Abstract
This heretofore-unpublished lecture focusses on the phenomenon of explanation by redescription in mathematics. In such cases, the explanandum is fitted into a new mathematical framework from which it can be easily derived or even seen to be self-evident.
Mark Steiner died before publication of this work was completed.
I am grateful to Jake Solomon, Sylvain Cappell, Barry Simon, and Shmuel Elitzur, for their help.
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Notes
- 1.
Editors’ Note: Though Steiner speaks here of a “book”, in the bibliography he cites the research article by Hsin and Xie. Naturally, this does not exclude the possibility that he had an additional source in mind.
- 2.
Editors’ Note: Source: Albert Einstein Archive 1–170. We are grateful to the Albert Einstein Archive (the Hebrew University) and to Chaya Becker for locating the source of this quote.
- 3.
See Raman, M. (2012). “Beauty as fit: An empirical study of mathematical proofs.” Proceedings of the British Society for Research into Learning Mathematics 32(3): 156–160.
- 4.
Editor’s note: The original sentence begins “Whether or not fitness ...”. We have removed the “or not” in order to preserve Steiner’s apparent intention.
- 5.
See Hardy, G. H. (1967). A mathematician’s apology. Cambridge, Cambridge University Press.
- 6.
See Wigner, E. (1967). The unreasonable effectiveness of mathematics in the natural sciences. Symmetries and Reflections. Bloomington, Indiana University Press: 222–237.
- 7.
Editors’ Note: Aesthetic considerations permeate the opening chapters of Steiner’s, The Applicability of Mathematics as a Philosophical Problem, Harvard U. Press, 1998. See in particular page 7.
- 8.
Editors’ Note: The original version of this sentence begins with “So I am uncertain...”. We have changed the “So” to “But” in order to preserve Steiner’s apparent intention.
- 9.
Raman, M. and L.-D. Öhman (2011). Two beautiful proofs of Pick’s theorem. Proceedings of Seventh Conference of European Research in Mathematics Education. Rzeszow, Poland: 9–13.
- 10.
Editors’ Note: Pick’s Theorem (first reported by G A. Pick in 1899) says that the area of a simple polygon whose vertices have integer coordinates is given by the number of interior points plus half the number of boundary points minus 1. (See Coxeter, Introduction to Geometry, Wiley 1969 section 13.4)) B. Grunbaum and G Shephard “Pick’s Theorem” (American Mat Monthly (v. 100, 1993, Feb. 1993, v. 100, number 2. Pp. 150–161) survey various methods of proof and generalizations. The proof using Euler’s formula is set in a larger context in A. Martin, G. Ziegler, “Three applications of Euler’s formula: Pick’s Theorem” in Proofs from THE BOOK, 6th edition, Springer, 2018 (published after Steiner gave the present lecture).
- 11.
Editors’ note: L. Euler. “Elementa Doctrinae Solidorum”, Novi commentarii academiae scientiarum Petropolitanae 4 (1752/3) 1758, p. 109–140, reprinted in Opera Omnia, Series I, Vol. 26, p. 71–93.
- 12.
Editor’s note: The original sentence reads “The torus, on the other hand, also has two one dimensional holes ,,,” We removed the word “also”, which is apparently unintended.
- 13.
I have adapted for my own purposes an example appearing in Atiyah, M. (1978). “The unity of mathematics.” Bulletin of the London Mathematical Society 10: 69–76. Editors’ remark: The following examples are well explained in the Atiyah paper that Steiner cites.
- 14.
Actually, 1 − y and 1 + y are tangent to the unit circle, as the parabola y = x 2 is to the x-axis at the origin. This counts as two zeros of the function.
- 15.
Editors’ note: R. Hartshorne, Algebraic Geometry (Springer-Verlag, 1977) Sections II.2 and II.3 contain a thorough treatment of the basic notion of a scheme.
- 16.
See Steiner, M. (2001). “Review of Margaret Morrison, Unifying Physical Theories: Physical Concepts and Mathematical Structures (Cambridge UP, 2000).” The Philosophical Quarterly 51(204): 405–408.
- 17.
Furstenberg, H. (1955). “On the infinitude of primes.” American Mathematical Monthly 62(5): 353.
- 18.
Editors’ Note: The text ends at this point, mid-sentence. We believe that his paragraph should be revised as follows:
-
1.
Change the order of the sentences in the text:
In this style of explanation, we begin with a proof that does not fit into its ostensible surroundings, but we then find the appropriate theory for the theorem to fit into.
-
2.
Add a concluding sentence:
In this way the initial phenomenon is shown to fit into the new surroundings and is thus “explained away.”
-
1.
References
Atiyah, M. (1978). The unity of mathematics. Bulletin of the London Mathematical Society, 10, 69–76.
Furstenberg, H. (1955). On the infinitude of primes. American Mathematical Monthly, 62(5), 353.
Hardy, G. H. (1967). A mathematician’s apology. Cambridge University Press.
Hsin, A., & Xie, Y. (2014). Explaining Asian Americans’ academic advantage over whites. Proceedings National Academy of Sciences, 111(23).
Raman, M. (2012). Beauty as fit: An empirical study of mathematical proofs. Proceedings of the British Society for Research into Learning Mathematics, 32(3), 156–160.
Raman, M., & Öhman, L.-D. (2011). Two beautiful proofs of Pick’s theorem. In Proceedings of seventh conference of European research in mathematics education, Rzeszow, Poland, pp. 9–13.
Steiner, M. (1978a). Mathematical explanation. Philosophical Studies, 34, 135–151.
Steiner, M. (1978b). Quine and mathematical reduction. Southwest Journal of Philosophy, 9, 133–143.
Steiner, M. (1983a). Mathematical realism. Nous, 17, 383–395.
Steiner, M. (1983b). The philosophy of mathematics of Imre Lakatos. Journal of Philosophy, 80, 502–521.
Steiner, M. (2001). Review of Margaret Morrison, unifying physical theories: Physical concepts and mathematical structures (Cambridge UP, 2000). The Philosophical Quarterly, 51(204), 405–408.
Wigner, E. (1967). The unreasonable effectiveness of mathematics in the natural sciences. In Symmetries and reflections (pp. 222–237). Indiana University Press.
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Steiner, M. (2023). Explaining and Explaining Away in Mathematics: The Role of “Fitness”. In: Posy, C., Ben-Menahem, Y. (eds) Mathematical Knowledge, Objects and Applications. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-031-21655-8_2
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