A lot of philosophical energy has been devoted recently in trying to determine if mathematics can contribute to our understanding of physical phenomena. Not many philosophers are interested, though, if the converse makes sense, i.e., if our cognitive interaction (scientific or otherwise) with the physical world can be helpful (in an explanatory or non-explanatory way) in our efforts to make sense of mathematical facts. My aim in this paper is to try to fill this important lacuna (...) in the recent literature. My answer to the question of this paper is negative. As I will argue, there are serious problems with the main reasons for believing in the first place that it is possible to have physical understanding of mathematical facts. (shrink)

There are lots of arguments for, or justifications of, mathematical theorems that make use of principles from physics. Do any of these constitute explanations? On the one hand, physical principles do not seem like they should be explanatorily relevant; on the other, some particular examples of physical justifications do look explanatory. In this article, I defend the idea that physical justifications can and do explain mathematical facts. 1 Physical Arguments for Mathematical (...) Truths2 Preview3 Mathematical Facts4 Purity5 Doubts about Purity: I6 Doubts about Purity: II7 How Physical Arguments Might Explain: I8 How Physical Arguments Might Explain: II9 Conclusion. (shrink)

We argue that two well-known examples (strawberry distribution and Konigsberg bridges) generally considered genuine cases of distinctively _mathematical_ explanation can also be understood as cases of distinctively _generic_ explanation. The latter answer resemblance questions (e.g., why did neither person A nor B manage to cross all bridges) by appealing to ‘generic task laws’ instead of mathematical necessity (as is done in distinctively mathematical explanations). We submit that distinctively generic explanations derive their explanatory force from their role (...) in ontological unification. Additionally, we argue that distinctively generic explanations are better seen as standardly mathematical instead of distinctively mathematical. Finally, we compare and contrast our proposal with the work of Christopher Pincock on abstract explanations in science and the views of Michael Strevens on abstract causal event explanations. (shrink)

This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal (...) structure of the physical world. The no-miracles argument is the primary motivation for scientific realism. It is a presupposition of this argument that unobservable entities are explanatory only when they determine the empirical phenomena they explain. I argue that mathematical entities should also be seen as explanatory only when they determine the empirical facts they explain, namely, the modal structure of the physical world. Thus, scientific realism commits us to a metaphysical determination relation between mathematics and physical modality that has not been previously recognized. The requirement to account for the metaphysical dependence of modal physical structure on mathematics limits the class of acceptable solutions to the applicability problem that are available to the scientific realist. (shrink)

In the debate over whether mathematical facts, properties, or entities explain physical events (in what philosophers call “extra-mathematical” explanations), Aidan Lyon’s (2012) affirmative answer stands out for its employment of the program explanation (PE) methodology of Frank Jackson and Philip Pettit (1990). Juha Saatsi (2012; 2016) objects, however, that Lyon’s examples from the indispensabilist literature are (i) unsuitable for PE, (ii) nominalizable into non-mathematical terms, and (iii) mysterious about the explanatory relation alleged to obtain (...) between the PE’s mathematical explanantia and physical explananda. In this paper, I propose a counterexample to Saatsi’s objections. My counterexample is Frank Jackson’s (1998a) program explanation for color experience, which I argue needs recasting as an extra-mathematical PE due to its implicit reliance on reflectance, a property that suffers conceptual regress unless redefined with Fourier harmonics. Pace Saatsi, I argue that this recast example is an authoritative PE, non-nominalizable, and minimally esoteric. Important for the indispensability debate at large, moreover, is that my counterexample reifies Fourier harmonics without the Enhanced Indispensability Argument (an argument to which Lyon applies PE as a premise). Indispensabilists have long overlooked the conditionalization of a limited mathematical realism on property realism, and my counterexample to Saatsi exploits this conditionalization. (shrink)

Several philosophers have argued that to capture the generality of certain scientific explanations, we must count mathematical facts among their explanantia. I argue that we can better understand these explanations by adopting a more nuanced stance toward mathematical representations, recognizing the role of mathematical representation schemata in representing highly abstract features of physical systems. It is by picking out these abstract but nonmathematical features that explanations appealing to mathematics achieve a high degree (...) of generality. The result is a rich conception of the role of mathematics in scientific explanations that does not require us to treat mathematical facts as explanantia. (shrink)

A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is (...) nothing mathematics can do to make an explanation mathematical ; realists think something can be done, but they are not clear about what that something is. I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's [1990] notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism. I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist. (shrink)

