5 ms·

Reminds me of this paper in Nature from some time back. Especially, the Euler's identity graph from the paper is pretty amazing. https://www.nature.com/news/equ

Reminds me of this paper in Nature from some time back. Especially, the Euler's identity graph from the paper is pretty amazing. https://www.nature.com/news/equations-are-art-inside-a-mathematician-s-brain-1.14825 https://www.nature.com/news/equations-are-art-inside-a-mathe...

notfashion 7y ago
Everybody who writes about Semir Zeki (PI on that paper) and his work is polite about him. I can only assume that this is because he is a) a likeable guy and b) the professional aestheticians and artists who could make negative comments think it wiser not to get involved in any way lest they be accused of punching down. Euler's identity is pretty obviously something elegant, and it's widely recognized. But Zeki seems to overlook the importance of acculturation in the appreciation of these things. The idea that a mind-blowing Ramanujan formula is ugly seems particularly problematic. Sure, it is surprising, complex, and apparently arbitrary, and most test subjects won't have any insight into how it works. But why should those people (let's say they are "ignorant" when it comes to Ramanujan's results) be expected to have any opinion on the formula? Any judgement they come up with will be superficial. This problem runs through most of Zeki's work. His experiments take test subjects who are, as it were, preloaded with certain aesthetic experiences and norms, and attempt to derive universal cognitive principles of beauty from the result. What he doesn't do is try to discover just what those aesthetic prejudices are. He doesn't try to extract from the test subjects the key features of their aesthetic indoctrination. In the math situation, it's totally conceivable that a group of test subjects who were educated to find every Ramanujan formula transcendently fascinating would rate those formulas as beautiful. The problem is more conspicuous when his research looks at art. Since he hasn't studied art himself, he really doesn't seem to appreciate the basic reality: that some people have learned to 'get' certain styles or genres of art, while others haven't had the good fortune to learn to appreciate those same styles. People walk around with extremely complex acquired aesthetic processing capabilities in their heads. Everyone's is different, and while Zeki may be able to find some universal regularities, (like Chomsky's hypothesized universal innate grammar) that wouldn't be a neural explanation of aesthetics any more than the success of Chomsky's programme would tell us about the nature of literature. It's a frivolous exercise from the point of view of anyone seriously interested in art (or math).
pnx 7y ago
I find it odd that people think that equation is beautiful. Rewriting it in polish notation/lisp you end up with: (= (+ (exp (* i pi)) 1) 0) Which is neither beautiful nor very enlightening.
- matt_j 7y ago
  There is more to a beautiful equation than the shape of the symbols used to describe it. The beauty of Euler's identity is the relationship between 5 fundamental constants (0, 1, e, i, pi). It's simple, elegant and far reaching. The relationship is the same regardless of the notation.
  - yesenadam 7y ago
    Mostly serious comment: I'm not sure why the form e^{i\pi}=-1 isn't better. Only it doesn't have 0, but is it worse, less beautiful? (It doesn't seem to have the same "relationship between 5 fundamental constants", although it adds the negative number realm, to the imaginary and transcendental–neat.) Would E-mc^2=0 be similarly be better than E=mc^2, because it has an additional "fundamental constant"?
    - thanatropism 7y ago
      People fetishize the formula but the beautiful idea is that multiplication/exponentiation can be expanded in such a way that it describes oscillatory relations. This is how eg. eigenvalues get to play a role in models of harmonic resonance. Or how AC impedance naturally generalizes DC resistance. Try to imagine complex interest rates. Now try to make them matrix-valued. It works. It all works.
    - mqduck 7y ago
      I don't want to start an argument about tau versus pi, but I like a tau form (or modification) of Eulor's Identy: [e^(ikτ) = 1] for all integer values of [k]. This gets across rather well that this formula expresses a complete turn around a unit circle. You can't get something quite equivalent using pi. I even more prefer the full form of Euler's Formula: [e^(ix) = cos(x) + isin(x)]. The real beauty of Euler's Formula I think is that it shows an equivalence between an algebraic function and a trigonometric function. (Note that I'm only a mildly learned laymen when it comes to mathematics. Any experts in math should feel very free to tell me why I'm wrong.)

