  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.
