21–30 of 104 posts
I love how well-spoken Tao is. I've enjoyed lots of his lectures before; even if you're not an expert in whatever he's discussing he knows how to explain it just right to get you up to speed as best as he can. His communication and math skills are phenomenal.
The first time I really felt I understood math in depth was my uni linear algebra course. Distance and orthogonality were replaced with a more abstract but better inner product. It behaved like an IT interface: As long as some basic properties were fulfilled, aal of linear algebra came along. Half o the examples were the usual numeric vectors and matrices, the others were integrals, etc...
I didn’t get much out of linear algebra. It felt too computational. I only really got it once I “relearned” it as part of abstract algebra
> One can roughly divide mathematical education into three stages: Similarly with programming. 1. Write programs that you think are cool 2. Learn about data structures and algorithms and complexity and software organization. 3. Write programs that you think are cool. But since you know more, you can write more cool programs. If things are working as they should, the end stage of mathematics and programming should be…
1. Programming in very concrete/practical terms because you do not know how to think in precise and abstract terms (do not know math)
2. Thinking more precisely and abstractly (more mathematical way)
3. Only do some key important abstractions, and being a bit hand-wawy again in terms of precision. The reason: important real-world problems are usually very complex, and complex problems resist most abstractions, and also being totally precise in all cases is impossible due to the complexity.
All-in-all it is due to increased complexity in my opinion.
Example: 1. Writing some fun geometry related programs 2. learn about geometry more seriously 3. write software based on a multiple hundred thousand line CAD kernel.
Other example: 1. Write fun games on C64 2. Learn about computer graphics in University 3. Contribute to the source code of Unreal Engine with multiple million lines of code with multiple thousand line class declaration header files.
Earlier quoted context omitted.
Absolutely. We have machines that can crank out true theorems, rigorously proven, all day. It takes a mathematician to know what is worth working on. And that is fundamentally an intuitive decision. Computers don't care whether a proof is interesting or not.
It’s a bit tautological since we are defining interesting as what human mathematicians work on. Perhaps if computers ran the show they wouldn’t agree with our definition.
Don't necessarily see a tautology here.
We need it to become much more common to operate at level 3, especially in fields like enterprise software development.
they say things like "Everything in math is a set," but then you ask them "OK, what's a theorem and what's a proof?" they'll either be confused by this question or say something like "It's a different object that exists in some unexplainable sidecar of set theory"
They don't know anything about type theory, implications of the law of excluded middle, univalent foundations, any of that stuff
“Before I learned the art, a punch was just a punch, and a kick, just a kick. After I learned the art, a punch was no longer a punch, a kick, no longer a kick. Now that I understand the art, a punch is just a punch and a kick is just a kick.” - Bruce Lee
The worst thing is when someone who thinks that "math is 100% infallible and all about rigor, you gotta show your work and include all the steps" yet they think that set theory is good enough and it doesnt have problems they say things like "Everything in math is a set," but then you ask them "OK, what's a theorem and what's a proof?" they'll either be confused by this question or say something like "It's a different…