Timeline for Lapses of "the early proponents of the doctrine of limits"
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 15, 2013 at 14:48 | comment | added | user112109 | Sorry, katz, but with respect to other "results" that I quoted above: I don't believe that you have to sit through any course in order to recognize that -1 < 1. And I maintain that these "results" have been addressed by Robinson. | |
| Apr 15, 2013 at 14:10 | comment | added | Mikhail Katz | Another elegant calculation. If you use an unlimited integer in place of $\infty$, the calculation can be formalized using modern infinitesimals. It is this kind of procedure that Patrick Reeder refers to as "proxy" to "Euler's deductive moves". The claim is not that Euler had our "nonstandard" objects (which would be absurd), but rather that formalisations in terms of modern infinitesimals mirror Euler's procedures more faithfully than epsilontic re-writes thereof. This, of course, can be confusing to a "newbie", but only if the latter already sat through a standard epsilontic course. | |
| Apr 15, 2013 at 13:11 | comment | added | user112109 | @katz: Your comment reminds me of how Euler calculated log2. I could add it to my answer but am too lazy. Certainly you know it anyway. $\log2 = \log2\infty - \log\infty$ . And again we have two harmonic series subtracted. | |
| Apr 15, 2013 at 11:43 | comment | added | Mikhail Katz | That's a good point. Euler's RESULTS are usually formulated in terms familiar to us, such as the series expansion of $e^x$. However, in proving such results, Euler exploited TECHNIQUES that routinely rely, for example, on infinite (and therefore not REAL) numbers. Thus, he obtains the expansion of $e^x$ by applying the binomial formula to an INFINITE exponent, and simplifying the coefficients. Moreover, Euler explicitly asserts that it is impossible to do calculus without infinitesimals and infinite numbers. | |
| Apr 15, 2013 at 10:08 | comment | added | user112109 | @katz: I have not looked into Euler's original paper. Did he explicitly say that his calculation is invalid in the real numbers? At least many of those who are reporting his results use the customary notation and do not mention any deviations from the real numbers. I find this misleading and incorrect. | |
| Apr 15, 2013 at 7:56 | comment | added | Mikhail Katz | My point was precisely that Euler was <em>not</em> working with our real number system, as argued by Detlef Laugwitz in Archive for History of Exact Sciences link.springer.com/article/10.1007%2FBF00329867, Patrick Reeder (see mathoverflow.net/questions/126986/…) as well as Bair et al. | |
| Apr 15, 2013 at 7:47 | comment | added | user112109 | @katz: With all respect, but today we have "interpretations" in mathematics (as well as in justice) which makes nearly anything correct or excusable - from a suitable point of view. In my opinion Euler has been one of the greatest, if not the greatest at all. Nevertheless, in the mathematics of real numbers as we know and teach it, his calculations are wrong and good for nothing but confusing newbies. | |
| Apr 15, 2013 at 7:19 | history | answered | Mikhail Katz | CC BY-SA 3.0 |