Never mind usefulness, it seems to me that "Evolution by natural selection occurs" and "God made the world and everything in it, but did so in such a way as to make it look exactly as if evolution by natural selection occured" are not the same hypothesis, that one of them is true and one of them is false, that it is simplicity that leads us to say which is which, and that we do, indeed, prefer the simpler of two theories that make the same predictions, rather than calling them the same theory.

While my post was pretty misguided (I even wrote an apology for it), your comment looks even more misguided to me. In effect, you're saying that between Lagrangian and Hamiltonian mechanics, at most one can be "true". And you're also saying that which of them is "true" depends on the programming language we use to encode them. Are you sure you want to go there?

0Perplexed9yHmmm. But the very first posting in the sequences says something about "making your beliefs pay rent in expected experience". If you don't expect different experiences in choosing between the theories, it seems that you are making an unfalsifiable claim. I'm not totally convinced that the two theories do not make different predictions in some sense. The evolution theory pretty much predicts that we are not going to see a Rapture any time soon, whereas the God theory leaves the question open. Not exactly "different predictions", but something close.
2cata9yI think there's a distinction that should be made explicit between "a theory" and "our human mental model of a theory." The theory is the same, but we rightfully try to interpret it in the simplest possible way, to make it clearer to think about. Usually, two different mental models necessarily imply two different theories, so it's easy to conflate the two, but sometimes (in mathematics especially) that's just not true.

A note on the description complexity of physical theories

by cousin_it 2 min read9th Nov 2010184 comments

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Followup to: The prior of a hypothesis does not depend on its complexity

Eliezer wrote:

In physics, you can get absolutely clear-cut issues.  Not in the sense that the issues are trivial to explain. [...] But when I say "macroscopic decoherence is simpler than collapse" it is actually strict simplicity; you could write the two hypotheses out as computer programs and count the lines of code.

Every once in a while I come across some belief in my mind that clearly originated from someone smart, like Eliezer, and stayed unexamined because after you hear and check 100 correct statements from someone, you're not about to check the 101st quite as thoroughly. The above quote is one of those beliefs. In this post I'll try to look at it closer and see what it really means.

Imagine you have a physical theory, expressed as a computer program that generates predictions. A natural way to define the Kolmogorov complexity of that theory is to find the length of the shortest computer program that generates your program, as a string of bits. Under this very natural definition, the many-worlds interpretation of quantum mechanics is almost certainly simpler than the Copenhagen interpretation.

But imagine you refactor your prediction-generating program and make it shorter; does this mean the physical theory has become simpler? Note that after some innocuous refactorings of a program expressing some physical theory in a recognizable form, you may end up with a program that expresses a different set of physical concepts. For example, if you take a program that calculates classical mechanics in the Lagrangian formalism, and apply multiple behavior-preserving changes, you may end up with a program whose internal structures look distinctly Hamiltonian.

Therein lies the rub. Do we really want a definition of "complexity of physical theories" that tells apart theories making the same predictions? If our formalism says Hamiltonian mechanics has a higher prior probability than Lagrangian mechanics, which is demonstrably mathematically equivalent to it, something's gone horribly wrong somewhere. And do we even want to define "complexity" for physical theories that don't make any predictions at all, like "glarble flargle" or "there's a cake just outside the universe"?

At this point, the required fix to our original definition should be obvious: cut out the middleman! Instead of finding the shortest algorithm that writes your algorithm for you, find the shortest algorithm that outputs the same predictions. This new definition has many desirable properties: it's invariant to refactorings, doesn't discriminate between equivalent formulations of classical mechanics, and refuses to specify a prior for something you can never ever test by observation. Clearly we're on the right track here, and the original definition was just an easy fixable mistake.

But this easy fixable mistake... was the entire reason for Eliezer "choosing Bayes over Science" and urging us to do same. The many-worlds interpretation makes the same testable predictions as the Copenhagen interpretation right now. Therefore by the amended definition of "complexity", by the right and proper definition, they are equally complex. The truth of the matter is not that they express different hypotheses with equal prior probability - it's that they express the same hypothesis. I'll be the first to agree that there are very good reasons to prefer the MWI formulation, like its pedagogical simplicity and beauty, but K-complexity is not one of them. And there may even be good reasons to pledge your allegiance to Bayes over the scientific method, but this is not one of them either.

ETA: now I see that, while the post is kinda technically correct, it's horribly confused on some levels. See the comments by Daniel_Burfoot and JGWeissman. I'll write an explanation in the discussion area.

ETA 2: done, look here.

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