Fermat实-幂零无穷小与Infinite Dimensional空间
Fermat Reals - Nilpotent Infinitesimals and Infinite Dimensional Spaces
摘要：
己.道：早上好，赫尔曼，我想和你谈谈无穷小。G.：告诉我皮埃尔。F.：我受够了所有这些关于我的态度不严谨的诽谤，所以我开始学习非标准分析（NSA）和综合微分几何（SDG）。G.：是的，我读过一些东西......己.道：好吧，没有问题关于他们的严格性。但是，当我看到NSA中无穷大的正弦无限接近于一个实数时，我很惊讶：这个数字的直观含义是什么？然后，我看到，要在SDG中工作，我必须学会在直觉主义逻辑中工作。你知道，我喜欢书的空白处，我不想浪费太多时间，我有很多事情要做... G.：在SDG中，他们还说每个无穷小同时是正的和负的，这一切是什么意思？为什么一个一阶无穷小的平方等于零，而两个一阶无穷小的乘积不一定是零？你知道从NSA中的任何一个无穷小都可以构造出一个可测集吗？不用选择公理！F.：是的，我知道，我知道...好，听着：为什么我们不能从一个实变量的标准实函数开始，然后使用.这项工作是理想的延续这一对话：理论的实际无穷小，不需要背景的形式逻辑理解，具有明确的直观意义和非平凡的应用微分几何的有限和无限维空间。
英文摘要：
F.: Good morning Hermann, I would like to talk with you about infinitesimals. G.: Tell me Pierre. F.: I'm fed up of all these slanders about my attitude to be non rigorous, so I've started to study nonstandard analysis (NSA) and synthetic differential geometry (SDG). G.: Yes, I've read something ... F.: Ok, no problem about their rigour. But, when I've seen that the sine of an infinite in NSA is infinitely near to a real number I was astonished: what is the intuitive meaning of this number, if any? Then, I've seen that to work in SDG I must learn to work in intuitionistic logic ... You know, I love margins of books, and I don't want to loose too much time, I have many things to do ... G.: In SDG they also say that every infinitesimal is at the same time positive and negative, what is the meaning of all these? And why does the square of a first order infinitesimal equal zero, whereas the product of two first order infinitesimals is not necessarily zero? And do you know that from any single infinitesimal in NSA is possible to construct a non measurable set? Without using the axiom of choice! F.: Yes, I know, I know ... Ok, listen: why cannot we start from standard real functions of one real variable and use ... This work is the ideal continuation of this dialogue: a theory of actual infinitesimals that do not need a background of formal logic to be understood, with a clear intuitive meaning and with non trivial applications to differential geometry of both finite and infinite dimensional spaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
