Patrick Reeder on Euler

Patrick Reeder (Dept of Philosophy, The Ohio State University), A `Non-standard Analysis' of Euler's Introductio in Analysin Infinitorum. Summary:

In Leonhard Euler's seminal work Introductio in Analysin Infinitorum (1748), he readily used infinite numbers and infinitesimals in many of his proofs. In this presentation, I aim to reformulate a pair of proofs from the 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. I will compare and contrast the epsilontic proofs with those I have developed by following Euler more closely through NSA.  I claim that NSA possesses the tools to provide appropriate proxies of the inferential moves found in the Introductio. With the remaining time, I will offer some preliminary discussion of the purity of the methods behind the proofs. Most notably, the theory behind NSA is conservative over the theory behind ordinary analysis (in effect, due to the crucial Transfer Principle of NSA.)  This peculiar feature of NSA raises special questions regarding purity.  Does the use of ideal elements count as impure when the theory that includes the ideal elements is conservative over the theory without ideal elements?  Do these methods capture the letter of purity even if they do not capture the spirit of purity?  These and closely related questions will be considered.

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