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I'm having trouble finding the source of this quote, but here: >Many mathematicians have the opposite opinion; they do not or cannot distinguish the beauty or

I'm having trouble finding the source of this quote, but here: >Many mathematicians have the opposite opinion; they do not or cannot distinguish the beauty or importance of a theorem from its proof. A theorem that is first published with a long and difficult proof is highly regarded. Someone who, preferably many years later, finds a short proof is "brilliant." But if the short proof had been obtained in the beginning, the theorem might have been disparaged as an "easy result." Erdős was a genius at finding brilliantly simple proofs of deep results, but, until recently, much of his work was ignored by the mathematical establishment. And here is an interesting study on what mathematicians consider beautiful: https://www.gwern.net/docs/math/1990-wells.pdf https://www.gwern.net/docs/math/1990-wells.pdf They found that mathematicians varied a great deal in what they considered beautiful. And that there isn't any consensus on what is considered beautiful. It also changes over time: >There was a notable number of low scores for the high rank theorems 9 Le Lionnais has one explanation [7]: "Euler's formula e^pi*i = - 1 establishes what appeared in its time to be a fantastic connection between the most important numbers in mathematics... It was generally considered 'the most beautiful formula of mathematics' . . . Today the intrinsic reason for this compatibility has become so obvious that the same formula now seems, if not insipid, at least entirely natural." I think the interestingness or surprisingness of a mathematical idea is a temporary thing. Once you really deeply understand something, it's no longer surprising or interesting. It should just feel obvious and simple. This goes along with Jurgen Schmidhuber's theory of art (http://people.idsia.ch/~juergen/creativity.html http://people.idsia.ch/~juergen/creativity.html). That humans crave novelty and unexpectedness in all areas, and that is what we consider beautiful. But once you get very familiar with something, it loses it's novelty.

jacobolus 9y ago
Euler’s formula says (in one interpretation) that a half-turn rotation in the plane is the same as a reflection through a point, a statement which is obvious to children who think about it for a bit. It’s phrased in a cute way, but “most beautiful formula of all time” etc. is really a stretch. The tricky part is thinking of rotations as a type of “number” with a multiplicative structure, in which case the [additive] angle measure becomes the logarithm of the rotation, letting us write our half-turn rotation using the fancy-looking exp(πi) instead of ½⥀ or rot(180°) or whatever. If you asked someone whether ⤺² = ⥀ (or similar) was the most beautiful formula in mathematics they would probably laugh or look at you funny.
- eru 9y ago
  It is beautiful that this interpretation maps (complex) numbers and geometry so well---even if the formula itself doesn't look so beautiful and surprising anymore.
  - jacobolus 9y ago
    Personally I take “similarity transformations which fix a point in the plane” (or if you like, “amplitwists of planar vectors”) to be the definition of complex numbers, from which a Cartesian coordinate representation (i.e. breaking them into scalar + bivector parts) can easily be derived. In this view the notion of “[mapping] complex numbers to geometry” is just tautological. Maybe better would be to say that Euler’s formula shows how classical trigonometry (angle measure &c.) is what you get when you apply logarithmic thinking to Euclidean vector geometry.
    - eru 9y ago
      If you see Geometry as flowing from something like Euclid's axioms, and (complex) numbers as flowing from a long line of advances in essentially bookkeeping/accounting, it is quite nice that they have such a neat synthesis.
      - jacobolus 9y ago
        A whole lot of the original bookkeeping/accounting was about adjudicating land boundary disputes and calculating land taxes based on area, or predicting/explaining astronomical events. In fact, the origins of both trigonometry and logarithms are in astronomy, though I’ll admit that the natural logarithm per se came from Bernoulli’s study of compound interest. But yes, the connection between geometry and arithmetic is something people have been noticing for at least 5000 years, and is certainly important and very interesting. I just think this particular expression of that connection is oversold as being uniquely insightful or beautiful.
        eru 9y ago
theoh 9y ago
Saw a lecture by Zeki (mentioned in the article) at an art college summer school last year. He showed "e to the i pi" and also a Ramanujan equation. Of course his results are that people see the elegance of e to the i pi but don't see beauty in the apparent arbitrariness of the Ramanujan equation. I have to admit I don't remember what his sample was -- definitely mathematicians (but of what level?). This is an issue that was raised by art colleagues -- would you determine the ultimate aesthetic value of an artist's work by using an unqualified audience? Zeki is a bit of a buffoon with a tendency to assume that he understands other people's domains well enough to talk about them. It may not be arrogance on his part but there's a definite sense that he doesn't let complexity and nuance get in the way of a good story.

jacobolus 9y ago

Euler’s formula says (in one interpretation) that a half-turn rotation in the plane is the same as a reflection through a point, a statement which is obvious to children who think about it for a bit. It’s phrased in a cute way, but “most beautiful formula of all time” etc. is really a stretch. The tricky part is thinking of rotations as a type of “number” with a multiplicative structure, in which case the [additive] angle measure becomes the logarithm of the rotation, letting us write our half-turn rotation using the fancy-looking exp(πi) instead of ½⥀ or rot(180°) or whatever. If you asked someone whether ⤺² = ⥀ (or similar) was the most beautiful formula in mathematics they would probably laugh or look at you funny.

eru 9y ago
It is beautiful that this interpretation maps (complex) numbers and geometry so well---even if the formula itself doesn't look so beautiful and surprising anymore.
- jacobolus 9y ago
  Personally I take “similarity transformations which fix a point in the plane” (or if you like, “amplitwists of planar vectors”) to be the definition of complex numbers, from which a Cartesian coordinate representation (i.e. breaking them into scalar + bivector parts) can easily be derived. In this view the notion of “[mapping] complex numbers to geometry” is just tautological. Maybe better would be to say that Euler’s formula shows how classical trigonometry (angle measure &c.) is what you get when you apply logarithmic thinking to Euclidean vector geometry.
  - eru 9y ago
    If you see Geometry as flowing from something like Euclid's axioms, and (complex) numbers as flowing from a long line of advances in essentially bookkeeping/accounting, it is quite nice that they have such a neat synthesis.
    - jacobolus 9y ago
      A whole lot of the original bookkeeping/accounting was about adjudicating land boundary disputes and calculating land taxes based on area, or predicting/explaining astronomical events. In fact, the origins of both trigonometry and logarithms are in astronomy, though I’ll admit that the natural logarithm per se came from Bernoulli’s study of compound interest. But yes, the connection between geometry and arithmetic is something people have been noticing for at least 5000 years, and is certainly important and very interesting. I just think this particular expression of that connection is oversold as being uniquely insightful or beautiful.
      - eru 9y ago

theoh 9y ago

Saw a lecture by Zeki (mentioned in the article) at an art college summer school last year. He showed "e to the i pi" and also a Ramanujan equation. Of course his results are that people see the elegance of e to the i pi but don't see beauty in the apparent arbitrariness of the Ramanujan equation. I have to admit I don't remember what his sample was -- definitely mathematicians (but of what level?). This is an issue that was raised by art colleagues -- would you determine the ultimate aesthetic value of an artist's work by using an unqualified audience? Zeki is a bit of a buffoon with a tendency to assume that he understands other people's domains well enough to talk about them. It may not be arrogance on his part but there's a definite sense that he doesn't let complexity and nuance get in the way of a good story.