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Mathematical explanation and the theory of why-questions.

1 Introduction

Philosophers of science have long described explanations as answers to why-questions.(1) The view that this is the most profitable way to study explanations has gained recent prominence. For instance, Alan Garfinkel writes, 'Attending to the questions rather than the answers and looking for the implicit question hiding behind the answer are a useful device for analysing explanations and understanding historical shifts' ([1981], p. 8). Recently, Jaakko Hintikka and Ilpo Halonen have written on the subject [1995]. Bas van Fraassen, the best-known supporter of this view, writes:

An explanation is not the same as a proposition, or an argument, or a list of propositions; it is an answer. (Analogously, a son is not the same as a man, even if all sons are men, and every man is a son.) An explanation is an answer to a why-question. So, a theory of explanation must be a theory of why-questions ([1980], p. 134, my emphasis).

I shall call the approach to explanation shared by these authors the why-question approach.(2) Its most distinctive claim concerns explanatory evaluation: explanations can only be properly evaluated with respect to why-questions. Garfinkel calls this thesis explanatory relativity. For him, an explanation must be evaluated in the context of a contrast space, specified in the why-question. He says, 'I mean this as a claim about how to explain explanations: the contrast space is a basic presupposition of the explanation context, an additional piece of structure necessary to explain how explanations function' ([1981], p. 28). Similarly, van Fraassen says, 'Since an explanation is an answer, it is evaluated vis-a-vis a question, which is a request for information' ([1980], p. 156). Following van Fraassen, Resnik and Kushner state this thesis strongly: '[N]othing is an explanation simpliciter but only relative to the context dependent why-question(s) that it answers' ([1987], p. 153). In this paper, I will argue against this approach, focusing on van Fraassen's theory of why-questions in particular. I will grant that explanations do answer why-questions and explanatory evaluations are context dependent, but claim that the why-question approach nonetheless misses crucial aspects of certain explanatory evaluations.

This essay will take the somewhat unusual approach of considering mathematical explanations. Though van Fraassen did not have mathematics in mind when he formulated his theory, an adequate why-question oriented theory of explanations should address mathematical, as well as empirical, why-questions. Analogous to the claim that scientific explanations are only distinguished by the fact that they invoke scientific information ([1980], pp. 155-6), mathematical explanations should differ from other types only in their subject-matter; not in any fundamental way. Since why-questions can be and are asked and answered about mathematical facts (for instance, after having been informed that 1 - 1/3 + 1/5 - 1/7 + . . . converges to [Pi]/4, it is certainly reasonable to ask why this is so), as well as empirical ones, the why-question approach should be adaptable to mathematical explanations. Resnik and Kushner [1987], in response to Steiner [1978], have also studied mathematical explanations, and though they do not fully endorse van Fraassen's theory, they endorse the why-question approach in general. We shall see, however, that mathematical explanations create special problems for van Fraassen's why-question theory.

After a brief review of van Fraassen's theory of why-questions, I will present an example of mathematical explanation, supplementing those already given by Steiner [1978], Resnik and Kushner [1987], and Kitcher [1983, 1989]. I will then give two arguments against van Fraassen's theory of explanation: (1) the methods he proposes to evaluate explanations are trivialized in the context of mathematical explanations; and (2) the structure of his theory makes it difficult to see how a proof could count as an explanation, though mathematical explanations often do take that form. Finally, I will argue that, though a stronger theory than van Fraassen's might be able to avoid those two points, no why-question theory can account for the way the explanation in the case study is evaluated.

2 Exposition of van Fraassen's theory

Van Fraassen's theory of why-questions is found in Chapter 5 of The Scientific Image [1980].(3) For him, why-questions consist of three items: a topic, a contrast class, and a relevance relation. When one asks, 'Why P?', the topic is P. That is, the topic is what we ordinarily would understand the subject of the request for an explanation to be. The contrast class, X, is a set of propositions: {[P.sub.1], [P.sub.2], . . ., [P.sub.n]}. The topic itself is some Pk from this class and all other [P.sub.i]'s are alternatives to P. That is, when one wants to know why P is true, one asks why P is true rather than some other possibility. The contrast class specifies the set of possibilities the questioner is willing to consider (including P itself). Finally, the relevance relation, R, is used to constrain admissible answers, by specifying what factors will count as explanatorily relevant. For instance, in many situations it would be inappropriate to answer 'Why does the blood circulate through the body?' in terms of the heart's mechanical pumping action. This question is often aimed instead at the function the circulation of the blood serves in keeping an organism alive. A more appropriate answer to the question in this sense is that the blood circulates to provide nourishment to the tissues. The relevance relation is meant to distinguish between different senses of such questions. In van Fraassen's words, the relevance relation encodes the 'respect-in-which a reason is requested' (p. 142). The only formal constraint on the relevance relation is that it obtains between proposed answers and topic/contrast-class pairs.

Van Fraassen's theory provides two types of evaluation of answers. First, a proposed explanation may or may not answer the question at all. For instance, if the question 'Why does the blood circulate through the body?' were intended to invoke a functional answer, a purely mechanical one would not answer the question in its original sense. Second, even if a proposed explanation answers the question in this sense, the answer may not be 'telling'. The proposed answer that the circulation of the blood functions to cool the tissues provides the right kind of information to answer the question as intended, but is a poorer explanation on factual grounds. The theory of what counts as an answer at all is developed in the spirit of Belnap and Steel, but the theory of telling answers has more in common with traditional studies of scientific explanation.

Answers are distinguished from non-answers by the definition of a direct answer:

B is a direct answer to a question Q = <[P.sub.k], X, R> exactly if there is some proposition A such that A bears relation R to <[P.sub.k], X> and B is the proposition which is true exactly if ([P.sub.k]; and for all i [not equal to] k, not [P.sub.i]: and A) is true (p. 144).

This definition also determines what, in Belnap and Steel's sense, the why-question presupposes - what every direct answer to the question must affirm as true. A why-question presupposes that its topic is true; that the other members of the contrast-class are not true; and that at least one proposition bearing the relevance relation to the topic and contrast-class is true. If a why-question has a false presupposition (for instance, if Pk is false), the best response is not a direct answer, for no direct answer could be true. Instead, the appropriate response is a corrective answer (such as '[P.sub.k] is not true!'). Van Fraassen takes this account of presupposition to solve what he considers one of the major problems for theories of explanation - accounting for rejections of explanation requests.

