# 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. I just noticed a related thread at Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite?

-
Reeder's point is that terms like "Euler's ideas" are not specific enough to give a proper evaluation of Euler. If "Euler's ideas" means his presentation of mathematical OBJECTS such as infinitesimals or infinite integers, then the presentation is lacking. If "Euler's ideas" means his techniques, or "inferential moves" in Reeder's terminology, then Reeder argues that the mapping to modern conceps is surprisingly exact. This aspect of Euler's ideas seems to have been overlooked by traditional scholars like Ferraro who are ill-equipped to deal with the mathematics beyond $\epsilon,\delta$. –  katz Apr 10 at 15:47
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. –  Peter L. Griffiths Jul 12 at 15:40
Laugwitz did some careful studies of Euler's work and proposed some analyses of Euler's proofs in terms of "hidden lemmas", i.e. assumptions made by Euler that can be justified using modern techniques. I can provide additional references if you are interested. –  katz Jul 21 at 8:08