Most of these happen at most scales, but is more to do with the classic laws of classic logic that we accept A priori because they are useful.
* PNC: at most one is true; both can be false
* PEM, at least one is true; both can be true
* PNC + PEM, exactly one is true, exactly one is false
If you stick to the familiar computational complexity classes P=co-P, but NP != co-NP, and those both relate to the accessibility of T/F, where both P and NP are by definition decision problems, specifically the ones that can be verified in poly time.
If you ignore the Heisenberg uncertainty principle to avoid that complexity, the standard model of QM is just mapping continuous functions to discrete space, and will have problems with the above.
This happens in math too, where we use the rationals over the reals or Cauchy sequences to construct the reals, because almost all reals are normal and non-computable, and even equality between to "real" real numbers is undecidable.
This is also related to why after Weierstrass show that almost all continuous functions are nowhere smooth, we moved to the Epsilon-Delta definition of a limits etc...
We have the lier's paradox, which is easier to understand than Berry's paradox, which relates to Chaitin proof that there is an upper limit to what any algorithm can prove.
Things at Quantum scale do act very different than our typical intuition, but lots of maps from a continuous space to discrete categories can exhibit the same behavior even at macro scale, we just can often use a model that lets us ignore that to accomplish useful work.
Often we can even use repeated approximation or other methods to reduce those problems to something that is practical, but that is still the map and not the territory.
Superposition is just that:
f(a + b) = f(a) + f(b)
And:
f(sa) = s*f(a)
If you have two vectors, one at i (0,1) and one at 1 (1,0), but a map that maps only to choice(1,i) and think of that as up and down, how you divide that continuous arc from that segment of the unit circle to UP or DOWN, almost all of quantum mechanics still works, and it will be the contradictions that conflict with our intuitions that will become the barrier. (ignoring Heisenberg)
If you think about Heisenberg uncertainty as indeterminacy instead, Independence in math (Like ZFC + CH) which are neither provable as false or true, Chatlin's complexity limits, the breakdown of Laplacian determinism, or even modal logic all apply at multiple levels.
The amazing thing in my mind is that we found useful models despite these limits and using methods which are effective.
But IMHO it is best to think that in the quantum world, we aren't so lucky rather than those limits aren't still lurking under the surface at the macro scale, which they very much are.
* PNC: at most one is true; both can be false
* PEM, at least one is true; both can be true
* PNC + PEM, exactly one is true, exactly one is false
If you stick to the familiar computational complexity classes P=co-P, but NP != co-NP, and those both relate to the accessibility of T/F, where both P and NP are by definition decision problems, specifically the ones that can be verified in poly time.
If you ignore the Heisenberg uncertainty principle to avoid that complexity, the standard model of QM is just mapping continuous functions to discrete space, and will have problems with the above.
This happens in math too, where we use the rationals over the reals or Cauchy sequences to construct the reals, because almost all reals are normal and non-computable, and even equality between to "real" real numbers is undecidable.
This is also related to why after Weierstrass show that almost all continuous functions are nowhere smooth, we moved to the Epsilon-Delta definition of a limits etc...
We have the lier's paradox, which is easier to understand than Berry's paradox, which relates to Chaitin proof that there is an upper limit to what any algorithm can prove.
Things at Quantum scale do act very different than our typical intuition, but lots of maps from a continuous space to discrete categories can exhibit the same behavior even at macro scale, we just can often use a model that lets us ignore that to accomplish useful work.
Often we can even use repeated approximation or other methods to reduce those problems to something that is practical, but that is still the map and not the territory.
Superposition is just that:
And: If you have two vectors, one at i (0,1) and one at 1 (1,0), but a map that maps only to choice(1,i) and think of that as up and down, how you divide that continuous arc from that segment of the unit circle to UP or DOWN, almost all of quantum mechanics still works, and it will be the contradictions that conflict with our intuitions that will become the barrier. (ignoring Heisenberg)If you think about Heisenberg uncertainty as indeterminacy instead, Independence in math (Like ZFC + CH) which are neither provable as false or true, Chatlin's complexity limits, the breakdown of Laplacian determinism, or even modal logic all apply at multiple levels.
The amazing thing in my mind is that we found useful models despite these limits and using methods which are effective.
But IMHO it is best to think that in the quantum world, we aren't so lucky rather than those limits aren't still lurking under the surface at the macro scale, which they very much are.