Abstract
Lately, philosophers of mathematics have been exploring the notion of mathematical explanation within mathematics. This project is supposed to be analogous to the search for the correct analysis of scientific explanation. I argue here that given the way philosophers have been using “explanation,” the term is not applicable to mathematics as it is in science.
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Notes
Lord Rayleigh, quoted in (Huntley 1970, 6).
Interesting debates from the mathematical community about the variety of positions on rigor in proof can be found in Bundy et al. (2005). Dawson (2006) addresses questions about the reasons different kinds of proofs are used in general. L. E. J. Brouwer’s work (1987) is the locus classicus for constructive proofs and mathematics; see Dalen (2004, Ch 5) for a recent mathematical discussion. “Analytic proof” was first used by Bolzano (1817). Brown (2008) contains discussions of various kinds of proofs as well.
The contemporary discussion of scientific explanation (and its terminology) begins with Carl Hempel’s Deductive-Nomological theory. Michael Scriven defended a causal theory; Michael Friedman and then Philip Kitcher defended a unification approach to explanation (the latter (Kitcher 1975 and 1989) also defended a mathematical application of this theory). Bas van Fraassen defended a constructive empiricist version of explanation to deal with the same problems. See Pitt (1988) for representative papers from each of these authors.
Our example appears in Nelsen (1993, 118). It is slightly simpler and less visually striking than Cellucci’s but makes the identical point.
Thus we do not address theories of explanation that describe, for instance, why honeycombs are comprised of hexagons or how group theory and symmetry considerations “explain” the Higgs Boson. We also do not address the repercussions for the Quine-Putnam indispensability thesis or Wigner’s problem of the unreasonable effectiveness of mathematics in science. See e.g. (Colyvan 2001) for more on the former subject and (Steiner 1998) for discussion of the latter.
Mathematics and Aristotle's Prime Mover are exceptions. The Prime Mover is purely final cause as it is immaterial, formless, and eternal. Mathematics is easily construed as being purely formal cause. But this kind of exception does not detract from the prima facie case that mathematics is not causal. Anyone who claims otherwise must justify this counterintuitive position. Thus it is still to have been expected that there is little discussion of explanation in mathematics.
Given the stock mathematicians put in the notion, there are surprisingly few philosophical studies of mathematical beauty. See however Netz (2005).
Until the twentieth century it was not at all clear how to even phrase questions about the length of proofs, and there are still open questions that we need not address here. However, Nickolas Pappas related an interesting anecdote to me that had Quine insisting to Warren Goldfarb, despite Goldfarb's strong resistance, that a particular six-line proof was superior to a seven-line proof of the same theorem, a priori, because of their respective sizes despite Goldfarb's argument that the longer proof had other virtues.
Recall that we are not talking about pedagogical applications of “explanation.” It is curious that some recent discussions of explanation in mathematics appeal to the literature on the nature of teaching mathematics to support the contention that some mathematics is explanatory, see e.g. Lange (2009, 205). Mancosu (2001) both appeals to (113) and cautions against it (101). (Many of the essays in e.g. Hanna et al. (2009) clearly address pedagogical claims about explanation.) I know of no comparable exploitation of pedagogical techniques to support any notions of scientific explanation. Equating what educators do to explain e.g. meteorology to their pupils and what philosophers mean when they ask for the structure of scientific explanation is an equivocation on “explanation.” To the extent that philosophers of mathematics do not see this as an equivocation, they are not talking about the same thing as philosophers of science.
Famous early counterexamples include 1) a barometer reading predicting a storm but not explaining it and 2) the pre-Newtonian ability to predict features of the tides given the position and phase of the moon, though being unable to explain them for lack of an adequate theory.
Cf. the discussion in Winch (1958, 91–94).
Note also Salmon's (1989, 104) negative remarks about all deductive arguments being considered explanatory.
Adapted from Cahn (1989).
E.g. Euclid's Elements Bk II prop. 14, Bk III prop. 1.
Cf. Dretske (1981, 7). Thanks to Michael Levin for bringing this and other arguments to my attention.
See Carnap (1966, 12–17).
In mathematics “theory” refers, say in the case of model theory, to sets of axioms, rules of inference, and the theorems they produce, or in the case of set theory to a body of knowledge within mathematics. In science on the other hand, “theory” refers to a complicated set of hypothesized rules that frame (sometimes causal) relationships among different pieces of empirical data together with tests of these theories, inferences and generalizations from the data, background ontology, and the explanations that all of these provide. Scientific theories are open to (among other things) verification, complete or partial revision, new evidence, and probabilistic arguments; mathematical theories are not.
