# Felix Klein and the mean value theorem

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. Why is Klein focusing specifically on the mean value theorem? Was this theorem a preoccupation of other mathematicians at the time?

• A useful resource, but no link with Klein. – Mauro ALLEGRANZA Jun 9 '16 at 9:47
• Klein's reference to Hilbert is to Grundlagen der geometrie; see page 21 of Engl.transl. of 1899 edition: §12. INDEPENDENCE OF THE AXIOM OF CONTINUITY. (NON-ARCHIMEDEAN GEOMETRY.). Here Hilbert produces a model based on a domain $\Omega(t)$ of "polynomials" such that "two numbers $n$ and $t$ of the domain, each of which is greater than zero and $n < t$, possess the property that any multiple whatever of the first always remains smaller than the second." – Mauro ALLEGRANZA Jun 9 '16 at 10:09
• I am not sure I follow this. Where does Klein mention Hilbert? The issue of the mean value theorem is not identical with the issue of the Archimedean axiom. – Mikhail Katz Jun 9 '16 at 12:04
• As for your third question: The Mean Value Theorem is perhaps the single most important theorem in single-variable calculus. It lays the foundations for everything else. For example, Taylor's theorem can be viewed as a natural generalization of it. Is this not a convincing enough reason for it to be a point of focus? – Will R Jun 10 '16 at 3:24
• @WillR, no, the importance of the theorem is not convincing enough reason. I don't think one can figure out the reason "from first principles". One needs to look at the historical context. Many other theorems are important: extreme value theorem, Taylor remainder formula, etc. – Mikhail Katz Jun 10 '16 at 7:12