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Philosophia Mathematica
, Volume Advance Article – Mar 14, 2018

12 pages

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- Oxford University Press
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- © The Authors [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
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- 0031-8019
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- 1744-6406
- D.O.I.
- 10.1093/philmat/nky004
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In his Moby Dick, Herman Melville writes that “to produce a mighty book you must choose a mighty theme”. Marc Lange’s Because Without Cause is definitely an impressive book that deals with a mighty theme, that of non-causal explanations in the empirical sciences and in mathematics. Blending a variety of insights originating within the philosophy of science and the philosophy of mathematics, the book has an ambitious goal: capturing what features, in the many examples of putative explanations offered by mathematicians and scientists, make them genuinely explanatory. The architecture of Because Without Cause reflects its mighty character. The book (489 pages) is divided into four parts and eleven chapters, preceded by a preface in which Lange accurately offers the reader an outline of what falls and does not fall within the scope of the book. Parts I and II are devoted to non-causal explanations in the empirical sciences, while in Part III Lange concentrates on explanation in pure mathematics. Part IV provides the reader with a broader view of Lange’s theory of explanation, bringing together his analysis of non-causal scientific explanation and explanation in mathematics. In each part Lange’s philosophical investigation is supported by a varied set of case studies taken from the empirical sciences (mainly physics) and mathematics. This approach well represents Lange’s “naturalistic metaphysics of science” [Saatsi, 2017] and considerably extends Lange’s previous work on non-causal explanation, laws and modality (e.g., [Lange, 2009; 2010; 2011; 2013].) Although the nature of non-causal explanations in mathematics and science is the main topic of the book, Lange’s analysis is not limited to these kinds of explanations. Causal explanations, as well as several topics connected to the debate on non-causal explanation, are treated and discussed throughout the book. In this review I shall not cover this huge spectrum of themes. Rather, I shall address several aspects of Lange’s book that I consider interesting for the readers of this journal. More precisely, I shall concentrate on those parts of Lange’s book that deal with mathematical explanation in the empirical sciences and mathematical explanation in pure mathematics (both are considered by Lange as types of non-causal explanation). Nevertheless, because of the very uniform picture of ‘explanation’ that Lange offers in his book, I shall also touch upon his analysis of non-causal non-mathematical explanations in science. In Part I (“Scientific explanations by constraint”) Lange focuses on a particular kind of non-causal scientific explanation, which he calls “explanation by constraint”. Depending on the nature of the constraints considered, explanations by constraint can be mathematical (when mathematical truths serve as constraints) or not (when the constraint is not a mathematical truth, as is the case when a conservation law or a symmetry principle acts as constraint). In both cases the explananda are facts about the natural, spatiotemporal world. And therefore explanations by constraint involving mathematical constraints (Lange calls these explanations “distinctively mathematical explanations”) are seen as a type of scientific explanations and are kept distinct from explanations in pure mathematics, where the explananda are mathematical theorems (or, more generally, mathematical facts; these explanations internal to mathematics are the subject of Part III of the book). How do Lange’s scientific explanations by constraint work? And what exactly makes something a constraint? The answers to these questions, according to Lange, lie in the modal strength that some mathematical and some non-mathematical facts possess and bring into the explanation: Explanations by constraint work not by describing the world’s causal relations but rather by describing how the explanandum arises from certain facts (“contraints”) possessing some variety of necessity stronger than ordinary laws of nature possess. (p. 10) Several examples are offered to the reader to illustrate the functioning of such modally stronger explainers. The first example given by Lange is an example of a distinctively mathematical explanation (cf. also [Lange, 2013] for a discussion of the same example). A mother has 23 strawberries and 3 children. Whenever she tries to distribute her strawberries evenly among her children, without cutting any (strawberry), she fails. Why? The straightforward reply to our question is that 23 is not divisible by 3. This is a mathematical fact. Therefore, according to Lange, what we have here is a “because without cause”. The mathematical fact ‘23 is not divisible by 3’ plays the role of constraint because