Certain scientific explanations of physical facts have recently been characterized as distinctively mathematical –that is, as mathematical in a different way from ordinary explanations that employ mathematics. This article identifies what it is that makes some scientific explanations distinctively mathematical and how such explanations work. These explanations are non-causal, but this does not mean that they fail to cite the explanandum’s causes, that they abstract away from detailed causal histories, or (...) that they cite no natural laws. Rather, in these explanations, the facts doing the explaining are modally stronger than ordinary causal laws or are understood in the why question’s context to be constitutive of the physical arrangement at issue. A distinctively mathematical explanation works by showing the explanandum to be more necessary than ordinary causal laws could render it. Distinctively mathematical explanations thus supply a kind of understanding that causal explanations cannot. 1 Introduction2 Some Distinctively Mathematical Scientific Explanations3 Are Distinctively Mathematical Explanations Set Apart by their Failure to Cite Causes? 4 Distinctively Mathematical Explanations do not Exploit Causal Powers5 How these Distinctively Mathematical Explanations Work6 Conclusion. (shrink)

Can there be mathematical explanations of physical phenomena? In this paper, I suggest an affirmative answer to this question. I outline a strategy to reconstruct several typical examples of such explanations, and I show that they fit a common model. The model reveals that the role of mathematics is explicatory. Isolating this role may help to re-focus the current debate on the more specific question as to whether this explicatory role is, as proposed here, also an (...) explanatory one. (shrink)

ABSTRACT Can there be mathematical explanations of physical phenomena? In this paper, I suggest an affirmative answer to this question. I outline a strategy to reconstruct several typical examples of such explanations, and I show that they fit a common model. The model reveals that the role of mathematics is explicatory. Isolating this role may help to re-focus the current debate on the more specific question as to whether this explicatory role is, as proposed here, also (...) an explanatory one. (shrink)

I identify and characterize a type of noncausal explanation in physics. I first introduce a distinction, between the physical properties of a system, and the representational properties of the mathematical expressions of the system’s physical properties. Then I introduce a novel kind of property, which I shall call a dual property. This is a special kind of representational property, one for which there is an interpretation as a physical property. It is these dual properties that, I (...) claim, are amenable to noncausal (mathematical, in fact) explanations. I discuss a typical example of such a dual property, and an example of an explanation as to why that dual property holds (the explanation of the quantization of the linear momentum). (shrink)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties (...) that the basic theory faces. The final view appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to (...) my title question impacts on broader issues in the philosophy of mathematics; in particular it may help platonists respond to a recent challenge by Joseph Melia concerning the force of the Indispensability Argument. (shrink)

In his influential 1960 paper ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’, Eugene P. Wigner raises the question of why something that was developed without concern for empirical facts—mathematics—should turn out to be so powerful in explaining facts about the natural world. Recent philosophy of science has developed ‘Wigner’s puzzle’ in two different directions: First, in relation to the supposed indispensability of mathematical facts to particular scientific explanations and, secondly, in connection with the (...) idea that aesthetic criteria track theoretical desiderata such as empirical success. An important aspect of Wigner’s article has, however, been overlooked in these debates: his worries about the underdetermination of physical theories by mathematical frameworks. The present paper argues that, by restoring this aspect of Wigner’s argument to its proper place, Wigner’s puzzle may become an instructive case study for the teaching of core issues in the philosophy of science and its history. (shrink)

In this paper I defend an intermediate position between the ‘bare mathematical results’ view and the ‘transmission’ view of mathematical explanations of physical phenomena (MEPPs). I argue that, in MEPPs, it is not enough to deduce the explanandum from the generalizations cited in the explanans. Rather, we must add information regarding why those generalizations obtain. However, I also argue that it is not necessary to provide explanatory proofs of the mathematical theorems that represent those generalizations. (...) I illustrate this with the bridges of Königsberg case. -/- . (shrink)

