back
19 comments
Fantastic article that sheds beautiful light on the core of modern digital computers.
What a lovely article! I happened to walk past his house in Cork last year, not having known he'd lived there. My partner couldn't work out why I was so excited, but then there's only 10 types of people in this world etc
From the article: "It is impossible that the same quality should both belong and not belong to the same thing … This is the most certain of all principles … For it is by nature the source of all the other axioms." Aristotle

And yet.. quantum mechanics managed to break the most certain of all principles. Or actually.. the man who says "I always lie", does he belong to the group of people who always lie, or does he not belong to it?

The universe we live in turns out to be an interesting place, at least.

> And yet.. quantum mechanics managed to break the most certain of all principles.

That's false. Quantum logic satisfies the principle of non-contradiction (the logical property described in the quotation from Aristotle).

The mistake you're making is misinterpreting the superposition of p and ~p, which is a combination of being p with some probability and ~p with some probability, with the simultaneous possession of p and ~p with certainty in both cases. Quantum mechanics doesn't say the latter is possible.

You are confusing "excluded middle" (¬p⋁p) with "non-contradiction" (¬(p∧¬p)). The former is not valid in e.g. constructive logic, whereas the latter is a basic statement of consistency.
My edit to correct that raced with your comment, but thanks. In any case, my point about quantum logic stands.
Or as Lewis Caroll puts it, through Tweedledee: "if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic."
Quantum Mechanics doesn't really allow for a particle to have many of a set of incompatible properties at the same time (like your example of both spin down and up).

What Quantum Mechanics actually postulates is that even talking about those properties makes little sense if somebody doesn't look at them (it's more of the tree falling with nobody to hear paradox), and once you look you may get one of the properties with some probability.

> And yet.. quantum mechanics managed to break the most certain of all principles.

Though natural language logic remains the same.

> Or actually.. the man who says "I always lie", does he belong to the group of people who always lie, or does he not belong to it?

The Russell-type paradox is more interesting than the liar-type paradoxes since it is not susceptible to the same resolutions (though can be avoided with a theory of types).

Similar to liar's paradox, may be "this page was intentionally left blank" paradox. Since the page contains this writing, it is not blank. So whoever wrote on that page "this page was intentionally left blank" was lying.

So, Aristotle holds, because the page is not "blank" and "not blank" at the same time.

That is an interesting parallel I hadn't thought about in that context. (It's better if we change the tense so that it's "this page is intentionally left blank", otherwise it could be trivially resolvable.) The "this sentence is false" type can be explained away, to an extent, but appealling to the fact that 'this sentence' mixes 'object language' and 'meta language' and we could say the paradox arises from self-reference (in a somewhat problematic recursion fashion). For "this page is intentionally left blank", 'this page' doesn't refer to itself, but indeed to the page, so it's doesn't seem susceptible to quite the same approach. (Though I suppose we could take 'this page' as elliptical for 'this page with this writing on it'.)
"I always lie" is easy to resolve: He can not always lie because that would be a contradiction. But he is currently lying. Sometimes he says the truth but currently he is not.
A slightly-less trivial version of the liar's paradox would be:

(3) Sentence (3) is false.

(Or, "What I am saying right now is a lie.")

So, is there consensus that QM does not violate the belong/not belong principle of Aristotle? Reading the comments, I wasn't sure.
I also was interested in the same quote. In what sense did quantum mechanics break this principle?
That depends a bit on how we look at it, I suppose.. usually we talk about about e.g. an electron that spins up and down at the same time, whereas in classical physics it could only do one of those things at the time. So it does two things that (classically) contradict each other. So when you look at it with a classical view it seems like the electron belongs to the group that spins up, thus it can't belong to the group that spins the opposite way.. but it does.
> usually we talk about about e.g. an electron that spins up and down at the same time, whereas in classical physics it could only do one of those things at the time. So it does two things that (classically) contradict each other.

That's because you're conflating "spin" for "angular motion" in classical physics. They just aren't the same thing, as particle spin clearly has different properties.

I'm not conflating it - I'm fully aware that it isn't angular spin. It's still called 'spin' simply because it's hard to find a term that explains what it really is. But as far as I know it's still considered separate states, in the same way that two locations are considered separate. In the quantum world it's still possible to be in both states or places at once - and that's where the quantum weirdness is. Nobody would care or talk about it if being in two spin states at the same time wasn't more strange than wearing a shoe and a tie at the same time.
> It's still called 'spin' simply because it's hard to find a term that explains what it really is.

No, it's called spin because it's a quantum mechanical form of angular momentum that simply has no analog in classical physics, and some physicists at the time suggested a physical interpretation that was consistent with an older form of quantum theory.

> In the quantum world it's still possible to be in both states or places at once - and that's where the quantum weirdness is. Nobody would care or talk about it if being in two spin states at the same time wasn't more strange than wearing a shoe and a tie at the same time.

Again, because they are conflating it with classical notions. Superposition has classical, deterministic interpretations which feature none of this apparent "weirdness".