However, even if the presupposition of a why-question is true, there may be no appropriate answers, because none is telling. For instance, in the time-honoured syphilis-paresis example, there is no telling answer why the mayor of the town developed paresis, given that he had latent untreated syphilis. There is nothing (so far as we can determine) that favours his developing paresis among this contrast class (p. 128).(4) Van Fraassen needs an evaluative component for answers beyond the relevance relation, because even if an answer gives the sort of information the questioner has in mind (i.e. satisfies the relevance relation), and all of that information is true, it may still not have any bearing on the topic with respect to the rest of the contrast class. As with other forms of question, one wishes not to give false answers, but for why-questions more is required. Van Fraassen introduces the theory of telling answers to specify what more is needed. His theory contains three elements. First, a telling answer must be probable in light of our background knowledge. Second, it must probabilistically favour the topic over the other members of the contrast class relative to background knowledge. Third, it must be comparatively better in these regards than other potential answers.

The first of these criteria is straightforward, but the other two lead to complications which van Fraassen took some pains to deal with. For our purposes, the most important problem is that when one asks a why-question, one generally knows that the topic is true, and other members of the contrast class false (indeed, the question presupposes this). Therefore, in the light of our full background knowledge there is no way that any answer could favour the topic. For the sake of evaluation, the topic must be probabilistically favoured in the light of some restricted part of the background knowledge, K(Q). This in addition to the topic, contrast class, and relevance relation, is a part of the context of the question. Van Fraassen said nothing about how this subset of the background knowledge is selected, merely noting that the problem is not unique to his theory.(5)

Van Fraassen mentioned several other problems concerning comparative evaluation of answers. I mention them here briefly for completeness, but they will not be important in what follows. First, determining when an answer favours the topic can be a subtle matter. Overly simple criteria can fail to properly delineate probabilistic relevance and some cases of favouring can be counterintuitive.(6) Second, there are additional complications in evaluating an answer with respect to other possible answers. Besides being better at the first two criteria than other direct answers to a question, a telling answer also must not be screened off, or rendered probabilistically irrelevant, by any other answer.

Because of these problems, van Fraassen did not put as much confidence in the details of his theory of telling answers as he did in his characterization of direct answers, so we should not ultimately hold him to this theory. I shall argue below that it cannot apply to mathematical explanation. But even further, I will argue against van Fraassen's entire picture of the relationship between why-questions and explanatory evaluations, and that any theory along the same lines will face serious challenges. Thus, I will spell out what I take to be the philosophical core of his theory, stripped of all technical details. Whatever the particular theoretical details, a theory within the why-question approach must follow these lines.

As van Fraassen points out, a why-question stated in English can often be understood in many different ways. The question 'Why did Adam eat the apple?' seems perfectly straightforward, until we consider it with different words emphasized: 'why did Adam eat the apple?' or 'Why did Adam eat the apple?' or 'Why did Adam eat the apple?. (In van Fraassen's theory, these readings of the question differ in their contrast classes.) Each requests a different sort of answer. The first requires that we say something about Adam among all people that could have eaten the apple. The second requires that we say something about the possible actions Adam could have taken with the apple. The third requires that we say something about the different things Adam could have eaten. If we say that Adam ate the apple because he was hungry, we will not have successfully answered 'Why did Adam eat the apple?' because he could have eaten something else and satisfied his hunger.

Thus, if explanations are regarded as answers to why-questions, they cannot just be evaluated with respect to the ordinary English statement of a question, which will often be ambiguous. Van Fraassen's logic of why-questions is an attempt to specify unambiguously what a why-question evaluation will involve. The natural-language why-question is then identified with one of these logical why-questions. This is where context steps in. It is the context in which the natural-language why-question is asked which is supposed to determine which logical why-question is being considered. In turn, this affects how an answer will be evaluated. The two evaluative components of van Fraassen's theory, the relevance relation and the theory of telling answers, both require that contextual matters be specified before they can be applied; the relevance relation depends directly on the context in which the question is asked, and the telling-answers theory depends on the contrast class, and also requires that K(Q) be specified. Therefore, for van Fraassen, explanatory evaluations are necessarily context-dependent.

This, then is the role of context in the why-question approach to explanation: A why-question is asked in natural language. The context of utterance determines which logical why-question is intended. Explanations are then evaluated with respect to this logical why-question. Context plays a role in the evaluation of explanations, but indirectly, through the specification of a logical why-question. We can ask why the context could not figure more directly in explanatory evaluations, instead of through the mediation of why-questions. My example will challenge the why-question approach on exactly this point: an explanation is given that is judged explanatory though it is a poor answer to the logical why-question that provoked it. It is a better answer to another logical why-question, but that question cannot be formulated in the original context. In contexts in which such a question can be formulated, however, the same explanation is not as effective. Thus, there appears to be something fundamentally wrong with the why-question theory's picture of context-dependent evaluation.

3 An example of mathematical explanation

Let us now examine a case of mathematical explanation. Supplementing those already given by other authors, it gives some reason to believe that there is something legitimately called 'explanation' in mathematics. After presenting the example, I will consider some of the why-questions that this explanation might be taken to address. Though this analysis will be somewhat useful, it will turn out that van Fraassen's theory fails to adequately account for mathematical explanation in general, and the explanation here is not best regarded as the answer to any why-question that could have preceded it.

On p. 147 of Patterns of Plausible Inference ([1968], Vol. 2),(7) George Polya gave an example of a proof which he finds profoundly unsatisfactory, though it is well within accepted standards of rigour. After giving a presentation intended to reflect how the proof would be presented in a textbook or journal article,(8) he supplemented the proof with additional explanatory material. This additional reasoning, though unnecessary for proving the theorem, enhances the proof's explanatory clarity. Since the reasoning is somewhat technical (though not beyond the grasp of a good advanced calculus student), I will only present a sketch of these arguments here.(9)

The theorem to be proved is: if the terms of the sequence [a.sub.1], [a.sub.2], [a.sub.3], . . . are non-negative real numbers, not all equal to 0, then

[summation of] [([a.sub.1][a.sub.2][a.sub.3] . . . [a.sub.n]).sup.1/n] where n = 1 to [infinity] [less than] e [summation of] [a.sub.n] where n = 1 to [infinity].

