shkrobius (shkrobius) wrote,
shkrobius
shkrobius

Kepler's ellipse, inspiration vs. perspiration, and useful errors

How do people arrive at new great ideas? Is it always inspiration? What if the idea is great but incorrect, though eventually leading to the correct one? Is this also inspiration?

These are not easy questions to ask. One of the difficulties is that all that I know with certainty is how ideas occur to me, but mine are not particularly good. The accounts of arriving at a great idea are surprisingly infrequent, because most people cannot explain how their best ideas come to them. One needs extreme lucidity of thought to capture this process; at least, the conscious part of it. The combination of great thought and great power of introspection complemented with the ability and the desire to express it all in full detail is exceedingly rare. My favorite account of this sort is Euler's essay of Pentagonal Number Theorem, where he explains, step by step, all of his guesses and logical leaps. The second favorite is Kepler's Astronomia Nova. Admittedly, it is not easy reading (no comparison with Euler), but it is very rewarding, because one can follow his thought exactly. Kepler was not good at exposition, but excruciatingly pedantic and truthful. He wrote just about everything he tried and thought. He had absolutely no idea how to write "scientifically." One gets used to the style after a while, but it takes time and effort. Some good people made extensively annotated, animated, very readable version of Kepler's classic
http://www.wlym.com/~animations/newastronomy.html
that allows to follow Kepler's thought closely without agonizing over his diary-turned-into-a-folio.

Now to the discussion: http://flying-bear.livejournal.com/602294.html?thread=10592694#t10592694


FB: Сделанное Кеплером воспринимается как куда менее тривиальное, чем сделанное Галилеем... Наблюдения Тихо Браге (на которых основывался Кеплер), естественно, сообщают нечто о движении Марса, видимом с Земли. Догадаться, что переход от кругов к эллипсам, и только он, позволяет обойтись без эпициклов (причем, только поместив Солнце не в центр, а в фокус эллипса)... Это непостижимо.

S: Kepler found his elliptical orbit by trial and error, and his greatest, most difficult and fateful step was not in the derivation of the law of motion per se, but in correcting the common error of using the mean Sun; with the correct trajectory it was possible to prove that there is no Ptolemaic equant. He then tried to find a way to match the "equant" that he still wanted to have with the position of the Sun, by playing with various orbital shapes. It was entirely logical, and his book is quite readable, although crushingly boring due to lots of primitive calculus. He had the idea that the motive power of the Sun weakens inversely with the distance (that followed from his own optical studies) and that should curve the orbit. He compared the apparent orbital velocities at the orbital extremes and that hinted at the connection between the distance and the velocity. Using the areal wedges was forced upon him: the accuracy of the orbit was insufficient and the areas helped Kepler to make better approximations, by integrating over the orbital imperfections. He then made (geometrical) calculations and discovered the law of equal areas. He then played with the orbit assuming it to be ovoid; that didn't work. After 5 years of trial and error with various ovoids, one day Kepler tried an ellipse, after doing his optical experiments on refraction, where ellises also occurred, and it had worked. His original motivation was to use the ellipse to approximate the oval! That's because the formula for the elliptical sectors was known from Archimedes, so it was saving calculation time, as the formula for the ovoids was not known and required summing up wedges by hand. It was typical science, when piling error upon error and wild guess upon wild guess serendipously yielded the correct result. It is very educational reading, too, because it gives the right idea how great discoveries are made.

