# Felix Klein on mean value theorem and infinitesimals

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?

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.

• I'd wager that the OP is already familiar with this paper ;-) – Ed Dean Feb 14 '14 at 0:52
• Still, I bet he finds the paper interesting. ;-) – Todd Trimble Feb 14 '14 at 0:55
• Oh, sorry. You might be right. – Kurisuto Asutora Feb 14 '14 at 2:19
• I am happy to see that the paper by Tall et al. is being read :-) – Mikhail Katz Feb 14 '14 at 9:21

Mikhail: In my monograph-length paper The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes, Arch. Hist. Exact Sci. 60 (2006) 1–121, which can be downloaded from my website (http://www.ohio.edu/people/ehrlich/), I made reference to the exact remark of Klein you quote when I wrote:

As we emphasized in the Introduction, and the above has already begun to show, while late nineteenth- and pre-Robinsonian twentieth-century mathematicians banished infinitesimals from mainstream analysis, they by no means banished them from mathematics. In fact, even the banishment of infinitesimals from analysis was never quite as complete as the standard histories (cf. [Boyer 1949; Edwards 1979; Bottazzini 1986]) might lead one to surmise. Indeed, as Robinson [1961, p. 433; 1974, p. 278] himself was well aware, from time to time during the period separating the arithmetization of analysis and the emergence of nonstandard analysis, there had been mainstream mathematicians who would not rule out the possibility of a logically satisfactory alternative foundation for analysis based on infinitesimals. Perhaps the best known examples of such mathematicians are Schmieden and Laugwitz in connection with their work Eine Erweiterung der Infinitesimalrechnung (An Extension of the Infinitesimal Calculus) [1958; also see Laugwitz 1961, 1961a]. However, to their voices one may add those of Neder [1941–43; 1941–43a], Fraenkel [1928, pp. 116–117; 1953, p. 165], Klein [1911/1939, pp. 218–219], Levi-Civita [1892–93], Bettazzi [1891] and, as we have already mentioned, Stolz. (page 15)

I have long planned on answering the very question you raise in Part II of my history of non-Archimedean mathematics. Unfortunately, that work is taking much longer to finish than I planned. However, you don't have to wait till it's finished to get my answer. I plan to provide it in the paper you invited me to give this summer in Tel Aviv.

I hope you'll forgive me for being such a tease.

• I don't see an answer to the question there. Whose work did Klein have in mind? – Matt F. Feb 14 '14 at 1:59

In our 2018 publication in Journal of Humanistic Mathematics we analyze the criterion of effectiveness as formulated by Klein and by Fraenkel, briefly summarize the controversy (over a proof of the MVT given by Camille Jordan) between Guiseppe Peano and Louis-Philippe Gilbert, give a related reference by Smorynski:

Smorynski, C. MVT: a most valuable theorem. Springer, Cham, 2017

and analyze the extent to which the criterion has been met by Abraham Robinson's framework for analysis with infinitesimals.