Most scientific models are not physical objects, and this raises important questions. What sort of entity are models, what is truth in a model, and how do we learn about models? In this paper I argue that models share important aspects in common with literary fiction, and that therefore theories of fiction can be brought to bear on these questions. In particular, I argue that the pretence theory as developed by Walton (1990, Mimesis as make-believe: on the foundations of the (...) representational arts. Harvard University Press, Cambridge/MA) has the resources to answer these questions. I introduce this account, outline the answers that it offers, and develop a general picture of scientific modelling based on it. (shrink)

In the mathematical documents which have come down to us from these peoples, there are no theorems or demonstrations, and the fundamental concepts of ...

Mimesis as Make-Believe is important reading for everyone interested in the workings of representational art.

The descriptions and theoretical laws scientists write down when they model a system are often false of any real system. And yet we commonly talk as if there were objects that satisfy the scientists’ assumptions and as if we may learn about their properties. Many attempt to make sense of this by taking the scientists’ descriptions and theoretical laws to define abstract or fictional entities. In this paper, I propose an alternative account of theoretical modelling that draws upon Kendall Walton’s (...) ‘make-believe’ theory of representation in art. I argue that this account allows us to understand theoretical modelling without positing any object of which scientists’ modelling assumptions are true. (shrink)

How do models represent reality? There are two conditions that scientific models must satisfy to be representations of real systems, the aboutness condition and the epistemic condition. In this article, I critically assess the two main fictionalist theories of models as representations, the indirect fiction view and the direct fiction view, with respect to these conditions. And I develop a novel proposal, what I call ‘the new fiction view of models’. On this view, models are akin to fictional stories; they (...) represent real-world phenomena if they stand in a denotation relation with reality; and they enable knowledge of reality via the generation of theoretical hypotheses, model–world comparisons and direct attributions. (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.

In “What Makes a Scientific Explanation Distinctively Mathematical?” (2013b), Lange uses several compelling examples to argue that certain explanations for natural phenomena appeal primarily to mathematical, rather than natural, facts. In such explanations, the core explanatory facts are modally stronger than facts about causation, regularity, and other natural relations. We show that Lange's account of distinctively mathematical explanation is flawed in that it fails to account for the implicit directionality in each of his examples. This inadequacy is remediable in each (...) case by appeal to ontic facts that account for why the explanation is acceptable in one direction and unacceptable in the other direction. The mathematics involved in these examples cannot play this crucial normative role. While Lange's examples fail to demonstrate the existence of distinctively mathematical explanations, they help to emphasize that many superficially natural scientific explanations rely for their explanatory force on relations of stronger-than-natural necessity. These are not opposing kinds of scientific explanations; they are different aspects of scientific explanation. (shrink)

Conflicting accounts of the role of mathematics in our physical theories can be traced to two principles. Mathematics appears to be both (1) theoretically indispensable, as we have no acceptable non-mathematical versions of our theories, and (2) metaphysically dispensable, as mathematical entities, if they existed, would lack a relevant causal role in the physical world. I offer a new account of a role for mathematics in the physical sciences that emphasizes the epistemic benefits of having mathematics around when we do (...) science. This account successfully reconciles theoretical indispensability and metaphysical dispensability and has important consequences for both advocates and critics of indispensability arguments for platonism about mathematics. (shrink)

Mark Colyvan uses applications of mathematics to argue that mathematical entities exist. I claim that his argument is invalid based on the assumption that a certain way of thinking about applications, called `the mapping account,' is correct. My main contention is that successful applications depend only on there being appropriate structural relations between physical situations and the mathematical domain. As a variety of non-realist interpretations of mathematics deliver these structural relations, indispensability arguments are invalid.

According to the indispensability argument, the fact that we quantify over numbers, sets and functions in our best scientific theories gives us reason for believing that such objects exist. I examine a strategy to dispense with such quantification by simply replacing any given platonistic theory by the set of sentences in the nominalist vocabulary it logically entails. I argue that, as a strategy, this response fails: for there is no guarantee that the nominalist world that go beyond the set of (...) sentences in the nominalist language such theories entail. However, I argue that what such theories show is that mathematics can enable us to express possibilities about the concrete world that may not be expressible in nominalistically acceptable language. While I grant that this may make quantification over abstracta indispensable, I deny that such indispensability is a reason for accepting them into our ontology. I urge that the nominalist should be allowed to quantify over abstracta whilst denying their existence and I explain how this apparently contradictory practice (a practice I call 'weaseling') is in fact coherent, unproblematic and rational. Finally, I examine the view that platonistic theories are simpler or more attractive than their nominalistic reformulations, and thus that abstract ought to be accepted into our ontology for the same sorts of reasons as other theoretical objects. I argue that, at least in the case of numbers, functions and sets, such arguments misunderstand the kind of simplicity and attractiveness we seek. (shrink)

This discussion note responds to Mark Colyvan’s claim that there is no easy road to nominalism. While Colyvan is right to note that the existence of mathematical explanations presents a more serious challenge to nominalists than is often thought, it is argued that nominalist accounts do have the resources to account for the existence of mathematical explanations whose explanatory role resides elsewhere than in their nominalistic content.

