gets out the ladder and climbs up to the scoreboard
5 posts without a tasteless and unnecessary torture reference
replaces the 5 with a 0
climbs back down
Years ago, before coming up with even crazier ideas, Wei Dai invented a concept that I named UDASSA. One way to think of the idea is that the universe actually consists of an infinite number of Universal Turing Machines running all possible programs. Some of these programs "simulate" or even "create" virtual universes with conscious entities in them. We are those entities.
Generally, different programs can produce the same output; and even programs that produce different output can have identical subsets of their output that may include conscious entities. So we live in more than one program's output. There is no meaning to the question of what program our observable universe is actually running. We are present in the outputs of all programs that can produce our experiences, including the Odin one.
Probability enters the picture if we consider that a UTM program of n bits is being run in 1/2^n of the UTMs (because 1/2^n of all infinite bit strings will start with that n bit string). That means that most of our instances are present in the outputs of relatively short programs. The Odin program is much longer (we will assume) than one without him, so the overwhelming majority of our copies are in universes without Odin. Probabilistically, we can bet that it's overwhelmingly likely that Odin does not exist.
If two theories imply different invisibles, they shouldn't be considered equivalent. That no evidence can tell them apart, and still they are not equal, is explained by them having different priors. But if two theories are logically (agent-provably, rather) equivalent, this is different, as the invisibles they imply and priors measuring them are also the same.
The thing about positivism is it pretends to be a down-to-earth common-sense philosophy, and then the more you think about it the more it turns into this crazy surrealist madhouse. So we can't measure parallel universes and there's no fact of the matter as to whether they exist. But people in parallel universes can measure them, but there's no fact of the matter whether these people exist, and there's a fact of the matter whether these universes exist if and only if these people exist to measure them, so there's no fact of the matter whether there is a fac...
This theory makes exactly the same observational predictions as your current best theory of physics, so it lies in the same equivalence class and you should give it the same credence.
You're blurring an important distinction between two types of equivalence:
If two theories ...
Boxes proofed against all direct and indirect observation, potential for observation mixed with concrete practicality of such observation, strictly-worse choices, morality... one would be hard-pressed to muddle your thought experiment more than that.
Let's try to make it a little more straightforward: assume that there exists a certain amount of physical space which falls outside our past light cone. Do you think it is equally likely that it contains galaxies and that it contains unicorns? More importantly, do you think the preceding question means anything?
I'm skeptical of the idea that the hypothesis "Odin created physics as we know it" would actually make no additional predictions over the hypothesis . I'm tempted to say that as a last resort we could distinguish between them by generating situations like "Omega asks you to bet on whether Odin exists, and will evaluate you using a logarithmic scoring rule and will penalize you that many utilons", though at this point maybe it is unjustified to invoke Omega without explaining how she knows these things.
But what do you think of some of th...
no matter how seductive it sounds, skipping over the is-ought distinction is not permitted
Yeah, some of us are still not convinced on that one.
Speaking of which, does anyone actually have something resembling a proof of this? People just seem to cast it about flippantly.
Is the Hulatioamiltonian formulation of classical mechanics apriori more probable than the Lagrangian formn?
They are both derivable from the same source, Newtonian mechanics plus the ZF Set theory. They are equivalent and therefore equally probable.
The shortest possible version of them all - mutually equivalent theories - is the measure how (equally) probable are they.
My favourite justification of the Occam razor is that even if two theories are equivalent in their explicit predictions, the simpler one is usually more likely to inspire correct generalisations. The reason may be that the more complicated the theory is, the more arbitrary constraints it puts on our thinking, and those constraints can prevent us from seeing the correct more general theory. For example, some versions of aether theory can be made eqivalent to special relativity, but the assumptions of absolute space and time make it nearly impossible to discover something equivalent to general relativity, starting from aether.
I personally think Occam's razor is more about describing what you know. If two theories are equally good in their explanatory value, but one has some extra bells and whistles added on, you have to ask what basis you have for deciding to prefer the bells and whistles over the no bells and whistles version.
Since both theories are in fact equally good in their predictions, you have no grounds for preferring one over the other. You are in fact ignorant of which theory is the correct one. However, the simplest one is the one that comes closest to describing th...
Adding premises like "Odin created everything" makes a theory less probable and also happens to make it longer; this is the entire reason why we intuitively agree with Occam's Razor in penalizing longer theories. Unfortunately, this answer seems to be based on a concept of "truth" granted from above - but what do differing degrees of truth actually mean, when two theories make exactly the same predictions?
and
...Imagine you have the option to put a human being in a sealed box where they will be tortured for 50 years and then incinerate
Theories that require additional premises are less likely to be true, according to the eternal laws of probability ... Unfortunately, this answer seems to be based on a concept of "truth" granted from above - but what do differing degrees of truth actually mean, when two theories make exactly the same predictions?