The proof begins with the definition of an auxiliary sequence, [c.sub.1], [c.sub.2], [c.sub.3], . . . by the formula [c.sub.1], [c.sub.2], [c.sub.3] . . . [c.sub.n] = [(n + 1).sup.n]. With the help of this sequence, the theorem follows straightforwardly by an unremarkable series of inequalities. The [c.sub.i] sequence is essential to the proof, but when it is first chosen, it seems quite arbitrary. It is not clear, even after one has studied the proof, why an auxiliary sequence is introduced in the first place, much less why this particular sequence is chosen. Polya called this a deus ex machina step, since it seems to come out of nowhere, but helps complete the proof once made. Because the reader is liable to be puzzled by the deus ex machina step, Polya considered this presentation of the proof to be insufficient, though it is adequately rigorous. He suggested that the author of the proof should inform the reader why the step was taken, beyond the simple reason that 'it makes the proof work'. He did the extra work in Patterns of Plausible Inference, and claimed that these additions to the original proof make the derivation more understandable (p. 152).

In his explanation, Polya first gave some motivation for the theorem itself - it is a lemma used to prove another theorem: if the series with positive terms [summation of] [a.sub.n] where n = 1 to [infinity] converges, then the series [summation of] [([a.sub.1][a.sub.2][a.sub.3] . . . [a.sub.n]).sup.1/n] where n = 1 to [infinity] also converges. Polya noted that the term [([a.sub.1][a.sub.2][a.sub.3] . . . [a.sub.n]).sup.1/n] on the left-hand side of the equation is rather complicated. It is, however, also the left-hand side of the well-known inequality between geometric and arithmetic means: [([a.sub.1][a.sub.2][a.sub.3] . . . [a.sub.n]).sup.1/n] [less than or equal to] [a.sub.1] + [a.sub.2] + [a.sub.3] + . . . + [a.sub.n]/n. Since the right-hand side of this inequality is simpler, it makes sense for Polya to base his strategy on this fact. However, if applied naively, it leads to immediate trouble:

[summation of] [([a.sub.1][a.sub.2][a.sub.3] . . . [a.sub.n]).sup.1/n] where n = 1 to [infinity] [less than or equal to] [summation of] [a.sub.1] + [a.sub.2] + [a.sub.3] + . . . + [a.sub.n]/n where n = 1 to [infinity]

= [summation of] [a.sub.k] where k = 1 to [infinity] [summation of] 1/n where n = k to [infinity]

The second step is a very natural way to collect terms, but since the inner sum is divergent, we have proved nothing. Polya noted that the problem is that since the series [summation of] [a.sub.n] where n = 1 to [infinity] converges by hypothesis, earlier terms will tend to be much larger than later terms (the sequence [a.sub.1], [a.sub.2], [a.sub.3], . . . [a.sub.n], . . . must eventually decrease at least as fast as 1/n). Therefore, the two sides of the inequality between the geometric and arithmetic means will tend to be somewhat unequal. Polya recommended trying to balance the two sides by making the terms in the inequality more equal. One way to do this is to multiply [a.sub.i] by some increasing factor. This is the core of the idea behind the [c.sub.i] sequence.

However it is not yet clear what sequence one should use. Polya did not begin by considering all possible auxiliary sequences. He first suggested replacing the terms [a.sub.1], [a.sub.2], [a.sub.3], . . . [a.sub.n], . . . with 1[a.sub.1], 2[a.sub.2], 3[a.sub.3], . . . n[a.sub.n] . . ., but quickly proposed a more general approach: [1.sup.[Lambda]][a.sub.1], [2.sup.[Lambda]][a.sub.2], [3.sup.[Lambda]][a.sub.3] . . ., [n.sup.[Lambda]][a.sub.n] . . . By considering the more general case, he hoped to find a value for [Lambda] that is most favourable for completing the proof. His initial attempt leads quickly to a term that cannot be calculated, but he was able to proceed by approximation, eventually winding up with the term [e.sup.[Lambda]][[Lambda].sup.-1] [summation of] [a.sub.k] where k = 1 to [infinity]. If not for the approximations, this would be very close to the desired result. Even so, he was at least in a position to choose a value for [Lambda]. Presumably, it is best to minimize [e.sup.[Lambda]][[Lambda].sup.-1], by choosing [Lambda] = 1. Polya's original first guess of auxiliary sequence, [c.sub.i] = i, was fairly good, but it now is more than a mere guess. Among the sequences considered, it is most favourable for Polya's purposes. However, even with [Lambda] = 1, he still came to a term he could not calculate, and so could not complete the proof.

At this point, Polya made an important observation, though he was not very explicit about its significance. He does not need to choose the [c.sub.i] sequence to be 1, 2, 3, . . . in order to realize the advantage of choosing [Lambda]. He could equally well use any sequence which is close to 1, 2, 3, . . . in the limit (or any constant multiple of such a sequence). Thus, any sequence asymptotically equivalent to 1, 2, 3, . . . will be as useful for balancing the inequality, but some other choice may lead to easier calculations.

To get an idea of what sort of sequence might aid in the calculation, Polya next considered the completely general case, where [c.sub.i] is unrestricted. He proceeded through a sequence of steps similar to that he used before, and wound up with the term

[summation of] 1/n[([c.sub.1][c.sub.2][c.sub.3] . . . [c.sub.n]).sup.1/n] where n = k to [infinity].

To proceed, he had to choose a [c.sub.i] sequence asymptotically proportional to n, and which also allows him to simplify this term. To do so, he resorted to a trick. He noted that

[summation of] 1/n(n + 1) = [summation of] (1/n - 1/n + 1),


[summation of] (1/n - 1/n + 1) where n = k [infinity] = 1/n.

