庞加莱定性思想的决定论意涵

The Implication of Poincaré’s Qualitative Thought on Determinism

常微分方程理论与决定论之间存在历史关联,二者相互影响。在经典数学物理中,决定论与微分方程的存在唯一性定理一致,随着微分方程理论和决定论不断取得发展,庞加莱的定性理论及其相关哲学思考开启了二者之间新的关系模式。在庞加莱之前,决定论或者采取线性的、形而上学的观点,或者并未受到同时代数学科学研究的实质性影响;而在庞加莱之后,微分方程理论成为讨论决定论问题不可或缺的思想资源,并为决定论的哲学讨论打开了新的视野。庞加莱的定性思想可被视为现代非完全决定论的起源。

There is a historical relationship between the theory of ordinary differential equations and determinism, and the two influence each other. In classical mathematical physics, determinism is consistent with the uniqueness theorem of differential equations, and with the continuous development of differential equation theory and determinism, Poincaré’s qualitative theory and related philosophical reflections have opened up a new relationship between the two. Before Poincaré, determinism either took a linear, metaphysical view or was not substantially influenced by the research of contemporary mathematical science;After Poincaré, the theory of differential equations became an indispensable intellectual resource for discussing deterministic problems and opened new horizons for the philosophical discussion of determinism. Poincaré’s qualitative ideas can be seen as the origin of modern incomplete determinism.