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Cambridge University Press eBooks

Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem

Geoffrey Paul Hellman

Published Jan 26, 2021• 0 citations
Publisher

Abstract

A major outstanding issue in the foundations and philosophy of mathematics concerns the indispensability of classical infinitistic mathematics for the empirical sciences. Claims of such indispensability form a modern cornerstone of mathematical platonism and alternative classical realist conceptions as well (e.g. modal structuralism) (cf. e.g., Quine [1953], Putnam [1967, 1971], and Hellman [1989a]). This has posed a corresponding challenge to constructivist views (intuitionistic, Bishop-constructivist [Bishop 1967], and related approaches). How much of the mathematics actually employed in the empirical sciences, especially physics, can be carried out constructively (in the various relevant senses)? It is probably no exaggeration to say that the viability of a constructivist philosophy of mathematics is here at stake (cf. Burgess [1984]).

DOI
10.1017/9781108657419.014

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A major outstanding issue in the foundations and philosophy of mathematics concerns the indispensability of classical infinitistic mathematics for the empirical sciences. Claims of such indispensability form a modern cornerstone of mathematical platonism and alternative classical realist conceptions as well (e.g. modal structuralism) (cf. e.g., Quine [1953], Putnam [1967, 1971], and Hellman [1989a]). This has posed a corresponding challenge to constructivist views (intuitionistic, Bishop-constructivist [Bishop 1967], and related approaches). How much of the mathematics actually employed in the empirical sciences, especially physics, can be carried out constructively (in the various relevant senses)? It is probably no exaggeration to say that the viability of a constructivist philosophy of mathematics is here at stake (cf. Burgess [1984]).
The article is devoted to the problem of time in modern science, where in recent years there have been major changes related to the latest discoveries in the field of the foundations of quantum theory. The author refers to works of K.-F. von Weizsacker (which works are not well-known in Russian-speaking field). Weizsacker deploys a large-scale program of building modern physics, while starting (not only as a physicist, but also a professional philosopher) with questions of philosophical interpretation of postulates of modern physics, especially quantum mechanics. His key thesis is that time in physics is fundamentally distinct, which represents the whole physics as an integral scientific discipline. Weizsacker comes from physical and philosophical reflections, that give a special value to his work. Analyzing the conclusions of theoretical physics, namely the issues of substantiation of statistical mechanics, thermodynamics, and, above all, quantum mechanics, Weizsacker comes to the key idea of the structure of time in the substantiation of physical theory. The author in this conclusion disagrees with the opinion of most modern physicists that time has a subjective character, and the modus of the past, present and future reflects its essential character. The allocation of the time structure requires special logic. Such a logic is temporal and quantum. Starting from temporal logic, introducing the concept of possibility and probability, he comes to the substantiation of quantum mechanics. Being modal logic, it is based on the concept of possibility and probability. Key concept for Weizsacker is a binary alternative (Uralternative), closely related to the principle of superposition in quantum mechanics. By introducing the concept of certain alternatives, essentially actualized alternatives, Weizsacker shows how one can get the theory of relativity with its space-time structure. Further, he shows that based on its structure, you can get the theory of relativity. The space-time structure in this case is secondary, relational, as understood by Leibniz, Mach, and Einstein.
The relationship between physical notions and the structures expressing them is an issue of primary importance the history and philosophy of pre-modern natural philosophy and of modern science. Historical evidence suggests that, when investigating the relationship between physics and mathematics, one should pay attention to the material, sensually perceivable forms mediating it: spoken or written words, drawings, artefacts, descriptions of experimental set-ups or, as the case discussed the present paper, symbolic notation. The reason for this is that, historically, physical notions and their expressions have very often been developed as one, and only distinguished from each other at a later stage. This was for example the case with what we today regard as mathematical and classical mechanics, which were developed the course of the eighteenth century as the discipline of the analysis of motion. When considered from a later point of view, both the physical notions and the concepts involved historical developments appear as being in flux, vague, and, as far as the side is concerned, non-rigorous. In such cases, verbal description, mechanical artefacts or symbols do not simply mediate between physical notions and structures, as might be the case at a later stage, but represent both of them one, allowing the historical actors to construct and manipulate physical-mathematical meaning by following, at times, their physical reasoning and, at times, their criteria of rigour. The history of quantum mechanics offers many examples of how new physics and new mathematics can emerge as one, with symbolic notation playing a central role shaping both. Among the many creators of quantum theory, Paul Dirac was particularly sensitive to questions of notation: