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Math became easier to read when I realized that there are relatively few ideas in math and most things are just permutations thereof. The main ones are adjoint
by adamnemecek 4y ago
Math became easier to read when I realized that there are relatively few ideas in math and most things are just permutations thereof.
The main ones are adjoint, norm, and fixed points. I wrote something about this https://github.com/adamnemecek/adjoint/ https://github.com/adamnemecek/adjoint/
- atq2119 4y agoThat's just one subfield of mathematics. During my PhD, I barely touched adjoints and fixed points. Though, many fields of math so indeed have this sort of repeating pattern. It's just not the same for all of them.
- adamnemecek 4y agoWhat subfield do you think I'm talking about? Which subfield doesn't involve adjoints and fixed points?
- bollu 4y agoMost "hard results" in any subfield of math cannot be derived from pure categorical nonsense like adjunctions. At some point, you have to roll up your sleeve and do the work. How would you prove the tychonoff theorem purely with adjunctions? How about the representation theorem of finitely presented abelian groups? And on and on and on...
- Tainnor 4y agoI would argue that something like induction is probably more universally relevant.
- gnulinux 4y agoInteresting post. In "Linear Algebra" section what do you mean by `a'`? Do you mean `inv(a)`? It seems like this was never explained in your own formalism.
- adamnemecek 4y agoIt’s some Julia pseudo code. inv is inverse. Tick is adjoint.
- quickthrower2 4y agoCategory theory?
- adamnemecek 4y agoConstructivism. Linear logic, geometry of interaction, transcendental syntax.
- galaxyLogic 4y agoI love Category theory, even though I know very little about it. But I've read the basic idea is there are "objects" and (directed) relations between them. The basic idea is you can use it with anything defining your own "nouns" which relate to each other via the "arrows". It is like Humpty Dumpty words only mean what you define them to mean. BUT to define them you must define how they relate to other words. What matters is the structure of relations/arrows between the concepts. Instead of leaving definitions in rather vague natural language everything becomes "concrete" when you can draw it as a graph of vertices and arcs. The best experience I had learning math was a linear algebra course during which I programmed most of the formulas that were taught into my programmable pocket calculator. I learned the things well and got straight A's on every test even though that usually didn't happen with me with other courses. So my suggestion for aspiring math students would be: Create programs that do the calculations.