"The intuitive mind is a sacred gift and the rational mind is a faithful servant.” - Einstein
There's more to mathematics than rigour and proofs (2007)
11–20 of 104 posts
Re: There's more to mathematics than rigour and proofs (2007)
#12I 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.
Re: There's more to mathematics than rigour and proofs (2007)
#13Could modern AI help amateur mathematicians to build proofs?
To an extent; they can give hints and suggest directions, but you need to treat them as an unreliable narrator: think of them as entities that can help or deceive you at random.
That being said, we are researching tailored LLMs and other architectures to assist mathematical research that are more geared towards accuracy at the expense of freedom ("imagination"). The Lean FRO has some related information and links.
Re: There's more to mathematics than rigour and proofs (2007)
#14> 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…
This is true but I think it's iterative, cyclic. It applies to any art and craft, really. You alternate between perceiving and projecting, receiving and creating.
Any skill really. You alternate between theory and practice.
For example in sports you play for fun, then do some coaching to get better, then play for fun using your new skills and so on.
Re: There's more to mathematics than rigour and proofs (2007)
#15I want to be Terrance Tao when I grow up
I think the ship sailed when you were 4..
Re: There's more to mathematics than rigour and proofs (2007)
#16The 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
Re: There's more to mathematics than rigour and proofs (2007)
#17Is there?
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.
Re: There's more to mathematics than rigour and proofs (2007)
#18"The intuitive mind is a sacred gift and the rational mind is a faithful servant.” - Einstein
The problem is though, that with half the data, your mind considers glueing another base to the seasaw, to balance things out and restore symmetry and intuitive beauty.
Re: There's more to mathematics than rigour and proofs (2007)
#19> The distinction between the three types of errors can lead to the phenomenon ... of a mathematical argument by a post-rigorous mathematician which locally contains a number of typos and other formal errors, but is globally quite sound, with the local errors propagating for a while before being cancelled out by other local errors
I was initially amazed at this when I was in graduate school, but with enough experience I started to do it myself. Handwaving can be a signal that someone doesn't know what they are doing or that they really know what they are doing and until you are far enough along it is hard to tell the difference.
Re: There's more to mathematics than rigour and proofs (2007)
#20Good article, but should mention that it's not new. First copy in the Wayback Machine is from 2018, but there are comments all the way back to 2009.
https://web.archive.org/web/20180301000000*/https://terrytao...