Element contents
Mathematics and Explanation
Published online by Cambridge University Press: 27 April 2023
Summary
This Element answers four questions. Can any traditional theory of scientific explanation make sense of the place of mathematics in explanation? If traditional monist theories are inadequate, is there some way to develop a more flexible, but still monist, approach that will clarify how mathematics can help to explain? What sort of pluralism about explanation is best equipped to clarify how mathematics can help to explain in science and in mathematics itself? Finally, how can the mathematical elements of an explanation be integrated into the physical world? Some of the evidence for a novel scientific posit may be traced to the explanatory power that this posit would afford, were it to exist. Can a similar kind of explanatory evidence be provided for the existence of mathematical objects, and if not, why not?
- Type
- Element
- Information
- Online ISBN: 9781009039154Publisher: Cambridge University PressPrint publication: 25 May 2023
References
Baker, A. (2005). Are There Genuine Mathematical Explanations of Physical Phenomena? Mind, 114(454), 223–238.CrossRefGoogle Scholar
Baker, A. (2009a). Mathematical Accidents and the End of Explanation. In Bueno, O. and Linnebo, Ø., eds., New Waves in the Philosophy of Mathematics, Palgrave Macmillan, pp. 137–159.Google Scholar
Baker, A. (2009b). Mathematical Explanation in Science. The British Journal for the Philosophy of Science, 60(3), 611–633.Google Scholar
Baker, A. (2021). Circularity, Indispensability, and Mathematical Explanation in Science. Studies in History and Philosophy of Science, 88, 156–163.Google Scholar
Balaguer, M. (1998). Platonism and Anti-Platonism in Mathematics, Oxford University Press.CrossRefGoogle Scholar
Baron, S. (2016). Explaining Mathematical Explanation. The Philosophical Quarterly, 66(264), 458–480.Google Scholar
Baron, S. (2019). Mathematical Explanation by Law. The British Journal for the Philosophy of Science, 70(3), 683–717.CrossRefGoogle Scholar
Baron, S. (forthcoming). Mathematical Explanation: A Pythagorean Proposal. The British Journal for the Philosophy of Science, 1–29.Google Scholar
Baron, S., Colyvan, M., and Ripley, D. (2017). How Mathematics Can Make a Difference. Philosophers’ Imprint, 17(3), 1–19.Google Scholar
Baron, S., Colyvan, M., and Ripley, D. (2020). A Counterfactual Approach to Explanation in Mathematics. Philosophia Mathematica, 28(1), 1–34.Google Scholar
Cain, A. J. (2010). Deus ex Machina and the Aesthetics of Proof. The Mathematical Intelligencer, 32, 7–11.Google Scholar
Colyvan, M. (2002). Mathematics and Aesthetic Considerations in Science. Mind, 111(441), 69–74.Google Scholar
Colyvan, M. (2010). There Is No Easy Road to Nominalism. Mind, 119(474), 285–306.CrossRefGoogle Scholar
Colyvan, M. (2012). Road Work Ahead: Heavy Machinery on the Easy Road. Mind, 121(484), 1031–1046.CrossRefGoogle Scholar
Colyvan, M., Cusbert, J., and McQueen, K. (2018). Two Flavours of Mathematical Explanation. In Reutlinger, A. and Saatsi, J., eds., Explanation beyond Causation: Philosophical Perspectives on Non-causal Explanations, Oxford University Press, pp. 231–249.Google Scholar
Conway, J. (2005). The Power of Mathematics. In Blackwell, A. and MacKay, D., eds., Power, Cambridge University Press. 71–86Google Scholar
Conway, J. (2014). On Morley’s Trisector Theorem. The Mathematical Intelligencer, 36, 3. DOI: https://doi.org/10.1007/s00283-014-9463-3.CrossRefGoogle Scholar