In books such as The World Within the World and The Anthropic Cosmological Principle, astronomer John Barrow has emerged as a leading writer on our efforts to understand the universe. Timothy Ferris, writing in The Times Literary Supplement of London, described him as "a temperate and accomplished humanist, scientist, and philosopher of science--a man out to make a contribution, not a show." Now Barrow offers the general reader another fascinating look at modern physics, as he explores the quest for a (...) single, unifying theory that will unlock nature's secrets. Theories of Everything is more than a history of science, more than a popular report on recent research and discoveries. Barrow provides a reflective, intelligent commentary on what a true Theory of Everything would be--its ingredients, its limitations, and what it could tell us about the universe. Never before, he writes, have physicists been so confident and so eager in the hunt for this "cosmic Rosetta Stone," as he calls it: "a single all-embracing picture of all the laws of nature from which the inevitability of all things seen must follow with unimpeachable logic." He lays out eight essential ingredients for a Theory of Everything and then explores each in turn, tracing how our knowledge has developed and how scientific discovery relates to our changing philosophy and religious thought in each area. Some of these ingredients are obvious--the laws of nature must be explained, for example, as well as its organizing principles--but others may be surprising, such as broken symmetries and selection biases. A Theory of Everything must account for the fact that the universe is "messy and complicated," he tells us, and for the limitations imposed by the questions we ask and the information we can obtain. The key lies in the remarkable capacity of mathematics to express the fundamental workings of the physical world--a language that the human mind is uniquely equipped to understand and manipulate. Barrow examines what mathematics actually is and describes how it makes the universe intelligible and provides a path to the underlying coherence in nature--which has led, in fact, to arguments that the universe itself is a vast computer. Yet even the most complete theory, even the most comprehensive mathematical explanation, cannot account for the uncomputable varieties of human experience and thought. "No non-poetic account of reality," he writes, "can be complete." In a field where the authorities converse in equations and mathematical notations, John Barrow speaks with the voice of thoughtful and knowledgeable humanist. Written with eloquence and expertise, Theories of Everything establishes a new perspective on humanity's efforts to explain the universe. (shrink)

In this thesis, I aim to motivate a particular philosophy of mathematics characterised by the following three claims. First, mathematical sentences are generally speaking false because mathematical objects do not exist. Second, people typically use mathematical sentences to communicate content that does not imply the existence of mathematical objects. Finally, in using mathematical language in this way, speakers are not doing anything out of the ordinary: they are performing straightforward assertions. In Part I, I argue (...) that the role played by mathematics in our scientific explanations is a purely expressive one, merely allowing us to say more about the physical world than we would otherwise be able to. Mathematical objects do not need to exist for mathematics to play this role. This proposal puts a normative constraint on our use of mathematical language: we ought to use mathematically presented theories to express belief only in the consequences they have for non-mathematical things. In Part II, I will argue that what the normative proposal recommends is in fact what people generally do in both pure and applied mathematical contexts. I motivate this claim by showing that it is predicted by our best general means of analysing natural language. I provide a semantic theory of applied arithmetical sentences that reveals they do not purport to refer to numbers, as well as a pragmatic theory for pure mathematical language use which reveals that pure mathematical utterances do not typically communicate content that implies the existence of mathematical objects. In conclusion, I show that the emerging hermeneutic fictionalist position is preferable to any alternative interpretation of mathematical discourse as aimed at describing a domain of independently existing abstract mathematical objects. (shrink)

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This (...) is because these explanations work not by appealing to what the world must be like as a matter of mathematical necessity but by appealing to various contingent causal facts. (shrink)

A proposal by Baron, Colyvan, and Ripley to extend the counterfactual theory of explanation to include counterfactual reasoning about mathematical explanations of physical facts is discussed. Their suggestion is that the explanatory role of mathematics can best be captured counterfactually. This paper focuses on their example with a number-theoretic antecedent. Incorporating discussions on the structure and de re knowledge of numbers, it is argued that the approach leads to a change in the structure of numbers. As (...) a result, the counterfactual is not about the natural numbers anymore. Linking the antecedent and consequent of the counterfactual also becomes problematic. (shrink)

We discuss a recent attempt by Chris Daly and Simon Langford to do away with mathematical explanations of physical phenomena. Daly and Langford suggest that mathematics merely indexes parts of the physical world, and on this understanding of the role of mathematics in science, there is no need to countenance mathematical explanation of physical facts. We argue that their strategy is at best a sketch and only looks plausible in simple cases. We also (...) draw attention to how frequently Daly and Langford find themselves in conflict with mathematical and scientific practice. (shrink)

This paper draws together two strands in the debate over the existence of mathematical objects. The first strand concerns the notion of extra-mathematical explanation: the explanation of physical facts, in part, by facts about mathematical objects. The second strand concerns the access problem for platonism: the problem of how to account for knowledge of mathematical objects. I argue for the following conditional: if there are extra-mathematical explanations, then the core thesis of (...) the access problem is false. This has implications for nominalists and platonists alike. Platonists can make a case for epistemic access to mathematical objects by providing evidence in favour of the existence of extra-mathematical explanations. Nominalists, by contrast, can use the access problem to cast doubt on the idea that mathematical objects play a substantive role in scientific explanation. (shrink)