it brings into the explanation a particular degree of necessity (mathematical necessity) that overcomes every contingent, nomological, and causal feature involved in this scenario. Consider, for instance, a potential array of causal attempts to divide the strawberries. Whatever the attempt, each of which involves different causal histories and relations, the impossibility for the mother to distribute the strawberries will always be constrained by the mathematical fact. Indeed, our explanation does not depend on the specific network of causal relations that is in place; rather, it crucially depends on a mathematical result, and more precisely on the modal strength of such result. The distinctively mathematical explanation therefore works not by describing the world’s network of causal relations (being thus non-causal), but by revealing the explanandum to be mathematically necessary (more necessary than ordinary natural necessity). As Lange puts it, this explanation shows the outcome to be inevitable to a stronger extent than facts merely about causal relations (actual and counterfactual) could make it. (p. 15) In acknowledging the existence of explanations of empirical phenomena that are mathematical and non-causal, Lange joins other philosophers, for instance Paolo Mancosu, Philip Kitcher, Alan Baker, Mark Colyvan, Mark Steiner, and Christopher Pincock. Nevertheless, Lange’s account of how these explanations work significantly departs from the (few, as Lange rightly observes) proposals that have been offered in the literature (cf. [Mancosu, 2011; Mancosu and Pincock, 2012] for a survey). For instance, Lange rejects Mark Steiner’s idea that every distinctively mathematical explanation should carry an explanatory proof of the mathematical result being used [Mark Steiner, 1978b]. More precisely, Lange observes that none of his examples of distinctively mathematical explanations incorporates an explanation in mathematics; in fact, they appeal to mathematical facts and their truths, as in the strawberries example, but not to their proofs. Moreover, for Lange a mathematical fact having no explanation in mathematics may appear in a distinctively mathematical explanation. Thus Steiner’s “transmission view” (i.e., explanatoriness in distinctively mathematical explanations is transmitted from a purely mathematical explanation), as Alan Baker [2012] has named it, is not endorsed by Lange. An important aspect of Lange’s account of explanation concerns its context-sensitivity. Depending on the way in which we model our why question, an empirical phenomenon may or may not receive a distinctively mathematical explanation. According to Lange, context sensitivity arises from the context influencing what particular sort of information an explanation has got to supply concerning the source of the especially strong necessity possessed by the fact to be explained: [...] an explanation by constraint derives its power to explain by virtue of providing information about where the explanandum’s especially strong necessity comes from. The context in which the why question is asked may influence what information about the origin of the explanandum’s especially strong necessity is relevant. Context plays a similar role in connection with causal explanations: by influencing what information about the explanandum’s causal history or the world’s network of causal relations is relevant. (p. 99) The strawberry example is offered by Lange as a simple example of a non-causal distinctively mathematical explanation that does not appeal to a contingent natural law. But are there cases of distinctively mathematical explanations that do appeal to a contingent law of nature? Lange’s reply is that such cases exist. In the first chapter he considers the case of a simple double pendulum, a physical system consisting of one pendulum attached to another, and tackles the following why question: why does a simple double pendulum have at least four equilibrium configurations? One way to answer this question is to identify the particular forces acting on the bobs and then, using Newton’s second law of motion, find the configurations in which the bobs feel zero net force. This amounts to finding when the potential energy $$U(\alpha ,\beta )$$ of the system is stationary ($$\alpha $$ and $$\beta $$ are the angles used to parameterize the system). Such a treatment provides a causal explanation. Nevertheless, we can also observe that the configuration space of this system (and, more generally, of any double pendulum) is the surface of a torus. We can therefore study our physical problem as a purely topological problem. By mathematics (compactness, to be precise), there must exist at least four stationary points for the surface considered. Thus there should be at least four equilibrium configurations for the physical system. This alternative explanation (of why the double pendulum has at least four equilibrium configurations) does appeal to a particular case of Newton’s second law and should be regarded as a distinctively mathematical explanation because No aspect of the particular forces operating on or within the system (which would make a difference to $$U(\alpha ,\beta )$$) matters to this explanation. Rather, the