Bell’s Theorem itself is agreed-upon academically as an experimental truth.
A theorem is a deductive truth. Only formally-proved mathematical results get to be called “theorems”. As such, a “theorem” can never be falsified by experiment.
The confusion is revealed a bit earlier in the post:
“Bell’s inequality” is that any theory of hidden local variables implies (1) + (2) >= (3). The experimentally verified fact that (1) + (2) < (3) is a “violation of Bell’s inequality”. So there are no hidden local variables. QED.
And that’s Bell’s Theorem.
Eliezer evidently thinks that “Bell’s inequality” and “Bell’s theorem” are two different things. They’re not. The theorem—the only thing that can be called such—is the mathematical statement that “any theory of hidden local variables implies (1) + (2) >= (3)”. The statement that “there are no hidden local variables [in the world as it actually exists]” is not purely mathematical—an empirical premise (“the experimentally verified fact that (1) + (2) < (3)”) was used to deduce it. -- and therefore cannot be labeled a “theorem”.
Actually, I lied slightly: there is a subtle difference between “Bell’s inequality” and “Bell’s theorem”. “Bell’s inequality” is just the bare formula “A+ B >= C” (where A,B, and C are whatever they are in the context), without the accompanying assertion that the inequality actually holds. “Bell’s theorem”, by contrast, is the statement that if A,B, and C are numbers satisfying [whatever conditions they are asserted to satisfy], then Bell’s inequality is true. As such, while it makes sense to speak of Bell’s inequality being “violated” (as it is for some triples of numbers), it does not make any logical sense to speak of “violations” of Bell’s theorem.
A theorem can never be violated. When people say, for example, “the Pythagorean theorem is violated in spherical geometry”, they are merely abusing language out of laziness. What they mean to say is that the conclusion of the Pythagorean theorem is violated. Strictly speaking, the Pythagorean theorem is not the statement that “the square of the hypotenuse is equal to the sum of the squares of the other two sides”; it is, rather, the statement “if a triangle is Euclidean and right-angled, then the square of the hypotenuse is equal to the sum of the squares of the other two sides.” The hypotheses of the theorem are not satisfied in spherical geometry in the first place, so spherical geometry does not “falsify” it or present “exceptions” to it. (Indeed, it exemplifies it as much as Euclidean geometry does, via the contrapositive: because the square of the hypotenuse of a spherical triangle does not equal the sum of the squares of the other two sides, we know, by the Pythagorean theorem, that spherical triangles are not Euclidean and right-angled!)
The violation of Bell’s inequalities rules out only theories where the state is completely local.
Theories where the the state is composed by both global and local variables, such as Bohmian mechanics, are not ruled out.
A terminological correction:
A theorem is a deductive truth. Only formally-proved mathematical results get to be called “theorems”. As such, a “theorem” can never be falsified by experiment.
The confusion is revealed a bit earlier in the post:
Eliezer evidently thinks that “Bell’s inequality” and “Bell’s theorem” are two different things. They’re not. The theorem—the only thing that can be called such—is the mathematical statement that “any theory of hidden local variables implies (1) + (2) >= (3)”. The statement that “there are no hidden local variables [in the world as it actually exists]” is not purely mathematical—an empirical premise (“the experimentally verified fact that (1) + (2) < (3)”) was used to deduce it. -- and therefore cannot be labeled a “theorem”.
Actually, I lied slightly: there is a subtle difference between “Bell’s inequality” and “Bell’s theorem”. “Bell’s inequality” is just the bare formula “A+ B >= C” (where A,B, and C are whatever they are in the context), without the accompanying assertion that the inequality actually holds. “Bell’s theorem”, by contrast, is the statement that if A,B, and C are numbers satisfying [whatever conditions they are asserted to satisfy], then Bell’s inequality is true. As such, while it makes sense to speak of Bell’s inequality being “violated” (as it is for some triples of numbers), it does not make any logical sense to speak of “violations” of Bell’s theorem.
A theorem can never be violated. When people say, for example, “the Pythagorean theorem is violated in spherical geometry”, they are merely abusing language out of laziness. What they mean to say is that the conclusion of the Pythagorean theorem is violated. Strictly speaking, the Pythagorean theorem is not the statement that “the square of the hypotenuse is equal to the sum of the squares of the other two sides”; it is, rather, the statement “if a triangle is Euclidean and right-angled, then the square of the hypotenuse is equal to the sum of the squares of the other two sides.” The hypotheses of the theorem are not satisfied in spherical geometry in the first place, so spherical geometry does not “falsify” it or present “exceptions” to it. (Indeed, it exemplifies it as much as Euclidean geometry does, via the contrapositive: because the square of the hypotenuse of a spherical triangle does not equal the sum of the squares of the other two sides, we know, by the Pythagorean theorem, that spherical triangles are not Euclidean and right-angled!)
This is incorrect.
The violation of Bell’s inequalities rules out only theories where the state is completely local. Theories where the the state is composed by both global and local variables, such as Bohmian mechanics, are not ruled out.