Ok, listen: why cannot we start from standard real functions of one real variable and use. Infinitesimals are often compared with other. Therefore, 'infinitesimal' means 'infinitesimally small' or less than any standard number of the real number when used as a mathematical adjective. 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. An infinitesimal object is an object less than any measurable size, but not so small or small that the available means cannot distinguish it from zero. You know, I love margins of books, and I don't want to loose too much time, I have many things to do. 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. 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).

F.: Good morning Hermann, I would like to talk with you about infinitesimals.

Positive and negative infinitesimals