If [c.sub.1][c.sub.2][c.sub.3] . . . [c.sub.n] = [(n + 1).sup.n], the problematic sum takes this form, and he could continue with the proof. Indeed, the sequence so defined allows him to proceed past the point at which he had been blocked, and complete the proof. Polya has reconstructed the original proof, but it is now easier to understand why the [c.sub.i] sequence appeared.

4 Why-question analysis of Polya's explanation

How well does the theory of why-questions account for this example? Certainly, the theorem and its initial proof seem to require some kind of further explanation. This need is naturally expressed in terms of why-questions, and the explanation superficially appears to address these questions. Thus, a why-question analysis initially appears to be a promising way to understand this case.

Polya's dissatisfaction with his original proof can be expressed naturally as a why-question: 'Why is it appropriate to introduce the [c.sub.i] sequence in the proof?' This question can be read in two ways: (1) 'Why should a sequence be introduced into the proof?' and (2) 'Why, of all sequences, should this particular one be chosen?' These two readings of the question are distinguished by different contrast classes. Thus the situation that prompts Polya's explanation seems quite amenable to a why-question analysis along van Fraassen's lines.

Polya's explanation is meant to account for the [c.sub.i] sequence, so we would expect that he answers one or both of the readings of the why-question. In answer to (1) we might say that an auxiliary sequence was used in order to replace a divergent series by a convergent one. In answer to (2) we might say that the particular sequence used was chosen for two reasons: it had a favourable growth behaviour. and it allows us to simplify a crucial term in the derivation. Prima facie, we might say then that Polya's additional exposition is a successful explanation because it provides a good answer to (1) and (2). We will see below that the situation is not so simple. Though this why-question analysis is useful in pointing out how Polya's explanation performs two distinct functions, it does not correctly account for what makes Polya's explanation good. But, before considering the case study in more detail, let us see how van Fraassen's theory might handle mathematical explanations in general.

5 Van Fraassen's theory and mathematical explanation

In the following sections I discuss two kinds of difficulty with the why-question approach. First, I discuss van Fraassen's theory in the general context of mathematical explanation. I show that his theory of explanatory evaluation cannot account for mathematical explanations. In the next section, I look at the Polya explanation in more detail from the general perspective of the why-question approach. This will lead to a more sensitive diagnosis of why-question theories of explanatory evaluation, and point to a potential alternative approach to context-dependent explanatory evaluations that avoids the problems uncovered by attention to mathematical explanation.

In this section I present two arguments against van Fraassen's theory of explanatory evaluations. First, it cannot account for mathematical answers to a common type of why-question, in which the members of the contrast class are mutually exclusive. Second, it cannot account for explanatory mathematical proofs, because proofs are not the sort of thing it recognizes as an answer, even though proofs are commonly offered as explanations in mathematics (see Steiner [1978] for numerous examples of explanatory and non-explanatory proofs). Though these arguments are directed at van Fraassen's theory of explanatory evaluation in particular, it will become clear that any theory drawn up along similar lines will encounter such difficulties, as I shall show below.

Consider a why-question for which the contrast class consists of mutually exclusive members, such as 'why does 1 - 1/3 + 1/5 - 1/7 + . . . converge to [Pi]/4, rather than some other real number?' Let us call such a question an exclusive-contrast question.(10) Under van Fraassen's theory, any proof of the topic at all will be a completely telling answer to such a question. The answer itself will be judged as maximally probable, as it follows from accepted mathematical propositions.(11) The topic itself, having been proven, will have probability 1. All other members of the contrast class, being incompatible with the topic, will have probability 0. No other answer can be more probable, favour the topic better, or screen it off. Therefore, van Fraassen's theory of evaluation of answers trivially recognizes any proof that establishes the truth of the topic of the question as completely telling. Thus, at least for exclusive-contrast questions, a proof must either be explanatory or not; there is no middle ground. But surely in mathematical cases, as in scientific ones, some explanations seem better than others. If mathematical explanations can be judged on more than a binary scale, the theory of telling answers must be revised. Clearly, it could no longer rely on the probability calculus, which favours all true mathematical propositions equally. The Polya case suggests that such a revision might be appropriate, because the judgements made there seem to form a continuum. Polya's explanation does not appear to uniquely favour the [c.sub.i] sequence (for masons spelled out in the next section). A (hypothetical) explanation that uniquely picked out the [c.sub.i] sequence would seem to be a more telling answer to the why-question.

More importantly for our purposes, only the relevance relation can distinguish explanatory from non-explanatory answers to exclusive-contrast why-questions, since it is the only evaluative component of van Fraassen's theory not left trivialized. This resembles a point made in Kitcher and Salmon [1987]. They argued that for any pair of true propositions, there is a way to 'explain' one by the other, as judged by van Fraassen's theory. Their argument also revolved around trivializing the theory of telling answers, because the answers they constructed were maximally telling - answers which, in conjunction with an appropriately selected subset of the background knowledge, imply the topic of the question and the negations of all other members of the contrast class. They were similarly able to trivialize the relevance relation. Because van Fraassen put no formal constraints on the relevance relation other than its relata, they had great leeway in constructing 'pathological' relevance relations. Between any fact to be 'explained' and any true statement, a relevance relation can be found such that the statement is a maximally telling answer to a why-question with that fact as its topic.

As an informal example, Kitcher and Salmon claim that van Fraassen's theory allows for an astrological explanation of John F. Kennedy's death on 22 November 1963. The topic is JFK's dying on that particular date, the contrast class consists of propositions stating that JFK died on each particular day of 1963, or survived the year, and the relevance relation allows only answers in terms of astral influence to be considered as explanations. This question can be naturally phrased as: 'Why, in terms of astral influences, did JFK die on 22 November 1963, rather than some other date?' The answer that Kitcher and Salmon propose consists in a true description of the configuration of stars and planets at the time of JFK's birth, which, in conjunction with an appropriate astrological theory, would imply that JFK died when he did. This information presumably would answer the question as asked, and in a maximally telling way. But surely the best answer to the question is a corrective one ('Astral influences had nothing to do with JFK's death'). This example resembles mathematical explanations in the way that it trivializes the telling-answers consideration. Mathematical explanations show that the difficulty arises in a large class of natural cases, not just for contrived examples. We cannot dismiss a non-explanatory proof as 'bad maths', as we might dismiss astrology as 'bad science'. The Kitcher-Salmon trivialization is a serious problem indeed.

