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In case you haven't read it; see "Science and Method by Henri Poincare" - https://archive.org/details/sciencemethod00poinuoft https://archive.org/details/scienc

In case you haven't read it; see "Science and Method by Henri Poincare" - https://archive.org/details/sciencemethod00poinuoft https://archive.org/details/sciencemethod00poinuoft Here is a summary - https://thetelos.org/science-and-method-science-et-methode-henri-poincare/ https://thetelos.org/science-and-method-science-et-methode-h... Here is a video discussion - https://www.youtube.com/watch?v=sQ-t8-igDZo https://www.youtube.com/watch?v=sQ-t8-igDZo The thesis is that Science is made for efficiency and hence does not include everything (which is infinite). It only looks for a hierarchy of interesting facts and focuses on simple foundational recurring phenomena from nature. It then uses the language of Mathematics to economize and impose order on the complexity to make it tractable. Finally, our subconscious prefers aesthetic attributes and hence we often find/create harmony/symmetry/beauty in our mathematical products.

podocarp 8d ago
Perhaps symmetry is overrated and it is not the beauty that works, but it's economy. If something is symmetric depending on how symmetrical you can remove half or more of the required work during calculation. It is also useful to work in abstractions where it does not related to the system 1:1 but is some transformed representation (e.g. phase space). And it may be useful when you find a way to transform it to some easily solvable (symmetrical?) state, solve it there, and reverse the transform. Think of working in Cartesian and polar coordinates. Just a change in coordinate representations can improve your quality of life immediately on some problems with different symmetry. There's no intrinsic reason other than its much simpler on one than the other. Beauty doesn't have to come in. If it's round it's round. If it's square it's square. So I think symmetry might just be one of the tricks that keep working and keeps on giving back and so we love it and call it "beautiful".

podocarp 8d ago

Perhaps symmetry is overrated and it is not the beauty that works, but it's economy. If something is symmetric depending on how symmetrical you can remove half or more of the required work during calculation. It is also useful to work in abstractions where it does not related to the system 1:1 but is some transformed representation (e.g. phase space). And it may be useful when you find a way to transform it to some easily solvable (symmetrical?) state, solve it there, and reverse the transform. Think of working in Cartesian and polar coordinates. Just a change in coordinate representations can improve your quality of life immediately on some problems with different symmetry. There's no intrinsic reason other than its much simpler on one than the other. Beauty doesn't have to come in. If it's round it's round. If it's square it's square. So I think symmetry might just be one of the tricks that keep working and keeps on giving back and so we love it and call it "beautiful".