I feel like one of the lucky 10,000 today! Thanks for asking.

Jump up an abstraction layer.

Multiplication and addition are each commutative, but performing multiplication followed by addition is not the same as addition followed by multiplication (in the real numbers), so they don't commute. Said another way, the operation of composing multiplication with addition isn't commutative.

Similarly, we can perform various operations on random variables, one being expectation value and another being multiplication (or conjunction): E(X•Y) ≠ E(x)•E(y) unless x and y are independent, so E and • don't commute.

When we say "commute" we often are directly or indirectly thinking of commutative diagrams, which capture a very general which of commutativity and allows us to precisely write down all the above.

Fun fact: associativity is also just commutativity of binary operator composition.

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ziofill 3 days ago | parent [-]

Yes I know addition and multiplication don't commute, but what does that have to do with the discussion above? Do you mean that repeated measurements on different systems might not be independent because they come from the same source?

	▲	xelxebar a day ago \| parent [-]
		> Do you mean that repeated measurements on different systems might not be independent because they come from the same source? More that the measurement settings could be correlated somehow. So called "cosmic Bell tests" try to push back how far said correlations would have to be, though, by determining measurement settings from distant antipodal astronomical objects, e.g. see the famous Rauch paper [0]. On the surface it seems a bit absurd to consider conspiracies from 7 billion years ago, but that's the whole deal with conservation laws, which introduce correlations between otherwise free parameters and constrain the state space to some submanifold. [0]:https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.12...