# Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in operation in their writings a conception of mathematics which is quite extraneous to that of Euler." Ferraro concludes that "the attempt to specify Euler's notions by applying modern concepts is only possible if elements are used which are essentially alien to them, and thus Eulerian mathematics is transformed into something wholly different"; see http://dx.doi.org/10.1016/S0315-0860(03)00030-2.

Meanwhile, P. Reeder writes: "I aim to reformulate a pair of proofs from [Euler's] "Introductio" using concepts and techniques from Abraham Robinson's celebrated non-standard analysis (NSA). I will specifically examine Euler's proof of the Euler formula and his proof of the divergence of the harmonic series. Both of these results have been proved in subsequent centuries using epsilontic (standard epsilon-delta) arguments. The epsilontic arguments differ significantly from Euler's original proofs." Reeder concludes that "NSA possesses the tools to provide appropriate proxies of the inferential moves found in the Introductio"; see http://philosophy.nd.edu/assets/81379/mwpmw_13.summaries.pdf (page 6).

Historians and philosophers thus appear to disagree sharply as to the relevance of modern theories to Euler's mathematics. Can one meaningfully reformulate Euler's infinitesimal mathematics in terms of modern theories?

Note 1. There is a related thread at Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite?

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Euler's mathematical theories are easily understood in their own right without any so called modern improvements. Euler's main fault is unsatisfactory acknowledgement and explanation of previous authorities. Solving the Basel problem needs help from the Newtonian formulae which are to be found in D.T. Whiteside's Mathematical Papers of Isaac Newton vol 5 pages 358-359. Euler was not able to give such a precise reference. –  user37007 Jul 12 '13 at 15:40
Given that "Historians and philosophers thus appear to disagree sharply," this would appear to be a question for which there are opinions rather than answers, thus poorly-suited to MO. And when OP writes in a comment about "scholars like Ferraro who are ill-equipped to deal with the mathematics beyond $\epsilon,\delta$, that tips the balance over for me. –  Gerry Myerson Feb 6 at 5:22
@GerryMyerson, deleting a question based on a comment (since deleted) does not seem too friendly a procedure. –  katz Feb 6 at 15:46
For example, would "synthetic differential geometry" count as an answer? –  Daniel Moskovich Feb 6 at 16:27
Dear @Daniel, I originally posed this question in the early stages of a current joint text on Euler's infinitesimal mathematics and its interpretation where we address some of the issues that came up here. The paper is currently being considered at a leading philosophy journal; I have recently submitted a revised draft. I can send you a current version if you are interested. –  katz Feb 6 at 16:28