鲁宾逊诞辰日思考有感

    今天是鲁宾逊的生日，思考有感。

当今，人们普遍接受的无穷小定义是：

“An innitesimal is a number whose magnitude exceeds zero yet remains smaller than every nite, positive number.”

    由此可见，无穷小是一种“数”，而不是什么“函数”。这是鲁宾逊诞辰日的第一感觉。

   对这个模糊不清附观念给出严格数学数学处理是鲁宾逊的最大历史功绩。

时至今日，国内菲氏徒子徒孙回避无穷小，胡说极限理论，心中一定有“鬼”。

袁萌   陈启清   10月6日

附：美国数学家Joel A. Tropp专著“Innitesimals: History & Application”的摘要全文：

Abstract. An innitesimal is a number whose magnitude exceeds zero but somehow fails to exceed any nite, positive number. Although logically problematic, innitesimals are extremely appealing for investigating continuous phenomena. They were used extensively by mathematicians until the late 19th century, at which point they were purged because they lacked a rigorous foundation. In 1960, the logician Abraham Robinson revived them by constructing a number system, the hyperreals, which contains innitesimals and innitely large quantities. This thesis introduces Nonstandard Analysis (NSA), the set of techniques which Robinson invented. It contains a rigorous development of the hyperreals and shows how they can be used to prove the fundamental theorems of real analysis in a direct, natural way. (Incredibly, a great deal of the presentation echoes the work of Leibniz, which was performed in the 17th century.) NSA has also extended mathematics in directions which exceed the scope of this thesis. These investigations may eventually result in fruitful discoveries.