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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?

Note 2. We challenged a reductionist view of Euler's infinitesimal mathematics in a recent article in The Mathematical Intelligencer here: http://dx.doi.org/10.1007/s00283-015-9565-6 Here we refute H. Edwards' reduction of Euler to an Archimedean framework in a way more consistent with Reeder's approach than Ferraro's.

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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 '14 at 5:22
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@GerryMyerson, deleting a question based on a comment (since deleted) does not seem too friendly a procedure. – Mikhail Katz Feb 6 '14 at 15:46
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For example, would "synthetic differential geometry" count as an answer? – Daniel Moskovich Feb 6 '14 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. – Mikhail Katz Feb 6 '14 at 16:28
up vote 5 down vote accepted

[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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@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. – Mikhail Katz Feb 6 '14 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 '14 at 17:44
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@PaulTaylor Name-calling is neither appropriate nor constructive. – S. Carnahan Feb 7 '14 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 '14 at 15:19
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@Todd, I hesitated to make this suggestion since it would likely prove to be time-consuming if I have to devote time to this again, so if you think this is not worth pursuing I certainly will not. At any rate what I wanted to suggest is that while the closing procedure in principle has a worthy goal, the way things turn out often does not match with the ideal. Perhaps a way can be found to ensure that proper explanation be given before a question can be "closed", and make sure this is not done for frivolous reasons such as an editor disliking another editor's "comment", as happened in this cas – Mikhail Katz Feb 9 '14 at 13:40

A somewhat delayed response is provided in our detailed study of Euler accepted for publication in Journal for General Philosophy of Science.

We apply Benacerraf's distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the 17th and 18th century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass's ghost behind some of the received historiography on Euler's infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a "primary point of reference for understanding the eighteenth-century theories." Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler's own.

Euler's use of infinite integers and the associated infinite products is analyzed in the context of his infinite product decomposition for the sine function. Euler's principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler.

We argue that Ferraro's assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler's work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.

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