[HN Gopher] The Sordid Past of the Cubic Formula
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The Sordid Past of the Cubic Formula
Author : nsoonhui
Score : 43 points
Date : 2022-07-01 02:50 UTC (1 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| gweinberg wrote:
| I 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.
| YouWhy wrote:
| Some fifth order equations have solutions that may be expressed
| as radicals. An example of such an equation is x^5=2, for which
| the fifth root of 2 as a solution.
|
| We know from school that there is a solution for _any_ equation
| of degree 1 (linear) with integer coefficients using plain
| arithmetic, and degree 2 using radicals and arithmetic. The
| linked article mentions that the same holds for degrees 3 and
| 4.
|
| What Abel proved for degree 5 and Galois for any degree >= 5,
| is that for _some_ equations of these degrees there 's no
| expression involving radicals and arithmetic (*) that is a
| solution.
|
| To reiterate, Abel and Galois' results are about the very
| existence of a specific form of a solution, not "findability".
|
| (*) More technically: any finite formula involving composition
| of radicals, arithmetic operations and natural numbers.
| contravariant wrote:
| It 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 wrote:
| If 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.
| contravariant wrote:
| Well, I didn't mean that, but sure.
|
| For what it's worth the numbers in question are computable.
| ogogmad wrote:
| That's technically correct. But all real numbers in
| practice are computable.
| pfdietz wrote:
| I looked at your statement for a while and I can't figure
| out how it could be made to make any sense.
| ogogmad wrote:
| This might not be the best written or best sourced
| article, but I would start here:
| 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.
| Someone wrote:
| > Most operations for defining real numbers are
| computable.
|
| Isn't that a tautology or at least selection bias?
| "Operation" seems to imply we can carry it out, which
| means the number can be computed.
|
| Even if we give "operation" a wider meaning to include
| those that can't be carried out, are we deceiving
| ourselves that most of them are computable because most
| of those we have thought of in the history of mathematics
| are?
|
| Or do we really know something about the cardinalities of
| those sets?
| tzs wrote:
| If you'd like to learn the math behind that result and have not
| studied abstract algebra, a good book is Pinter's "A Book of
| Abstract Algebra" [1]. That will take you from beginning group
| theory through rings and fields culminating in the
| insolvability of the quintic.
|
| The chapters are usually short, maybe 8-12 pages, with about
| half that being lots of exercises to help cement your
| understanding of the material. This greatly helps with self-
| study. And it's a Dover edition so is inexpensive.
|
| [1] https://www.amazon.com/Book-Abstract-Algebra-Second-
| Mathemat...
| Hani1337 wrote:
| linearity 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.
| [deleted]
| mturmon wrote:
| Fantastic 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 wrote:
| These 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.
| SilasX wrote:
| Oh wow I haven't seen those. Any good examples/links?
| joachimma wrote:
| Veritasium had a good video about this a few months back.
|
| https://www.youtube.com/watch?v=cUzklzVXJwo
| [deleted]
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