Craver, C. F. (2014). The Ontic Account of Scientific Explanation. In Kaiser, M. and Scholz, O., eds., Explanation in the Special Sciences, Springer, pp. 27–52.CrossRefGoogle Scholar
Craver, C. F., and Povich, M. (2017). The Directionality of Distinctively Mathematical Explanations. Studies in History and Philosophy of Science Part A, 63, 31–38.Google Scholar
D’Alessandro, W. (2020). Mathematical Explanation beyond Explanatory Proof. The British Journal for the Philosophy of Science, 71, 581–603.CrossRefGoogle Scholar
D’Alessandro, W. (2021). Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth. Synthese, 198(9), 8621–8664.Google Scholar
Detlefsen, M., and Arana, A. (2011). Purity of Methods. Philosophers’ Imprint, 11(2), 1–20.Google Scholar
Field, H. (1980). Science without Numbers: A Defence of Nominalism, Princeton University Press.Google Scholar
Franklin, J. (2008). Aristotelian Realism. In Irvine, A., ed., Handbook of the Philosophy of Science. Philosophy of Mathematics, Elsevier, pp. 101–153.Google Scholar
Gorjian, I., Karamzadeh, O. A. S., and Namdari, M. (2015). Morley’s Theorem Is No Longer Mysterious! The Mathematical Intelligencer, 37, 6–7.CrossRefGoogle Scholar
Hafner, J., and Mancosu, P. (2005). The Varieties of Mathematical Explanation. In Mancosu et al., P., eds., Visualization, Explanation and Reasoning Styles in Mathematics, Springer, pp. 215–250.Google Scholar
Hafner, J., and Mancosu, P. (2008). Beyond Unification. In Mancosu, P., ed., The Philosophy of Mathematical Practice, Oxford University Press, pp. 151–178.CrossRefGoogle Scholar
Hempel, C. (1965). Aspects of Scientific Explanation and Other Essays in the Philosophy of Science, Free Press.Google Scholar
Ingram, V. M. (1957). Gene Mutations in Human Hæmoglobin: The Chemical Difference between Normal and Sickle Cell Hæmoglobin. Nature, 180(4581), 326–328.CrossRefGoogle ScholarPubMed
Jackson, F., and Pettit, P. (1990). Program Explanation: A General Perspective. Analysis, 50, 107–117.CrossRefGoogle Scholar
Karamzadeh, O. A. S. (2014). Is John Conway’s Proof of Morley’s Theorem the Simplest and Free of a Deus ex Machina? The Mathematical Intelligencer, 36, 4–7.CrossRefGoogle Scholar
Karamzadeh, O. A. S. (2018). Is the Mystery of Morley’s Trisector Theorem Resolved? Forum Geometricorum, 18, 297–306.Google Scholar
Kasirzadeh, A. (forthcoming). Counter Countermathematical Explanations. Erkenntnis, 1–24.Google Scholar
Kitcher, P. (1989). Explanatory Unification and the Causal Structure of the World. In Kitcher, P. and Salmon, W., eds., Scientific Explanation, University of Minnesota Press, pp. 410–505.Google Scholar
Knowles, R. (2021a). Platonic Relations and Mathematical Explanations. The Philosophical Quarterly, 71(3), 623–644.CrossRefGoogle Scholar
Knowles, R. (2021b). Unification and Mathematical Explanation. Philosophical Studies, 178(12), 3923–3943.Google Scholar
Knowles, R., and Saatsi, J. (2021). Mathematics and Explanatory Generality: Nothing but Cognitive Salience. Erkenntnis, 86(5), 1119–1137.CrossRefGoogle Scholar
Koslicki, K. (2012). Varieties of Ontological Dependence. In Correia, F. and Schnieder, B., eds., Metaphysical Grounding: Understanding the Structure of Reality, Cambridge University Press, pp. 186–213.Google Scholar
Kostić, D., and Khalifa, K. (2021). The Directionality of Topological Explanations. Synthese, 199(5), 14143–14165.CrossRefGoogle Scholar
Kuorikoski, J. (2021). There Are No Mathematical Explanations. Philosophy of Science, 88(2), 189–212.CrossRefGoogle Scholar
Lange, M. (2009). Why Proofs by Mathematical Induction Are Generally Not Explanatory. Analysis, 69(2), 203–211.CrossRefGoogle Scholar
