Hm. I think you think the robot works differently than it actually does.
My guess for what you were going to say was that you wanted the robot to work in this different way, but nopes.
A robot that predicts the next number in a series is something like a minimum message-length predictor. The robot outlined here actually can’t do that.
Instead, the robot attacks each problem based on theorems that require or rule out different combinations of relevant possibilities. So, for example, in the question “how many smooth structures are there in n dimensions” (thank you for making that example clear btw), the robot would for each separate case try to prove what was going on, and if it failed, it would likely do something dumb (like run out of time, or pick the first thing it hadn’t proven a theorem about) because it doesn’t handle infinity-option problems very well. What it wouldn’t do is try and predict the answer based on the answer for smaller dimensions, unless it could prove theorems connecting the two.
Okay, but say the robot in ten seconds manages to prove the theorem: “For all dimensions n not equal to 4, there is exactly one smooth structure on R^n.” But is unable to produce a result regarding n=4? Or more realistically, say it is able to construct 1 smooth structure on R^n for any n, and can prove that no others exist unless n=4. How does it make guesses in the case n=4?
If we want to live in the least convenient possible world assume that in second 1 it constructs the smooth structures; it takes three seconds to prove that there are no more for n>5, three seconds to prove no more for n=1,2, and three more seconds to prove there are no more for n=3 and runs out of time. These results are obtained incidentally from inequalities that arise when pursuing a proof for n=4, which is the central value of some equation at the core of the proofs. (so the proofs really say “if another smooth structure exists, it exists for n<5, 2<n<5, 3<n<5.”)
If it really can’t prove any theorems that directly include the translation of “the number of smooth structures for n=4 is,” it simply won’t ever update that.
Hm. I think you think the robot works differently than it actually does.
My guess for what you were going to say was that you wanted the robot to work in this different way, but nopes.
A robot that predicts the next number in a series is something like a minimum message-length predictor. The robot outlined here actually can’t do that.
Instead, the robot attacks each problem based on theorems that require or rule out different combinations of relevant possibilities. So, for example, in the question “how many smooth structures are there in n dimensions” (thank you for making that example clear btw), the robot would for each separate case try to prove what was going on, and if it failed, it would likely do something dumb (like run out of time, or pick the first thing it hadn’t proven a theorem about) because it doesn’t handle infinity-option problems very well. What it wouldn’t do is try and predict the answer based on the answer for smaller dimensions, unless it could prove theorems connecting the two.
Okay, but say the robot in ten seconds manages to prove the theorem: “For all dimensions n not equal to 4, there is exactly one smooth structure on R^n.” But is unable to produce a result regarding n=4? Or more realistically, say it is able to construct 1 smooth structure on R^n for any n, and can prove that no others exist unless n=4. How does it make guesses in the case n=4?
If we want to live in the least convenient possible world assume that in second 1 it constructs the smooth structures; it takes three seconds to prove that there are no more for n>5, three seconds to prove no more for n=1,2, and three more seconds to prove there are no more for n=3 and runs out of time. These results are obtained incidentally from inequalities that arise when pursuing a proof for n=4, which is the central value of some equation at the core of the proofs. (so the proofs really say “if another smooth structure exists, it exists for n<5, 2<n<5, 3<n<5.”)
If it really can’t prove any theorems that directly include the translation of “the number of smooth structures for n=4 is,” it simply won’t ever update that.
Well it can prove “the number of structures for n=4 is at least 1.”