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)

What makes human beings intentional and thinking subjects? How does their intentionality and thought connect with their social nature and their communal experience? How do the answers to these questions shape the assumptions which it is legitimate to make in social explanation and political evaluation? These are the broad-ranging issues which Pettit addresses in this novel study. The Common Mind argues for an original way of marking off thinking subjects, in particular human beings, from other intentional systems, natural and artificial. (...) It holds by the holistic view that human thought requires communal resources while denying that this social connection compromises the autonomy of individuals. And, in developing the significance of this view of social subjects--this holistic individualism--it outlines a novel framework for social and political theory. Within this framework, social theory is allowed to follow any of a number of paths: space is found for intentional interpretation and decision-theoretic reconstruction, for structural explanation and rational choice derivation. But political theory is treated less ecumenically. The framework raises serious questions about contractarian and atomistic modes of thought and it points the way to a republican rethinking of liberal commitments. (shrink)

Some properties are causally relevant for a certain effect, others are not. In this paper we describe a problem for our understanding of this notion and then offer a solution in terms of the notion of a program explanation.

Frank Jackson and Robert Pargetter (1987)2 have argued for a version of reductive physicalism about color which they claim can accommodate the basic intuitions that have led others to embrace dispositionalism or subjectivism about color. Jackson (1996) has further developed the view and provided responses to some objections to its original statement. While Jackson and Pargetter do not have much company in endorsing their specific form of color physicalism, elements of their view have shown up in other realist accounts, including (...) the relativized account of color offered by John Spackman (2002), the disjunctivism of color properties endorsed by Peter Ross (2000), and the subjectivist strain present in Sydney Shoemaker’s (1994) discussion of color.3 Additionally, Mark Johnston (1992) has used Jackson and Pargetter’s view as a principal target in his arguments against color physicalism. (shrink)

Colour has often been supposed to be a subjective property, a property to be analysed orretly in terms of the phenomenological aspects of human expereince. In contrast with subjectivism, an objectivist analysis of color takes color to be a property objects possess in themselves, independently of the character of human perceptual expereince. David Hilbert defends a form of objectivism that identifies color with a physical property of surfaces - their spectral reflectance. This analysis of color is shown to provide a (...) more adequate account of the features of human color vision than its subjectivist rivals. The author's account of colro also recognises that the human perceptual system provides a limited and idiosyncratic picture of the world. These limitations are shown to be consistent with a realist account of colour and to provide the necessary tools for giving an analysis of common sense knowledge of color phenomena. (shrink)

The target article is an attempt to make some progress on the problem of color realism. Are objects colored? And what is the nature of the color properties? We defend the view that physical objects (for instance, tomatoes, radishes, and rubies) are colored, and that colors are physical properties, specifically types of reflectance. This is probably a minority opinion, at least among color scientists. Textbooks frequently claim that physical objects are not colored, and that the colors are "subjective" or "in (...) the mind." The article has two other purposes: first, to introduce an interdisciplinary audience to some distinctively philosophical tools that are useful in tackling the problem of color realism and, second, to clarify the various positions and central arguments in the debate. (shrink)

Frank Jackson champions the cause of conceptual analysis as central to philosophical inquiry. In recent years conceptual analysis has been undervalued and widely misunderstood, suggests Jackson. He argues that such analysis is mistakenly clouded in mystery, preventing a whole range of important questions from being productively addressed. He anchors his argument in discussions of specific philosophical issues, starting with the metaphysical doctrine of physicalism and moving on, via free will, meaning, personal identity, motion, and change, to ethics and the philosophy (...) of color. In this way the book not only offers a methodological program for philosophy, but also casts new light on some much-debated problems and their interrelations. (shrink)

We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content to key (...) intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. (shrink)

How can we think about things in the outside world? There is still no widely accepted theory of how mental representations get their meaning. In light of pioneering research, Nicholas Shea develops a naturalistic account of the nature of mental representation with a firm focus on the subpersonal representations that pervade the cognitive sciences.

