Online research in philosophy

Most contemporary philosophy of mathematics focuses on a small segment of mathematics, mainly the natural numbers and foundational disciplines like set theory. While there are good reasons for this approach, in this paper I will examine the philosophical problems associated with the area of mathematics known as applied mathematics. Here mathematicians pursue mathematical theories that are closely connected to the use of mathematics in the sciences and engineering. This area of mathematics seems to proceed using different methods and standards when compared to much of mathematics. I argue that applied mathematics can contribute to the philosophy of mathematics and our understanding of mathematics as a whole.

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 of that 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.

Fictionalism in the philosophy of mathematics is the view that mathematical statements, such as ‘8+5=13’ and ‘π is irrational’, are to be interpreted at face value and, thus interpreted, are false. Fictionalists are typically driven to reject the truth of such mathematical statements because these statements imply the existence of mathematical entities, and according to fictionalists there are no such entities. Fictionalism is a nominalist (or anti-realist) account of mathematics in that it denies the existence of a realm of abstract mathematical entities. It should be contrasted with mathematical realism (or Platonism) where mathematical statements are taken to be true, and, moreover, are taken to be truths about mathematical entities. Fictionalism should also be contrasted with other nominalist philosophical accounts of mathematics that propose a reinterpretation of mathematical statements, according to which the statements in question are true but no longer about mathematical entities. Fictionalism is thus an error theory of mathematical discourse: at face value mathematical discourse commits us to mathematical entities and although we normally take many of the statements of this discourse to be true, in doing so we are in error (cf. error theories in ethics).

The theoretical physicist Paul Dirac rejected, explicitly on aesthetic grounds, a successful theory known as quantum electrodynamics (QED), which is the prototype for the family of theories known as quantum field theories (QFTs). Remarkably, the theoretical physicist Steven Weinberg, also largely on aesthetic grounds, supports QED and other QFTs. In order to evaluate these opposing aesthetic views a short introduction to the physical properties of QFTs is presented together with a detailed analysis of the aesthetic claims of Dirac and Weinberg. It turns out that Dirac rejected QED, without regard to its success, because this theory fails to yield to what he perceived as beautiful mathematics, whereas Weinberg's support of QFTs is founded primarily on the physical concepts of the theories. In particular, he relies on symmetries that are the basis for the construction of the extremely successful current fundamental theories of particles physics. This success was decisive in leading to Weinberg's conviction of the beauty of QFTs. As a result of the evaluation of these approaches, the factors causing scientists to perceive a theory as being a fundamentally beautiful theory are discussed in detail.

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. Introduction: Mathematical Explanation Indispensability and Explanation Is the Mathematics Indispensable to the Explanation? 3.1 Object-level arbitrariness 3.2 Concept-level arbitrariness 3.3 Theory-level arbitrariness Is the Explanandum ‘Purely Physical’? Is the Mathematics Explanatory in Its Own Right? Does Inference to the Best Explanation Apply to Mathematics? 6.1 Leng's first argument 6.2 Leng's second argument 6.3 Leng's third argument Conclusions CiteULike Connotea Del.icio.us What's this?

This paper considers the role of mathematics in the process of acquiring new knowledge in physics and astronomy. The defining of the notions of continuum and discreteness in mathematics and the natural sciences is examined. The basic forms of representing the heuristic function of mathematics at theoretical and empirical levels of knowledge are studied: deducing consequences from the axiomatic system of theory, the method of generating mathematical hypotheses, “pure” proofs for the existence of objects and processes, mathematical modelling, the formation of mathematics on the basis of internal mathematical principles and the mathematical theory of experiment.

Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the Bolzano-Weierstrass, Hahn-Banach, and intermediate value theorems, and then the implications of these arguments for such “crown jewels” of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser [51] in his extension of Gödel’s incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him.

This paper is devoted to examining the relevance of Dirac's view on the use of transformation theory and invariants in modern physics --- as it emerges from his 1930 book on quantum mechanics as well as from his later work on singular theories and constraints --- to current reflections on the meaning of physical symmetries, especially gauge symmetries.

: Although Dirac rarely participated in the interpretational debates over quantum theory, it is traditionally assumed that his views were aligned with Heisenberg and Bohr in the so-called Copenhagen-Göttingen camp. However, an unpublished—and apparently unknown—lecture of Dirac's reveals that this view is mistaken; in the famous debate between Einstein and Bohr, Dirac sided with Einstein. Surprisingly, Dirac believed that quantum mechanics was not complete, that the uncertainty principle would not survive in the future physics, and that a deterministic description of the microworld would be recovered. In this paper I show how we can make sense of this unpublished lecture in the context of Dirac's broader philosophy of quantum mechanics, and how our present understanding of Dirac's philosophical views must be revised.

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.