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out (...) relative to such domains; puzzles about ‘large categories’ and ‘proper classes’ are handled in a uniform way, by relativization, sustaining insights of Zermelo. (shrink)

Non-actual model systems discussed in scientific theories are compared to fictions in literature. This comparison may help with the understanding of similarity relations between models and real-world target systems. The ontological problems surrounding fictions in science may be particularly difficult, however. A comparison is also made to ontological problems that arise in the philosophy of mathematics.

Not all scientific explanations work by describing causal connections between events or the world's overall causal structure. In addition, mathematicians regard some proofs as explaining why the theorems being proved do in fact hold. This book proposes new philosophical accounts of many kinds of non-causal explanations in science and mathematics.

Wigner famously referred to the `unreasonable effectiveness' of mathematics in its application to science. Using Wigner's own application of group theory to nuclear physics, I hope to indicate that this effectiveness can be seen to be not so unreasonable if attention is paid to the various idealising moves undertaken. The overall framework for analysing this relationship between mathematics and physics is that of da Costa's partial structures programme.

We outline Ladyman's 'metaphysical' or 'ontic' form of structuralrealism and defend it against various objections. Cao, in particular, has questioned theview of ontology presupposed by this approach and we argue that by reconceptualisingobjects in structural terms it offers the best hope for the realist in thecontext of modern physics.

We defend Joseph Melia's thesis that the role of mathematics in scientific theory is to 'index' quantities, and that even if mathematics is indispensable to scientific explanations of concrete phenomena, it does not explain any of those phenomena. This thesis is defended against objections by Mark Colyvan and Alan Baker.

In this sequence of philosophical essays about natural science, the author argues that fundamental explanatory laws, the deepest and most admired successes of modern physics, do not in fact describe regularities that exist in nature. Cartwright draws from many real-life examples to propound a novel distinction: that theoretical entities, and the complex and localized laws that describe them, can be interpreted realistically, but the simple unifying laws of basic theory cannot.

We examine, from the partial structures perspective, two forms of applicability of mathematics: at the “bottom” level, the applicability of theoretical structures to the “appearances”, and at the “top” level, the applicability of mathematical to physical theories. We argue that, to accommodate these two forms of applicability, the partial structures approach needs to be extended to include a notion of “partial homomorphism”. As a case study, we present London's analysis of the superfluid behavior of liquid helium in terms of Bose‐Einstein (...) statistics. This involved both the introduction of group theory at the top level, and some modeling at the “phenomenological” level, and thus provides a nice example of the relationships we are interested in. We conclude with a discussion of the “autonomy” of London's model. (shrink)

A number of people have recently argued for a structural approach to accounting for the applications of mathematics. Such an approach has been called "the mapping account". According to this view, the applicability of mathematics is fully accounted for by appreciating the relevant structural similarities between the empirical system under study and the mathematics used in the investigation ofthat system. This account of applications requires the truth of applied mathematical assertions, but it does not require the existence of mathematical objects. (...) In this paper, we discuss the shortcomings of this account, and show how these shortcomings can be overcome by a broader view of the application of mathematics: the inferential conception. (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)

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)

From the beginning, the implications of quantum theory for our most general understanding of the world have been a matter of intense debate. Einstein argues that the theory had to be regarded as fundamentally incomplete. Its inability, for example, to predict the exact time of decay of a single radioactive atom had to be due to a failure of the theory and not due to a permanent inability on our part or a fundamental indeterminism in nature itself. In 1964, John (...) Bell derived a theorem which showed that any deterministic theory which preserved locality would have certain consequences for measurements performed at a distance from one another. An experimental check seems to show that these consequences are not, in fact, realized. The correlation between the sets of events is much stronger than any local deterministic theory could allow. What is more, this stronger correlation is precisely that which is predicted by quantum theory. The astonishing result is that local deterministic theories of the classical sort seem to be permanently excluded. Not only can the individual decay not be predicted, but no future theory can ever predict it. The contributors in this volume wrestle with this conclusion. Some welcome it; others leave open a return to at lease some kind of deterministic world, one which must however allow something like action-at-a distance. How much lit it? And how can one avoid violating relativity theory, which excludes action-at-a-distance? How can a clash between the two fundamental theories of modern physics, relativity and quantum theory, be avoided? What are the consequences for the traditional philosophic issue of causality explanation and objectivity? One thing is certain; we can never return to the comfortable Newtonian world where everything that happened was, in principle, predictable and where what happened at one measurement site could not affect another set of measurements being performed light-years away, at a distance that a light-signal could not bridge. Contributors: James T. Cushing, Abner Shimony, N. David Mermin, Jon P. Jarrett, Linda Wessels, Bas C. van Fraassen, Jeremy Butterfield, Michael L. G. Redhead, Henry P. Stapp, Arthur Fine, R. I. G. Hughes, Paul Teller, Don Howard, Henry J. Folse, and Ernan McMullin. (shrink)

Thirteen promising young researchers write on what they take to be the right philosophical account of mathematics and discuss where the philosophy of mathematics ought to be going. New trends are revealed, such as an increasing attention to mathematical practice, a reassessment of the canon, and inspiration from philosophical logic.

