From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti

Masanao Ozawa

doi:10.3390/math9040397

Abstract

Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory that is based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. As quantum logic is intrinsic and empirical, the results of the quantum set theory can be experimentally verified by quantum mechanics. In this paper, we analyze Takeuti’s mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic. We outlook the present status of his program, and envisage the further development of the program, by which we would be able to take a huge step forward toward unraveling the mysteries of quantum mechanics that have persisted for many years.

Highlights

In 1982, Gaisi Takeuti published a book entitled Mathematical Worldview: Ideas and Prospects of Modern Mathematics [1]. He states that mathematics based on classical logic is an absolute truth, and it has the characteristic that all possible statements always belong to only one of the two, either true or false, and yet in the future, the progress of human culture may lead to the birth of a “new mathematics” that is based on a “new logic”
This was advocated by Brouwer as one of the positions of mathematics, but Takeuti has proposed a “new mathematics” that should be called “quantum mathematics” based on quantum logic
“Quantum logic” is the logic of quantum mechanics first discovered by Birkhoff and von Neumann [2] in 1936

Summary

Introduction

In 1982, Gaisi Takeuti published a book entitled Mathematical Worldview: Ideas and Prospects of Modern Mathematics [1] In this book, he states that mathematics based on classical logic is an absolute truth, and it has the characteristic that all possible statements always belong to only one of the two, either true or false, and yet in the future, the progress of human culture may lead to the birth of a “new mathematics” that is based on a “new logic”. In intuitionistic logic, the distributive law holds, but the law of excluded middle, the law of double negation, and De Morgan’s law do not hold In this sense, quantum logic is a logic that contrasts with intuitionistic logic, and Takeuti compared the relationship between classical logic, intuitionistic logic, and quantum logic to “God’s logic”, “human logic”, and “the logic of things”.

Set Theoretical Worldview

Modern Mathematics and ZFC Set Theory

Set Theory Based on Multi-Valued Logic

Boolean Valued Analysis

Quantum Set Theory