Mathematical models provide explanations of limited power of specific aspects of phenomena. One way of articulating their limits here, without denying their essential powers, is in terms of contrastive explanation.

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.

According to a causal model of explanation, we explain phenomena by giving their causes or, where the phenomena are themselves causal regularities, we explain them by giving a mechanism linking cause and effect. If we explain why smoking causes cancer, we do not give the cause of this causal connection, but we do give the causal mechanism that makes it. The claim that to explain is to give a cause is not only natural and plausible, but it also avoids many (...) of the objections to other accounts of explanation, such as the views that to explain is to give a reason to believe the phenomenon occurred, to somehow make the phenomenon familiar, or to give a Deductive-Nomological argument. Unlike the reason for belief account, a causal model makes a clear distinction between understanding why a phenomenon occurs and merely knowing that it does, and the model does so in a way that makes understanding unmysterious and objective. Understanding is not some sort of super-knowledge, but simply more knowledge: knowledge of the phenomenon and knowledge of its causal history. A causal model makes it clear how something can explain without itself being explained, and so avoids the regress of whys, since we can know a phenomenon's cause without knowing the cause of the cause. It also accounts for legitimate self-evidencing explanations, explanations where the phenomenon is an essential part of the evidence that the explanation is correct, so the explanation can not supply a non-circular reason for believing the phenomenon occurred. There is no barrier to knowing a cause through its effects and also knowing that it is their cause. The speed of recession of a star explains its observed red-shift, even though the shift is an essential part of the evidence for its speed of recession. The model also avoids the most serious objection to the familiarity view, which is that some phenomena are familiar yet not understood, since a phenomenon can be perfectly familiar, such as the blueness of the sky or the fact that the same side of the moon always faces the earth, even if we do not know its cause. Finally, a causal model avoids many of the objections to the Deductive-Nomological model. Ordinary explanations do not have to meet the requirements of the Deductive-Nomological model, because one does not need to give a law to give a cause, and one does not need to know a law to have good reason to believe that a cause is a cause. As for the notorious over-permissiveness of the Deductive-Nomological model, the reason recession explains red-shift but not conversely, is simply that causes explain effects but not conversely, and the reason a conjunction of laws does not explain its conjuncts is that conjunctions do not cause their conjuncts. (shrink)

A new edition of this well-established monograph, this volume provides a comprehensive overview over the still fascinating field of chaos research. The authors include recent developments such as systems with restricted degrees of freedom but put also a strong emphasis on the mathematical foundations. Partly illustrated in color, this fourth edition features new sections from applied nonlinear science, like control of chaos, synchronisation of nonlinear systems, and turbulence, as well as recent theoretical concepts like strange nonchaotic attractors, on-off intermittency and (...) spatio-temporal chaotic motion. (shrink)

Over the past two decades, academic economics has undergone a mild revolution in methodology. The language, concepts and techniques of noncooperative game theory have become central to the discipline. This book provides the reader with some basic concepts from noncooperative theory, and then goes on to explore the strengths, weaknesses, and future of the theory as a tool of economic modelling and analysis. The central theses are that noncooperative game theory has been a remarkably popular tool in economics over the (...) past decade because it allows analysts to capture essential features of dynamic competition and competition where some parties have proprietary information. The theory is weakest in providing a sense of when it - and equilibrium analysis in particular - can be applied and what to do when equilibrium analysis is inappropriate. Many of these weaknesses can be addressed by the consideration of individuals who are boundedly rational and learn imperfectly from the past. Written in a non-technical style and working by analogy, the book, first given as part of the Clarendon Lectures in Economics, is readily accessible to a broad audience and will be of interest to economists and students alike. Knowledge of game theory is not required as the concepts are developed as the book progresses. (shrink)