explanation exploits merely the fact that by virtue of the system’s being a double pendulum, its configuration space is the surface of a torus. (p. 27) The double-pendulum example is important because it highlights three key aspects of Lange’s conception of explanation. First, an empirical phenomenon may have a causal and a non-causal (in the case of the double pendulum, a distinctively mathematical) explanation. Second, a distinctively mathematical explanation may appeal to a contingent natural law. Third, in some cases the distinctively mathematical explanation has an extra feature with respect to the causal explanation because it not only applies to a particular case (the double-pendulum system), but also to several cases that share a relevant feature (the configuration space of any double-pendulum system is the surface of a torus). For instance, if we consider a compound square double pendulum and a complex double pendulum under the influence of various springs forcing its oscillations, both these pendulums have the very same configuration space. This latter consideration elucidates another important aspect of Lange’s account of non-causal explanations, namely the role played by the notion of “coincidence”. In Lange’s account, the notion of coincidence arises in connection with both scientific explanations (i.e., causal and non-causal explanations in science) and explanation in mathematics. I will return to this topic when reviewing Lange’s account of explanations in pure mathematics. Nevertheless, let me just illustrate the significance of this notion for the case of the double pendulum. Consider, as Lange does, two distinct double-pendulum systems, one simple and the other with inhomogeneous, extended masses and oscillations driven by various springs (p. 31). We want an answer to the following question: why do both of these pendulums have at least four equilibrium configurations? If we analyze each pendulum independently and we give a causal explanation for the equilibrium configurations of each (in the same way as suggested above), what we get is an explanation that applies separately to each distinct case. But this explanation does not highlight the reason why both pendulums have at least four equilibrium configurations, thus misconstruing our explanandum as a coincidental fact: this derivation would portray the explanandum as a coincidence since this derivation would fail to identify some important feature common to the two pendulums as responsible for their both having at least four equilibrium configurations. (p. 31) On the other hand, by offering a unified explanatory treatment for the two distinct systems, the explanation using a mathematical result as a constraint shows the common feature (i.e., their having at least four equilibrium configurations) to be no coincidence: every double pendulum has this property for the same reason, namely, because of the kind of configuration space they all have in virtue of being double pendulums. (p. 278) Distinctively mathematical explanations are for Lange a specific kind of non-causal scientific explanation. In these explanations mathematical facts play the role of constraints by virtue of having a stronger variety of necessity than ordinary natural laws possess. Nevertheless, Lange argues, there are also other varieties of necessity in science that are stronger than ordinary natural necessity. And therefore there are constraints that are not mathematical but that play the same role that mathematical facts play in distinctively mathematical explanations. These non-causal non-mathematical scientific explanations by constraint are the subject of Chapters 2, 3, and 4. In Chapter 2 Lange focuses on conservation laws and symmetry principles as possible constraints that transcend the various force laws, while in Chapter 3 he looks at the Galilean and Lorentzian coordinate transformations as possible constraints. Chapter 2, and more precisely Section 2.2, is extremely interesting because it deals with some of the “intermediate cases” (i.e., explanations that have a causal as well as a non-causal component) that Lange mentions at the beginning of Chapter 1. For instance, Lange takes as explanandum the fact that for a classical pendulum having initially been released from rest at height $$h$$, the bob rises no higher than $$h$$ on any subsequent upswing. This fact, Lange argues, admits a “hybrid” explanation composed by a non-causal part (appealing to energy conservation as a constraint) and a causal one (appealing to the fact that the bob’s potential energy is entirely gravitational and the potential-energy function is associated with the gravitational force). Chapter 4 is probably the most historically oriented part of Because Without Cause (it is here that historians of physics and mathematics can take their short break from metaphysics). Lange presents a controversy that took place in the nineteenth century and that concerned the explanation of the parallelogram law for the composition of forces. The point at issue in the dispute was whether the parallelogram of forces is explained by statics or dynamics. Lange’s aim is to specify what the laws of statics would have to be like in order for them to explain the parallelogram