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there. Like that you can treat i or something else like regular numbers when taking derivatives, or the fact you can just throw a negative into a logarithm and manipulate it algebraically to get an answer that makes sense, etc. No references at the moment but I think I saw it mentioned in multiple places when I was reading about the history of Euler's formula somewhere (maybe a JSTOR article or a PDF from somewhere, idk)...
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interprtetions made by
Knowledge editors. At the moment, the article is hopelessly vague and says nothing useful about its subject. Of course, if the phrase has no specific meaning and was simply a throw-away description used once or twice by Cauchy, then it is probably not sufficiently notable to merit an article.
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infinite (or impossible to interpret), in spite of the conclusion being finite (and correct). Euler was particularly known for this style of "proof", and there are some well-known series and product identities (which naturally I can't remember at the moment) where he employed precisely this sort of argument.
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says that a primary source may be used to support "straightforward, descriptive statements that any educated person, with access to the source but without specialist knowledge, will be able to verify are supported by the source" - citing a source as an instance of the use of a phrase is an example of
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I think I may have also seen generality of algebra used to refer to more than just infinite series but also justification that things that *aren't regular* numbers could be treated like a number and therefore manipulated according to the same rules even if the operations and functions did not exist
426:
Thanks for your input. As I mentioned above, unfortunately historians of science seem to be less interested in participating in wiki, so we are all dealing in guesswork. I find it surprising that the term is attributed to Cauchy. I had the impression it was already used earlier. If anyone has
664:
OK, thanks, I was not familiar with Grandi's method in detail. In other cases, I think, generality of algebra does "work" by assigning to a series a value provided by the analytic continuation of the sum, but I don't have examples right now. Perhaps we could follow up the suggestion above to
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Yes, the article needs some examples of what the "generality of algebra" principle actually is - together, of course, with sources that demonstrate that these examples are actually referred to in the mathematical literature as demonstrating the "generality of algebra", and are not guesses and
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I agree that the analogy with analyticity is somewhat obscure. My understanding is that "generality of algebra" is not just a general lack of rigor, but that refers to a particular class of proof techniques where expressions would be manipulated past values that, if taken literally, would be
173:
then the connection is too weak for me to see it: please explain. It looks to me like someone has misunderstood: the "generality of algebra" is (as far as I can interpret that book) a statement about standards of proof, and how (informally) one may generalise results and formulae
380:, which actually contains the phrase "Examples 1 through 4 illustrate the principle of generality of algebra" preceded by 4 examples. There's no preview on google books, but there is on amazon.com. Starting at page 206 seems to be a nice source on the "generality of algebra".
502:
Hi G, thanks for your edits. I am not sure why you deleted my references to analytic continuation. Obviously, if you want to use the formula beyond the radius of convergence, you have to extend it first. How do you propose to extend it if not by analytic continuation?
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examine Euler's arguments related to values of the zeta function at negative integers? At any rate, this is not a major point, and one of the historians (Smithies, if I recall correctly) specifically warns against taking the analogy with analytic continuation too far.
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Could you please add this in? We are all reluctant to act as historians of mathematics but until we can get some professional historians interested in editing wiki pages (which does not seem to be the case on a large scale), it will have to be done by mathematicians.
646:- the power series plainly does not converge at that point, therefore it has no limit value. Grandi's arguments were ingenious but wrong because he and his contemporaries did not have a rigorous understanding of limits or convergence. This was exactly Cauchy's point.
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This deletion was explained in an edit comment. If you want to look neutral, you should acknowledge that. If L says something interesting and relevant, then add it to the article, or put it on the talk page for discussion. The article is not, currently, over-long
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I think you are likely correct. OK, so the best-guess here is that the original article author has somewhat misunderstood the subject. That leaves open the question of whether this is a good title to discuss / describe these ideas under
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access to
Koetsier's book, could they please check his 4 examples? Another idea is to look for the French expression "generalite d'algebre" (I am missing some accents here), which might be more common than the English term.
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Not really. I am just explaining what he did in terms that a modern reader can understand. Obviously these techniques anticipate analytic continuation to a certain extent, but there is no need to emphasize this point.
731:
At wiki the test of an assertion is not its truth but it being sourced. If a reliable historian says they were used before Cauchy, we can rely on that. On the contrary, we generally avoid relying on primary sources.
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An author appealing to this principle would usually use some such phrase as "the generality of algebra" or "the generality of analysis"; many such appeals can be found in the writings of CAUCHY'S immediate
522:
because I believe you are misusing the term. Analytic continuation does not magically extend the domain of convergence of a power series - instead, it says that an analytic function may be represented by
823:
Euler's vision of a general partial differential calculus for a generalized kind of function. J Lützen, in
Sherlock Holmes in Babylon and Other Tales of Mathematical History, 2004, page 354.
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I agree with that with the exception that I would replace the word "demonstrate" by "illustrate": generality of algebra being a heuristic principle, it is not something one can demonstrate.
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that says "As for methods I have sought to give them all the rigour that one requires in geometry, so as never to have recourse to the reasons drawn from the generality of algebra" -
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power series in different parts of the complex plane. Analytic continuation forms part of the more rigorous approach to analysis introduced by Cauchy and
Weierstrass as a reaction
566:. He did not use analytic continuation or anything like it. A more fundamental problem with your claims is that you just cannot use analytic continuation to assign a value to
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The analysts of the 18th century shared certain convictions... and the standards of proof. When he referred to them later, Cauchy spoke aptly of the
Generality of algebra
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as a source. As far as I can see, the phrase was coined and mainly used by Cauchy; it is mentioned in our article on Cauchy, in the last sentence of the section
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Lutzen is a distinguished historian, and if there is anyone competent to speak about the generality of algebra, it would be Lutzen. This deletion is odd.
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the less watertight arguments of earlier mathematicians. When you imply that Grandi used analytic continuation, you are rewriting mathematical history.
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I understand that you left an "edit comment", and I look forward to a productive collaboration on this page, but I don't accept the "edit comment".
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such a straightforward, descriptive statement. However, I am bored with this pointless debate, so I have changed "coined" to "used" to placate you.
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The rewrite makes sense, so that is great. I agree with you re notability, but OTOH it is now harmless, so don't care much. Q: we now have
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specific example, with source, of a mathematical author who used the term before Cauchy. Assertions without sources are worthless.
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Contrary to what is currently written in the article, the term was in use well before Cauchy, as the references above testify.
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Cauchy's conception of rigour in analysis. F Smithies - Archive for history of exact sciences, 1986 - Springer. ...
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be an example, I don't actually know; though from what follows, it seems likely. A quick search turns up this book
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Well, if the paper is indeed important, could you perhaps quote some of the important observations made in it?
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Another example could be Euler's computation of the value of the
Riemann zeta function at negative integers.
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More specifically, I feel that the paper is important and should be visible to all editors working on
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of the term "generality of algebra" to the article, I have finally done this myself, using Jahnke's
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The only specific quotation that I can find for the phrase is a passage from
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One example (not really the best illustration of the concept, though) is
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Happily, S accidentally pasted in his google books search
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Lakatos' philosophy of mathematics, Volume 3 by
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There is a detailed description of Grandi's methods at
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