Quantum physics and principle of non-contradiction

Question

We all know that mathematics is based on axioms in the science of logic. These axioms we proceed through to prove the validity or falsehood of any statement. One of the most important of these axioms is the principle of non-contradiction, which is to assume that something false is true , then we reach a contradiction, so we conclude that it is wrong. Many of those who introduce popular science and some physicists say that quantum physics allow the contradiction to occur, for example in the so-called Schrödinger's cat,As they say the cat is alive and dead together at the same time which oppose the law of non-contradiction. And alot of similar examples. Also when they talk about a superposition Which is the sum of two vectors in a Hilbert space they also say the electron spin up and spin down at the same time and in the same sense as an interpretation for quantum superposition , so how true are these allegations? and Can any experiment contradict the laws of mathematical logic?

Answer 1

Schrodinger's paradox is resolved by the fact that the cat exists in a superposition of states which contradicts determinism. We cannot determine whether or not the cat is alive until we observe the cat. This is not the same as a mathematical inconsistency which would allow us to prove that the cat is alive and to prove that the cat is dead.

You asked "Can any experiment contradict the laws of mathematical logic?". The answer is no. What can be determined by an experiment is whether or not a mathematical model is a valid description of a physical phenomenon.

Suppose we lived in a universe where we could prove, experimentally, that straight lines don't exist. Every theorem of Euclidean geometry would still be true. What may no longer be true is that Euclidean geometry is a good model for physical space.

Answer 2

It is incorrect to assert that Schrödinger's cat or some subatomic particle is in a state $|\phi\rangle$of being $|0\rangle$ and $|1\rangle$ at the same time. Every such cat or particle is in a state that is a linear combination of two distinct vectors: $|0\rangle$ and $|1\rangle$. To say the cat is both fully dead and fully alive is to assert the following two statements:

$1.$ $|\phi\rangle = 1|0\rangle + 0|1\rangle = |0\rangle + |0\rangle = |0\rangle$

$2.$ $|\phi\rangle = 0|0\rangle + 1|1\rangle = |0\rangle + |1\rangle = |1\rangle$

which would lead to the contradiction $|0\rangle = |1\rangle$ by substitution. Hence, at least one of the above statements is false.

When the cat is in a state of superposition, its state is represented as follows

$ |\phi\rangle = \alpha|0\rangle + \beta|1\rangle$

such that $\alpha \neq 0, \beta \neq 0$ and $|\alpha|^2 + |\beta|^2 = 1$. According to the math, the best description of the cat's state is that the cat is not both fully dead and fully alive but partially dead and partially alive, and the extent to which it is either one depends on the corresponding amplitude $\alpha$ or $\beta$. In other words, the state of the cat is some combination of dead and alive. Now, this does not imply the cat is in the process of dieing or recovering. Whatever it means, it's definitely unfamiliar because the cat partially occupies two states we normally consider mutually exclusive. To make the weirdness more apparent, imagine partially eating a hamburger in America while talking to your mom while also partially sipping soup in China while watching a movie... and you're doing all of this at the same time. How can this be?

I don't think anyone truly understands what the math is pointing at. If you're familiar with Plato's allegory of the cave, the math is a shadow on the wall. It happens to be a good description of whatever is actually there (i.e. a model of reality), and it's even good at making predictions, but the math is not the substance. The true forms casting the shadows are out of sight.