law. In Part II (“Two other varieties of non-causal explanation in science”) Lange examines two other species of non-causal scientific explanations that are not explanations by constraint. One instance, labeled by Lange ‘really statistical’ (RS) explanation, is presented in Chapter 5. A really statistical explanation “exploits merely the fact that some process is chancy, and so an RS explanation shows the result to be just a statistical fact of life” (p. 192). Among Lange’s preferred examples illustrating how RS work are explanations in population biology appealing to drift. In Chapter 6 a different kind of non-causal scientific explanation is analyzed: “dimensional explanation”. Although this form of explanation should be considered as an additional kind of non-causal explanation, Lange observes how some dimensional explanations can be explanations by constraints (cf. the example given by Lange in Section 6.1). Various examples are provided to illustrate how dimensional explanations work. In Part III (“Explanation in mathematics”) Lange concentrates on explanation(s) in pure mathematics. Before discovering Lange’s take on this topic, it is important to note that the study of explanatory practices internal to mathematics is still “in early adolescence” [Tappenden, 2008, p. 260] and little work has been done in the direction of providing an account of how explanations in mathematics work. Nevertheless, several philosophers of mathematics have recently offered a set of examples from mathematical practice that show how explanation is considered by many mathematicians as an important component of their activity ([Mancosu, 2011] and [Mancosu and Pincock, 2012] provide a survey and extensive references). These authors agree that it is possible to have a better comprehension of explanation in mathematics by focusing on particular case studies and taking the case studies themselves as a starting point for philosophical considerations. This practice-driven approach is endorsed by Lange as well. The main goal of Part III is to provide a way to distinguish between explanatory and non-explanatory proofs in mathematics. Nevertheless, according to Lange, not all explanatory practices internal to mathematics converge to the case of proofs that explain why mathematical theorems hold. Besides a theorem, what is subject to explanation in mathematics also includes ‘mathematical facts’ that are not theorems, such as the fact that one given result is so much more difficult to prove than another given result. Furthermore, Lange argues that not all mathematical explanations consist of proofs. In Chapter 7 we are presented with several examples of proofs that, according to (some) working mathematicians, explain why a mathematical theorem holds. Lange argues that all these proofs share a particular feature that makes them explanatory. This feature concerns the way in which the proof connects some aspects regarding the content of the theorem to be explained with the information contained in the axioms that are used to prove it. More precisely, […] for a theorem of the form “All F’s are G”, I will argue that the distinction between explanatory and non-explanatory proofs rests on differences between proofs in their ways of extracting the property of being G from the property of being F. A proof’s explanatory power depends on the theorem being expressed in terms of a “setup” or “problem” (involving F’s instantiation) — such as the “problem” of Zeitz’s coin (discussed in the following section) — and in terms of a “result” or “answer” (G’s instantiation). Furthermore, as I will also argue, the manner in which a theorem is expressed may call attention to a particular feature of the theorem, where that feature’s salience helps to determine what a proof must do in order to explain why the theorem holds. (p. 232) To achieve a better grip on Lange’s idea, let us consider one of his examples, specifically an explanation of d’Alembert’s theorem on conjugate roots of polynomials with real coefficients. The theorem, first proved by d’Alembert in 1746, states that if a polynomial $$P(x)$$ with real coefficients has a non-real root $$x=a+ib$$, then it also has a root $$x=a-ib$$ (i.e., non-real roots of real polynomials always come in conjugate pairs). We can observe that the theorem exhibits an interesting symmetrical property: the equation’s non-real solutions all come in pairs when one member of the pair can be transformed into the other by the replacement of $$i$$ with $$-i$$. Now, in order to show why the theorem is true, we can follow a merely brute-force approach: we arrive at the result by using all the available information and carrying out all the necessary calculations. The resulting brute-force proof, although formally impeccable, is not regarded by Lange as a proof that explains why the theorem holds (and mathematicians generally agree in considering brute-force proofs as explanatorily weak). Indeed, it depicts the $$i/-i$$ symmetry arising as an “algebraic miracle”. Alternatively, we can concentrate on the particular symmetrical feature of the problem (the “setup”) and look for a proof that exploits this feature. The proof sought draws on the fact that $$i$$ and $$-i$$ share the very same definition