P and L are idealized and thus may not be doing science or mathematics in a way we recognize. Importantly for us however, the relevant difference between P and L, and us, is that L is not distracted by psychological concerns. L simply runs through all the logical implications and knows all via brute force.
I believe that Mellor (2003) is expressing similar concerns in his reply to Koslow (2003, 223). He contrasts statements about chance with those of seemingly non-contingent statements. He writes: “… because ‘P is contingent’ and hence ‘~P is possible’ are necessary if true, they need no truthmakers. Similarly for the sense in which truth or falsity are the possible truth values of any ‘P’ and ‘~P’. Similarly again … of possible cases invoked in mathematical proofs.”.
A similar argument is attributed to Thomas Forester as follows: There is no explanation without counterfactuals. There are no counterfactuals without contingency. All of mathematics is necessary. Therefore there are no explanations in mathematics.
References
Avigad, J. (2006). Mathematical method and proof. Synthese, 153, 105–159.
Benacerraf, P. (1965). What numbers could not be. In P. Benacerraf & H. Putnam (Eds.). (1983), Philosophy of mathematics (pp. 77–89). Cambridge: Cambridge University Press.
Bolzano, B. (1810). Beiträge Zu Einer Begründeteren Darstellung der Mathematik. Prague: Caspar Widtmann. English translation by Steven Russ in W. Ewald (Ed.). (1996). From Kant to Hilbert: A source book in the foundations of mathematics (Vol. 1, pp. 174–224). New York: Oxford, as “Contributions to a better-grounded presentation of mathematics”.
Bolzano, B. (1817). Rein Analytischer Beweis Des Lehrsatzes, Dass Zwischen Je Zwei Werten, Die ein Entgegengesetztes Resultat Gewähren, Wenigstens eine Reelle Wurzel der Gleichung Liege. Gottlieb Haase, Prague. English translation by Steven Russ in W. Ewald (Ed.). (1996). From Kant to Hilbert: a source book in the foundations of mathematics (Vol. 1, pp. 225–248). New York: Oxford, as “Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one, real root of the equation”.
Brouwer, L. E. J. (1987). Intuitionism and formalism. In P. Benacerraf, H. Putnam (Eds.), Philosophy of mathematics: Selected readings. (pp. 77–89). New York: Cambridge.
Brown, J. R. (2008). Philosophy of mathematics: A contemporary introduction to the world of proofs and pictures (2nd ed.). New York: Routledge.
Bundy, A., Atiyah, M., Macintyre, A., & MacKenzie, D. (2005). The nature of mathematical proof. Philosophical Transactions of the Royal Society A, 363(1835), 2329–2461. doi:10.1098/rsta.2005.1660
Butchart, S. J. (2001). Evidence and explanation in mathematics. PhD thesis. Monash University.
Cahn, S. (1989). Philosophical explorations. New York: Promethus Books.
Carnap, R. (1966). Philosophical foundations of physics: An introduction to the philosophy of science. New York: Basic Books.
Cellucci, C. (2008). The nature of mathematical explanation. Studies in History and Philosophy of Science Part A, 39(2), 202–210.
Colyvan, M. (2001). The indispensability of mathematics. New York: Oxford University Press.
Dalen, D. (2004). Logic and structure (4 edition). Berlin: Springer.
Dawson, J. W, Jr. (2006). Why do Mathematicians Re-prove Theorems? Philosophia Mathematica, 14(3), 269–286.
Dretske, F. (1981). Knowledge and the flow of information. Cambridge: MIT Press.
Gale, D. (1990). Proof as explanation. The Mathematical Intelligencer, 12(1), 4.
Giaquinto, M. (2005). Mathematical activity. In P. Mancosu, K. F. Jørgensen, & S. A. Pedersen (Eds.), Visualization, explanation, and reasoning styles in mathematics (pp. 75–87). Dordrecht: Springer.
Gleick, J. (1988). Chaos: The making of a new science. New York: Penguin Books.
Grosholz, E. R. (2000). The partial unifcation of domains, hybrids, and the growth of mathematical knowledge. In E. Grosholz & H. Breger (Eds.), The growth of mathematical knowledge (pp. 81–91). Dordrecht: Kluwer Academic Publishers.
Hafner, J., & Mancosu, P. (2005). The varieties of mathematical explanation. In K. F. Jørgensen, P. Mancosu, and S. A. Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp 1–32). Dordrecht: Springer.
Hanna, G., Jahnke, H. N., & Pulte, H. (Eds.). (2009). Explanation and proof in mathematics: Philosophical and educational perspectives. Dordrecht: Springer.
Hempel, C. G. (1963). Explanation and prediction by covering laws. In B. Baumrin (Ed.), Philosophy of science: The Delaware seminar (pp. 107–133). New York: Wiley.