Dispositional theories of colour have been attacked by McGinn and others on the ground that ‘Colours do not look like dispositions’. Langsam has argued that on the contrary they do, in ‘Why Colours Do Look Like Dispositions’, The Philosophical Quarterly, 50 , pp. 68–75. I make three claims. First, neither side has made its case. Secondly, it is true, at least on one interpretation, that colours do not look like dispositions. Thirdly, this does not show that dispositionalism about colours is (...) false. (shrink)

Joshua Gert presents an original account of color properties, and of our perception of them. He employs a general philosophical strategy - neo-pragmatism - which challenges an assumption made by virtually all other theories of color: he argues that colors are primitive properties of objects, irreducible to physical or dispositional properties.

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)

Metaphysics and science have a long but troubled relationship. In the twentieth century the Logical Positivists argued metaphysics was irrelevant and that philosophy should be guided by science. However, metaphysics and science attempt to answer many of the same, fundamental questions: What are laws of nature? What is causation? What are natural kinds? -/- In this book, Markus Schrenk examines and explains the central questions and problems in the metaphysics of science. He reviews the development of the field from the (...) early modern period through to the latest research, systematically assessing key topics including -/- dispositions, counterfactual conditionals, laws of nature, causation, natural kinds, essence, necessity. -/- With the addition of chapter summaries and annotated further reading, Metaphysics of Science is a much-needed, clear and informative survey of this exciting area of philosophical research. It is essential reading for students and scholars of philosophy of science and metaphysics. (shrink)

The literature on the indispensability argument for mathematical realism often refers to the ‘indispensable explanatory role’ of mathematics. I argue that we should examine the notion of explanatory indispensability from the point of view of specific conceptions of scientific explanation. The reason is that explanatory indispensability in and of itself turns out to be insufficient for justifying the ontological conclusions at stake. To show this I introduce a distinction between different kinds of explanatory roles—some ‘thick’ and ontologically committing, others ‘thin’ (...) and ontologically peripheral—and examine this distinction in relation to some notable ‘ontic’ accounts of explanation. I also discuss the issue in the broader context of other ‘explanationist’ realist arguments. (shrink)

This paper is about metaphysical ‘infinitism’, the view that there are, or could be, infinite chains of ontological dependence. Its main aim is to show that, contrary to widespread opinion, metaphysical infinitism is a coherent position. On the basis of this, it is then additionally argued that metaphysical infinitism need not fare worse than the more canonical ‘foundationalist’ alternatives when it comes to formulating metaphysical explanations. In the course of the discussion, a rather unexplored parallel with the debate concerning infinitism (...) about justification is suggested. (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)

Idealized scientific representations result from employing claims that we take to be false. It is not surprising, then, that idealizations are a prime example of allegedly inconsistent scientific representations. I argue that the claim that an idealization requires inconsistent beliefs is often incorrect and that it turns out that a more mathematical perspective allows us to understand how the idealization can be interpreted consistently. The main example discussed is the claim that models of ocean waves typically involve the false assumption (...) that the ocean is infinitely deep. While it is true that the variable associated with depth is often taken to infinity in the representation of ocean waves, I explain how this mathematical transformation of the original equations does not require the belief that the ocean being modeled is infinitely deep. More generally, as a mathematical representation is manipulated, some of its components are decoupled from their original physical interpretation. (shrink)

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, and I discuss (...) potential future directions of research for each side in the debate over the existence of abstract mathematical objects. (shrink)

This is a welcome reprint of a book that continues to grow in importance.

John Barrow is increasingly recognized as one of our most elegant and accomplished science writers, a brilliant commentator on cosmology, mathematics, and modern physics. Barrow now tackles the heady topic of impossibility, in perhaps his strongest book yet. Writing with grace and insight, Barrow argues convincingly that there are limits to human discovery, that there are things that are ultimately unknowable, undoable, or unreachable. He first examines the limits on scientific inquiry imposed by the deficiencies of the human mind: our (...) brain evolved to meet the demands of our immediate environment, Barrow notes, and much that lies outside this small circle may also lie outside our understanding. Barrow investigates practical impossibilities, such as those imposed by complexity, uncomputability, or the finiteness of time, space, and resources. Is the universe finite or infinite? Can information be transmitted faster than the speed of light? The book also examines the deeper theoretical restrictions on our ability to know, including Godel's theorem--which proved that there were things that could not be proved--and Arrow's Impossibility theorem about democratic voting systems. Finally, having explored the limits imposed on us from without, Barrow considers whether there are limits we should impose upon ourselves. For instance, if the secrets of the atom are to be found only by recreating extreme environments at great financial cost, just how much should we devote to that quest? Weaving together this intriguing tapestry, he illuminates some of the most profound questions of science, from the possibility of time travel to the very structure of the universe. (shrink)

