Quantum mechanics seems to suggest that there really is such a thing as a random number, yet all of philosophy and logic point to a reason or cause for everything, perhaps beyond our understanding. Is this notion of a random number just another demonstration of limited human understanding?

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I guess I'd have to disagree

I guess I'd have to disagree with the idea that "all of philosophy and logic point to a reason or cause for everything." There's certainly no argument from logic as such; it's perfectly consistent to say that some events are genuinely random. Some philosophers have held that there's a reason (not necessarily a cause in the physical sense, BTW) for everything, but the arguments are not very good.

On the other hand... quantum mechanics is a remarkably well-confirmed physical theory that, at least as standardly interpreted, gives us excellent reason to think that some things happen one way rather than another with no reason or cause for which way they turned out.

An example: suppose we send a photon (a quantum of light) through a polarizing filter pointed in the vertical direction. We let the photon travel to a second polarizing filter, oriented at 45 degrees to the vertical. Quantum theory as usually understood says that there's a 50% chance that the photon will pass this filter and a 50% chance that it won't. But quantum theory itself provides no account whatsoever of which will actually happen. And on the usual interpretation, there is no reason or cause; it's really random.

Now the usual interpretation of quantum mechanics could be wrong. There are deterministic interpretations, most notably "many worlds" or Everettian quantum mechanics, and Bohmian mechanics. No one is in a position to rule either of those out; all I can say is that neither of those approaches is to my taste. But even though both of them restore determinism, that's not really their motivation. Most people who work in foundations of physics are not bothered by the very idea of indeterminism and in fact, indeterminism wasn't by any means Einstein's biggest issue with quantum mechanics.

So to sum up: I don't think there are any good general arguments against randomness. I think the concept is coherent, and that it's a plausible fit for our best physical theory. It also happens to suit my own prejudices about quantum mechanics, but that's just icing on the cake. ;-)

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