This is a reference request prompted by some intriguing comments made by Felix Klein.
In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be successful. Namely, one must be able to prove a mean value theorem (MVT) for arbitrary intervals, including infinitesimal ones:
The question naturally arises whether … it would be possible to modify the traditional foundations of infinitesimal calculus, so as to include actually infinitely small quantities in a way that would satisfy modern demands as to rigor; in other words, to construct a non-Archimedean system. The first and chief problem of this analysis would be to prove the mean-value theorem $$ f(x+h)-f(x)=h \cdot f'(x+\vartheta h) $$ from the assumed axioms. I will not say that progress in this direction is impossible, but it is true that none of the investigators have achieved anything positive.
This comment appears on page 219 in the book (Klein, Felix Elementary mathematics from an advanced standpoint. Arithmetic, algebra, analysis) originally published in 1908 in German.
Question 1: When Klein writes that none of the current investigators have achieved, etc., who is he referring to? There were a number of people working "in this direction" at the time, and it would be interesting to know whose work Klein had in mind: Stolz, Paul du Bois-Raymond (somewhat earlier), Hahn, Hilbert, etc.
Question 2: Did Klein elaborate in this direction in other works of his?
Question 3: As noted in this article, A. Fraenkel formulated a similar criterion to Klein's for what it would take for a theory of infinitesimals to be successful (also in terms of the mean value theorem). Did other authors express related sentiments of measuring success in terms of being able to implement the MVT?
Best Answer
You may be interested in the following paper:
"THE TENSION BETWEEN INTUITIVE INFINITESIMALS AND FORMAL MATHEMATICAL ANALYSIS, Mikhail G. Katz & David Tall, Abstract: we discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating them to the natural cognitive development of mathematical thinking and imaginative visual interpretations of axiomatic proof. http://arxiv.org/ftp/arxiv/papers/1110/1110.5747.pdf
It has quite some history on the subject.