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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
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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
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@GerryMyerson, deleting a question based on a comment (since deleted) does not seem too friendly a procedure. –  katz Feb 6 at 15:46
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For example, would "synthetic differential geometry" count as an answer? –  Daniel Moskovich Feb 6 at 16:27
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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

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[Converted from comment to answer per Yemon Choi's suggestion.]

From a casual run-through of the Ferraro paper, it seems like Euler's ideas about infinitesimals were, unsurprisingly, not formalized to modern standards and therefore don't map exactly onto modern concepts. He apparently didn't think of a line segment as a point set, which would be more similar to smooth infinitesimal analysis than to NSA. But other aspects of Ferraro's description do seem more like NSA than SIA. Infinite numbers are imagined as infinitely increasing sequences, whereas not all models of SIA have invertible infinitesimals. I assume Euler used Aristotelian logic.

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"He apparently didn't think of a line segment as a point set." Sounds good, please tell us more. –  Paul Taylor Feb 6 at 9:12
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@PaulTaylor, Ben has apparently not been around recently, so I would comment that most historians agree that Euler was not working with a punctiform continuum that we are used to in the post-Cantor era. Rather, points were locations marked on an unanalyzed continuum. Some have argued that this makes the Eulerian continuum closer to an intuitionistic continuum. I am somewhat sceptical about this claim. –  katz Feb 6 at 16:53
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I am of the view that the "punctiform continuum that we are used to in the post-Cantor era" was vandalism on his part and hope to see the end of his "era". So I would like to hear more of how Euler saw the continuum prior to this damage. I would also like to see the answers to your question, but the Thought Police have moved in again. –  Paul Taylor Feb 6 at 17:44
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@PaulTaylor Name-calling is neither appropriate nor constructive. –  S. Carnahan Feb 7 at 0:44
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@PaulTaylor I am joining Scott Carnahan to ask you to please stop referring to those who vote to close questions as 'Thought Police'. Closing and reopening questions is a normal part of the operation of MO, and is not done to shut down 'thought' but rather to help bring questions into the form for which MO was created. This has been explained before. MO is not a board for posting people's opinions and getting in arguments; it is a site for people to ask focused questions and get focused and definitive answers. Anyway, further rude references to 'Thought Police' will henceforth be pruned out. –  Todd Trimble Feb 7 at 15:19

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