A new trend in the philosophical literature on scientific explanation is that of starting from a case that has been somehow identified as an explanation and then proceed to bringing to light its characteristic features and to constructing an account for the type of explanation it exemplifies. A type of this approach to thinking about explanation – the piecemeal approach, as I will call it – is used, among others, by Lange (2013) and Pincock (2015) in the context of their (...) treatment of the problem of mathematical explanations of physical phenomena. This problem is of central importance in two different recent philosophical disputes: the dispute about the existence on non-causal scientific explanations and the dispute between realists and antirealists in the philosophy of mathematics. My aim in this paper is twofold. I will first argue that Lange (2013) and Pincock (2015) fail to make a significant contribution to these disputes. They fail to contribute to the dispute in the philosophy of mathematics because, in this context, their approach can be seen as question begging. They also fail to contribute to the dispute in the general philosophy of science because, as I will argue, there are important problems with the cases discussed by Lange and Pincock. I will then argue that the source of the problems with these two papers has to do with the fact that the piecemeal approach used to account for mathematical explanation is problematic. (shrink)

Some scientific explanations appear to turn on pure mathematical claims. The enhanced indispensability argument appeals to these ‘mathematical explanations’ in support of mathematical platonism. I argue that the success of this argument rests on the claim that mathematical explanations locate pure mathematical facts on which their physical explananda depend, and that any account of mathematical explanation that supports this claim fails to provide an adequate understanding of mathematical explanation.

Lange argues that some natural phenomena can be explained by appeal to mathematical, rather than natural, facts. In these “distinctively mathematical” explanations, the core explanatory facts are either modally stronger than facts about ordinary causal law or understood to be constitutive of the physical task or arrangement at issue. Craver and Povich argue that Lange’s account of DME fails to exclude certain “reversals”. Lange has replied that his account can avoid these directionality charges. (...) Specifically, Lange argues that in legitimate DMEs, but not in their “reversals,” the empirical fact appealed to in the explanation is “understood to be constitutive of the physical task or arrangement at issue” in the explanandum. I argue that Lange’s reply is unsatisfactory because it leaves the crucial notion of being “understood to be constitutive of the physical task or arrangement” obscure in ways that fail to block “reversals” except by an apparent ad hoc stipulation or by abandoning the reliance on understanding and instead accepting a strong realism about essence. (shrink)

This paper discusses Baker’s Enhanced Indispensability Argument for mathematical realism on the basis of the indispensable role mathematics plays in scientific explanations of physical facts, along with various responses to it. I argue that there is an analogue of causal explanation for mathematics which, of several basic types of explanation, holds the most promise for use in the EIA. I consider a plausible case where mathematics plays an explanatory role in this sense, but argue that such (...) use still does not support realism about mathematical objects. (shrink)

Several philosophers have suggested that some scientific explanations work not by virtue of describing aspects of the world’s causal history and relations, but rather by citing mathematical facts. This paper investigates what mathematical facts could be in order for them to figure in such “distinctively mathematical” scientific explanations. For “distinctively mathematical explanations” to be explanations by constraint, mathematical language cannot operate in science as representationalism or platonism describes. It can (...) operate as Aristotelian realism describes. That is because Aristotelian realism enables mathematics to apply to the physical world without a morphism’s mediation. (shrink)

Mathematics clearly plays an important role in scientific explanation. Debate continues, however, over the kind of role that mathematics plays. I argue that if pure mathematical explananda and physical explananda are unified under a common explanation within science, then we have good reason to believe that mathematics is explanatory in its own right. The argument motivates the search for a new kind of scientific case study, a case in which pure mathematical facts and physical (...) class='Hi'>facts are explanatorily unified. I argue that it is possible for there to be such cases, and provide some toy examples to demonstrate this. I then identify a potential source of scientific case studies as a guide for future work. (shrink)

Explanations of three different aspects of the rainbow are considered. The highly mathematical character of these explanations poses some interpretative questions concerning what the success of these explanations tells us about rainbows. I develop a proposal according to which mathematical explanations can highlight what is relevant about a given phenomenon while also indicating what is irrelevant to that phenomenon. This proposal is related to the extensive work by Batterman on asymptotic explanation with special reference (...) to Batterman’s own discussion of the rainbow. (shrink)

