Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

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)

It is proposed that we use the term “approximation” for inexact description of a target system and “idealization” for another system whose properties also provide an inexact description of the target system. Since systems generated by a limiting process can often have quite unexpected, even inconsistent properties, familiar limit systems used in statistical physics can fail to provide idealizations, but are merely approximations. A dominance argument suggests that the limiting idealizations of statistical physics should be demoted to approximations.

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)

In this paper, the issues of computability and constructivity in the mathematics of physics are discussed. The sorts of questions to be addressed are those which might be expressed, roughly, as: Are the mathematical foundations of our current theories unavoidably non-constructive: or, Are the laws of physics computable?

It is commonly believed that there is no equivalent of the Church–Turing thesis for computation over the reals. In particular, computational models on this domain do not exhibit the convergence of formalisms that supports this thesis in the case of integer computation. In the light of recent philosophical developments on the different meanings of the Church–Turing thesis, and recent technical results on analog computation, I will show that this current belief confounds two distinct issues, namely the extension of the notion (...) of effective computation to the reals on the one hand, and the simulation of analog computers by Turing machines on the other hand. I will argue that it is possible in both cases to defend an equivalent of the Church–Turing thesis over the reals. Along the way, we will learn some methodological caveats on the comparison of different computational models, and how to make it meaningful. (shrink)

We discuss the legacy of Alan Turing and his impact on computability and analysis.

This dissertation is an investigation into the degree to which the mathematics used in physical theories can be constructivized. The techniques of recursive function theory and classical logic are used to separate out the algorithmic content of mathematical theories rather than attempting to reformulate them in terms of "intuitionistic" logic. The guiding question is: are there experimentally testable predictions in physics which are not computable from the data? ;The nature of Church's thesis, that the class of effectively calculable functions on (...) the natural numbers is identical to the class of general recursive functions, is discussed. It is argued that this thesis is an example of an explication of the very notion of an effectively calculable function. This is contrary to a view of the thesis as a hypothesis about the limitations of the human mind. ;The extension to functions of a real variable of the notion of effective calculability is discussed, and it is argued that a function of a real variable must be continuous in order to be considered effectively calculable . The relation between continuity and computability is significant for the problem at hand. The results of a well-designed experiment do not depend critically upon the precise values of the relevant parameters. Accordingly, if the solution to a problem in mathematical physics depends discontinuously upon the data, it cannot be regarded as an experimentally testable prediction of the theory. The principle that the testable predictions of a physical theory cannot be singular is known as the principle of regularity. This principle is significant, because discontinuities generate non-computability, but they also disqualify a prediction from being experimentally testable. ;A mathematical framework is set up for discussing computability in physical theories. This framework is then applied to the case of quantum mechanics. It is found that, due to the use of unbounded operators in the theory, noncomputable objects appear, but predictions which satisfy the principle of regularity are nevertheless computable functions of the data. (shrink)

This dissertation examines aspects of the interplay between computing and scientific practice. The appropriate foundational framework for such an endeavour is rather real computability than the classical computability theory. This is so because physical sciences, engineering, and applied mathematics mostly employ functions defined in continuous domains. But, contrary to the case of computation over natural numbers, there is no universally accepted framework for real computation; rather, there are two incompatible approaches --computable analysis and BSS model--, both claiming to formalise algorithmic (...) computation and to offer foundations for scientific computing. -/- The dissertation consists of three parts. In the first part, we examine what notion of 'algorithmic computation' underlies each approach and how it is respectively formalised. It is argued that the very existence of the two rival frameworks indicates that 'algorithm' is not one unique concept in mathematics, but it is used in more than one way. We test this hypothesis for consistency with mathematical practice as well as with key foundational works that aim to define the term. As a result, new connections between certain subfields of mathematics and computer science are drawn, and a distinction between 'algorithms' and 'effective procedures' is proposed. -/- In the second part, we focus on the second goal of the two rival approaches to real computation; namely, to provide foundations for scientific computing. We examine both frameworks in detail, what idealisations they employ, and how they relate to floating-point arithmetic systems used in real computers. We explore limitations and advantages of both frameworks, and answer questions about which one is preferable for computational modelling and which one for addressing general computability issues. -/- In the third part, analog computing and its relation to analogue (physical) modelling in science are investigated. Based on some paradigmatic cases of the former, a certain view about the nature of computation is defended, and the indispensable role of representation in it is emphasized and accounted for. We also propose a novel account of the distinction between analog and digital computation and, based on it, we compare analog computational modelling to physical modelling. It is concluded that the two practices, despite their apparent similarities, are orthogonal. (shrink)