in the axioms of complex arithmetic (their squares both equal $$-1$$): Whatever the axioms of complex arithmetic say about one can also be truly said about the other. Since the axioms remain true under the replacement of $$i$$ with $$-i$$, so must the theorems — for example, any fact about the roots of a polynomial with real coefficients. (The coefficients must be real so that the transformation of $$i$$ into $$-i$$ leaves the polynomial unchanged.) The symmetry expressed by d’Alembert’s theorem is thus grounded in the same symmetry in the axioms. (pp. 240–241) This is how an explanatory proof works: it picks out a particular interesting (“salient”, in Lange’s terminology) feature of the explanandum and shows how it is responsible for the result being explained (the salient feature does not count as the explanandum, otherwise we would have an uninteresting circularity here: an explanandum explaining itself). In d’Alembert’s theorem’s case, the salient feature is a symmetry that both the result and the proof share. Nevertheless, Lange argues, there may be other interesting features (besides symmetry) that figure in mathematical explanations, as for instance “simplicity” (p. 257) or the property that a theorem may have to identify a similarity among mathematical facts that otherwise seem unrelated (p. 280). Moreover, there may also be cases where a mathematical result has no explanation at all (despite having a proof): when the result exhibits no salient feature or when it exhibits some interesting feature but no proof traces the result to a similar feature in the setup. As Lange points out: The distinction between proofs that explain why some theorem is true and proofs that merely show that the theorem is true exists only when some feature of the result being proved is salient. The feature’s salience makes certain proofs explanatory. A proof is accurately characterized as an explanation (or not) only in a context where some feature of the result being proved is salient. (p. 255) The end of the previous quotation highlights an important facet of Lange’s account of explanations in mathematics. Similarly to what we have seen in the case of non-causal scientific explanations, the way in which we frame the explanandum really matters to the goal of having (or not having) an explanation for that (mathematical or empirical) fact. In the case of mathematical explanations, We can make a feature of some result more or less salient by the way we express it. What’s salient is a feature of the result under a certain representation. (p. 266) In Chapter 7 Lange also offers some examples of mathematical explanations that do not involve proofs and whose explananda are not just theorems but also mathematical facts. This topic is treated again, with more philosophical sophistication, in Section 9.6. Let me just mention one of these examples to illustrate what sort of cases Lange has in mind. Consider again d’Alembert’s theorem. Furthermore, consider the following two results: the solutions of $$z^3+6z-20=0$$ are $$2,-1+3{i}$$, and $$-1-3{i}$$; the solutions of $$z^2 -2z+2=0$$ are $$1-i$$ and $$1+i$$. Why is it that for both these equations the non-real solutions are pairs of complex conjugates? An explanation, according to Lange, is given by d’Alembert’s theorem. Indeed, the theorem makes it not coincidental that, for both of these polynomials, all their non-real roots fall into complex conjugate pairs. Therefore this is a simple example of a mathematical explanation in which the explanans is not a proof (but a theorem). In Chapter 8 Lange elaborates further his account of explanation in mathematics. More precisely, he concentrates on the coincidental (and non-coincidental) character of some mathematical facts and how this is related to explanations that unify many of them. In this chapter Lange also confronts his proposal with the accounts of mathematical explanation in mathematics proposed by Mark Steiner [1978a; b] and Philip Kitcher [1984; 1989]. I have already considered the notion of coincidence during my presentation of Lange’s non-causal distinctively mathematical explanations. More precisely, I have pointed out how in the example of the double pendulums Lange maintains that it is no coincidence that all double pendulums are alike in having at least four equilibrium configurations. What makes this fact non-coincidental is that all double-pendulum systems have the very same configuration space, which can be treated mathematically. There is, in other words, a common explainer that both the systems share. Similarly, Lange argues, a mathematical fact is no coincidence when its components have a salient similarity, but also a common proof (cf. also [Lange, 2010]). On the other hand, Lange points out, the components of a mathematical coincidence have no such common proof. For instance, we can consider the fact that the two Diophantine equations $$2x^2(x^2-1)=3(y^2-1)$$ and $$x(x-1)/2=2^n-1$$ have exactly the same five positive solutions for $$x$$. This fact, according to Lange, is a mere coincidence: As far as I know, this is just a coincidence. If it is in fact a coincidence, then that is because there is no proof that in a uniform way traces this similarity in the equations to some