For an ‘explanation' of physical facts by laws of nature, we have to establish a relation between physical facts and laws of nature. It is an open question, whether the laws of nature govern the facts with necessity or whether the laws are related to the facts merely by supervenience. In addition, it is not quite clear, whether the known laws of physics describe only artificially simplified cases, e.g. isolated situations, or whether the laws (...) of physics actually grasp real facts. Known solutions of these problems refer to situations where laws of classical physics are applied to phenomena of classical physics. However, if the same laws were applied to matter of facts of the domain of modern physics, then in many cases there would be no ‘explanation' in the sense mentioned. These new problems can be treated either by additional ‘interpretations' of the theories in question, or by a radical change of the ontological preconditions of classical physics. (shrink)

In The Explanatory Dispensability of Idealizations, Sam Baron suggests a possible strategy enabling the indispensability argument to break the symmetry between mathematical claims and idealization assumptions in scientific models. Baron’s distinction between mathematical and non-mathematical idealization, I claim, is in need of a more compelling criterion, because in scientific models idealization assumptions are expressed through mathematical claims. In this paper I argue that this mutual dependence of idealization and mathematics cannot be read in terms of symmetry (...) and that Baron’s non-causal notion of mathematical difference-making is not effective in justifying any symmetry-breaking between mathematics and idealization. The function of making a difference that Baron attributes to mathematics cannot be referred to physical facts, but to the features of quantities, such as step lengths or time intervals taken into account in the models. It appears, indeed, that it does not follow from Baron’s argument that idealizations do not help to carry the explanatory load at least for two reasons: mathematics is not independent of idealizations in modelling and idealizations help mathematics to carry the explanatory load of a model in different degrees. (shrink)

Standard approaches to counterfactuals in the philosophy of explanation are geared toward causal explanation. We show how to extend the counterfactual theory of explanation to non-causal cases, involving extra-mathematical explanation: the explanation of physical facts by mathematical facts. Using a structural equation framework, we model impossible perturbations to mathematics and the resulting differences made to physical explananda in two important cases of extra-mathematical explanation. We address some objections to our approach.

In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an (...) area, D—the challenge to justify our D-beliefs—with the reliability challenge for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism. (shrink)

Lange’s collection of expanded, mostly previously published essays, packed with numerous, beautiful examples of putatively non-causal explanations from biology, physics, and mathematics, challenges the increasingly ossified causal consensus about scientific explanation, and, in so doing, launches a new field of philosophic investigation. However, those who embraced causal monism about explanation have done so because appeal to causal factors sorts good from bad scientific explanations and because the explanatory force of good explanations seems to derive from revealing the (...) relevant causal (or ontic) structures. The taxonomic project of collecting examples and sorting their types is an essential starting place for a theory of non-causal explanation. But the title of Lange’s book requires something further: showing that the putative explanations are, in fact, explanatory and revealing the non-causal source of their explanatory power. This project is incomplete if there are examples of putative non-causal explanations that fit the form but that nobody would accept as explanatory (absent a radical revision of intuitions). Here we provide some reasons for thinking that there are such examples. (shrink)

According to one popular nominalist picture, even when mathematics features indispensably in scientific explanations, this mathematics plays only a purely representational role: physical facts are represented, and these exclusively carry the explanatory load. I think that this view is mistaken, and that there are cases where mathematics itself plays an explanatory role. I distinguish two kinds of explanatory generality: scope generality and topic generality. Using the well-known periodical-cicada example, and also a new case study involving bicycle gears, (...) I argue that what is picked out by the mathematics are structural facts that go beyond any specific physical facts. (shrink)

Although all mathematical truths are necessary, mathematicians take certain combinations of mathematical truths to be ‘coincidental’, ‘accidental’, or ‘fortuitous’. The notion of a ‘ mathematical coincidence’ has so far failed to receive sufficient attention from philosophers. I argue that a mathematical coincidence is not merely an unforeseen or surprising mathematical result, and that being a misleading combination of mathematical facts is neither necessary nor sufficient for qualifying as a mathematical coincidence. I argue (...) that although the components of a mathematical coincidence may possess a common explainer, they have no common explanation ; that two mathematical facts have a unified explanation makes their truth non-coincidental. I suggest that any motivation we may have for thinking that there are mathematical coincidences should also motivate us to think that there are mathematical explanations, since the notion of a mathematical coincidence can be understood only in terms of the notion of a mathematical explanation. I also argue that the notion of a mathematical coincidence plays an important role in scientific explanation. When two phenomenological laws of nature are similar, despite concerning physically distinct processes, it may be that any correct scientific explanation of their similarity proceeds by revealing their similarity to be no mathematical coincidence. (shrink)

