Abstract
This paper argues that in mathematical practice, conjectures are sometimes confirmed by “Inference to the Best Explanation” (IBE) as applied to some mathematical evidence. IBE operates in mathematics in the same way as IBE in science. When applied to empirical evidence, IBE sometimes helps to justify the expansion of scientists’ ontological commitments. Analogously, when applied to mathematical evidence, IBE sometimes helps to justify mathematicians' in expanding the range of their ontological commitments. IBE supplements other forms of non-deductive reasoning in mathematics, avoiding obstacles sometimes faced by enumerative induction or hypothetico-deductive reasoning. Both platonist and non-platonist interpretations of mathematics ought to accommodate explanation in mathematics and ought to recognize IBE in mathematics, though these interpretations disagree on the ontological commitments that mathematicians ought to have. This paper offers an inductive account of why mathematical IBE tends to lead to mathematical truths.
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Notes
This suggestion is not new. It is famously (though briefly) made by Gödel (1964, p. 265). In Sect. 2, I contrast my proposal with the accounts (whether of IBE in mathematics or of the non-deductive reasoning used in mathematics to support the expansion of mathematicians’ ontological commitments) given by some other philosophers. Pincock’s proposals are much nearer to mine than these other philosophers’ proposals are. Like me, Pincock (2012, pp. 295–299) proposes that IBE in mathematics is used to support the expansion of mathematicians’ ontological commitments. Like me, Pincock (2012, pp. 210–220) sharply distinguishes this sort of IBE (where the fact being explained is purely mathematical) from IBE in mathematical indispensability arguments in philosophy (where the explanandum is a fact about the physical world). As I am about to do, Pincock (2012, pp. 270–275) discusses the expansion of mathematicians’ ontological commitments to complex numbers—though whereas I emphasize the role of IBE arguments in underwriting this expansion (with complex numbers providing the explanations of some facts exclusively about the reals), Pincock emphasizes arguments from the ways that real-number functions could be extended to complex numbers. Of course, these are not incompatible views of how the recognition of complex numbers was historically underwritten; I believe that both of these sorts of considerations were influential.
I am not original in emphasizing the role played by explanatory considerations in the mathematicians’ expansion of their ontological commitments to include complex numbers. That mathematicians’ recognition of complex numbers was motivated significantly by mathematicians’ awareness that complex numbers would nicely explain various facts about the reals has been emphasized by, for instance, MacLane (1986, pp. 118–119): “The complete acceptance of complex numbers came primarily in their many uses in helping to understand other parts of Mathematics. … [T]here are phenomena with real numbers which can be properly explained only with complex numbers.” Furthermore, whereas Pincock (2015, p. 13) appeals to IBE in mathematics in connection with explanations of the unsolvability of the quintic, I would look further back (and two exponential powers lower) for IBE in mathematics having historically underwritten the expansion of mathematicians’ ontological commitments to include complex numbers. As Birkhoff and MacLane (2010, p. 476) emphasize, it was an important discovery (the so-called “irreducible case” of the cubic) that a general (unificatory, explanatory) formula for finding the real roots of cubic equations must use complex numbers. (For more on this expansion of mathematicians’ ontological commitments to complex numbers, see note 8; for more on Pincock’s views, see note 11.).
On the other hand, while my approach is not unprecedented, appeal to IBE in mathematics is hardly uncontroversial. For instance, Dummett (1994, p. 13) writes that “there is nothing in mathematics that could be described as inference to the best explanation.”.
My presentation here of this example closely follows my presentations in Lange (2010, pp. 329–331; 2017, pp. 290–292, 331, 344–345; 2019, pp. 3–4, 13; forthcoming1:4–5; forthcoming2:8–9); I have previously used this example in different places to make different points about explanation in mathematics. In Lange (forthcoming1), I use it (in an informal, non-philosophical essay) to motivate the importance of IBE to mathematics. But I have not previously devoted particular attention to the central philosophical topics of this paper, such as the role of IBE in expanding the range of mathematicians’ ontological commitments.
Here is an example of such an IBE argument from the mathematics literature: “If a theory explains much data, then perhaps the theory is true. … Are there a set of random facts that P ≠ NP would help explain? Yes. The obvious one: P ≠ NP explains why we have not been able to solve all of those NP-complete problems any faster!” (Gasarch, 2014, p. 258) Thanks to Bill D’Alessandro for this example.
Also note the important parenthetical reference in the main text to IBE permitting relevant collateral evidence and background knowledge to override explanatory considerations. IBE is not best understood as some mechanical rule of confirmation. (This point has been emphasized even by van Fraassen (1989, pp. 145–146), a notable critic of IBE.) Rather, IBE involves considerations of explanatory quality serving as a guide (not the sole guide) to likeliness. Therefore, IBE requires that explanatory considerations be combined with other information, which sometimes outweighs explanatory considerations or even renders them irrelevant. Unfortunately, IBE is sometimes formulated as purporting to be a mechanical rule of confirmation. For instance, Boyce (2021) shows that collateral information can produce counterexamples to the following principle: If H is part of the “best explanation” of E and C is an indispensable part of that explanation, then E confirms C. However, such counterexamples are accommodated by IBE when IBE is understood in the way that I just mentioned.