Hempel, C. G. (1965). Aspects of scientifc explanation and other essays in the philosophy of science. New York: Free Press.
Hirsch, M. W. (1989). Chaos, rigor, and hype. The Mathematical Intelligencer, 11(3), 6–9. With a reply by James Gleick.
Hume, D. (1980). Dialogues concerning natural religion. R. H. Popkin (Ed.). Indianapolis: Hackett.
Humphreys, P. (1993). Greater unification equals greater understanding? Analysis, 53(3), 183–188.
Huntley, H. E. (1970). The divine proportion: A study in mathematical beauty. New York: Dover.
Jacquette, D. (2006). Review of Mancosu, P., Jørgensen, K. F., and Pedersen, S. A. (Eds.), Visualization, explanation and reasoning styles in mathematics. The Mathematical Intelligencer, 28(2), 79–81.
Kitcher, P. (1975). Bolzano’s ideal of algebraic analysis. Studies in the History and Philosophy of Science, 6, 229–269.
Kitcher, P. (1989). Explanatory unification and the causal structure of the world. In P. Kitcher & W. C. Salmon (Eds.), Scientific explanation, Minnesota studies in the philosophy of science (pp. 410–505). Minneapolis: University of Minnesota Press.
Koslow, A. (2003). Laws, explanations and the reduction of possibilities. In G. Rodriguez-Pereyra, H. Lillehammer (Eds.), Real metaphysics: Essays in honour of Hugh Mellor (chap. 11, pp. 169–183). New York: Routledge. With a reply by D. H. Mellor, pp 232–234.
Lange, M. (2009). Why Proofs by mathematical induction are generally not explanatory. Analysis, 69(2), 201–211.
Leng, M. (2005). Mathematical explanation. In C. Cellucci & D. Gillies (Eds.), Mathematical reasoning and heuristics (pp. 167–189). London: King’s College Publications.
Mach, E. (1893). The science of mechanics: A critical and historical account of its development. La Salle, IL: Open Court.
Maddy, P. (1990). Realism in mathematics. New York: Oxford University Press.
Mancosu, P. (2000). On mathematical explanation. In E. Grosholz & H. Breger (Eds.), The growth of mathematical knowledge (pp. 103–119). Dordrecht: Kluwer Academic Publishers.
Mancosu, P. (2001). Mathematical explanation: Problems and prospects. Topoi, 20, 97–117.
Mancosu, P., Jørgensen, K. F., & Pedersen, S. A. (Eds.). (2005). Visualization, explanation and reasoning styles in mathematics. Dordrecht: Springer.
Mellor, D. H. (2003). Real metaphysics: Replies. In G. Rodriguez-Pereyra & H. Lillehammer (Eds.), Real metaphysics: Essays in honour of hugh mellor (chap. 14, pp. 212–238). New York: Routledge.
Mill, J. S. (1941). A system of logic (8th ed.). London: Longmans, Green and Co.
Nelsen, R. B. (1993). Proofs without words: Exercises in visual thinking. Oberlin: The Mathematical Association of America.
Netz, R. (2005). The aesthetics of mathematics: A case study. In P. Mancosu, K. F. Jørgensen, & S. A. Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 251–293). Dordrecht: Springer.
Pitt, J. (1988). Theories of explanation. New York: Oxford University Press.
Polya, G. (1949). With, or without, motivation? American Mathematical Monthly, 56, 684–691.
Popper, K. R. (1959). The logic of scientifc discovery. New York: Harper & Row.
Priest, G. (2002). Beyond the limits of thought. New York: Oxford University Press.
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7, 5–41.
Salmon, W. C. (1989). Four decades of scientific explanation. In P. Kitcher, & W. C. Salmon (Eds.), Scientific explanation. Minnesota studies in the philosophy of science (pp. 3–219). Minneapolis: University of Minnesota Press. References are to the reprinted 1990 edition.
Sandborg, D. A. (1997). Explanation in mathematical practice. PhD thesis, University of Pittsburgh.
Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34, 135–151.
Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Cambridge, MA: Harvard University Press.
van Fraassen, Bas C. (1980). The scientific image. New York: Oxford University Press.
Winch, P. (1958). The idea of a social science and its relation to philosophy. London: Routledge & Kegan Paul.
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Thanks to Rohit Parikh, Noson Yanofsky, Mike Koss and the Midwest Workshop in PhiloSTEM at Indiana Purdue, and especially to the editor and two astute and helpful anonymous reviewers for this Journal.
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Zelcer, M. Against Mathematical Explanation. J Gen Philos Sci 44, 173–192 (2013). https://doi.org/10.1007/s10838-013-9216-6
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DOI: https://doi.org/10.1007/s10838-013-9216-6