数学における革命とはどういうものか? --トーマス・S・クーンの科学哲学の光のもとでみた数学的真理 (数学史の研究)

Abstract

In the early 1960s, Thomas S. Kuhn (1922~1996) proposed a novel view of science and developed his idea on the revolution in science. But, he deliberately avoided discussions about mathematics which was not his specialty. In 1975, Michael J. Crowe concluded that "Revolutions never occur in mathematics." The author was a graduate student of Prof. Kuhn at Princeton University during the late 1970s, and had a discussion with him on the revolution in mathematics. After the discussion with me, he began to express his opinion implying there must be revolutions in mathematics. In what follows, the author examines the origins of the fundamental theorem of the differential and integral calculus and pays attention to the fact that the algebraic mode played very crucial role in formulating the algorithmic form of the theorem. In his opinion, Leibniz's formulation was the most radical and most revolutionary, and Leibniz can be compared to Lavoisier in the Chemical Revolution. Mathematical truth is less constrained ontologically by the real natural world than the truth in natural sciences, and characterized to be conditional. Consequently, although mathematics has not so drastic transformations as natural sciences, there exist revolutionary changes in mathematics which has evolved historically, as well as in natural sciences. Changes in mathematics is either gradual or revolutionary, and both contentual and institutional. The author concludes that mathematical knowledge as an intellectual activity having a certain hermeneutic basis has experienced revolutions.