3 ms·

I feel like "closed form" becomes a lot less special when you realize that things as simple as sin, cos, log, or even just sqrt don't have "closed forms" in the

by alecbz 6y ago

I feel like "closed form" becomes a lot less special when you realize that things as simple as sin, cos, log, or even just sqrt don't have "closed forms" in the sense of "able to be expressed in terms of 'simpler' functions".

analog31 6y ago
Indeed, there's always a convention as to what the building blocks are. Like chemists don't look for how the quarks are configured -- they are satisfied to know how the atoms are arranged. In some cases, finding out what the building blocks are is an interesting problem in itself, for instance what things are computable with geometric construction.
- ehvatum 6y ago
  > chemists don’t look for how the quarks are configured Neutron magnetic moment results from quark configuration, in terms of where the up and down quarks like to orbit each other within this gigantic neutron hadron thing. This is useful for understanding the magnetic properties of materials up to a hundred angstroms thick! Fire cold neutrons into your multi-angstrom Great Wall and see how they scatter.
tikej 6y ago
Oh it is special precisely because of that! We can almost always get the answer with numerical methods. Yet, numerical solutions are good almost only for certain values of parameters. Analytic solutions on the other hand provide more insight. It can be differentiated, integrated, compared and maybe (big maybe) transform to some other/more familiar form. Finally it can be connected to whole body of mathematical knowledge. Analytic solutions are very satisfying for that reasons. Closed solutions are small, very narrow subset of analytic ones, that’s why they feel special. But I suppose they become special only after some time spent with problems where closed form solutions are rare and unexpected. I suppose high school/college mathematics makes us way too used to rare special cases, such as closed form solutions.

by alecbz 6y ago

analog31 6y ago

Indeed, there's always a convention as to what the building blocks are. Like chemists don't look for how the quarks are configured -- they are satisfied to know how the atoms are arranged. In some cases, finding out what the building blocks are is an interesting problem in itself, for instance what things are computable with geometric construction.

ehvatum 6y ago
> chemists don’t look for how the quarks are configured Neutron magnetic moment results from quark configuration, in terms of where the up and down quarks like to orbit each other within this gigantic neutron hadron thing. This is useful for understanding the magnetic properties of materials up to a hundred angstroms thick! Fire cold neutrons into your multi-angstrom Great Wall and see how they scatter.

tikej 6y ago

Oh it is special precisely because of that! We can almost always get the answer with numerical methods. Yet, numerical solutions are good almost only for certain values of parameters. Analytic solutions on the other hand provide more insight. It can be differentiated, integrated, compared and maybe (big maybe) transform to some other/more familiar form. Finally it can be connected to whole body of mathematical knowledge. Analytic solutions are very satisfying for that reasons. Closed solutions are small, very narrow subset of analytic ones, that’s why they feel special. But I suppose they become special only after some time spent with problems where closed form solutions are rare and unexpected. I suppose high school/college mathematics makes us way too used to rare special cases, such as closed form solutions.