notfashion 7y ago

Everybody who writes about Semir Zeki (PI on that paper) and his work is polite about him. I can only assume that this is because he is a) a likeable guy and b) the professional aestheticians and artists who could make negative comments think it wiser not to get involved in any way lest they be accused of punching down. Euler's identity is pretty obviously something elegant, and it's widely recognized. But Zeki seems to overlook the importance of acculturation in the appreciation of these things. The idea that a mind-blowing Ramanujan formula is ugly seems particularly problematic. Sure, it is surprising, complex, and apparently arbitrary, and most test subjects won't have any insight into how it works. But why should those people (let's say they are "ignorant" when it comes to Ramanujan's results) be expected to have any opinion on the formula? Any judgement they come up with will be superficial. This problem runs through most of Zeki's work. His experiments take test subjects who are, as it were, preloaded with certain aesthetic experiences and norms, and attempt to derive universal cognitive principles of beauty from the result. What he doesn't do is try to discover just what those aesthetic prejudices are. He doesn't try to extract from the test subjects the key features of their aesthetic indoctrination. In the math situation, it's totally conceivable that a group of test subjects who were educated to find every Ramanujan formula transcendently fascinating would rate those formulas as beautiful. The problem is more conspicuous when his research looks at art. Since he hasn't studied art himself, he really doesn't seem to appreciate the basic reality: that some people have learned to 'get' certain styles or genres of art, while others haven't had the good fortune to learn to appreciate those same styles. People walk around with extremely complex acquired aesthetic processing capabilities in their heads. Everyone's is different, and while Zeki may be able to find some universal regularities, (like Chomsky's hypothesized universal innate grammar) that wouldn't be a neural explanation of aesthetics any more than the success of Chomsky's programme would tell us about the nature of literature. It's a frivolous exercise from the point of view of anyone seriously interested in art (or math).

pnx 7y ago

I find it odd that people think that equation is beautiful. Rewriting it in polish notation/lisp you end up with: (= (+ (exp (* i pi)) 1) 0) Which is neither beautiful nor very enlightening.

matt_j 7y ago
There is more to a beautiful equation than the shape of the symbols used to describe it. The beauty of Euler's identity is the relationship between 5 fundamental constants (0, 1, e, i, pi). It's simple, elegant and far reaching. The relationship is the same regardless of the notation.
- yesenadam 7y ago
  Mostly serious comment: I'm not sure why the form e^{i\pi}=-1 isn't better. Only it doesn't have 0, but is it worse, less beautiful? (It doesn't seem to have the same "relationship between 5 fundamental constants", although it adds the negative number realm, to the imaginary and transcendental–neat.) Would E-mc^2=0 be similarly be better than E=mc^2, because it has an additional "fundamental constant"?
  - thanatropism 7y ago
    People fetishize the formula but the beautiful idea is that multiplication/exponentiation can be expanded in such a way that it describes oscillatory relations. This is how eg. eigenvalues get to play a role in models of harmonic resonance. Or how AC impedance naturally generalizes DC resistance. Try to imagine complex interest rates. Now try to make them matrix-valued. It works. It all works.
  - mqduck 7y ago
    I don't want to start an argument about tau versus pi, but I like a tau form (or modification) of Eulor's Identy: [e^(ikτ) = 1] for all integer values of [k]. This gets across rather well that this formula expresses a complete turn around a unit circle. You can't get something quite equivalent using pi. I even more prefer the full form of Euler's Formula: [e^(ix) = cos(x) + isin(x)]. The real beauty of Euler's Formula I think is that it shows an equivalence between an algebraic function and a trigonometric function. (Note that I'm only a mildly learned laymen when it comes to mathematics. Any experts in math should feel very free to tell me why I'm wrong.)