We consider the nature of quantum randomness and how one might have empirical evidence for it. We will see why, depending on one's computational resources, it may be impossible to determine whether a particular notion of randomness properly characterizes one's empirical data. Indeed, we will see why even an ideal observer under ideal epistemic conditions may never have any empirical evidence whatsoever for believing that the results of one's quantum-mechanical experiments are randomly determined. This illustrates a radical sort of empirical underdetermination faced by fundamentally stochastic theories like quantum mechanics.Isn't this obvious?

A lot of people say that quantum mechanics shows that the world is intrinsically random, or objectively random, or some such nonsense. There is no empirical support for such statements. For one thing, there could be a superdeterminism that makes nothing random.

We say that coin tosses are random, because nobody goes to the trouble of tracking all the variables needed to predict the outcome.

We say radioactive decay is random, because there is no known way of predicting the precise decay time. But it seems possible that we could, if we knew more about about the state of nucleus in question.

The paper discusses tests for coin toss sequences to appear random, but we have no way of recognizing intrinsic randomness even if we saw it.

It is a strange day for irony, when one can use an argument that basically goes:

"If only I knew EVERYTHING about something then I could have certainty in my randomness."

... and keep a straight face.

Roger, the man you love to ridicule had it right.

" ... as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

Geometry and Experience, Lecture before the Prussian Academy of Sciences, January 27, 1921

Correct. Randomness is an abstract, mathematical notion that does not directly exist.

ReplyDeleteThe idealization involved in the concept of randomness starts from the idea of an absence of *some* form of order, and then, in the limit, it goes to the idea of an absence of *every* form of order.

Taken in this ``purest'' form, randomness is a purely mathematical concept; it is a concept of method alone. It cannot concretely exist for a similar set of reasons that objects of zero or infinite size cannot.

Thus, first, it's best to begin with relative degrees of randomness.

For instance, a complete polynomial of degree 7 is more complex than another with a degree 3. Hence, predicting the next number in a discrete data-series generated using a 7-degree polynomial, is more difficult; it takes more computational effort; the data series generated shows less order in the successive numbers. Hence, we can say that a 7-degree polynomial-generated series shows more randomness than a 3-degree one.

From this viewpoint, generation of a purely random sequence would require an infinite-degree polynomial.

Those who say that perfect randomness *physically* exists---whether at the level of stat. mech. or QM, or QFT---are welcome to keep themselves engaged with infinite-degree polynomials, and shut their mouth up in the meanwhile.

(Guess the same argument can also be made using the Fourier transform, but polynomials are simpler, and I guess, equally powerful (if not more) for the purpose.)

Best,

--Ajit