Isn’t it about

empirical evidencethat these problems are hard, not “predictions”? They’re considered hard because many people have tried to solve them for a long time and failed.

No, this is Preemption 1 in the Original Post.

“hard” doesn’t mean “people have tried and failed”, and you can only witness the latter *after the fact*. If you prefer, even if have empirical evidence for the problem being “level n hard” (people have tried up to level n), you;d still do not have empirical evidence for the problem being “level n+1 hard” (you’d need people to try more to state that if there’s nothing you can say about it ahead of time). Ie, no predictive power.

An expert can estimate how hard a problem is by eyeballing how distant the abstractions needed to solve it feel from the known ones

Great! We’re getting closer to what I care about.

Then what I am saying is that there is a heuristic that the experts are using to eyeball this, and I want to know what that is, start ingwith those 2 conjectures!

I am also saying that the more distant “the abstraction need to solve it feel from the known ones”, the easier it should be to do so.

They’re able to do this because they’ve developed strong intuitions for their professional domain: they roughly know what’s possible, what’s on the edge of the possible, and what’s very much not.

Exactly, but those intuitions are implemented *somehow*. How?

Also, the more experts agree on a judgment, and the stronger their judgment, the easier you expect to be to explain that intuition.

But there’s no

objectiveproperty that makes these problemsintrinsicallyhard,only subjectively hard from the point of view of our conceptual toolbox.

I was confused very confused when I read this. For instance, the part in bold is already reflected in the Original Post.

There are many theoretical problems that are considered to be obviously far far outside

our problem solving ability.

If you prefer, interpret “*intrinsically* hard” as “having an *intrinsic* property that makes it *subjectively hard* for us. To model how that would look, consider the following setup:

The space of problems is just the real plane, and our ability to solve problems is modeled by a unit disk in the plane (if in the disk, solved, and the closer it is to the disk, the easier it is to solve). Then, the difficulty of a problem is subjective, it depends on where the disk is.

But let’s say the disk is somewhere on the x axis, then the *intrinsic* property of a problem being far having a high y coordinate, makes it *subjectively hard*.

I’ll make some edits to the post. I thought most of this was clear because of Preemption #1, but it was not.

Coin = problem

Flipping head = not being solved

Flipping tail = being solved

More flips = more time passing

Then,

yes. Because you had many other coins that had started flipping tail at some point, and there is no easily discernable pattern.By your interpretation, the Solomonoff induced prior for that coin is basically “it will never flip tail”. Whereas, you do expect that most problems that have not been solved now will be solved at some point, which does mean that you are incorporating more knowledge.

Experts from many different fields of Maths and CS have tried to tackle the Collatz’ Conjecture and the P vs NP problem. Most of them agree that those problems are way beyond what they set out to prove. I mostly agree with you on the fact that each expert’s intuition vaguely tracks one specific dimension of the problem.

But any simplicity prior tells you that it is more likely for there to be a general reason for why those problems are hard along

all those dimensions, rather than a whole bunch of ad-hoc reasons.What makes you think that? I see you repeating this, but I don’t see why that would be the case.

Good question, thanks! I tried to hint at this in the Original Post, but I think I should have been more explicit. I will make a second edit that incorporates the following.

The first reason is that many different approaches have been tried. In the case where only a couple of specific approaches have been tried, I expect the reason for why it hasn’t been solved to be ad-hoc and related to the specific approaches that have been tried. The more approaches are tried, the more I expect a general reason that applies to all those approaches.

The second reason is that the problems are simple. In the case of a complicated problem, I would expect the reason for why it hasn’t been solved to be ad-hoc. I have much less of this expectation for simple problems.