other respect in which they are alike. Such a proof would have explained why the two equations’ positive solutions are exactly the same. In the absence of such a “common proof”, this similarity has no explanation; it is coincidental. (pp. 278–279) In Chapter 9 Lange applies his account of explanation in mathematics to the case of Desargues’ theorem in Euclidean and projective geometry. Lange considers various proofs of the theorem and investigates why projective rather than Euclidean geometry is regarded as the natural home for Desargues’ theorem. In this chapter Lange also elaborates his notion of ‘natural property’ in mathematics and connects it to his account of mathematical explanation. Since the discussion of natural properties in mathematics is quite elaborate, I will give here just a sketch of how the subject of natural mathematical properties arises in connection with mathematical explanation. Consider again the example of the two polynomial equations (both have their non-real solutions coming in complex conjugate pairs). D’Alembert’s theorem explains this case because it identifies a genuine similarity between the two equations. More precisely, according to Lange, d’Alembert’s theorem identifies a natural property that both the cases possess: for both equations, their non-real solutions form complex-conjugate pairs (p. 342). In Part IV (“Explanations in mathematics and non-causal scientific explanations — together”) Lange offers a homogeneous treatment of non-causal explanations, also showing how several topics that have been explored throughout the previous parts provide a connection between different kinds of non-causal and even causal explanation. In Chapter 10 he focuses on a fourth kind of non-causal scientific explanations (besides explanations by constraint, RS explanations, and dimensional explanations). The target of these explanations, which draw on the notion of mathematical coincidence elaborated in Chapter 8, is the similarity exhibited by certain physically unrelated laws of nature. Such explanations, according to Lange, explain by revealing the laws to be no mathematical coincidence. Chapter 11 explores more closely the connections not only between the various kinds of non-causal explanations, but also between non-causal explanations and causal explanations. These connections hinge on notions, such as that of (mathematical or physical) coincidence or (mathematical or physical) natural property, that have been put forward in the previous chapters but that are given here a more comprehensive role within Lange’s framework. Apart from the technical details, which I leave to the interested reader, the last part of Because Without Cause really provides a key to understanding, from a broader perspective, the picture of explanation Lange has in mind. His approach to explanation is not monistic as other proposals in the literature (e.g., Steiner’s, or Kitcher’s). Lange does not want to capture the nature of scientific and mathematical explanations by providing a single general model to which the variety of explanations can be reduced. Rather, he identifies different kinds of non-causal explanations and provides an account of, first, what makes these explanations genuinely explanatory in scientific practice and, second, what relevant features these explanations share. As has already been stressed in other reviews of this book [Saatsi, 2017; Reutlinger, 2017], Lange adopts a pluralist view towards explanations in mathematics and in science. In his words: In this book, I have identified several kinds of non-causal explanations in mathematics and science. I have not argued that every example of explanation in math or every example of non-causal scientific explanation falls into one of the kinds of non-causal explanation that I have identified. I have also not tried to force all of the explanations that I have examined into a single narrow mold. […] However, I have tried to group the examples that I have studied into various kinds based on how the explanations work, and I have also tried to highlight some of the affinities among these kinds of explanation. (p. 371) A pluralist standpoint is not a novelty in the debate on the nature of scientific explanation, and many authors have adopted this attitude towards non-causal explanations in mathematics and in science. Nevertheless, there is an important respect in which Lange’s pluralism considerably adds to these proposals. Lange’s account provides a conceptually homogeneous framework that applies, in a very organic way, to an impressive variety of non-causal explanations that have been subject to unrelated philosophical analysis. This framework accommodates a sophisticated metaphysical analysis with a more practice-driven approach that strongly hinges on mathematical and scientific practice. The resulting picture of explanation is therefore extremely balanced and convincing. As Lange observes in the Preface, non-causal explanations have been largely neglected by philosophers of science, and explanation in mathematics has never been among the central topics in the philosophy of mathematics. I agree. Nevertheless, I think it is important to note that times are changing and that the topic of mathematical explanation (in science and in pure mathematics) has recently received increasing attention from philosophers of mathematics. This interest has stemmed primarily from the pivotal role that the notion of mathematical explanation plays in the so-called enhanced indispensability argument, an improved version of the Quine-Putnam indispensability argument that appeals to inference to the best explanation and to the notion of mathematical explanation in science (see [Molinini et al., 2016] for various analyses of the role played by the notion of mathematical explanation in the context of the enhanced indispensability argument). It is therefore natural, for those philosophers that are involved in the ontological dispute between platonists and nominalists, to draw attention to the nature of mathematical explanation. But I take Lange’s observation in the Preface also as an indicator of a different, more profound, issue that affects the study of mathematical explanation. The few accounts of mathematical explanation (in science and in pure mathematics) that have been proposed are quite patchy and do not offer a clear idea of how to characterize explanatory practices in mathematics (besides the few toy cases that are often cited in the literature). This fact has been recognized as a major deficit of the current debate on mathematical explanation [Saatsi, 2016; Molinini, 2016]. I therefore think that the important role that this book has in filling this gap is the because of Because Without Cause. Lange gives a broad and metaphysically robust account of what a mathematical explanation is (or, better, what makes some mathematical explanations genuinely explanatory). This account not only seems to fit the practice of scientists and mathematicians very well. It also fruitfully interacts with philosophy of science and metaphysics, thus paving the way for new exciting connections. The debate on the existence and functioning of non-causal explanations is extremely multifaceted and it is reasonable to expect that some readers will resist many of Lange’s ideas. On the other hand, I think that even the (partially or totally) unconvinced reader will agree on the exceptional quality and ‘mighty’ character of Because Without Cause. With this book Lange really raises the bar for those involved in the study of explanation(s) in science and in mathematics. REFERENCES Baker, Alan [ 2012]: ‘Science-driven mathematical explanation’, Mind 121, 243– 267 Google Scholar CrossRef Search ADS Kitcher, Philip [ 1984]: The Nature of Mathematical Knowledge . Oxford University Press. Kitcher, Philip [ 1989]: ‘Explanatory unification and the causal structure of the world’, in Kitcher Philip and Salmon, Wesley eds, Scientific Explanation , pp. 410– 505. Minnesota Studies in the Philosophy of Science; 13. Minneapolis: University of Minnesota Press. Lange, Marc [ 2009]: Laws and Lawmakers. Oxford University Press. Lange, Marc [ 2010]: ‘What are mathematical coincidences (and why does it matter)?’, Mind 119, 307– 340 Google Scholar CrossRef Search ADS Lange, Marc [ 2011]: ‘Conservation laws in scientific explanations: Constraints or coincidences?’, Philosophy of Science 78, 333– 352 Google Scholar CrossRef Search ADS Lange, Marc [ 2013]: ‘What makes a scientific explanation distinctively mathematical?’, The British Journal for the Philosophy of Science 64, 485– 511 Google Scholar CrossRef Search ADS Mancosu, Paolo [ 2011]: ‘Explanation in mathematics’, in Zalta, Edward N. ed., The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/archives/sum2015/entries/mathematics-explanation Accessed January 2018. Mancosu, Paolo, and Pincock Christopher [ 2012]: ‘Mathematical explanation’, in Pritchard, D. ed., Oxford Bibliographies Online: Philosophy. Oxford University Press. https://doi.org/10.1093/obo/9780195396577-0029. Accessed January 2018. Molinini, Daniele [ 2016]: ‘Evidence, explanation and enhanced indispensability’, Synthese 193, 403– 422 Google Scholar CrossRef Search ADS Molinini, D., Pataut, F. and Sereni A. [ 2016]: ‘Indispensability and explanation: An overview and introduction’, Synthese 193, 317– 332 Google Scholar CrossRef Search ADS Reutlinger, Alexander [ 2017]: Review of Marc Lange’s Because Without Cause, Notre Dame Philosophical Reviews. http://ndpr.nd.edu/news/because-without-cause-non-causal-explanations-in-science-and-mathematics/ Accessed January 2018. Saatsi, Juha [ 2016]: ‘On the “Indispensable explanatory role” of mathematics’, Mind 125, 1045– 1070 Google Scholar CrossRef Search ADS Saatsi, Juha [ 2017]: ‘A pluralist account of non-causal explanation in science and mathematics’, Metascience , https://doi.org/10.1007/s11016-017-0249-z. Steiner, Mark [ 1978a]: ‘Mathematical explanation’, Philosophical Studies 34, 135– 151 Google Scholar CrossRef Search ADS Steiner, Mark [ 1978b]: ‘Mathematics, explanation, and scientific knowledge’, Noûs 12, 17– 28 Google Scholar CrossRef Search ADS Tappenden, Jamie [ 2008]: ‘Mathematical concepts and definitions’, in Mancosu, Paolo ed., The Philosophy of Mathematical Practice , pp. 256– 275. Oxford University Press. Google Scholar CrossRef Search ADS © The Authors [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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