Structural analysis of infinite and conceptual analysis of dynamical system: from Newton and Leibniz to theory of operator algebra

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K02017
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Philosophy/Ethics
• Research Institution: Seisen University.
• Principal Investigator: HARADA Masaki 清泉女子大学, 付置研究所, 教授 (90453357)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 概念の哲学 / 概念史 / 作用素環論 / 非可換幾何学 / 無限次元

Outline of Final Research Achievements

While quantum mechanics has made visible the noncommutativity of physical quantity, problems concerning noncommutative infinite groups and measure theory, such as the Banach-Tarsky paradox, have been found. Von Neumann, who devised the theory of operator algebras aiming at providing a mathematical foundation for quantum mechanics, introduced a good property of groups, named amenability, which enables avoidance of the paradox. The theory of operator algebras was originally derived from functional analysis, but the dynamical systems such as ergodic theory, noncommutatively modified, join to it automorphic group and its normal subgroup, which originated in Galois theory. As a result, amenability has become better understood, as is deeply related to infinite dimensional operators that can be approximated by finite dimensional operators. This matter provides an interesting contribution to the philosophy of mathematics, which is related to the developement of concepts in the sciences.

Free Research Field

科学哲学

Academic Significance and Societal Importance of the Research Achievements

数学の哲学というと、その基礎付けを巡って論理学、集合論、そして最近では圏論について論じられることが多く、その中で、構造やパターンといった概念を用いながら数学的対象の実在性や本質が論じられる。しかし、数学の哲学は、実際に行われている数学や数学史と離れてしまっている。その中で、フランスの「概念の哲学」に連なる数学の概念史をたどりながら、そこに哲学的分析の鍬を入れる方法論は、近年、英米圏の哲学においても重要性を帯びてきている。本研究は、その方法論を作用素環論における無限次元の構造分析に適用したものである。このような方法による数学の哲学、そして科学哲学が実行されることが期待される。