This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.A unique feature of the book is its emphasis on applications. These include (...) mechanical vibrations, lasers, biological rhythms, superconducting circuits, insect outbreaks, chemical oscillators, genetic control systems, chaotic waterwheels, and even a technique for using chaos to send secret messages. In each case, the scientific background is explained at an elementary level and closely integrated with the mathematical theory.Richly illustrated, and with many exercises and worked examples, this book is ideal for an introductory course at the junior/senior or first-year graduate level. It is also ideal for the scientist who has not had formal instruction in nonlinear dynamics, but who now desires to begin informal study. The prerequisites are multivariable calculus and introductory physics. (shrink)

This paper examines explanations that turn on non-local geometrical facts about the space of possible configurations a system can occupy. I argue that it makes sense to contrast such explanations from ‘geometry of motion’ with causal explanations. I also explore how my analysis of these explanations cuts across the distinction between kinematics and dynamics.

The goal of this paper is to develop a counterfactual theory of explanation. The CTE provides a monist framework for causal and non-causal explanations, according to which both causal and non-causal explanations are explanatory by virtue of revealing counterfactual dependencies between the explanandum and the explanans. I argue that the CTE is applicable to two paradigmatic examples of non-causal explanations: Euler’s explanation and renormalization group explanations of universality.

This paper examines explanations that turn on non-local geometrical facts about the space of possible configurations a system can occupy. I argue that it makes sense to contrast such explanations from "geometry of motion" with causal explanations. I also explore how my analysis of these explanations cuts across the distinction between kinematics and dynamics.

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)

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)

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)

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)

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

David Malament argued that Hartry Field's nominalisation program is unlikely to be able to deal with non-space-time theories such as phase-space theories. We give a specific example of such a phase-space theory and argue that this presentation of the theory delivers explanations that are not available in the classical presentation of the theory. This suggests that even if phase-space theories can be nominalised, the resulting theory will not have the explanatory power of the original. Phase-space theories thus raise problems for (...) nominalists that go beyond Malament's initial concerns. Thanks to Mark Steiner, Jens Christian Bjerring, Ben Fraser, John Mathewson, and two anonymous referees for helpful comments on an earlier draft of this paper. CiteULike Connotea Del.icio.us What's this? (shrink)

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)

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)

In this paper, I discuss the prospects of nominalistic scientific realism and show that it fails on many counts. In section 2, I discuss what is required for NSR to get off the ground. In section 3, I question the idea that theories have well-defined nominalistic content and the idea that causal activity is a necessary condition for commitment to the reality of an entity. In section 4, I challenge the notion of nominalistic adequacy of theories.

This book is concerned with `the problem of existence in mathematics'. It develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. It explores the philosophical implications of such an approach through an examination of the writings of Field, Burgess, Maddy, Kitcher, and others.

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.

The big news about chaos is supposed to be that the smallest of changes in a system can result in very large differences in that system's behavior. The so-called butterfly effect has become one of the most popular images of chaos. The idea is that the flapping of a butterfly's wings in Argentina could cause a tornado in Texas three weeks later. By contrast, in an identical copy of the world sans the Argentinian butterfly, no such storm would have arisen (...) in Texas. The mathematical version of this property is known as sensitive dependence. However, it turns out that sensitive dependence is somewhat old news, so some of the implications flowing from it are perhaps not such “big news” after all. Still, chaos studies have highlighted these implications in fresh ways and led to thinking about other implications as well. -/- In addition to exhibiting sensitive dependence, chaotic systems possess two other properties: they are deterministic and nonlinear (Smith 2007). This entry discusses systems exhibiting these three properties and what their philosophical implications might be for theories and theoretical understanding, confirmation, explanation, realism, determinism, free will and consciousness, and human and divine action. (shrink)

In his recent book, Explaining Chaos, Peter Smith presents a new problem in the foundations of chaos theory. Specifically, he argues that the standard ways of justifying idealizations in mathematical models fail when it comes to the infinite intricacy found in strange attractors. I argue that Smith's analysis undermines much of the explanatory power of chaos theory. A better approach is developed by drawing analogies from the models found in continuum mechanics.

A natural way to think of models is as abstract entities. If theories employ models to represent the world, theories traffic in abstract entities much more widely than is often assumed. This kind of thought seems to create a problem for a scientific realist approach to theories. Scientific realists claim theories should be understood literally. Do they then imply the reality of abstract entities? Or are theories simply—and incurably—false? Or has the very idea of literal understanding to be abandoned? Is (...) then fictionalism towards scientific theories inevitable? This paper argues that scientific realism can happily co-exist with models qua abstracta. (shrink)

We reassess Woodward’s counterfactual account of explanation in relation to regularity explananda. Woodward presents an account of causal explanation. We argue, by using an explanation of Kleiber’s law to illustrate, that the account can also cover some noncausal explanations. This leads to a tension between the two key aspects of Woodward’s account: the counterfactual aspect and the causal aspect. We explore this tension and make a case for jettisoning the causal aspect as constitutive of explanatory power in connection with regularity (...) explananda. (shrink)