In non-causal explanations, some non-causal facts (such as mathematical, modal or metaphysical) are used to explain some physical facts. However, precisely because these explanations abstract away from causal facts, they face two challenges: 1) it is not clear why would one rather than the other non-causal explanantia be relevant for the explanandum; and 2) why would standing in a particular explanatory relation (e.g., “counterfactual dependence”, “constraint”, “entailment”, “constitution”, “grounding”, and so on), and not (...) in some other, be explanatory. I develop an explanatory relevance account which is based on erotetic constraints and show how it addresses these two challenges. (shrink)

Philosophers intent upon characterizing the difference between physics and biology often seize upon the purported fact that physical explanations conform more closely to the covering law model than biological explanations. Central to this purported difference is the role of laws of nature in the explanations of these two sciences. However, I argue that, although certain important differences between physics and biology can be highlighted by differences between physical and biological explanations, these differences are not (...) differences in the degree to which those explanations conform to the covering law model, which fits biology about as well as it does physics. (shrink)

Debates between contemporary platonist and nominalist conceptions of the metaphysical status of mathematical objects have recently included discussions of explanations of physical phenomena in which mathematics plays an indispensable role, termed mathematical explanations in science (MES). I will argue that MES requires an ontology that can (1) ground claims about mathematical necessity as distinct from physical necessity and (2) explain how that mathematical necessity applies to the physical world. I contend that (...) nominalism fails to meet the first criterion and platonism the second. I then articulate an alternative, Aristotelian approach to mathematical objects and defend such a view as meeting both criteria. (shrink)

According to a grand narrative that long ago ceased to be told, there was a seventeenth century Scientific Revolution, during which a few heroes conquered nature thanks to mathematics. This grand narrative began with the exhibition of quantitative laws that these heroes, Galileo and Newton for example, had disclosed: the law of falling bodies, according to which the speed of a falling body is proportional to the square of the time that has elapsed since the beginning of its fall; the (...) law of gravitation, according to which two bodies are attracted to one another in proportion to the sum of their masses and in inverse proportion to the square of the distance separating them -- according to his own preferences, each narrator added one or two quantitative laws of this kind. The essential feature was not so much the examples that were chosen, but, rather, the more or less explicit theses that accompanied them. First, mathematization would be taken as the criterion for distinguishing between a qualitative Aristotelian philosophy and the new quantitative physics. Secondly, mathematization was founded on the metaphysical conviction that the world was created pondere, numero et mensura, or that the ultimate components of natural things are triangles, circles, and other geometrical objects. This metaphysical conviction had two immediate consequences: that all the phenomena of nature can be in principle submitted to mathematics and that mathematical language is transparent; it is the language of nature itself and has simply to be picked up at the surface of phenomena. Finally, it goes without saying that, from a social point of view, the evolution of the sciences was apprehended through what has been aptly called the "relay runner model," according to which science progresses as a result of individual discoveries. Grand narratives such as this are perhaps simply fictions doomed to ruin as soon as they are clearly expressed. In any case, the very assumption on which this grand narrative relies can be brought into question: even in the canonical domain of mechanics, the relevant epistemological units crucial to understanding the dynamics of the Scientific Revolution are perhaps not a few laws of motion, but a complex set of problems embodied in mundane objects. Moreover, each of the theses just mentioned was actually challenged during the long period of historiographical reappraisal, out of which we have probably not yet stepped. Against the sharp distinction between a qualitative Aristotelian philosophy and the new quantitative physics, numerous studies insist that Rome wasn't built in a day, so to speak. Since Antiquity, there have always been mixed sciences; the emergence of pre-classical mechanics depends on both medieval treatises and the practical challenges met by Renaissance engineers. It is indeed true that, for Aristotle, mathematics merely captures the superficial properties of things, but the Aristotelianisms were many during the Renaissance and the Early Modern period, with some of them being compatible with the introduction of mathematics in natural philosophy. In addition, the gap between the alleged program of mathematizing nature and its effective realization was underlined as most natural phenomena actually escaped mathematization; at best they were enrolled in what Thomas Kuhn began to rehabilitate under the appellation of the "Baconian sciences," i.e., empirical investigations aiming at establishing isolated facts, without relating them to any overarching theory. Hence, mathematization of nature cannot pretend to capture a historical fact: at most, it expresses an indeterminate task for generations to come. On top of these first two considerations, and against the thesis of the neutrality of the mathematical language, it was urged that mathematics is not "only a language" and that, exactly as other symbolic means or cognitive tools, it has its own constraints. For example, it has been thoroughly explained that the Euclidean theory of proportions both guides and frustrates the Galilean analysis of motion; its shortages were particularly clear with respect to the expression of continuity, which is crucial in the case of motion. Consequently, when calculus was invented and applied to the analysis of motion, it was not a transposition that left things as they stood. Even more clearly than in the case of a translation from one natural language to another, the shift from one symbolic language to another entails that certain possibilities are opened while others are closed. The cognitive constraints imposed by established mathematical theories, as seen in the theory of proportions or calculus, were not the only ones to be studied in relation to mathematization. Certain schemes dependent on the grammar of natural languages, e.g., the scheme of contrariety, or certain symbolic means of representation, e.g. geometrical diagrams and numerical tables, were also subject to such scrutiny. Lastly, it was insisted that, even if we concede the existence of scientific geniuses, mathematics is largely produced by intellectual communities and embedded within social practices. More attention was consequently paid to the forms of communication in given mathematical networks, or to the teaching of the discipline in, for example, Jesuit colleges and universities. The set of mathematical practices specific to specialized craftsmen, highly-qualified experts and engineers began to be studied in its own right. All these reflections may have helped us change our perspectives on the question of mathematization. It seems, however, that they were instead set aside, both because of a general distrust towards sweeping narratives that are always subject to the suspicion that they overlook the unyielding complexity of real history, and because of a shift in our interests. The more obscure and idiosyncratic they are, an alchemist, a patron of the sciences or a lunatic collector is nowadays honored in journals of the history of sciences. As for the general issues involved in the question of mathematization, they are rejected as obsolete, or reserved for specialized journals in the history of mathematics. Consequently, before presenting the essays of this fascicle, I would like to say a few words in favor of a renewed study of the forms of mathematization in the history of the early sciences. (shrink)