Social sciences face a well-known problem, which is an instance of a general problem faced as well by psychological and biological sciences: the problem of establishing their legitimate existence alongside physics. This, as will become clear, is a problem in metaphysics. I will show how a new account of structural explanations, put forward by Frank Jackson and Philip Pettit, which is designed to solve this metaphysical problem with social sciences in mind, fails to treat the problem in any importantly new (...) way. Then I will propose a more modest approach, and show how it does not deserve the criticism directed at a prototype by Jackson and Pettit. (shrink)

In The Subjective View,' I defended (unoriginally) a dispositional theory of color and drew out some consequences of that theory. The dispositional theory (DT) maintains, roughly speaking, that for an object to instantiate a color property is for it to have a disposition to cause experiences as of an object having that property in normal perceivers in normal conditions. This theory has notable merits in capturing (assuming one wants them captured) the subjectivity and relativity of ascriptions of color, while allowing (...) that it is external ob- jects themselves that are colored. It makes colors both sense depen- dent and object qualifying.2 But it runs into prima facie problems in giving a plausible account of the phenomenology of color percep- tion, as I ruefully observed in my earlier book (op. cit., pp. 132-37). (shrink)

Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the linguistic (...) pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics both sensitive to mathematical practice, and to the ontological and epistemological issues that concern philosophers. (shrink)

Chaotic dynamics has been hailed as the third great scientific revolution in physics this century, comparable to relativity and quantum mechanics. In this book, Peter Smith takes a cool, critical look at such claims. He cuts through the hype and rhetoric by explaining some of the basic mathematical ideas in a clear and accessible way, and by carefully discussing the methodological issues which arise. In particular, he explores the new kinds of explanation of empirical phenomena which modern dynamics can deliver. (...) Explaining Chaos will be compulsory reading for philosophers of science and for anyone who has wondered about the conceptual foundations of chaos theory. (shrink)

Science and philosophy both express, and attempt to quench, the distinctively human thirst for knowledge. Today, their mutual relationship has become one of conflict or indifference rather than cooperation. At the same time, scientists and philosophers alike have moved away from at least some of our ordinary beliefs. But what can scientific and philosophical theories tell us about the world, in isolation from each other? And to what extent does a sophisticated investigation into the nature of things force us to (...) question commonsense beliefs? This book defends a form of naturalism which preserves the autonomy of metaphysics – the part of philosophy that aims to uncover the fundamental features of reality - while also keeping science centre stage. The proposed methodological perspective allows one to seek the right equilibrium between science, metaphysics and common sense. This is illustrated via three case studies, which provide a clear discussion of key philosophical issues. (shrink)

A Metaphysics for Scientific Realism: Knowing the Unobservable has two primary aims. The first is to extract the most promising refinements of the idea of scientific realism to emerge in recent decades and assemble them into a maximally defensible realist position, semirealism. The second is to demonstrate that, contra antirealist scepticism to the contrary, key concepts typically invoked by realists in expounding their views can be given a coherent and unified explication. These concepts include notions of causation, laws of nature, (...) scientific kinds, and approximate truth, and consequently, the demonstration undertaken includes a metaphysical study of ideas more commonly employed unreflectively in epistemological assessments of the sciences. In this paper, I answer searching critiques of this project by Steven French, Michel Ghins, and Stathis Psillos. (shrink)

This expanded edition of C L Hardin's ground-breaking work on colour features a new chapter, 'Further Thoughts: 1993', in which the author revisits the dispute ...

I first explore the notion of the world's being such that everything in it is a proper part. I then explore the notion of the world's being such that everything in it both is and has a proper part. Given two well recognized assumptions, I argue that both notions represent genuine metaphysical possibilities. Finally I consider, but dismiss, some possible objections.

