[...] But what does the integral over squared moduli have to do with anything? On a straight reading of the data, you would always find yourself in both blobs, every time. How can you find yourself in one blob with greater probability? What are the Born probabilities, probabilities of? Here’s the map—where’s the territory?
This problem is even worse than it looks, because the squared-modulus business is the only non-linear rule in all of quantum mechanics. Everything else—everything else—obeys the linear rule that the evolution of amplitude distribution A, plus the evolution of the amplitude distribution B, equals the evolution of the amplitude distribution A + B.
When you think about the weather in terms of clouds and flapping butterflies, it may not look linear on that higher level. But the amplitude distribution for weather (plus the rest of the universe) is linear on the only level that’s fundamentally real.
Does this mean that the squared-modulus business must require additional physics beyond the linear laws we know—that it’s necessarily futile to try to derive it on any higher level of organization?
[...] But, said Scott, we might encounter future evidence in favor of single-world quantum mechanics, and many-worlds still has the open question of the Born probabilities.
This is indeed what I would call the fallacy of privileging the hypothesis. There must be a trillion better ways to answer the Born question without adding a collapse postulate that would be the only non-linear, non-unitary, discontinous, non-differentiable, non-CPT-symmetric, non-local in the configuration space, Liouville’s-Theorem-violating, privileged-space-of-simultaneity-possessing, faster-than-light-influencing, acausal, informally specified law in all of physics. Something that unphysical is not worth saying out loud or even thinking about as a possibility without a rather large weight of evidence—far more than the current grand total of zero.
But because of a historical accident, collapse postulates and single-world quantum mechanics are indeed on everyone’s lips and in everyone’s mind to be thought of, and so the open question of the Born probabilities is offered up (by Scott Aaronson no less!) as evidence that many-worlds can’t yet offer a complete picture of the world. Which is taken to mean that single-world quantum mechanics is still in the running somehow.
In the minds of human beings, if you can get them to think about this particular hypothesis rather than the trillion other possibilities that are no more complicated or unlikely, you really have done a huge chunk of the work of persuasion. Anything thought about is treated as “in the running,” and if other runners seem to fall behind in the race a little, it’s assumed that this runner is edging forward or even entering the lead.
[… O]ur uncertainty about where the Born statistics come from should be uncertainty within the space of quantum theories that are continuous, linear, unitary, slower-than-light, local, causal, naturalistic, et cetera—the usual character of physical law. Some of that uncertainty might slop outside the standard space onto theories that violate one of these standard characteristics. It’s indeed possible that we might have to think outside the box. But single-world theories violate all these characteristics, and there is no reason to privilege that hypothesis.
The main claims Eliezer is criticizing in the QM sequence are that (1) reifying QM’s complex amplitudes runs afoul of Ockham’s Razor, (2) objective collapse is a plausible explanation for the Born probabilities, (3) QM shows that reality is ineffable, and (4) QM shows that there’s no such thing as reality. I don’t know what question of fact you think the Quantum Bayesians and Eliezer disagree about, or what novel factual claim QB is making. (I assume we agree ‘physical formalisms can be useful tools’ and ‘we can use probability theory to think about strength of belief’ aren’t novel claims.)
No, Eliezer talks about this at some length. See The Born Probabilities:
And Privileging the Hypothesis:
The main claims Eliezer is criticizing in the QM sequence are that (1) reifying QM’s complex amplitudes runs afoul of Ockham’s Razor, (2) objective collapse is a plausible explanation for the Born probabilities, (3) QM shows that reality is ineffable, and (4) QM shows that there’s no such thing as reality. I don’t know what question of fact you think the Quantum Bayesians and Eliezer disagree about, or what novel factual claim QB is making. (I assume we agree ‘physical formalisms can be useful tools’ and ‘we can use probability theory to think about strength of belief’ aren’t novel claims.)