A note on the description complexity of physical theories

No.

What I am saying is to compute the answer using quantum mechanics.

The way to do it correctly, Copenhagen style, is to say "okay, the electron goes through the plate with two holes in it. But since, from the perspective of the electron, it can't go through two holes, the state of the electron on the other side should be entangled something like |10> + |01>. If we fast-forward to the screen, we get an interference pattern"

The way to do it correctly, MW style, is to say "okay, the electron has equal probability of going through each hole, so let's split into two worlds with equal phase. The detector will then observe the superposition of the two worlds, something like |10> + |01>, except fast-forwarded to the screen, so there should be an interference pattern."

If these two approaches look similar, there's a reason. And it's not that one is cribbing off the other! As you can see, introducing entanglement in the Copenhagen interpretation was definitely not arbitrary, but it is conceptually trickier than thinking through the same process using the MWI.

The analogous question is "when does branching occur?" Your predictions in MWI depend on what notion of "observation" you use, it is only less apparent in the formulation. To obtain some meaningful prediction, at least you have to specify what are the observables and what subsystem of the world corresponds to the observer and his mind states.

But I am not completely sure what you are speaking about. Maybe you can give a concrete example where MWI gives a unique answer while collapse formulation doesn't?

My understanding is that, whatever version of Copenhagen you adhere to, you no longer believe that there is another version of the scientist somewhere who has measured 1.

Depends. From the outside observer's point of view, he can be in a superposition of [has measured 0] and [has measured 1]. From the scientist's point of view, the collapse has happened.

...but then I have absolutely no idea what distinguishes Copenhagen from many worlds.

That's why it's called "interpretation". It's the way how we speak about it, and some untestable statements about consciousness perhaps with some philosophical implications, which make the whole difference. Of course, an objective collapse is a different thing, but I don't believe much Copenhagenists today believe in an objective collapse.

According to many worlds, there are multiple versions of earth---one in which the scientist observed 0, and one in which the scientist observed 1.

Such statement can be misleading. There is one version of Earth, but the individual observers see only certain projections. The difference between MWI and single-world interpretations is that MWI says that all projections are experienced.

A note: I am pretty sure that paul's claim (that MWI predicts more interference than typical QM) is false, and comes from not considering entanglement (which is understandable, because entanglement is hard). For example, if you collapsed the wavefunction of an electron in the two slit experiment improperly (by not keeping the state entangled after going through the slits), you would predict no interference.

if you have an arbitrary program X from the space of all programs that generate an output Y, there can only be a finite number of programs that generate that output more quickly.

Amusingly, this statement is false. If a program Z is faster than X, then there exist infinitely many versions of Z that also run faster than X: just add some never-executed code under an if(false) branch. I'm not sure whether your overall argument can be salvaged.

Ah, but which Hamiltonian mechanics is the true one: the one that says real numbers are infinite binary expansions, or the one that says real numbers are Dedekind cuts? I dunno, your way of thinking makes me queasy.

Sorry - I wrote an incorrect reply and deleted it. Let me think some more.

That point of view has far-reaching implications that make me uncomfortable. Consider two physical theories that are equivalent in every respect, except they use different definitions of real numbers. So they have a common part C, and theory A is the conjunction of C with "real numbers are Dedekind cuts", while theory B is the conjunction of C with "real numbers are infinite binary expansions". According to your and Eliezer's point of view as I understand it right now, at most one of the two theories can be "true". So if C (the common part) is "true", then ordinary logic tells us that at most one definition of the real numbers can be "true". Are you really, really sure you want to go there?

Clever. But ...

If theories A and B pay rent on the same house, then the theory (A OR B) pays enough rent so that the stronger theory A need pay no additional rent at all. Yet you seem to prefer A to B, and also to (A OR B).

Which brings us back to an issue which I was debating here a couple of weeks ago: Is there a difference between an event being impossible, and an event being of measure zero?

Orthodox Bayesianism says there is no difference and strongly advises against thinking either to be the case. I'm wondering whether there isn't some way to make the idea work that there is a distinction to be made - that some things are completely impossible given a theory, while other things are merely of infinitesimal probability.

...if the length of the prefix is non-negligible compared to the length of the program, then we can get misleading results.

For the purposes of complexity comparisons the prefix should be held constant across the algorithms. You should always be comparing algorithms in the same language.