Reflectance physicalists define reflectance as the intrinsic disposition of a surface to reflect finite-duration light pulses at a given efficiency per wavelength. I criticize the received view of dispositional reflectance (David R. Hilbert’s) for failing to account for what I call “harmonic dispersion,” the inverse relationship of a light pulse's duration to its bandwidth. I argue that harmonic dispersion renders reflectance defined in terms of light pulses an extrinsic disposition. Reflectance defined as the per-wavelength efficiency to reflect the superimposed, infinite-duration, (...) Fourier harmonics of pulses can be an intrinsic disposition of surfaces. This conclusion raises questions about mathematical realism, about which I nevertheless remain neutral. (shrink)

This article offers a limited defense of metaphysical “infinitism,” the view that there are, or might be, infinite chains of ontological dependence. According to a widespread presupposition, there must be an ultimate ground of being—most likely, a plurality of fundamental atoms. Contrary to this view, this article shows that metaphysical infinitism is internally coherent. In particular, a parallel with the debate concerning infinitism about epistemic justification is suggested, and an “emergence model” of being is put forward. According to the emergence (...) model, the being of any given entity gradually arises out of an infinite series of progressively less dependent entities—it is not wholly transmitted, as it were, from a basic, ungrounded level to all the dependent ones in a step-by-step fashion. Some objections are considered and rebutted. (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)

The Enhanced Indispensability Argument (Baker [ 2009 ]) exemplifies the new wave of the indispensability argument for mathematical Platonism. The new wave capitalizes on mathematics' role in scientific explanations. I will criticize some analyses of mathematics' explanatory function. In turn, I will emphasize the representational role of mathematics, and argue that the debate would significantly benefit from acknowledging this alternative viewpoint to mathematics' contribution to scientific explanations and knowledge.

Aidan Lyon has recently argued that some mathematical explanations of empirical facts can be understood as program explanations. I present three objections to his argument.

Our aim in this paper is to evaluate Frank Jackson and Philip Pettit’s ‘program explanation’ framework as an account of the autonomy of the special sciences. We argue that this framework can only explain the autonomy of a limited range of special science explanations. The reason for this limitation is that the framework overlooks a distinction between two kinds of properties, which we refer to as ‘higher-level’ and ‘higher-order’ properties. The program explanation framework can account for the autonomy of special (...) science explanations that appeal to higher-level properties but it does not account for the autonomy of most of those explanations that appeal to higher-order properties. (shrink)

I argue on the basis of an example, Fourier theory applied to the problem of vibration, that Field's program for nominalizing science is unlikely to succeed generally, since no nominalistic variant will provide us with the kind of physical insight into the phenomena that the standard theory supplies. Consideration of the same example also shows, I argue, that some of the motivation for mathematical fictionalism, particularly the alleged problem of cognitive access, is more apparent than real.

It is often held that identity properties like the property of being identical to Paris are intrinsic. It is also often held that, while some logically uninstantiable properties are intrinsic, some logically uninstantiable properties are non-intrinsic. The combination of these views, however, raises a problem, since virtually every existing account of intrinsicality fails to analyse a notion of intrinsicality on which both these views are true. In this paper, I argue that, given the orthodox theory of counterlogicals, there is no (...) notion of intrinsicality on which both these views are true. If this argument is sucessful, then, at least given orthodoxy about counterlogicals, we face no challenge in analysing a notion of intrinsicality on which both these views are true, since there is no such notion. In addition, if we are also convinced that there is a notion of intrinsicality on which one of these views is true and a notion of intrinsicality on which the other of these views is true, we should hold that there is more than one notion of intrinsicality. (shrink)

This book includes detailed critical analysis of a wide variety of versions of the indispensability argument, as well as a novel approach to traditional views about mathematics.

Robert Koons and Matthew O’Brien have leveled a number of objections against the New Natural Law account of human action and intention. In this paper, I discuss five areas in which I believe that the Koons-O’Brien criticism of the New Natural Law theory is mistaken, or in which their own view is problematic. I hope to show, inter alia, that the New Natural Law approach is not committed to a number of theses attributed to it by Koons and O’Brien; that (...) their own view suffers from many ambiguities and difficulties; that passages from St. Thomas on which they draw to support their own view are in fact fully compatible with the New Natural Law account; and that neither the New Natural Law account of the controversial Phoenix abortion case, nor their account of the casuistry surrounding the acceptance of side-effects, is deficient in the ways asserted by Koons and O’Brien. (shrink)