Kepler had all kinds of peculiar ideas, but few of these informed his derivation of the first two laws of orbital motion. The third law is a different story, there his mysticism mattered significantly. I've read the New Astronomy in excerpts and up to Chapter 60 it is quite readable (not much worse than Newton's Principia). After that it becomes inpenetrable, but Kepler himself confessed in his introduction that he was not able to understand what he wrote by the time he published his treatise, so I do not feel guilty on this account. Here is a brief summary of the final leg of his derivation: http://math.berkeley.edu/~robin/Kepler/index.html

FB: Дело в том, что я не верю в перебор вариантов. "По существу, когда речь идет об источнике гениальных озарений, можно предложить только два существенно различных ответа — перебор вариантов (в духе дарвиновского естественного отбора для идей) и — «Господь надоумил». Даже люди, в целом далекие от религии, часто переходят на вполне мистический язык, пытаясь понять эту тайну. Австрийский философ К. Поппер говорил о «третьей вселенной», откуда берутся еще не написанные (но уже существующие!) художественные тексты и не сделанные научные открытия, которые переводятся во вторую вселенную — человеческую культуру; первая вселенная — это материальный (физико-астрономический) мир. Аргументом в пользу этой идеи служит то общеизвестное обстоятельство, что такой «перенос» часто делается несколькими людьми одновременно, когда настало время (например, периодическая система элементов, создание неевклидовой геометрии, формулировка специальной теории относительности...). Бывает, что отдельные гении опережают свое время, а затем, спустя долгие годы, их непонятые результаты переоткрываются целой группой ученых (например, история законов наследственности Менделя). В действительности идея нематериального внешнего источника всех содержательных идей является вполне традиционной и уже у Платона выражена с полной отчетливостью. Если угодно, это вопрос терминологический. Естественно назвать «содержательной» ту идею, которая не может быть получена перебором вариантов, то есть, выражаясь «операционалистски», не может быть сформулирована компьютером, в который загружена «вся сумма знаний, накопленных человечеством». Вопрос тогда в том — существуют ли содержательные (в этом смысле!) идеи. Материалист по определению должен ответить — нет." (http://lit.lib.ru/i/irhin_w_j/fenix.shtml).

S: Do you believe Kepler? He confided how he went through various shapes of ovoids over the years, how tedious were his calculations, and how he fit the ellipse to the ovoid for no other reason than to make the estimates of the wedge areas simpler. He then jumped to the conclusion that the ellipse fits the data best, that the orbit is elliptical, not only for Mars, for which he had at least some data, but for celestial bodies in general, for which he had no data whatsoever. His reliable pattern for Mars was 12(!) points. There was no way to prove that the trajectory is, in fact, elliptical. Would he had five times more accurate data, he would find that this is not true at all; it is not an ellipse. It was delusion and proliferation of error on his part, and he was indeed sifting through possibilities. The ellipse was chosen for no other reason that it made calculations easier and it was broadly consistent with the imperfect observations; it was parsimony argument, and it was incorrect. He himself never made a claim that his first two laws have been inspired, or were a mystical revelation, unlike his third (that clearly was). His book showcases the first two laws as the result of exhaustive elimination of possibilities and settling on a parsimonous solution. Your argument is correct in a sense that the possibilities are countless. But in Kepler's case the solution was dictated by poor development of calculus and the quality of observations rather than inspiration. I think that should not be tossed away. There is still plenty of this going on, when the exhaustion of possibilities is stopped not because those are exhausted or the solution is correct, but simply because one of these has "worked." I would be hard pressed to call this process inspiration.

FB: Забавно, как одни и те же "факты" (кавычки в смысле: "фактов", отдельных от итерпретации, просто не бывает) могут пониматься прямо протовоположным образом. Все это действительно очень знакомо по многим случаям: уверенность в правильности какой-то идеи, когда она вдруг приходит, всегда основана на крайне недостаточной и не слишком точной информации (двенадцать точек, если бы точность была выше, отклонения от эллипса стали бы заметными)... Типичное проявление веры в то, что Эйнштейн много, много позже назвал критерием "внутреннего совершенства" физической теории. Каковая вера для меня бесспорно является мистической. Возможно, я, с Вашей точки зрения, понимаю слово "мистический" в слишком широком смысле - или Вы - в слишком узком, с моей. Не говоря уже о том, бесспорно мистическом, факте, что к правильным результатам можно приходить совершенно ошибочным путем.