However, I don't agree that there is a principled way of choosing the right representation.

You are correct. I only use phrases such as "the essential entities and relationships of the domain" to outline the nature of the problem.

The problem with comparing the complexity of QM interpretations is that our representation of those interpretations is arbitrary. We can only guess at the proper representation of QM. By choosing different representations we could favor one theory or the other as the most simple.

This is related to a point that I attempted to make previously. You can measure complexity, but you must pick the context appropriately.

If he shares all the data and priors with external observers, his posterior probability of MWI being true should end up the same as theirs.

No; I think you're using the Aumann agreement theorem, which can't be used in real life. It has many exceedingly unrealistic assumptions, including that all Bayesians agree completely on all definitions and all category judgements, and all their knowledge about the world (their partition functions) is mutual knowledge.

In particular, to deal with the quantum suicide problem, the reasoner has to use an indexical representation, meaning this is knowledge expressed by a proposition containing the term "me", where me is defined as "the agent doing the reasoning". A proposition that contains an indexical can't be mutual knowledge. You can transform it into a different form in someone else's brain that will have the same extensional meaning, but that person will not be able to derive the same conclusions from it, because some of their knowledge is also in indexical form.

(There's a more basic problem with the Aumann agreement theorem - when it says, "To say that 1 knows that 2 knows E means that E includes all P2 in N2 that intersect P1," that's an incorrect usage of the word "knows". 1 knows that E includes P1(w), and that E includes P2(w). 1 concludes that E includes P1 union P2, for some P2 that intersects P1. Not for all P2 that intersect P1. In other words, the theorem is mathematically correct, but semantically incorrect; because the things it's talking about aren't the things that the English gloss says it's talking about.)

I was actually going off the idea that the vast majority - 100% minus pr(survive all suicides) - of worlds would have the subject dead at some point, so all those worlds would not be convinced. Sure, people in your branch might believe you, but in (100 - 9.3x10^-302) percent of the branches, you aren't there to prove that quantum suicide works. This means, I think, that the chance of you existing to prove that quantum suicide proves MWI to the rest of the world, the chance is equal to the chance of you surviving in a nonMWI universe.

I was going to say well, if you had a test with a 1% chance of confirming X and a 99% chance of disconfirming X, and you ran it a thousand times and made sure you presented only the confirmations, you would be laughed at to suggest that X is confirmed - but it is MWI that predicts every quantum event comes out every result, so only under MWI could you run the test a thousand times - so that would indeed be pretty convincing evidence that MWI is true.

Also: I only have a passing familiarity with Robin's mangled worlds, but at the power of negative three hundred, it feels like a small enough 'world' to get absorbed into the mass of worlds where it works a few times and then they actually do die.

If the chance of survival is p, then in both cases P(I survive) = p, P(I survive | I'm there to observe it) = 1.

Indeed, the anthropic principle explains the result of quantum suicide, whether or not you subscribe to the MWI. The real question is whether you ought to commit quantum suicide (and harness its anthropic superpowers for good). It's a question of morality.

Related (somewhat): The Hero With A Thousand Chances.

One of you would. The huge number of other diverging yous would run into an unpleasant surprise sooner or later.

First, not to be nit-picky but MWI != QI. Second, if your suicide attempts are well documented the branch in which you survived would be populated by Bayesians who agreed with you, no?

Flip a quantum coin.

The observation that you survived 1000 good suicide attempts is much more likely under MWI than under Copenhagen.

Isn't that like saying "Under MWI, the observation that the coin came up heads, and the observation that it came up tails, both have probability of 1"?

The observation that I survive 1000 good suicide attempts has a probability of 1, but only if I condition on my being capable of making any observation at all (i.e. alive). In which case it's the same under Copenhagen.

Nah. QI relies on your "subjective thread" coming to an end in some worlds and continuing in others. In your experiment I'd be pretty certain to get knocked out and wake up after 24 hours.

There are two problems with this test.

First, the result of a coin flip is almost certainly determined by starting conditions. With enough knowledge of those conditions you could predict the result. Instead you should make a measurement on a quantum system, such as measuring the spin of an electron.

Second the result of this test does not distinguish between QI and not-QI. The probability of being knocked out or left awake is the same in both cases.

I suppose you could be assuming that your consciousness can jump arbitrarily between universes to follow a conscious version of you.... but no that would just be silly.