6 ms·

I feel when it comes to motivation in math, I just want to know why people got so excited about a particular theorem in the first place. I'm ok with the answer

by cf 9y ago

I feel when it comes to motivation in math, I just want to know why people got so excited about a particular theorem in the first place. I'm ok with the answer "before this theorem we assumed all these different things we proved in similar ways were different. Now we know about their commonality and it allows us to borrow mathematical machinery and use it here". For example, take something like measure theory. A reasonable motivation for measure theory to me is remembering your introductory probability class how you had to learn a probability mass function for discrete spaces and a probability density function for continuous spaces. Now these two ideas are obviously nearly the same idea. I mean the notation used is a pretty big hint. But you need measure theory to have the right concepts to describe how they are the same. Where are all those kinds of motivation for things like topology and cohomology?

drostie 9y ago
I asked the ##math chat room on Freenode about cohomology recently, more precisely I expressed that it was for me the scariest math-word-that-I-don't-know-yet, and I received the explanation "it isn't that scary once you get used to it; it's just a way to repair exactness of sequences." Maybe that helps? For topology, I feel like there's a sort of meeting-of-two-different-things; one starts with being very frustrated with the delta-epsilon-definition of "limit" and its one-dimensional nature, and the limit-definition of "continuous" and its clumsiness. The other starts from wanting to play with spheres and Möbius strips and knots and the like. When you're playing with these shapes a bijective mapping between two surfaces is not a fine-grained-enough idea because it is not continuous; adding continuity gives "homeomorphisms" which also aren't a fine-grained-enough idea because they do not make reference to the space an object is embedded in; wrap a torus about itself in a pretzel knot and you have something which is homeomorphic to a torus but in 3D you can't get there without tearing part of the surface through the other one, but in 4D you can. So finally we come to the idea of an isotopy, which bumps "continuous" to the next level by saying "Just like you can have a continuous path of points in space, you can have a continuous path of homeomorphisms from one to another," and that's where the pretzel knot becomes finally distinct from the torus in 3D, there is no continuous path from the homeomorphism of the pretzel knot to the torus, to the identity homeomorphism of the torus to itself. Or something like that. So this path is then an "isotopy" and then certain things are nicely isotopy-invariant and so forth.
- cf 9y ago
  The topology explanation is about what I had in mind for motivation. I think to appreciate cohomology, I need appreciate what problems in homology it makes easy to solve. To appreciate that I need have the vocabulary of algebraic topology. One way to maybe then to describe homology is it gives you way to take about shapes and surfaces in the language of algebra.
mathperson 9y ago
I like quanta for general purpose math articles. they typically take a fairly recent paper and explain it qualitatively but still pretty accurately. almost like war stories. https://www.quantamagazine.org/ https://www.quantamagazine.org/

by cf 9y ago

drostie 9y ago

I asked the ##math chat room on Freenode about cohomology recently, more precisely I expressed that it was for me the scariest math-word-that-I-don't-know-yet, and I received the explanation "it isn't that scary once you get used to it; it's just a way to repair exactness of sequences." Maybe that helps? For topology, I feel like there's a sort of meeting-of-two-different-things; one starts with being very frustrated with the delta-epsilon-definition of "limit" and its one-dimensional nature, and the limit-definition of "continuous" and its clumsiness. The other starts from wanting to play with spheres and Möbius strips and knots and the like. When you're playing with these shapes a bijective mapping between two surfaces is not a fine-grained-enough idea because it is not continuous; adding continuity gives "homeomorphisms" which also aren't a fine-grained-enough idea because they do not make reference to the space an object is embedded in; wrap a torus about itself in a pretzel knot and you have something which is homeomorphic to a torus but in 3D you can't get there without tearing part of the surface through the other one, but in 4D you can. So finally we come to the idea of an isotopy, which bumps "continuous" to the next level by saying "Just like you can have a continuous path of points in space, you can have a continuous path of homeomorphisms from one to another," and that's where the pretzel knot becomes finally distinct from the torus in 3D, there is no continuous path from the homeomorphism of the pretzel knot to the torus, to the identity homeomorphism of the torus to itself. Or something like that. So this path is then an "isotopy" and then certain things are nicely isotopy-invariant and so forth.

cf 9y ago
The topology explanation is about what I had in mind for motivation. I think to appreciate cohomology, I need appreciate what problems in homology it makes easy to solve. To appreciate that I need have the vocabulary of algebraic topology. One way to maybe then to describe homology is it gives you way to take about shapes and surfaces in the language of algebra.

mathperson 9y ago

I like quanta for general purpose math articles. they typically take a fairly recent paper and explain it qualitatively but still pretty accurately. almost like war stories. https://www.quantamagazine.org/ https://www.quantamagazine.org/