The moral Kitcher and Salmon drew is that van Fraassen's theory needs an additional condition on why-questions, namely that the relevance relation used is a genuine relevance relation (rather than something gerrymandered). However, there are several reasons why this does not seem to be an appropriate solution. Van Fraassen's discussion of explanatory asymmetries ([1980], pp. 130ff.) indicates that he favours a somewhat unrestricted relevance relation. In situations where the traditional asymmetries of explanation are reversed, a somewhat atypical relevance relation may be called for. I suspect that he would also resist such restrictions on the relevance relation because they would not be logical constraints. Whether astral influence is relevant to people's deaths is not a logical matter, but a fact about our universe, and doesn't belong in a theory of the logic of why-questions. Also, the reasons we do not consider some proofs to be explanatory don't seem to have to do with relevance at all, or at least not relevance that the questioner can specify in advance. An early unintelligible proof of a result (such as one using 'brute force' calculational techniques) may be obviously relevant to that result, even if considered non-explanatory, while an explanatory proof employing more abstract mathematical resources may not be so obviously relevant; it may not be immediately clear that the resources used are appropriate to the problem. Indeed, discovering these resources may be a large step towards solving the problem in the first place. No relevance relation specifiable at the outset would be able to distinguish between explanatory and non-explanatory proofs, because what is relevant to the explanandum is not known prior to the explanation itself. This argument will be further developed in the next section, where a new way to view explanations will be proposed that will not be subject to such trivialization.

Thus, van Fraassen's theory seems to have inadequate resources to avoid trivialization in mathematical cases. Neither the theory of telling answers nor the relevance relation seems to do the trick; the theory of telling answers recognizes any mathematical proof as equally good in a wide range of realistic cases, but a priori considerations of relevance do not seem to correctly distinguish between explanatory and non-explanatory proofs. Thus, mathematical explanations call van Fraassen's theory of explanatory evaluation into question.

As it stands, there is another reason why van Fraassen's theory would be difficult to apply to mathematical cases. He regards an explanation as a presentation of descriptive information relevant to the topic: '[I]f you ask a scientist to explain something to you, the information he gives you is not different in kind (and does not sound or look different) from the information he gives you when you ask for a description' ([1980], p. 155). This attitude is also implicit in the theory of telling answers. An answer is evaluated by considering what contribution it makes to favouring the topic when it is used to supplement some subset of our background knowledge, K(Q).(12) Thus, the answer presents a piece of crucial descriptive information that makes the topic more credible than it would otherwise have been.

Suppose we attempt to extend this theory to mathematical cases. The natural thing to take as the background knowledge K in a mathematical context is either the set of axioms of some mathematical theory, or their deductive closure. The topic of the explanation (a theorem, for instance) will then be either implied by or contained in K. Therefore, as with empirical explanations, K must be further restricted to some suitable K(Q). The natural way to do so is to consider a smaller set of axioms, or its deductive closure. Then one would expect an answer to be some proposition which, in conjunction with this K(Q), implied the topic of the why-question. This could either be a missing axiom, or some weaker statement that still implies the theorem in question. Van Fraassen's theory thus suggests that mathematical explanations would have to be analyses of the preconditions required for a theorem to be true, such as showing that a theorem depended on the axiom of choice.

Van Fraassen is not alone in regarding answers to why-questions in this way. Though the model of why-questions presented by Hintikka and Halonen [1995] is significantly different from van Fraassen's, it also turns out that they implicitly regard explanations as providing additional logical information. This is clear when examining the assumptions behind their model; that P(b) can be derived from (T&A), but not T alone, where T is an 'initial premise', A the answer, and P(b) the explanandum. They consider the assumption that P(b) cannot be derived from T to rule out one possible kind of trivialization (p. 648), but it appears that many explanatory proofs are 'trivial' in exactly that way; they are alternate ways to derive a theorem from the same set of premises. Under their model, as under van Fraassen's, mathematical explanations would take the form of analyses of axioms required for theorems.

Though such analysis certainly goes on in - and is an important part of - mathematics,(13) it is neither a clearly explanatory activity, nor does our example (or many others which could be mentioned) take this form at all. Polya does not single out any proposition that, in conjunction with some other part of background mathematics, leads to the result in question. Further, van Fraassen's picture of explanation makes it mysterious why one should ever offer a proof as an explanation.(14) A proof does not fill in any missing information, but instead draws out consequences from previously given propositions. Ideally, a proof is taken to add no information at all. Yet some proofs are said to explain their conclusions.

Thus it seems that van Fraassen's theory of explanatory evaluation will not easily be extended to mathematical cases. Indeed, suitably modified they highlight the difficulty of extending many theories of explanation to mathematical explanations. For any such extension will have to recognize that some proofs explain their conclusions, while others don't. But any theory that uses probabilistic tools to evaluate explanations is liable to regard all proofs as explanatory, for proofs will tend to attain maximal scores in such evaluations. On the other hand, any theory that takes explanations to increase our propositional knowledge will tend not to regard any proofs as explanatory, since they do not do this, but at best display the consequences of what we have already accepted as given.

6 The why-question approach

Rather than pursue the above reasoning further, I now turn to an argument against any theory, not just van Fraassen's formulation, that reconstructs the idea of 'explanatory relativity' in terms of why-questions. The problem is that some context-dependent factors that go into our evaluation of explanations cannot be understood in terms of why-questions. Though the why-question approach is right in recognizing the importance of context in such evaluations, the correct account of the context-dependent evaluations in the Polya example cannot properly be called a why-question account. The resolution to this argument will also point the way to an account of explanation that is not subject to the objections in the previous section.

My argument resembles one given by Paul Humphreys ([1989], pp. 137-8). I will show that the Polya explanation represents a concrete example of a problem Humphreys raised abstractly. The mathematical example more sharply delineates the problem, and demonstrates that it is pervasive to the why-question approach, extending far beyond the scope of the singular causal explanations to which Humphreys restricts his attention.

