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Comment by oconnor663 | Hacker News Reader

A neat comment on the accepted answer:

> From OP's point of view this could be viewed as glass half-full rather than glass half-empty. Their dissertation results hold unequivocally on the sphere and might hold on the torus, though it is an open problem if they do. It is certainly legitimate to study what follows from a given conjecture being true. It could even be spun as a feature rather than a bug of the dissertation. If the results in fact fail on the torus then you know that the conjecture must be false. Potentially, it could open up a fruitful avenue of attack.

Kind of reminds me of Terence Tao's post on what solving big problems looks like: https://terrytao.wordpress.com/career-advice/be-sceptical-of...

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When Terence Tao writes stuff like this, I'm always very happy that I got to experience the Moore-method for learning math (at UT Austin). A group of us would be dumped into a class with a common topic and we'd just have to prove things (topology, algebra, analysis) ... on the blackboard, in front of everyone. The best work we did was when something started going wrong and then we'd all start arguing about the proof, building count-conjectures on the fly and riffing on the math. The worst work was when someone went and found a proof ahead of time and just showed the answer. There's so much to learning where the sharp bits of math are; proofs are the razor-thin path through the briar patch.

It was only later that I found out that history, the study of art & literature, and philosophy can all teach you the same thing. The important part is that you're interested in the topic.

> There's so much to learning where the sharp bits of math are; proofs are the razor-thin path through the briar patch.

This is an excellent metaphor using beautiful language. Did you just coin this yourself or is it pre-existing?

Just felt the same way about the portion after the semicolon; it's quite catchy as well! Please give a reference or else if I use this later I'll have to say it came from "some guy on hackernews".

Cite it as "thechao on HN", not just as "some guy". (Unless thechao gives you something more specific.)

I guess this is mine? I would hesitate to believe I made it up, but I don't remember it from anywhere. If you don't feel comfortable with that, then credit it to Michael Starbird, who was my first Moore-method professor 20-odd years ago.

> > There's so much to learning where the sharp bits of math are; proofs are the razor-thin path through the briar patch.

As a student representative for my undergraduate mathematics course, I got really pissed off at lecturers for exactly this reason: they'd write out a perfect correct proof on the whiteboard, but wouldn't explain where it had come from or how people had arrived at the solution. We were left to figure that out on our own.

They then complained that students were rote-learning for exams, rather than coming to a full understanding of the material. I'm not sure what they were expecting, given that that's exactly how they were teaching it.

My discrete mathematics professor was like that. He would regurgitate a proof onto the whiteboard. Then he'd do it a few more times with proofs of other things.

He has an identical twin brother, who is also a math professor at the same college. The regular professor was out for a day, and his brother came in to teach the class. His teaching style was completely different. "Ok, we need to prove X. Where should we start?" and would sit on the table and look at us with an inquisitive look on his face. Then learning happened.

Everyone's mind was blown. Most people didn't realize it was a different person. Then on Thursday it was back to same-old same-old.

I still don't know shit about discrete math.

I am not getting anywhere with the latter approach when TAing

I ask them such a question, and then wait for 15 minutes, and no one says anything.

If you're taking a class with proofs, you're being prepared for research. Doing your own research into, and reverse engineering, proofs is an important skill.

Proofs in math journals are given 'as-is' and you learn the intuition through social means and discussions.

i think general math courses would benefit from teaching that skill _at all_. you'd be hard pressed to find people who took that lesson away from a course.

And I think math journals would benefit from a wiki approach.

Anyone could modify the proofs and add comments/explanations.

My math teacher used to only have a tiny piece of paper with the thing he had to talk about during the lecture; since we had to prove everything we learned during this class, more often than once he couldn't remember how to prove some thing and usually happily sent a student to the blackboard to think together about how to prove the proposition. I thought that was a great way of teaching maths.

This is what Putnam seminars are like, the whole class goes through problems together. A few of them are on Youtube, maybe more will show up as everything is remote now.

As an aside, looks like one of the comments in the Math Overflow link above was made by Terence Tao himself

Oh neat!

It reminds me of this math overflow answer: https://mathoverflow.net/a/338620

I suspect the reason OP's thesis worked out okay is because his intuition wrt the problem is correct, even if his formulation was a bit off. Very cool, sounds like a good mathematician to me

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