Lange, M. (2013). What Makes a Scientific Explanation Distinctively Mathematical? The British Journal for the Philosophy of Science, 64(3), 485–511.Google Scholar
Lange, M. (2017). Because without Cause: Non-causal Explanation in Science and Mathematics, Oxford University Press.Google Scholar
Lange, M. (2018a). A Reply to Craver and Povich on the Directionality of Distinctively Mathematical Explanations. Studies in History and Philosophy of Science Part A, 67, 85–88.Google Scholar
Lange, M. (2018b). Mathematical Explanations That Are Not Proofs. Erkenntnis, 83(6), 1285–1302.Google Scholar
Lange, M. (2019). Ground and Explanation in Mathematics. Philosophers’ Imprint, 19(33), 1–18.Google Scholar
Lange, M. (2021a). Asymmetry as a Challenge to Counterfactual Accounts of Non-causal Explanation. Synthese, 198(4), 3893–3918.Google Scholar
Lange, M. (2021b). What Could Mathematics Be for It to Function in Distinctively Mathematical Scientific Explanations? Studies in History and Philosophy of Science Part A, 87, 44–53.Google Scholar
Lange, M. (2022a). Challenges Facing Counterfactual Accounts of Explanation in Mathematics. Philosophia Mathematica, 30(1), 32–58.Google Scholar
Lange, M. (2022b). Inference to the Best Explanation as Supporting the Expansion of Mathematicians’ Ontological Commitments. Synthese, 200(146), 1–26.Google Scholar
Leng, M. (2020). Mathematical Explanation Doesn’t Require Mathematical Truth. In Dasgupta, S., Dotan, R., and Weslake, B., eds., Current Controversies in Philosophy of Science, Routledge, pp. 51–59.Google Scholar
Leng, M. (2021). Models, Structures, and the Explanatory Role of Mathematics in Empirical Science. Synthese, 199(3), 10415–10440.Google Scholar
Lewis, D. (2004). Causation as Influence. In Collins, J., Hall, N., and Paul, L. A., eds., Causation and Counterfactuals, Massachusetts Institute of Technology Press, pp. 75–106.Google Scholar
Lyon, A. (2012). Mathematical Explanations of Empirical Facts, and Mathematical Realism. Australasian Journal of Philosophy, 90, 559–578.CrossRefGoogle Scholar
Machamer, P., Darden, L., and Craver, C. F. (2000). Thinking about Mechanisms. Philosophy of Science, 67(1), 1–25.Google Scholar
Mancosu, P. (2000). On Mathematical Explanation. In Grosholz, E. and Breger, H., eds., The Growth of Mathematical Knowledge, Kluwer, pp. 103–119.Google Scholar
Mancosu, P. (2001). Mathematical Explanation: Problems and Prospects. Topoi, 20(1), 97–117.CrossRefGoogle Scholar
Mancosu, P. (2008). Mathematical Explanation: Why It Matters. In Mancosu, P., ed., The Philosophy of Mathematical Practice, Oxford University Press, pp. 134–150.Google Scholar
Mancosu, P. (2018). Explanation in Mathematics. In Zalta, E. N., ed., The Stanford Encyclopedia of Philosophy, Summer 2018, Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/sum2018/entries/mathematics-explanation.Google Scholar
Morrison, M. (2000). Unifying Scientific Theories: Physical Concepts and Mathematical Structures, Cambridge University Press.Google Scholar
Nahin, P. J. (2004). When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible, Princeton University Press.Google Scholar
Nandor, M., and Helliwell, T. (1996). Fermat’s Principle and Multiple Imaging by Gravitational Lenses. American Journal of Physics, 64(1), 45–49.CrossRefGoogle Scholar
Pincock, C. (2015a). Abstract Mathematical Explanations in Science. British Journal for the Philosophy of Science, 66(4), 857–882.Google Scholar
Pincock, C. (2015b). The Unsolvability of the Quintic: A Case Study in Abstract Mathematical Explanation. Philosopher’s Imprint, 15(3).Google Scholar