Consider again the situation after Polya has presented his initial proof. Though we agree it is successful as a proof, certain aspects of the proof remain puzzling. Polya's explanation largely dissolves this puzzlement; it appears to be a largely satisfactory explanation. According to the why-question theory, this explanatory success must be due to Polya's having answered some why-question. Indeed, our initial puzzlement seems to be well expressed in terms of a why-question, namely 'Why is it appropriate to introduce the [c.sub.i] sequence in the proof?' Therefore, it seems natural to suppose that Polya's explanation would satisfactorily answer this question.

As we saw above, this why-question can be given two different readings. Polya seems to address both readings in his explanation. In response to reading (1), he observes that an auxiliary sequence sharpens a crucial inequality used in the proof. The observation may be a bit vague, but it is not unsatisfactory. However, this observation is only the beginning of Polya's explanation. The rest appears aimed at reading (2). Moreover, if he had simply stopped at this point, the explanation would not have been as satisfying. Therefore, it appears that much of the virtue of Polya's explanation must be in answering reading (2).

However, Polya's explanation, on close examination, is not a very successful response to this reading of our initial why-question. It shows how the growth rate of the [c.sub.i] sequence makes it favourable for completing the proof, but this is far from enough to show that this sequence is the appropriate one to use. There are many other sequences with the same growth rate, some seemingly more natural. The [c.sub.i] sequence is chosen in part because it allows a crucial simplification of the term

[summation of] 1/n[([c.sub.1][c.sub.2][c.sub.3] . . . [c.sub.n]).sup.1/n] where n = k to [infinity].

But this choice still feels like a 'trick', even after Polya's expanded exposition. Besides the raw fact that the chosen sequence enables this simplification, there is no reason for using that sequence instead of another with similar growth behaviour. We are shown that a particular [c.sub.i] sequence works; we are not shown why. The explanation thus does not tellingly distinguish between the sequence used in the proof and a large class of others, and cannot be considered a telling answer to the original why-question.(15)

Thus, we appear to have a good explanation that doesn't provide a good answer to its motivating why-question. This presents a serious problem for the why-question approach because it seems to imply that the explanation is evaluated on other grounds than having answered why-questions. But there are several options open to the why-question advocate. First, the claim that Polya does not satisfactorily address the (second reading of the) motivating why-question can be questioned. Second, the argument relies on our intuitive judgement that Polya's explanation is actually satisfactory. Perhaps it is really not as effective as we initially thought. Finally, we might claim that the explanation is effective because it answers some other why-question, not this particular one. Let us address these points in turn. The last one will be particularly important. It will turn out that Polya's explanation indeed does answer a different why-question satisfactorily, but we will also see that its virtue as an explanation is not primarily in answering this why-question.

The argument that Polya's exposition does not tellingly answer the (second reading of the) initial why-question may be felt as a bit too quick. Perhaps a more sophisticated use of van Fraassen's telling answers theory will allow us to reject its conclusion. Indeed, van Fraassen's theory has some resources which one might think would allow him to claim this. He allows an explanation to be judged as telling not only by how well it favours the topic over the other members of the contrast class, but also by how well it fares in comparison to other possible answers. One could then claim that although Polya's explanation doesn't fully spell out why this particular sequence helps complete the proof, no other answer would be able to do any better. Though we can't be certain of this judgement without considering all possible answers to the initial why-question, it is plausible. We suspect that the [c.sub.i] sequence takes such a seemingly unusual form because it allows one to simplify the term

[summation of] 1/n[([c.sub.1][c.sub.2][c.sub.3] . . . [c.sub.n]).sup.1/n].

This feature of the sequence may not be amenable to any further principled analysis; it would then be a sort of mathematical coincidence. Though Polya's answer doesn't seem too good on its own, it may be the best we could expect.

Even if this reasoning is correct, it simply does not make the explanation a good answer. That there is no better alternative doesn't add to the positive qualities of the explanation, yet Polya's explanation strikes one as good by what it accomplishes, not by comparison to unspecified alternative attempts. The comparative aspect of van Fraassen's theory of telling answers is best used to rule out seemingly good explanations that are dominated by some other explanation (such as probabilistic explanations that are 'screened off' by other, better explanations). If an explanation we judge to be good is dominated by another one, van Fraassen's theory will correctly lead us to reject it. It is less clear that by the same principle we should reconsider an initially negative judgement of explanatory quality in light of the fact that there is no better explanation available. That something is the best available answer does not make it a good answer. To say otherwise is to resort to the force majeure argument that van Fraassen himself rejects in Laws and Symmetry ([1989], pp. 144-5). Therefore, it appears that we must accept that Polya's explanation does not successfully answer its motivating why-question.

The why-question advocate might then say that this shows Polya's exposition is not a good explanation alter all. That seems jarring in light of our admittedly pretheoretical but clear intuitions about the example; and more motivated by an a priori adherence to the why-question theory than by attention to the case study. Certainly, the explanation does manage to clear up a lot of confusion about the [c.sup.i] sequence. Though one cannot be said to have received a fully illuminating answer to the question, 'Why is it appropriate to use this particular sequence in this proof?' one is far less puzzled about the sequence than one was beforehand. Furthermore, we are not confined to pretheoretic intuitive judgements; we can say at least something positive about what this explanation contributes. A key part of the explanation seems to be the use of growth rates. The concept of growth rate did not explicitly arise in the original proof, but it seems to be the best way to understand the selection of the [c.sub.i] sequence. Therefore, our initial puzzlement about the proof seems to stem from the fact that it did not highlight an explanatorily crucial property of the sequence, and the virtue of the explanation seems to be that it does highlight this property. This is an important observation, which we will return to below.

Given that Polya's explanation is a good one, but that it doesn't answer the motivating why-question, there appears to be one other response available to the why-question theorist - to seek a different why-question that Polya's explanation does answer. The focus on growth rates points us to such a question: 'Why is it appropriate to introduce a sequence asymptotically proportional to 1, 2, 3, . . . into the proof?' The contrast class has shifted. Rather than sequences, it now contrasts different growth rates. Polya's supplementary exposition successfully shows why the asymptotic growth rate selected is the best candidate to solve the problem. The augmented exposition points to a unique growth rate.

