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One thing about Euler is that for much of his life[1] he was basically able to publish whatever he liked without worrying about how many pages it would be and s
by dan-robertson 6y ago
One thing about Euler is that for much of his life[1] he was basically able to publish whatever he liked without worrying about how many pages it would be and suchlike. He would often explain what things he thought of or tried and how he came to his answer. Compare this to Gauss who didn’t have this ability and kept his papers mostly limited to the definitions, propositions and proofs, although this could have been a difference in personality too.
[1] I’m using Erdős’ definition of life here being the time when one is awake and capable of doing mathematics.
- JoeyBananas 6y agoEuler had the advantage of being alive during a time when a lot less things had been already discovered.
- Koshkin 6y agoMathematics is infinite.
- ianmcgowan 6y agoBut asymptotic..
- postalrat 6y agoProof?
- barry-cotter 6y agoTrue but publication standards change over time. If none of Euler’s publications are “trivial” in the sense that processors will gloss over the proof during lectures that would be surprising.
- pmiller2 6y agoThat would be surprising. But, my experience has generally been that when a result of Euler comes up in class, it's more frequently preceded with "this was published by Euler but not really rigorously proven." Standards of proof were obviously different 250 years ago.
- mannykannot 6y agoIndeed, but it is done on an expanding boundary. Nevertheless, I do not agree with the spirit of GP’s point, as one could equally argue that Euler accomplished so much while working from what was, from today’s perspective, an impoverished starting point. There is a term, ‘Whig history’, which has come to epitomize the attitude of evaluating historical characters in accordance with current standards. It is not a helpful mode of analysis.
- dan-robertson 6y agoWhiggishness is just a massive problem in the history of science and mathematics. There’s so much history that sees it as inevitable as if all the contemporaries of Newton and Leibniz were desperately trying to invent the calculus if only they could figure out how to do it.
- 0-_-0 6y agoBut the edges of known math are finite
- dan-robertson 6y agoCompared to Gauss or Erdős? The thing about mathematics is that it doesn’t become particularly easy if you don’t know anything. If you imagine a growing sphere of mathematical knowledge then things in the middle might have been discovered but the surface area of the edge grows as that happens. Sure Euler could do random crazy algebra things that no one had thought of but I don’t think he was in a position to invent Gaussian curvature. Erdős did a lot of combinatorics and graph theory and Ramsay theory which, apart from the latter, are full of questions that are easy to state and which plenty of people might have thought of, and yet which weren’t solved until he came along. There’s also the possibility of discovering a new proof of an old result which he did a bit.
- liveoneggs 6y ago1) you think that was an advantage? 2) that is a small way to think
- asdfasgasdgasdg 6y agoA recognition of diminishing returns on effort and low-hanging fruit is not "a small way to think." You might not agree that these things exist but that is no reason to insult someone who believes a fairly mainstream thing about scientific progress.
- maxerickson 6y agoNewton and Leibniz figured out calculus at about the same time. Only one guy published a paper outlining a method for basic integration in the 20th century. It obviously got harder to discover calculus.
- Godel_unicode 6y agoThat's not how that works.
- maxerickson 6y agoHow what works? https://academia.stackexchange.com/questions/9602/rediscovery-of-calculus-in-1994-what-should-have-happened-to-that-paper https://academia.stackexchange.com/questions/9602/rediscover...
- Godel_unicode 6y agoIn order to prove that discovering calculus got harder, you would obviously need to show that the population of people who could discover it but didn't have prior knowledge of it was at least as large as when it was discovered. In addition, in order for there to be published (or even submitted) papers, you would need those people to not only not have any prior knowledge but to not encourter anyone with a highschool understanding of calculus in the review and publishing process. As that is equally obviously not the case, I repeat; that's not how that works.
- asdfasgasdgasdg 6y agoIf Gauss published less, I'd be inclined to attribute it to differences in scientific publishing culture in the 1700s vs. 1900s. In the sense of, hey, paper and books and such were far more expensive in the 1700s.
- pmiller2 6y agoGauss received his doctorate in 1799. He was primarily active in the 19th century. https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#Career_and_achievements https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#Career_an...