Pincock, C. (2018). Accommodating Explanatory Pluralism. In Reutlinger, A. and Saatsi, J., eds., Explanation beyond Causation, Oxford University Press, pp. 39–56.Google Scholar
Povich, M. (2020). Modality and Constitution in Distinctively Mathematical Explanations. European Journal for Philosophy of Science, 10(3), 1–10.Google Scholar
Povich, M. (2021). The Narrow Ontic Counterfactual Account of Distinctively Mathematical Explanation. The British Journal for the Philosophy of Science, 72(2), 511–543.Google Scholar
Povich, M. (forthcoming). A Scheme Foiled: A Critique of Baron’s Account of Extra-mathematical Explanation. Mind. DOI: https://doi.org/10.1093/mind/fzac019.Google Scholar
Putnam, H. (1975). Philosophy and Our Mental Life. In Mind, Language and Reality: Philosophical Papers, Cambridge University Press, pp. 291–303.Google Scholar
Reutlinger, A. (2016). Is There a Monist Theory of Causal and Noncausal Explanations? The Counterfactual Theory of Scientific Explanation. Philosophy of Science, 83(5), 733–745.CrossRefGoogle Scholar
Reutlinger, A. (2017). Does the Counterfactual Theory of Explanation Apply to Non-causal Explanations in Metaphysics? European Journal for Philosophy of Science, 7(2), 239–256.Google Scholar
Reutlinger, A. (2018). Extending the Counterfactual Theory of Explanation. In Reutlinger, A. and Saatsi, J., eds., Explanation beyond Causation: Philosophical Perspectives on Non-causal Explanations, Oxford University Press, pp. 74–95.Google Scholar
Reutlinger, A., Colyvan, M., and Krzyżanowska, K. (2022). The Prospects for a Monist Theory of Non-causal Explanation in Science and Mathematics. Erkenntnis, 87, 1773–1793.Google Scholar
Rice, C., and Rohwer, Y. (2021). How to Reconcile a Unified Account of Explanation with Explanatory Diversity. Foundations of Science, 26(4), 1025–1047.Google Scholar
Ronan, M. (2007). Symmetry and the Monster: One of the Greatest Quests of Mathematics, Oxford University Press.Google Scholar
Ryan, P. J. (2021). Szemerédi’s Theorem: An Exploration of Impurity, Explanation, and Content. The Review of Symbolic Logic, 1–40. DOI: https://doi.org/10.1017/S1755020321000538.Google Scholar
Saatsi, J. (2016). On the “Indispensable Explanatory Role” of Mathematics. Mind, 125(500), 1045–1070.Google Scholar
Salmon, W. (1984). Scientific Explanation and the Causal Structure of the World, Princeton University Press.Google Scholar
Salmon, W. (1989). Four Decades of Scientific Explanation. In Kitcher, P. and Salmon, W., eds., Scientific Explanation, University of Minnesota Press, pp. 3–219.Google Scholar
Senior, A. W., Evans, R., Jumper, J. et al. (2020). Improved Protein Structure Prediction Using Potentials from Deep Learning. Nature, 577(7792), 706–710.Google Scholar
Solomon, R. (2001). A Brief History of the Classification of the Finite Simple Groups. Bulletin of the American Mathematical Society, 38(3), 315–352.Google Scholar
Steiner, M. (1978b). Mathematics, Explanation, and Scientific Knowledge. Noûs, 12(1), 17.Google Scholar
Taylor, E. (forthcoming). Explanatory Distance. The British Journal for the Philosophy of Science, 1–33.Google Scholar
Woodward, J. (2003). Making Things Happen: A Theory of Causal Explanation, Oxford University Press.Google Scholar
Woodward, J. (2018). Some Varieties of Non-Causal Explanation. In Reutlinger, A. and Saatsi, J., eds., Explanation beyond Causation, Oxford University Press, pp. 117–137.Google Scholar
Zelcer, M. (2013). Against Mathematical Explanation. Journal for General Philosophy of Science, 44(1), 173–192.Google Scholar
- 9
- Cited by