Thus, the why-question advocate can claim that Polya's explanation is good not because it answers the initial why-question, but because it answers a new question. It would appear that this situation is a variation of rejection of the why-question as described by van Fraassen. Instead of simply rejecting the initial why-question with a corrective answer, Polya goes on to address a somehow more appropriate why-question. In this case we actually have a satisfactory explanation, even in a case where the initial why-question had a false presupposition.

But for this analysis of the situation not to be ad hoc, the why-question theory must be supplemented somehow. Once the initial why-question is rejected, we cannot expect to answer just any other why-question and still have a successful explanation. For one thing, this would put the why-question approach in danger if a trivial why-question can be constructed for any purported 'explanation', as Kitcher and Salmon have done for van Fraassen's particular formulation. Secondly, such a response appears to minimize the role of context in the evaluation of explanations, since the apparent context, specified by the initial why-question, would no longer be an important part of the evaluation. If we are to account for why Polya's explanation was effective in this situation, we must understand how it responds to our initial state of puzzlement. It seems therefore that the why-question theory still owes us a story of how Polya's explanation relates to the initial why-question (which is presumably the way the why-question theory would find that puzzlement best expressed), even if it does not answer it.

But even if the why-question approach can answer this challenge, it will still miss an important way in which Polya's explanation responds to our initial state of puzzlement after we have first seen the original proof. If the above response were correct, the problem with the initial question was that it made an incorrect assumption - that there was a telling reason to favour the [c.sub.i] sequence over some other with the same growth rate. The questioner could have and should have asked a different question instead. Once the questioner is 'put on the right track', so to speak, the explanation can proceed. However, to formulate this new question, van Fraassen's logical apparatus demands that the contrast class consist of statements of the form: 'The growth rate of sequence [c.sub.i] is the most appropriate for an auxiliary sequence that will lead to a successful proof.' If the concept of growth rate is not available to the questioner in the first place, we cannot do this. The explanation, if it gets its virtue by answering a why-question at all, does so by answering a question that couldn't even be asked prior to the explanation.

Suppose, however. that the questioner could formulate the question 'Why is it appropriate to introduce a sequence asymptotically proportional to 1, 2, 3, . . . into the proof?' The ability to formulate this question presupposes some familiarity with mathematical analysis; at least enough familiarity to understand (a) what a growth rate is, and (b) that it might be important to choosing the [c.sub.i] sequence. A person able to ask this question is less likely to be puzzled in the first place by the [c.sub.i] sequence than someone more naive in mathematics. The augmented exposition is therefore less likely to be explanatorily effective for a person in a position to ask the question than for one who cannot. Whether the concept of growth rate is available to the questioner is a crucial part of the context in which the need for an explanation arises. But since the explanation answers the same why-question regardless of whether the questioner could formulate it or not, the why-question approach cannot distinguish between the sophisticated and naive questioners. The why-question correctly acknowledges the role of context in evaluating explanations, but misses an important aspect of that context - the conceptual resources the questioner has available to analyse the situation. Much of what makes Polya's exposition successful is that it shows that growth rate is important; not because it picks out a particular growth rate rather than another. Our reactions to Polya's explanation are not best understood as stemming from how he answers why-questions, but from the conceptual resources for analysing the situation he brings to our attention.

Since I mentioned the contrast class above, I might be accused of still attacking only van Fraassen's theory rather than the general why-question approach. However, the argument doesn't depend upon the details of his theory at all, but only on the model of context-dependent evaluation that is characteristic of the why-question approach. The key point is that a why-question is taken to implicitly fix the way an answer must regard its topic. In van Fraassen's theory. this is done by specifying a contrast class and relevance relation, which must therefore be antecedently specifiable in the context in which the question is considered. But any why-question-based evaluation should at least be able to distinguish between asking about sequences and asking about growth rates. It must then allow an answer in terms of growth rates to be judged explanatory by a questioner who couldn't even talk about growth rates until the explanation had been given.(16) For the why-question theory, an explanation must respond to a question, which implies a fixed way of looking at the topic. But our initial state of puzzlement may be due to not even knowing how best to regard the topic. An explanation can gain most of its virtue by responding to this state of affairs - showing us an effective way to understand the subject-matter - rather than through any particular why-questions it happens to answer. In so far as asking a why-question fixes a way of looking at the explanandum and demands an explanation in those terms, the why-question approach will be subject to this problem. But this seems to be the very point of 'explanatory relativity', a central feature of the why-question approach. Thus it is hard to see how the why-question approach can address this problem successfully without being watered down beyond recognition.

This problem is highlighted by attention to mathematical explanations, but is not restricted to them. A physical explanation may also introduce a completely new way of looking at its topic. For instance, Isaac Newton provided an explanation for the movements of the planets, but not one which answers the question posed by his predecessors; they demanded mechanical explanations without reference to action at a distance. Newton's answer would have failed to resolve the question the Cartesians had in mind. Indeed, it would have been impossible to specify a Newtonian answer as an appropriate answer to a question posed before the Principia; the pertinent concepts couldn't yet be given to indicate that kind of answer was appropriate.

These considerations suggest an alternate way to look at theoretical explanations than the why-question approach. An explanation may be significant because it deploys relevant conceptual resources not previously available. This still allows for context-dependent explanatory evaluations, since what is considered 'previously available' will depend upon the context in which the need for explanation arises; but the role that context plays is different from that recognized in the why-question approach. Even though only programmatic, this picture of explanation-evaluation has a number of virtues. In particular, it avoids both problems with van Fraassen's theory of explanatory evaluation mentioned in the previous section. First, it avoids trivialization, because perfect 'favouring' of the explanandum, as so often happens in mathematical cases, is no longer sufficient for explanation. When an explanation does not look at the explanandum in a new way, it need not be considered explanatory. Second, our picture makes clear why one might offer a proof of a result as an explanation of it. Though a proof may not give us access to new propositional information (certainly not if we presume logical omniscience), it may invoke new conceptual resources that were not previously available. It is this, rather than establishing propositional facts, that makes such a proof explanatory.

