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For centuries we struggled with the concept of spatial rotations. We used to represent them in many different ways: mostly, Euler Angles and matrices. Those all had drawbacks and failed in specific edge cases that required manual/special treatment. Suddenly, we found out about quaternions. Those allow us to describe spatial rotations in a much more natural way. Formulas using quaternions require no special treatments, no "ifs". They clearly have lower kolmogorov complexity.

I feel like number themselves are in the state spatial rotations were. We can describe and represent them in many different ways; yet, those reps are complex and seemingly fail in some edge cases - infinities and division by 0, for example, has required all sorts of special treatments through the whole history of math. Those days, I've found about graphical linear algebra, which is basically a graphical, not syntactical, treatment for different kinds of numbers. Not only it looks to be a much more elegant description of numbers, but it also seems to cover edge cases: division by 0 works, has meaningful representations and requires no special treatment. Algorithms written for that representation clearly have lower Kolmogorov Complexity.

I'm looking for insights.

  1. Is it meaningful to look for "elegant" representation of mathematical objects?

  2. Is it possible to measure elegance quantitively? I.e., can we formalize such search? Is Kolmogorov Complexity related?

  3. Am I wrong in thinking this graphical view is truly more elegant and than the traditional representations?

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  • $\begingroup$ Why people keep making comments and deleting shortly after? $\endgroup$ – MaiaVictor Jun 6 '16 at 12:29
  • $\begingroup$ Looking for more elegant representations is great, but Kolmogorov complexity of computational algorithms is definitely the wrong criterion to use. Math isn't a series of algorithms, and text-length isn't the main criterion of elegance for a proof or an algorithm. $\endgroup$ – anomaly Jun 6 '16 at 17:17
  • $\begingroup$ @anomaly what is, though? What else would you use to numerically quantify the elegance of a proof for example? $\endgroup$ – MaiaVictor Jun 6 '16 at 18:47
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  1. Is it meaningful to look for "elegant" representation of mathematical objects?

OMG, yes. I would replace "meaningful" with "useful" though

  1. Is it possible to measure elegance quantitively? I.e., can we formalize such search? Is Kolmogorov Complexity related?

Low K(x) is useful, but the lowest K(x) is random, and that's not useful or meaningful. I like to think of "elegance" as somewhere just short of K(x) (say 0.1K(x)) but this is of course subjective. Alternatively, you can define "elegance" as including "robustness" where you find K(x1 + x2 + ...) where x_i is a set of math problems.

  1. Am I wrong in thinking this graphical view is truly more elegant and than the traditional representations?

I'm not sure if you're wrong, but the fact that K(x) is not computable, gives me pause. Also, there is no "best" compressor for all strings. This suggests that graphical representations may be good in some cases, but not in others.

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