Abstract
Explanation in mathematics has recently attracted increased attention from philosophers. The central issue is taken to be how to distinguish between two types of mathematical proofs: those that explain why what they prove is true and those that merely prove theorems without explaining why they are true. This way of framing the issue neglects the possibility of mathematical explanations that are not proofs at all. This paper addresses what it would take for a non-proof to explain. The paper focuses on a particular example of an explanatory non-proof: an argument that mathematicians regard as explaining why a given theorem holds regarding the derivative of an infinite sum of differentiable functions. The paper contrasts this explanatory non-proof with various non-explanatory proofs (and non-explanatory nonproofs) of the same theorem. The paper offers an account of what makes the given non-proof explanatory. This account is motivated by investigating the difficulties that arise when we try to extend Mark Steiner’s influential account of explanatory proofs to cover this explanatory non-proof.
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Notes
The article appears (unsigned, as a “gleaning”) on p. 283 of the December 1986 issue.
Of course, there are many senses of “mathematical explanation” that do not involve proofs; for instance, I can explain to my class what “uniform convergence” means or how to evaluate an integral or why so many students got the wrong answer on their test. None of these is a “mathematical explanation” in the sense of this paper (and of the recent literature I just mentioned). The same point arises in connection with scientific explanation; as Hempel (2001: 80) pointed out, an account of scientific explanation does not aim to cover “the vastly different senses of ‘explain’ involved when we speak of explaining the rules of a game, or … when we ask someone to explain to us how to repair a leaking faucet”.
Of course, a mathematical explanation may fail to prove the fact it explains because it merely sketches such a proof rather than giving it fully—just as some “scientific explanations” are merely explanation sketches. My concern in this paper is with some mathematical explanations that do not work by proving or even by sketching a proof of their explanandum.
D’Alessandro (forthcoming) also discusses mathematical non-proofs that explain. He focuses on non-arguments that explain in mathematics; in particular, he argues that various mathematical theorems explain, independent of their proofs. By contrast, I will focus on mathematical arguments that are not proofs but nevertheless explain.
The mathematical discussion here and below closely follows Bressoud (1994: 73–74). Note that if each of the fi is differentiable on [a, b] and f1′(x) + f2′(x) + f3′(x)··· converges uniformly on [a, b], then f1′(x) + f2′(x) + f3′(x)··· = (f1 + f2 + f3 + ···)′(x) for all x in [a, b].
May explain—but may not. As we will see, the explanation requires certain further conditions to be satisfied.
Nevertheless, talk of “mechanism” is sometimes used by those who study math education in order to gesture toward the difference between a mathematical explanation of some fact and a proof of it that does not explain why it holds. For instance:
Specific counter-examples are examples that merely satisfy the task of refuting a statement, and do not contribute to the understanding of the general case …. On the other hand, general examples uncover the crucial mechanism involved in the situation. This mechanism is both an explanation to the fact that the claim can be refuted, as well as a manifestation that counter-examples can be generated. (Peled and Zaslavsky 1997: 58–59, cf. 50)
This point applies to the explanation of the sum rule’s failing in the infinite case and how this explanation differs in explanatory power from the specific counterexample (also given in Sect. 2) to the sum rule for the infinite case.
This argument could equally well have been expressed in terms of mathematical induction.
The principle of mathematical induction (see previous footnote) does not allow the argument to extend to the infinite case, but does not specify any difference between the finite and infinite cases (other than that the latter is infinite) that is standing in the way. The infinite case simply fails to fall within the principle’s scope.
As we saw at the very start of the paper, mathematicians regard the fact that two calculator numbers are both divisible by 37 as no coincidence, and that is because of the proof that explains why every calculator number is divisible by 37. I have suggested that when a result’s salient feature is that it identifies a respect in which various cases are alike, then a proof explains that result exactly when the proof exploits some other respect in which those cases are alike and from there arrives at the result by treating all of the cases in the same way. I now suggest that if the result with such a salient feature has no such explanation, then it is a mathematical coincidence. Likewise, suppose we had not yet proved that the sum rule holds of every pair of differentiable functions and suppose we found that for four particular functions f1, …, f4, f1′(x) + f2′(x) = (f1 + f2)′(x) and f3′(x) + f4′(x) = (f3 + f4)′(x). We might then wonder whether or not this fact is a coincidence. The general proof of the sum rule for two functions makes it no coincidence. However, let’s add to those two facts an example where the sum rule holds for the sum of infinitely many functions. (There are such examples; see footnote 6.) That the sum rule holds for those three cases is a coincidence since the sum rule’s holding of those three cases cannot be derived from the respects in which those three cases are alike (e.g., that each is the sum of differentiable functions on the real numbers).
This is how I would reply to the objection that the explanatory non-proof I have been examining does not explain why the sum rule fails in the infinite case; it explains merely why a given proof in the finite case does not carry over to the infinite case. I agree that it explains why the proof does not carry over. But when the salient feature of the sum rule’s failure in the infinite case is that by this failure, the infinite case stands in contrast to all finite cases, then we can explain the sum rule’s failure in the infinite case by identifying another difference between finite and infinite cases that stops a proof of the sum rule in the finite case from generalizing to the infinite case. In this respect, the example is analogous to a standard case from the literature on scientific explanation. Suppose tertiary syphilis is necessary for the development of paresis (though very few patients with tertiary syphilis develop paresis). Suppose we ask why Jones developed paresis—and we pose this question in a context where the salient contrast is with Smith, who did not develop paresis. That Jones had tertiary syphilis (where Smith did not) answers the question; it explains more than merely why the mechanism that kept Smith from getting paresis failed to carry over to Jones.
Resnik and Kushner (1987: 147–152) use this example to argue that Steiner’s requirement that an explanatory proof generalize is too high since this proof of the IVT fails to generalize to the rationals.
This sum is ζ(3), where ζ(s) = \( \sum\nolimits_{n = 1}^{\infty } {\frac{1}{{n^{s} }}} \) is the Riemann zeta function. ζ(s) is intractable for any positive odd integer (other than s = 1, for which the sum diverges).
As another example where a mathematician explains why one problem turns out to be so much more difficult than another, seemingly closely related one, consider these remarks that Hardy wrote in his copy of a letter he had written to Ramanujan: “When r is even \( {\text{f}}({\text{x}}e^{ - p\pi i/q} ) \) is an elementary function. The same sort of method applies but is much easier. Hence we see why the odd case is so much harder” (Berndt and Rankin 1995: 152).
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Lange, M. Mathematical Explanations that are Not Proofs. Erkenn 83, 1285–1302 (2018). https://doi.org/10.1007/s10670-017-9941-z
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DOI: https://doi.org/10.1007/s10670-017-9941-z