S: I am beginning to understand your point. You are saying that the inspiration and the mystical experience of Kepler were not in the derivation of the laws per se, which was perspiration, but in Kepler's bold assertion that it is THE universal and true law, as it did not follow either from his methods or his logic. You have a point there, but it reveals something that I find very uncomfortable, extremely troubling in fact, namely that the inspired thought can be false. I wonder how do you deal with this problem. There are three (perhaps more) possibilities there. (i) The falsity creeps as the inspired thought is rationalized, (ii) the thought has been falsified so that we can understand it. (iii) what is taken as inspired thought may not always be inspired. In Kepler's case I gravitate towards (iii), but you exclude it. I think we would easily agree on what can be the source of great true ideas. But what is the source of great wrong - but eventually useful - ideas?

FB: Вы сейчас уцепили самое, мне кажется, важное. Величайшая гносеологическая проблема для меня - это способность приходить к правильным выводам из неправильных предпосылок. Ну, мое объяснение вполне мистическое, если не считать - мракобесное. Я действительно считаю, что идеи не вырабатываются в наших головах, а воспринимаются... ну, допустим, из платоновского мира идей. Главное - настроиться. Когда вы говорите "киса, киса, киса..." - и кошка приходит, ведь неважно, какой теорией кошки вы руководствуетесь. Известно, что кошек надо подманивать так. Все их так подманивают, они привыкли.

S: I apologize for pedantry, but I did not quite follow your answer. Are you saying that Platonic world of ideal forms contains not only true ideas but also false ones? That true ideas avoid us whereas false ones, ennobled by their contact with true ones, may be domesticated? That would be an interesting view, but Plato, to the best of my knowledge, believed that the ideal world is populated by true ideas only. Or do you envision some layered worlds of ideas, with people catching only the corrupt ones? Or perhaps your view falls into class (i): true ideas become falsified when reflected by human mind. I myself think that (ii) could also be an option, i.e. the ideas are corrupted intentionally when communicated. Kepler is a good example. Would he do a better job of observations, invent calculus instead of fooling around with numerical integrations, etc. he would never make his deductions which proved to be so useful for later developments. He would end up fitting epicycles to the ovoids or some other nonsense. He'd still be euphoric about it, of course, but it will be of little use to the posterity. Is it possible to be divinely deluded and misguided?

FB: "Имена, которые даны вещам земным, заключают великое заблуждение, ибо они отвлекают сердце от того, что прочно, к тому, что не прочно, и тот, кто слышит [слово] Бог, не постигает того, что прочно, но постигает то, что не прочно. Также подобным образом [в словах] Отец, и Сын, и Дух святой, и жизнь, и свет, и воскресение, и церковь, [и] во всех остальных — не постигают того, что [прочно], но постигают, что не прочно, [разве только] познали то, что прочно. [Имена, которые были] услышаны, существуют в мире [для обмана. Если бы они были] в эоне, их и день не называли бы в мире и не полагали бы среди вещей земных. Они имеют конец в эоне. ... Единственное имя не произносится в мире — имя, которое Отец дал Сыну. Оно превыше всего. Это — имя Отца. Ибо Сын не стал бы Отцом, если бы он не облачился во имя Отца. Те, кто обладает этим именем, постигают его, но не произносят его. Те же, кто не обладает им, не постигают его. Но истина породила имена в мире из-за того, что нельзя познать ее без имен. Истина едина, она является множеством, и [так] ради нас, чтобы научить нас этому единству посредством любви через множество" (Филипп 11—12) http://lit.lib.ru/i/irhin_w_j/fenix.shtml

S: I take it that your answer is a nuanced "yes," and you take confounding of language as an example of such a delusion. This reminds of another mystic: Rabbi Isaac ben Samuel (a Kabbalist in the 13th century). He estimated, using Gematra and the verse about the tower of Babel, that the sky is 13 AU away, which happens to be within a factor of 3 from the actual size of the solar system! He also used the book of Psalms to estimate that the Universe is 15 Gyr old. Another remarkable case of correct delusions, on a par with Kepler's!

Is it possible to be Divinely misguided?
Tags: forgotten topics, memorable exchanges
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