Could the truths of mathematics have been different than they in fact are? If so, which truths could have been different? Do the contingent mathematical facts supervene on physical facts, or are they free floating? I investigate these questions within a framework of higher-order modal logic, drawing sometimes surprising connections between the necessity of arithmetic and analysis and other theses of modal metaphysics: the thesis that possibility in the broadest sense is governed by a logic of (...) S5, that what is possible holds in some maximally specific possibility, and that every property can be rigidified. The investigation will distinguish sharply between platonic contingency---contingency about whether particular abstract ``platonic'' mathematical objects are arranged in a certain way (e.g. in a natural number or real number structure)---from a deeper variety of structural contingency concerning what holds of objects whenever they *are* arranged in that way. (shrink)

If physicalism is true, everything is physical. In other words, everything supervenes on, or is necessitated by, the physical. Accordingly, if there are logical/mathematical facts, they must be necessitated by the physical facts of the world. The aim of this paper is to clarify what logical/mathematical facts actually are and how these facts can be accommodated in a purely physical world.

This paper explores whether there is any relation between mathematical proofs that specify the grounds of the theorem being proved and mathematical proofs that explain why the theorem obtains. The paper argues that a mathematical fact’s grounds do not, simply by virtue of grounding it, thereby explain why that fact obtains. It argues that oftentimes, a proof specifying a mathematical fact’s grounds fails to explain why that fact obtains whereas any explanation of the fact does not (...) specify its ground. The paper offers several examples from mathematical practice to illustrate these points. These examples suggest several reasons why explaining and grounding tend to come apart, including that explanatory proofs need not exhibit purity, tend not to be brute force, and often unify separate cases by identifying common reasons behind them even when those cases have distinct grounds. The paper sketches an account of what makes a proof explanatory and uses that account to defend the morals drawn from the examples already given. (shrink)

The present paper advances an analogy between cases of extra-mathematical explanation and cases of what might be termed ‘extra-logical explanation’: the explanation of a physical fact by a logical fact. A particular case of extra-logical explanation is identified that arises in the philosophical literature on time travel. This instance of extra-logical explanation is subsequently shown to be of a piece with cases of extra-mathematical explanation. Using this analogy, we argue extra-mathematical explanation is part of a broader (...) class of non-causal explanation. This has important implications for extra-mathematical explanation, for time travel and for theories of explanation more generally. (shrink)

