Suppose I take out a coin and flip it 100 times in front of your eyes, and it lands heads every time. Will you have no ability to predict how it lands the next 30 times? Will you need some special domain knowledge of coin aerodynamics to predict this?
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 have a precise model of the known subset of the concept-space of their domain, and they can make vague high-level extrapolations on how that domain looks outside the known subset, and where in the wider domain various unsolved problems are located, and how distant they are from the known domain”
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.
The way I see it, that’s it. This statement isn’t reducible to something more neat and simple.
For any given difficult problem, you can walk up to an expert and ask them why it’s considered hard, but the answers they give you won’t have any unifying theme aside from that. It’s all ad hoc.
What makes you think that? I see you repeating this, but I don’t see why that would be the case.
Why would you think there’s something else?
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.
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.