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The Sordid Past of the Cubic Formula
- mturmon 4y agoFantastic short article with some tidbits and puzzle pieces of this oft-told story that I had not known about before. Also, due to an outside writer, kind of a respite from the typical Quanta breathless tone. (They do an excellent job, but sometimes I feel they push too hard on the significance of discoveries.) One such interesting tidbit is the notion of a different mathematical culture at the time, which valued “duels” — exchanges of mathematical puzzles. Whoever solves the most, wins. This practice (we read) incentivized keeping some clever problem solutions secret, as ammo for a future contest.
- kayamon 4y agoThese mathematical duels still happen today. The field of cryptoeconomics is full of online debaters trying to prove/disprove who amongst them understands the most economic game theory.
- gweinberg 4y agoI knew that it was in general impossible to analytically solve equations of degree 5 or higher, but this is the first time I heard you can't even express the solutions in terms of sums of nth roots. I had assumed that a fifth order polynomial would have solutions which were 5th roots, you just couldn't find them.
- contravariant 4y agoIt goes a bit further than that even, it's not just sums of nth roots it's all formulae involving just sums, products and roots (basically any of the familiar algebraic operations). This includes nested roots etc. Of course you can write them as a limit of such operations, but that's true for all numbers.
- pfdietz 4y agoIf by "write them as a limit" you mean "give an algorithm for computing the i-th element of a sequence of rational numbers whose limit is the desired real number", then this is not true for almost all real numbers, since there are only a countable number of such algorithms.
- ogogmad 4y agoThat's technically correct. But all real numbers in practice are computable.
- pfdietz 4y agoI looked at your statement for a while and I can't figure out how it could be made to make any sense.
- ogogmad 4y agoThis might not be the best written or best sourced article, but I would start here: https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis Most operations for defining real numbers are computable. The reason for this is that, in practice, continuous functions and functionals are the same thing as computable functions and functionals. Continuous functions which aren't computable are generally pathological curiosities. One consequence of this heuristic is that since the Riemann integral is a continuous functional, one might guess that it is also computable. Indeed it is (but the proof isn't obvious). This then implies the computability of the constant pi because pi is equal to the integral 1 ⌠ ⎮ ________ ⎮ ╱ 2 ⎮ 2⋅╲╱ 1 - x dx ⌡ -1 The computability of integration implies the computability of root-finding because one can use the argument principle to do it.
- joachimma 4y agoVeritasium had a good video about this a few months back. https://www.youtube.com/watch?v=cUzklzVXJwo https://www.youtube.com/watch?v=cUzklzVXJwo
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- Hani1337 4y agolinearity is an illusion created by the bias of being trapped at one level of cognition and perception. if something can not work with abstract geometrical vector rules in an exponential way, then we got something wrong at the local level. even if it makes sense from that local perspective, in that local frame of reference. biggest example of this is our lack of theory of time that is compatible with quantum mechanics.
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