无穷小量的命运及对数学发展动力的思考

近代力学的需要催生了无穷小量.历史上的数学家对无穷小量作了各种解读,试图论证它存在的合法性,以使微积分严密化,都没有成功.在19世纪末的数学公理化运动中,极限取代了无穷小量成为微积分的基础.无穷小量的曲折历史使人们认识到:数学有效性并不必须由其真理性来保证,数学家的信仰是数学发展的精神动力.近代数学家经历了从信仰上帝到信仰自然再到信仰数学内部的逻辑美的过程,数学家的任务也经历了从解决上帝、自然和数学三者之间的矛盾到解决自然和数学两者之间的矛盾,再到解决数学内部的矛盾过程.哥德尔不完备性定理抽走了数学家的逻辑美信仰,数学界出现了信仰危机.

The Fate of Infinitesimal and Ponderation on Mathematical Motive Force

The need of modern mechanics brought about infinitesimal. Mathematicians in history, who gave different explanations on infinitesimal, failed to verify its legal status to define it well. The limit became the basis of differential in place of infinitesimal in the movement of 19th mathematical axiom. In the winding history of infinitesimal, we have realized that mathematical effectiveness doesn't have to depend on its truth, and that mathematicians' belief is the spiritual motive force of mathematical development.Modern mathematicians have changed their beliefs from having belief in god to nature, and then to logic beauty. They also altered their tasks in the same time, from resolving the contradictions among god, nature, and mathematics to the contradictions between nature and mathematics, and then to mathematical inner contradictions. Godel theorem made mathematicians lose their belief of logic beauty, the crisis of belief appeared in mathematical field.