Okay, that is probably not that good a characterization. However, I do like when someone figures out a simple way of looking at problems that have gone unsolved and so thought to be very difficult, so therefore must be really complicated.

Okay, that is probably not that good a characterization.

I appreciate the caveat, but I’m actually not seeing the connection at all. What is the relationship you see between common sense and surprisingly simple solutions to problems?

In my experience the phrase is generally used in situations where everyone is approaching a situation as if it must have some complicated and complex solution. They seem to get caught up with thinking just because we have not seen the solution it’s not ultimately can have a rather simple solution approach.

The common sense is to not over complicate and get in to all kinds of convoluted thinking.

Another case of “common sense” not being common?

Short version:

If there were enough problems like this, where it’s “just common sense to solve” then the time it takes could be indicative of:

Few people with common sense

Plenty of common sense dispersed over lots of problems = slow progress on average, but lots of total progress

Problems aren’t not widely heard of (until they’re solved)

Problems are encoded in technical jargon which makes understanding them, much less solving them, more time consuming.

Theorems are encoded in technical jargon which makes knowing which one is the tool you’re looking for hard.

Long version:

Yes and no.
Yes, offhand I can’t think of a website which has a list of open problems (particularly ones that seem solvable[3]) and a brief summary of the latest research/work on them which outlines what one or two things need to be done to solve it, cross-referenced with a list of what different theorems/math tools do, at a high enough level that this match up would be easy. [1]

No, in that this ‘problem’ probably isn’t easy to for one person[2] to solve—prior to reading that I had

a) never heard of Boolean sensitivity,

b) the other things on it I googled quickly, that existed before this result did not explain it well (whereas that paper did an excellent job),

a) I had never heard of the the Cauchy interlace/interlacing theorem, and

b) searching it got similar results to 1b. - that paper did an excellent job of explaining one thing you can use that theorem for, whereas other stuff was about eigenvectors, compression, orthogonal projection and Hilbert Spaces.

(I don’t know what Hilbert Spaces are, but if that’s just a fancy way of saying “a circuit” or “a graph”, *rather than a consequence of Hilbert Spaces being a more general category that includes them, then I am surprised/disappointed that such vague terminology is in use.
I probably learn more about uses for high level theorems by reading papers in quantamagazine.org than I will by trying to understand them more straightforwardly.)

[1] It’s entirely possible that this isn’t a case of “this theorem says that a totally black surface outputs radiation” + “the event horizon of a black hole is a totally black surface” but is more of a “We need to prove A” + “A is a consequence of this theorem that isn’t about circuits, but a way more general class that includes circuits”.

[2] The pool of people who have that knowledge isn’t as wide as it could be.

[3] This might be the kicker—when someone finds an open math problem and goes “I could solve this”, do they share it?

Huang added the sensitivity conjecture to a “secret list” of problems he was interested in, and whenever he learned about a new mathematical tool, he considered whether it might help. “Every time after I’d publish a new paper, I would always go back to this problem,” he said. “Of course, I would give up after a certain amount of time and work on some more realistic problem.”

No. To be fair, if I was in his shoes, I probably wouldn’t either—and he knows more about how one goes about publishing results like this than I do, and the incentives. (Which is to say, if I did have such a list[5], I probably wouldn’t go around telling people what was on it. Part of having a comparative advantage[4] on a problem is there not being lots of other people working on it.)

The key thing here is a desire to solve/be known as the solver of a/the problem. A desire for it to be answered would drive one to both try to solve it, and spread/share info. about it. This is in contrast to problems with applications or open bounties, where it may be an extrinsic reward driving interest—although considering he got a grant to solve it, in academia, the line may be less clear.

[4] This might not be the right term—it’s not about being better at producing a good, but at achieving a goal first.

[5] Especially if it was a short list. The key thing is a desire to solve/be known as the solver of a/the problem.

## [Question] Another case of “common sense” not being common?

Okay, that is probably not that good a characterization. However, I do like when someone figures out a simple way of looking at problems that have gone unsolved and so thought to be very difficult, so therefore must be really complicated.

If you didn’t see this:

https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/

I appreciate the caveat, but I’m actually not seeing the connection at all. What is the relationship you see between common sense and surprisingly simple solutions to problems?

In my experience the phrase is generally used in situations where everyone is approaching a situation as if it must have some complicated and complex solution. They seem to get caught up with thinking just because we have not seen the solution it’s not ultimately can have a rather simple solution approach.

The common sense is to not over complicate and get in to all kinds of convoluted thinking.

It was sounds something like a pre-existing solution (as opposed to the start of a completely new field).

Short version:

If there were enough problems like this, where it’s “just common sense to solve” then the time it takes could be indicative of:

Few people with common sense

Plenty of common sense dispersed over lots of problems = slow progress on average, but lots of total progress

Problems aren’t not widely heard of (until they’re solved)

Problems are encoded in technical jargon which makes understanding them, much less solving them, more time consuming.

Theorems are encoded in technical jargon which makes knowing which one is the tool you’re looking for hard.

Long version:

Yes and no. Yes, offhand I can’t think of a website which has a list of open problems (particularly ones that seem solvable[3]) and a brief summary of the latest research/work on them which outlines what one or two things need to be done to solve it, cross-referenced with a list of what different theorems/math tools do, at a high enough level that this match up would be easy. [1]

No, in that this ‘problem’ probably isn’t easy to for one person[2] to solve—prior to reading that I had

a) never heard of Boolean sensitivity,

b) the other things on it I googled quickly, that existed before this result did not explain it well (whereas that paper did an excellent job),

a) I had never heard of the the Cauchy interlace/interlacing theorem, and

b) searching it got similar results to 1b. - that paper did an excellent job of explaining one thing you can use that theorem for, whereas other stuff was about eigenvectors, compression, orthogonal projection and Hilbert Spaces.

(I don’t know what Hilbert Spaces are, but if that’s just a fancy way of saying “a circuit” or “a graph”, *rather than a consequence of Hilbert Spaces being a more general category that includes them, then I am surprised/disappointed that such vague terminology is in use. I probably learn more about uses for high level theorems by reading papers in quantamagazine.org than I will by trying to understand them more straightforwardly.)

[1] It’s entirely possible that this isn’t a case of “this theorem says that a totally black surface outputs radiation” + “the event horizon of a black hole is a totally black surface” but is more of a “We need to prove A” + “A is a consequence of this theorem that isn’t about circuits, but a way more general class that includes circuits”.

[2] The pool of people who have that knowledge isn’t as wide as it could be.

[3] This might be the kicker—when someone finds an open math problem and goes “I could solve this”, do they share it?

No. To be fair, if I was in his shoes, I probably wouldn’t either—and he knows more about how one goes about publishing results like this than I do, and the incentives. (Which is to say, if I did have such a list[5], I probably wouldn’t go around telling people what was on it. Part of having a comparative advantage[4] on a problem is there not being lots of other people working on it.)

The key thing here is a desire to solve/be known as the solver of a/the problem. A desire for it to be answered would drive one to both try to solve it, and spread/share info. about it. This is in contrast to problems with applications or open bounties, where it may be an extrinsic reward driving interest—although considering he got a grant to solve it, in academia, the line may be less clear.

[4] This might not be the right term—it’s not about being better at producing a good, but at achieving a goal

first.[5] Especially if it was a short list. The key thing is a desire to solve/be known as the solver of a/the problem.

You make many good observations/points that I agree with.

The quoted bullet is really related to the main reaction I had.