Hey, I just peeked at your MathOverflow page and saw that you are interested in "spatial and visual arguments". So I tell you something else (which you didn't ask for):
There also is another version of analysis with nilpotent infinitesimals, i.e. elements which are not zero, but some power of which is zero. In classical logic this contradicts the field axioms, but in intuitionistic logic it can be done. J.L. Bell's Primer of Infinitesimal Anlysis develops basic analysis on these grounds, by assuming (axiomatically) that you have something like the real numbers with nilpotents. Proofs become much easier even than in Nonstandard Analysis. Only in an appendix he addresses the existence of models for his axioms - they live in toposes.
As is very nicely laid out in the preface of Moerdijk/Reyes' "Models for Smooth Infinitesimal Analysis", it is these infinitesimals which were (implicitly) used by classical geometers like Cartan, and are (implicitly) used by physicists until today. They illustrate their point with a visual proof of Stokes' theorem using nilpotents.
In the settings of Moerdijk/Reyes (which are certain toposes) there also exist real numbers which combine the two kinds of infinitesimals, nilpotents and invertibles.
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