The further study of mathematical cases will be important to developing this new approach to explanation. First, mathematical cases must be considered in order to avoid the trivialization arguments raised in the previous section. Second, the role of new conceptual resources, rather than new propositional knowledge, will be most clearly seen in mathematical examples, where typically no new propositional knowledge is generated. I therefore claim that in order to improve our understanding of explanation, we must seriously consider mathematical, as well as empirical, explanations.

We can now respond to van Fraassen's short argument, 'An explanation is an answer to a why-question. So, a theory of explanation must be a theory of why-questions.' Even though an explanation may be offered as an answer to a why-question, it may not be in virtue of its being the answer to a why-question that it is explanatory. Though a theory of why-questions may aid the theory of explanation, the theory of explanation must go beyond it.


I would like to thank Wesley Salmon, David Rudge, and the anonymous reviewers for this journal for their helpful comments on earlier versions of this paper. I would also like to thank Kenneth Manders for his criticism and guidance throughout this project.

Department of History and Philosophy of Science University of Pittsburgh Pittsburgh, PA 15260 USA

1 For instance, see Hempel ([1965], p. 334). Sylvain Bromberger [1962, 1966] also developed an approach to explanation centred on this insight about question-answering (though he did not believe that why-questions exhausted explanatory requests).

2 Achinstein's illocutionary theory of explanation [1983] represents a similar approach, but not restricted to why-questions. Achinstein focuses on the act of explanation, not just the product of such an act. An act of explanation is a response, so that an explanation must be evaluated with respect to the utterance it responds to.

3 Van Fraassen's general approach is inspired by Belnap and Steel's The Logic of Questions and Answers [1976], though he introduces some additional innovations. Belnap and Steel proceed from the dictum: 'Knowing what counts as an answer is equivalent to knowing the question' (p. 35, attributed to Hamblin [1958]). However, as Bromberger [1962] points out, somebody requesting an explanation may not even be able to state one adequate answer. Van Fraassen's approach seems to turn this dictum around, by understanding the answers (explanations) in terms of the questions they are answers to.

4 Van Fraassen states on p. 146 that he feels that the problem of rejections of explanation requests is solved without recourse to the theory of telling answers, but the syphilis-paresis example indicates that some questions can be rejected, not because they have no admissible answers, but because they have no telling answers.

5 Salmon ([1989], p. 145) regards the problem of determining the appropriate K(Q) as equivalent to the problems leading to Hempel's requirement of maximal specificity and his own objective homogeneity.

6 For instance, Hannson's initially plausible criterion - the probability of the topic given the answer is higher than the average probability of members of the contrast class given the answer - falls prey to simple counterexamples. (van Fraassen [1980], p. 128). On the other hand, one might think at least that any answer that decreased the probability of the topic could not favour it. However, if an answer decreases the probability of its likeliest competitors in the contrast class even more, this may count as favouring (p. 148).

7 This example also appeared in substantially the same form in his [1949], 'With, or Without, Motivation?'

8 Substantially the same proof appears in a 1924 article by Polya in the Proceedings of the London Mathematical Society. The theorem also appears as Theorem 334 in Hardy, Littlewood, and Polya's [1934] Inequalities (with some, but not all, of the additional explanation).

9 The complete details can be found in Polya [1968] or Sandborg [1997].

10 Note that a contrast class need not in general consist of mutually exclusive members. For instance, when we ask why the mayor has paresis, in contrast to the other members of the country club, we accept that more than one member of the country club could have had paresis. Indeed, if we were to ask 'why does 1 - 1/3 + 1/5 - 1/7 + . . ., rather than sonic other sequence, converge to [Pi]/4?', we would have to accept that other sequences could also converge to the same number. Moreover, in the Polya case, the question 'Why does this sequence serve to complete the proof?' is not an exclusive-contrast question, since other sequences might serve as well as the one actually used. Lipton [1991] discusses this issue further. However, exclusive-contrast questions form an important subset of why-questions. Many of van Fraassen's examples use exclusive contrast classes (i.e. why did the sample burn green, as opposed to some other colour); and Garfinkel too considers explanatory contrasts to be mutually exclusive.

11 Technically, it makes little sense to assign a probability to a proof. However, it is clearly in the spirit of van Fraassen's theory to consider proofs as maximally acceptable on this criterion - we have complete confidence in the proof. I will return below to the question of how well proofs can be regarded as answers under van Fraassen's theory.

12 Note that the limitation of the background knowledge to K(Q) for the sake of explanatory evaluation (p. 147) is an attempt to avoid a trivialization like that noted above. In general, the explanandum and the negation of all other members of the contrast class will be completely telling information. The above trivialization argument suggests that there is a more fundamental problem that the invocation of K(Q) has failed to fix.

13 Such analyses fall under the heading of 'reverse mathematics'.

14 I do not wish to claim that all mathematical explanations are proofs. Indeed, the example shows that mathematical explanation is not the same thing as proof. Proof is neither necessary (Polya's original proof is not explanatory), nor sufficient (Polya's explanation is not itself a proof) for explanation. None the less, proofs are often vehicles for mathematical explanation. For the purposes of my argument, it is enough to observe that some answers to mathematical why-questions are proofs, and van Fraassen's theory cannot account for answers of that form.

15 Here I do not necessarily assume the particular theory of telling answers van Fraassen has given, but presume we can make intuitive judgements about tellingness not subject to the trivialization argument given in the previous section.

16 How any why-question theory could do this is unclear, remembering Belnap and Steel's dictum that to understand a question is to know what would count as an answer.


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Bromberger, S. [1966]: 'Why-questions', in R. G. Colodny (ed.), Mind and Cosmos: Essays in contemporary science and philosophy, Pittsburgh, University of Pittsburgh Press, pp. 86-111.

Garfinkel. A. [1981]: Forms of Explanation: Rethinking the Questions in Social Theory, New Haven, Yale University Press.

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Author:Sandborg, David
Publication:The British Journal for the Philosophy of Science
Date:Dec 1, 1998
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