It depends on the framework. I can tell a geometric figure is a square because it’s a quadrilateral with right angles and sides of equal length. You could ask me a question like “Is a rectangle with a side of length 1 and a diagonal of root 2 a square?” and I can tell it is.
Ask me “Was 1 1 1 1 produced by a random process?” and it’s impossible to tell in the way I did with the square.
> It depends on the framework. I can tell a geometric figure is a square because it’s a quadrilateral with right angles and sides of equal length.
You're claiming to be able to craft a mathematical proof with 100% certainty. Although the thing you are proving appears to be obviously true (assuming a certain mathematical framework), the probability that you made a mistake is not 0%. You might falsely believe that the probability of making a mistake in a simple proof like this is 0%, but you would be wrong, and we have plenty of historical examples of mathematicians "proving" something and thinking that there is 0% chance of errors in the proof, only later being shown that they were incorrect.
You’re mixing up two layers of uncertainty. There’s an outer uncertainty. This would include things like I made a mistake, this is all a dream, etc. This outer uncertainty pervades all problems.
It’s often useful to ignore that outer uncertainty. We create a framework where we take certain things as true (shared reality, mathematical axioms). This framework may or may not have uncertainty inside of it, which we could call inner uncertainty.
Questions of probability have inner uncertainty. Questions of geometry do not. This makes them qualitatively different.
If you frame the initial task as something like “do your best to lead people to believe your sequence is random”, that makes sense. If the task is “make it so they can’t tell if it’s random”, that’s a bit off in some way. At the very least, it’s because you’ve presented the spotting of randomness as something that can truly be done to a logical conclusion (random/not or true/false). This violates both the outer and inner uncertainties of randomness.
Interestingly, the article computes the odds incorrectly: "... hit the same number on seven consecutive spins [...] the odds of which happening are 114billion to one...", which actually are the odds of having 7 consecutive 19s or the same (unspecified) number on 8 consecutive spins.
Ask me “Was 1 1 1 1 produced by a random process?” and it’s impossible to tell in the way I did with the square.