'Explanation, Quantity and Law' is a sustained elaboration and defence of a theory of explanation, called the instance view, that is able to deal with the characteristic aspects of physical science, such as the use of mathematics, the fact that errors of measurement are ubiquitous, and so forth. The book begins with a summary of 'new directions' in the theory of explanation and continues with a systematic account of the view that to explain is to show that something is (...) an instance of a law of nature. This embraces topics such as the following - Salmon’s casual-mechanistic account and the ontic conception of explanation; theories of quantity from Ellis to Bigelow-Pargetter to quantum mechanics; Armstrong on numerical laws and laws of nature as relations between quantities; explanation and the quantum state. (shrink)

The aim of this dissertation is to clarify the debate over the explanation of quantum speedup and to submit, for the reader's consideration, a tentative resolution to it. In particular, I argue, in this dissertation, that the physical explanation for quantum speedup is precisely the fact that the phenomenon of quantum entanglement enables a quantum computer to fully exploit the representational capacity of Hilbert space. This is impossible for classical systems, joint states of which must always be representable as (...) product states. I begin the dissertation by considering, in Chapter 2, the most popular of the candidate physical explanations for quantum speedup: the many worlds explanation of quantum computation. I argue that, although it is inspired by the neo-Everettian interpretation of quantum mechanics, unlike the latter it does not have the conceptual resources required to overcome objections such as the so-called `preferred basis objection'. I further argue that the many worlds explanation, at best, can serve as a good description of the physical process which takes place in so-called network-based computation, but that it is incompatible with other models of computation such as cluster state quantum computing. I next consider, in Chapter 3, a common component of most other candidate explanations of quantum speedup: quantum entanglement. I investigate whether entanglement can be said to be a necessary component of any explanation for quantum speedup, and I consider two major purported counter-examples to this claim. I argue that neither of these, in fact, show that entanglement is unnecessary for speedup, and that, on the contrary, we should conclude that it is. In Chapters 4 and 5 I then ask whether entanglement can be said to be sufficient as well. In Chapter 4 I argue that despite a result that seems to indicate the contrary, entanglement, considered as a resource, can be seen as sufficient to enable quantum speedup. Finally, in Chapter 5 I argue that entanglement is sufficient to explain quantum speedup as well. (shrink)

Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.

It is often supposed that we can use mathematics to capture the time evolution of any physical system. By this, I mean that we can capture the basic truths about the time evolution of a physical system with a set of mathematical assertions, which can then be used as premises in arbitrary mathematical arguments to deduce more complex properties of the system. ;I would like to argue that this picture of the role of mathematics in physics (...) is incorrect. Specifically, I shall assert: ;The Deduction Failure Thesis: Bodies of knowledge in physics are generally not closed under otherwise valid mathematical argument forms. ;The Representation Failure Thesis: We cannot assume that the state of any system, together with its fundamental laws, can be captured by some set of mathematical assertions or equations. In fact, it is more likely that the world is not representable by a set of mathematical assertions or equations than that it is. ;The dissertation largely consists of arguments for these two theses. (shrink)

A. N. Whitehead's approach to physical theorizing contrasts with that of mainstream or official physics in being centrally concerned with articulating a background explanation of physical facts and phenomena in general that would take the place of the “ether” of classical physics, a project otherwise unpursued by the science in its modern period. Unlike Einstein's, Whitehead's approach to relativity primarily seeks explanation rather than utility ; also, it avoids the philosophical problems with Einstein's theory alleged by Whitehead (...) and a range of other philosophers. This stops short of a finding as to the comparative worth of Whitehead's alternative basis for quantitative formulations. (shrink)

Some mathematical proofs explain why the theorems they prove hold. This paper identifies several challenges for any counterfactual account of explanation in mathematics (that is, any account according to which an explanatory proof reveals how the explanandum would have been different, had facts in the explanans been different). The paper presumes that countermathematicals can be nontrivial. It argues that nevertheless, a counterfactual account portrays explanatory power as too easy to achieve, does not capture explanatory asymmetry, and fails to (...) specify why certain proofs are explanatory and others are not. Greater informativeness about counterfactual dependence can even yield less explanatory power. (shrink)