. & Craver, C. (2011). The Explanatory Force of Dynamical and Mathematical Models in Neuroscience: A Mechanistic Perspective. Philosophy of Science, 78(4): 601-627.

Kaplan, D. (2011). Explanation and Description in Computational Neuroscience. Synthese, 183(3): 339-373.

Kim, J. (1964) Inference, Explanation, and Prediction. Journal of Philosophy, 61(12): 360-368.

Lange, M. (2013). What Makes a Scientific Explanation Distinctively Mathematical? British Journal for the Philosophy of Science, 64(3): 485-511.

Laudan, L. (1977). Progress and its Problems: Toward a Theory of Scientific Growth. Berkeley: University of California Press.

Lipton, P. (2004). Inference to the Best Explanation. London: Routledge/Taylor and Francis Group.

Lipton, P. (2011). Mathematical Understanding. In Meaning in Mathematics, ed. John Polkinghorne, 49-54. Oxford: Oxford University Press.

Lyon, A. & Colyvan, M. (2008). The Explanatory Power of Phase Spaces. Philosophia Mathematica, 16(2): 227-243.

Melia, J. (2000). Weaseling Away the Indispensability Argument. Mind, 109(435): 455-480.

Melia, J. (2002). Response to Colyvan. Mind, 111(441): 75-80.

Molinini, D. (2016). Evidence, Explanation and Enhanced Indispensability. Synthese, 193(2): 403-422.

Morgan, F. (1996). What Is a Surface? The American Mathematical Monthly, 103(5): 457-469.

Nerlich, G. (1994). What Spacetime Explains: Metaphysical Essays on Space and Time. Cambridge: Cambridge University Press.

Peirce, C. (1908/1968). Retroduction and Genius, in B. Brody and N. Capaldi (eds), Science: Men, Methods, Goals, first published 1908, New York: W. A. Benjamin, pp. 143–9.

Pincock, C. (2015). Abstract Explanations in Science. British Journal for the Philosophy of Science, 66(4): 857-882.

Rescher, N. (1957) On Prediction and Explanation. British Journal for the Philosophy of Science, 8(32): 281-290.

Rice, C. (2012). Optimality Explanations: A Plea for an Alternative Approach. Biology and Philosophy, 27(5): 685-703.

Rice, C. (2015). Moving Beyond Causes: Optimality Models and Scientific Explanation. Noûs, 49(3): 589-615.

Saatsi, J. (2011). The Enhanced Indispensability Argument: Representational Versus Explanatory Role of Mathematics in Science. British Journal for the Philosophy of Science, 62(1): 143-154.

Salmon, W. (1989). Four decades of scientific explanation. Pittsburgh: University of Pittsburgh.

Schupbach, J. & Sprenger, J. (2011). The Logic of Explanatory Power. Philosophy of Science, 78(1): 105-127.

Schurz, G. (1995). Scientific Explanation: A Critical Survey. Foundations of Science, 1(3): 429-465.

Sklar, L. (1993). Idealization and Explanation: A Case Study From Statistical Mechanics. Midwest Studies in Philosophy, 18(1): 258-270.

Sober, E. (1983). Equilibrium Explanation. Philosophical Studies, 43(2): 201 - 210.

Steiner, M. (1978). Mathematics, Explanation, and Scientific Knowledge. Noûs, 12(1): 17-28.

Strevens, M. (2008). Depth: An Account of Scientific Explanation. Harvard, MA: Harvard University Press.

Teller, P. (2001). Twilight of the Perfect Model Model. Erkenntnis, 55(3): 393-415.

Trout, J. D. (2002). Scientific Explanation and the Sense of Understanding. Philosophy of Science, 69(2):212-233.

Trout, J. D. (2007). The Psychology of Scientific Explanation. Philosophy Compass, 2(3):564–591.

Weber, E., Van Bouwel, J. & De Vreese, L. (2013). Scientific Explanation. Berlin: Springer.

Weinberg, S. (1994). Dreams of a Final Theory. New York: Vintage Books.

Woodward, J. (2003). Making Things Happen: A Theory of Causal Explanation. New York: Oxford University Press.

Yudell, Z. & Wong, Wai-Hung (2015). A Normative Account of the Need for Explanation. Synthese, 192(9):2863-2885.


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