Let me emphasize that even a fictionalist should recognize explanations in mathematics. The fact that p would have been the case, had there been certain platonistic mathematical entities, can have explanatory proofs and non-explanatory proofs. (Of course, fictionalism interprets those proofs as proceeding from other mathematical facts understood in this same fictionalist way.).
I am invoking here the distinction between what Armstrong (1978, pp. 38–41) and Lewis (1999, pp. 10–13) call “natural” (or “sparse”) properties—that is, respects in which things may genuinely resemble one another—on the one hand, and mere shadows of predicates (i.e., “abundant” properties), on the other hand. For instance, an arbitrary disjunction of natural properties is not a natural property since, for instance, being five grams or positively electrically charged is not a genuine respect in which objects may resemble one another. For more on the distinction in mathematics between natural and non-natural properties, see Corfield (2003), Lange (2017), and Tappenden (2008).
I take this to be part of the point behind Kitcher’s (1989, p. 564) rhetorical question: “Can we assume that invoking entities that satisfy constraints we favor is a legitimate strategy of recognizing hitherto neglected objects that exist independently of us?”
Mancosu (2008, p. 140) also points out that an explanatory proof cannot be replaced with a proof that avoids additional mathematical ontology without loss of explanatory power.
The case of the two Taylor series is just one of many examples where facts about complex numbers beyond the reals were recognized as having the potential to explain facts exclusively about reals. For instance, Euler (Elements of Algebra, Part II, §193) saw that complex numbers could explain why the only integral solutions to the elliptic curve x3 = y2 + 2 are (x,y) = (3, ± 5).
It might be objected that frictionless planes, ideal gases, and so forth can help to explain the behavior of actual physical objects even if they are not on an ontological par with those objects. In my view, however, the explaining is not done by facts about actual frictionless planes (obviously) or even by facts about what frictionless planes would have been like, had there been any. Rather, what explains are simply facts that certain actual physical objects are certain ways. Since their being those ways makes them (to a sufficient degree of approximation) like frictionless planes, it is helpful to describe those objects in terms of what frictionless planes would be like.
One way to elaborate this objection to the Quinean IBE is that mathematical objects are not serving as causes according to the proposed scientific explanation and so the evidence for that explanation does not confirm their existence. This objection to the Quinean IBE would apply to the mathematical IBE since the mathematics there is not describing causes of the explanandum (a purely mathematical fact having no causes). However, this is not a good objection. Evidence for a scientific explanation often counts as evidence for non-causes it posits. For instance, laws of nature (and features of spacetime structure) are not causes, are often posited by putative scientific explanations, and are often confirmed by evidence for the explanation.
Clarke-Doane (2012, p. 314, 326) gives more references to biological and philosophical literature on natural selection for a disposition to generate true simple mathematical beliefs. Pincock (2012, pp. 297–298), in taking IBE in mathematics as having supported expansions in the range of mathematical domains to which mathematicians have justified commitments (see note 1), also proposes a non-IBE rationale for the commitments to (e.g.) ordinary arithmetic from which the expansion takes place.
Lange (forthcoming3) presents an approach along these lines as justifying IBE in science.
My own account of explanation in mathematics (Lange, 2014, pp. 506–507; 2017, pp. 254–268; 2018, pp 1296–127; 2019, pp. 12–13; forthcoming1:3) emphasizes that many (though not all) explanatory proofs derive their explanatory power by tracing a salient similarity among the cases in the explanandum back to some analogous similarity identified by the explanans. In this paper, I have no need to presuppose that my account correctly identifies the source of these proofs’ explanatory power. It suffices that this feature is often loveliness-enhancing.
The article appears (unsigned, as a “gleaning”) on p. 283 of the December 1986 issue.
For example, it is just a coincidence that 31, 331, 3 331, …, and 33 333 331 are all prime. (The next number in the sequence is composite.) Similarly, consider these two Diophantine equations (that is, equations where the variables can take only integer values):
2x2(x2 − 1) = 3(y2 − 1)
and
x(x − 1)/2 = 2n − 1.
As it happens, each equation has exactly the same five positive solutions for x: 1, 2, 3, 6, and 91 (Guy, 1988, p. 704). Mathematicians believe that this is just a coincidence—that there is no explanation. (I gave this example in Lange (2010, p. 316; 2017, pp. 278–279, 289; forthcoming2:9).).
My proposed account of why potential explanatory loveliness is a good guide to truth (see note 12) is in the spirit of Sober’s (1990) account of why simplicity is a good guide to truth: because the background beliefs that guide us when we are being guided by simplicity are true. (For instance, simpler phylogenetic trees posit fewer mutations, and we know that mutations are rare, so in being guided by simplicity in arriving at phylogenetic trees, we tend to be guided toward the truth.) The accuracy of those background beliefs is, in turn, no great mystery (bracketing Hume’s problem) considering that they were arrived at inductively.
In (Lange, 2018, pp. 1288–1290), I gave this example for a different purpose.
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Lange, M. Inference to the best explanation as supporting the expansion of mathematicians’ ontological commitments. Synthese 200, 146 (2022). https://doi.org/10.1007/s11229-022-03656-4
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DOI: https://doi.org/10.1007/s11229-022-03656-4