This book offers a defence of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at (...) least approximately) true. But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true. Previous responses to the indispensability argument have focused on arguing that mathematical assumptions can be dispensed with in formulating our empirical theories. This book, by contrast, offers an account of the role of mathematics in empirical science according to which the successful use of mathematics in formulating our empirical theories need not rely on the truth of the mathematics utilized. (shrink)

This study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. Not only does (...) this engaging book present the Platonist-Naturalist debate over mathematics in a comprehensive fashion, but it also sheds considerable light on non-mathematical aspects of a dispute that is central to contemporary philosophy. (shrink)

Thirteen up-and-coming researchers in the philosophy of mathematics have been invited to write on what they take to be the right philosophical account of mathematics, examining along the way where they think the philosophy of mathematics is and ought to be going. A rich and diverse picture emerges. Some broader tendencies can nevertheless be detected: there is increasing attention to the practice, language and psychology of mathematics, a move to reassess the orthodoxy, as well as inspiration from philosophical logic.

Mathematics plays a central role in much of contemporary science, but philosophers have struggled to understand what this role is or how significant it might be for mathematics and science. In this book Christopher Pincock tackles this perennial question in a new way by asking how mathematics contributes to the success of our best scientific representations. In the first part of the book this question is posed and sharpened using a proposal for how we can determine the content of a (...) scientific representation. Several different sorts of contributions from mathematics are then articulated. Pincock argues that each contribution can be understood as broadly epistemic, so that what mathematics ultimately contributes to science is best connected with our scientific knowledge. In the second part of the book, Pincock critically evaluates alternative approaches to the role of mathematics in science. These include the potential benefits for scientific discovery and scientific explanation. A major focus of this part of the book is the indispensability argument for mathematical platonism. Using the results of part one, Pincock argues that this argument can at best support a weak form of realism about the truth-value of the statements of mathematics. The book concludes with a chapter on pure mathematics and the remaining options for making sense of its interpretation and epistemology. Thoroughly grounded in case studies drawn from scientific practice, this book aims to bring together current debates in both the philosophy of mathematics and the philosophy of science and to demonstrate the philosophical importance of applications of mathematics. (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.

What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions.

Classical mechanics and quantum mechanics are two of the most successful scientific theories ever discovered, and yet how they can describe the same world is far from clear: one theory is deterministic, the other indeterministic; one theory describes a world in which chaos is pervasive, the other a world in which chaos is absent. Focusing on the exciting field of 'quantum chaos', this book reveals that there is a subtle and complex relation between classical and quantum mechanics. It challenges the (...) received view that classical and quantum mechanics are incommensurable, and revives another, largely forgotten tradition due to Niels Bohr and Paul Dirac. By artfully weaving together considerations from the history of science, philosophy of science, and contemporary physics, this book offers a new way of thinking about intertheory relations and scientific explanation. It will be of particular interest to historians and philosophers of science, philosophically-inclined physicists, and interested non-specialists. (shrink)

Defends an account of mathematical knowledge in which mathematical knowledge is a kind of modal knowledge. Leng argues that nominalists should take mathematical knowledge to consist in knowledge of the consistency of mathematical axiomatic systems, and knowledge of what necessarily follows from those axioms. She defends this view against objections that modal knowledge requires knowledge of abstract objects, and argues that we should understand possibility and necessity in a primative way.

As Aristotle stated, scientific explanation is based on deductive argument--yet, Wesley C. Salmon points out, not all deductive arguments are qualified explanations. The validity of the explanation must itself be examined. _Four Decades of Scientific Explanation_ provides a comprehensive account of the developments in scientific explanation that transpired in the last four decades of the twentieth century. It continues to stand as the most comprehensive treatment of the writings on the subject during these years. Building on the historic 1948 essay (...) by Carl G. Hempel and Paul Oppenheim, "Studies in the Logic of Explanation,” which introduced the deductive-nomological model on which most work on scientific explanation was based for the following four decades, Salmon goes beyond this model's inherent basis of describing empirical knowledge to tells us “not only _what,_ but also _why_.” Salmon examines the predominant models in chronological order and describes their development, refinement, and criticism or rejection. _Four Decades of Scientific Explanation_ underscores the need for a consensus of approach and ongoing evaluations of methodology in scientific explanation, with the goal of providing a better understanding of natural phenomena. (shrink)

The orthodox, empiricist covering-law account of scientific explanation, as developed by C. G. Hempel and others, has long dominated philosophical discussions of scientific explanation. In recent years it has met overwhelming critical resistance. We should give up this account of scientific ex.