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Inference to the Best Explanation Is an Important Form of Reasoning in Mathematics

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Figure 1

Notes

  1. Brown [1, p. 164], [2, p. 48], and Franklin [3, p. 12] have pointed this out.

  2. The arithmetic–geometric mean agm(x,y) is the limit of both of the linked sequences ai and bi, where an+1 = (an + bn)/2, bn+1 = \(\sqrt{{a}_{n}{b}_{n}}\), a0 = x, b0 = y, 1/agm(1,\(\sqrt{2}\)) = 0.8346268… . Borwein and Devlin [4, pp. 4–5] and Rodriguez Villegas [5, p. ix] portray Gauss as justified in regarding his evidence as strongly confirmatory. (My thanks to James Robert Brown for suggesting the former book and several other useful references.) This example of Gauss’s approach is not unusual. In 1817, Gauss wrote (quoted by Gray [6, p. ix]: “It is characteristic of higher arithmetic that many of its most beautiful theorems can be discovered by induction with the greatest of ease but have proofs that lie anywhere but near at hand… .” (Regarding “induction,” see note 3.)

  3. By “inductive,” Pólya is not referring to mathematical induction, which is a form of proof (i.e., deductive reasoning). Rather, Pólya means the kind of reasoning by which hypotheses are confirmed but not proved.

  4. Pólya [7, p.4]. Franklin [8, p. 16] likewise takes this pattern of reasoning to be central.

  5. The philosophy of science literature is full of examples in which a generalization’s making some successful (and no unsuccessful) predictions fails to confirm some of its other, as yet unverified, predictions. Two classic discussions are Goodman [9, pp. 66–83] and Salmon [10].

  6. For the tradition of characterizing some confirmation as operating by IBE, see the surveys and references in Lipton [11] and Lycan [12]. The label “inference to the best explanation” was popularized by Harman [13]. Peirce [14, pp. 5.180–5.212] referred to (very roughly) the same sort of reasoning as “abduction.” IBE has also had its share of opponents, notably van Fraassen [15], Salmon [16], and (more recently) Roche and Sober [17], to which Lange [18] replies.

  7. Darwin [19, p. 421], my emphasis.

  8. The gleaning appeared on page 283 of volume 70, number 454. I am grateful to Roy Sorensen for calling my attention to this example in a lovely unpublished essay.

  9. In Leavitt [21, p. 182], for instance. (This example is also mentioned by Sorensen op. cit.) I will not try here to spell out what it would take for a proof to give a unified uniform treatment of the various cases falling under the theorem being proved. To do so would require an account of natural mathematical properties and kinds (as Baker [22, p. 144] puts it) so that an arbitrary pair of proofs cobbled together does not count as a unified uniform proof of the conjunction of the two theorems proved. My argument for IBE’s role in mathematics will fortunately not depend on any particular account of what it is for a proof in mathematics to be explanatory. I discuss all of these topics (mathematical coincidence, natural mathematical properties and kinds, explanation in mathematics, unification, …) in greater detail in Lange [23].

  10. I discuss this example more fully in Lange [23], where I also refer briefly to some earlier discussions of this example’s methodological lessons.

  11. For instance, suppose one circle is centered at (0, 20) with radius 16 and the other circle is centered at (0, –15) with radius 9. These circles intersect nowhere in the Euclidean plane. But their equations are x2 + (y – 20)2 = 162 and x2 + (y + 15)2 = 92, which have common solutions (12i, 0) and (–12i, 0). This example generalizes (though the confirmation that I am characterizing as IBE does not depend on the mathematicians’ having already proved that this example generalizes, since the mathematicians could merely have confirmed that it does). Let the two circles be centered at (x1, y1) and (x2, y2) with radii r1 and r2, respectively. To keep the following expressions relatively compact, let L be the distance between the centers and let A be the area of a “triangle” with sides of lengths r1, r2, and L. Then the circles’ two points of intersection (x, y) are \(x = \frac{{x_{1} + x_{2} }}{2} + \frac{{r_{1}^{2} - r_{2}^{2} }}{{2L^{2} }} \left( {x_{2} - x_{1} } \right) \pm 2A\frac{{y_{2} - y_{1} }}{{L^{2} }} ,\) \(y = \frac{{y_{1} + y_{2} }}{2} + \frac{{r_{1}^{2} - r_{2}^{2} }}{{2L^{2} }} \left( {y_{2} - y_{1} } \right) \mp 2A\frac{{x_{2} - x_{1} }}{{L^{2} }}.\) These expressions apply whether the two circles intersect at two points in the Cartesian plane or none. Imaginary coordinates for points of intersection arise when there is no Euclidean triangle with sides of lengths r1, r2, and L, which occurs when L > r1 + r2 (case (b) in the figure) or L < |r1 \(-\) r2| (case (c) in the figure). In all cases, A is to be evaluated by Heron’s formula: \(A = \frac{1}{4} \sqrt {\left( {L + r_{1} + r_{2} } \right)\left( {L + r_{1} - r_{2} } \right)\left( {L - r_{1} + r_{2} } \right)\left( { - L + r_{1} + r_{2} } \right)} .\) This formula gives imaginary A for an impossible triangle.

  12. For another example, see the discussion of Desargues’s theorem in Lange [23, pp. 314–346].

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    Marc Lange

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Lange, M. Inference to the Best Explanation Is an Important Form of Reasoning in Mathematics. Math Intelligencer 44, 32–38 (2022). https://doi.org/10.1007/s00283-021-10112-7

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