Reading this and going back to my post to work out what I was thinking, I have a sort-of clarification for the issue in the quote. The original argument was that, before experiencing the universe, all premises are a priori equally lik...
I wonder if this can't be considered more pragmatically? There was a passage in the MIT Encyclopedia of Cognitive Sciences in the Logic entry that seems relevant:
...Johnson-Laird and Byrne (1991) have argued that postulating more imagelike MENTAL MODELS make better predictions about the way people actually reason. Their proposal, applied to our sample argument, might well help to explain the difference in difficulty in the various inferences mentioned earlier, because it is easier to visualize “some people” and “at least three people” than it is to visualiz
What if Tegmark's multiverse is true? All the equivalent formulations of reality would "exist" as mathematical structures, and if there's nothing to differentiate between them, it seems that all we can do is point to appropriate equivalence class in which "we" exist.
However, the unreachable tortured man scenario suggests that it may be useful to split that class anyway. I don't know much about Solomonoff prior - does it make sense now to build a probability distribution over the equivalence class and say what is the probability mass of its part that contains the man?
Theories that require additional premises are less likely to be true, according to the eternal laws of probability. Adding premises like "Odin created everything" makes a theory less probable and also happens to make it longer; this is the entire reason why we intuitively agree with Occam's Razor in penalizing longer theories. Unfortunately, this answer seems to be based on a concept of "truth" granted from above -
Not to me it doesn't. (Though I may not understand what you mean by "truth" here.) Bayesian probability theory...
Another intriguing answer came from JGWeissman. Apparently, as we learn new physics, we tend to discard inconvenient versions of old formalisms. So electromagnetic potentials turn out to be "more true" than electromagnetic fields because they carry over to quantum mechanics much better. I like this answer because it seems to be very well-informed!
I don't like this explanation- while potentials are useful calculation tools both macroscopically and quantum mechanically, fields have unique values whereas potentials have non-unique values. It's no...
Your thought experiment of the person in the sealed torture box ignores the question of what evidence I have to believe that such a box exists and what evidence I have that the physical theory you've outlined is true (in the thought experiment).
The fact that a theory makes the same predictions as some other theory is irrelevant if I don't have good reason for thinking the theory might be true in the first place. The problem with "Odin created physics" is that I have no good reasons to believe in the existence of Norse gods and that the universe w...
I like this argument. But in this case I think there's another argument that doesn't rely on morality so much.
Your belief that the two theories in question will always make the same predictions is conditional on the box being perfectly sealed, and the universe continuing to expand forever. There's a small chance that these things are not true, and if that turns out to be the case, you may or may not expect to see the guy again, depending on what physical theory you believe in.
I think Matt Simpson is getting at this when he talks about counterfactual predic...
In other words. You can't make a theory less/more probable just by expressing it differently, with more/less words.
Only the shortest known formulation counts.
Doesn't the human inside qualify as an observer? For all we know, WE outside the box could be the ones tortured for 50 years and then incinerated once the button is pushed.
It probably has no consequences in reality, because no matter how seductive it sounds, skipping over the is-ought distinction is not permitted.
What, we can't just assume for now that torture is bad without getting into metaethics?
I don't think that argument is even valid. After all, I have the option of putting a human in a box. If I do, one hypothesis states that the human will be tortured and then killed. The other hypothesis states that the human will "vanish"; it's not precisely clear what "vanish" means here, but I'm going to assume that since this state is supposed to be identical in my experience to the state in the first hypothesis, the human will no longer exist. (Alternative explanations, such as the human being transported to another universe which I ...
This post is a summary of the different positions expressed in the comments to my previous post and elsewhere on LW. The central issue turned out to be assigning "probabilities" to individual theories within an equivalence class of theories that yield identical predictions. Presumably we must prefer shorter theories to their longer versions even when they are equivalent. For example, is "physics as we know it" more probable than "Odin created physics as we know it"? Is the Hamiltonian formulation of classical mechanics apriori more probable than the Lagrangian formulation? Is the definition of reals via Dedekind cuts "truer" than the definition via binary expansions? And are these all really the same question in disguise?
One attractive answer, given by shokwave, says that our intuitive concept of "complexity penalty" for theories is really an incomplete formalization of "conjunction penalty". Theories that require additional premises are less likely to be true, according to the eternal laws of probability. Adding premises like "Odin created everything" makes a theory less probable and also happens to make it longer; this is the entire reason why we intuitively agree with Occam's Razor in penalizing longer theories. Unfortunately, this answer seems to be based on a concept of "truth" granted from above - but what do differing degrees of truth actually mean, when two theories make exactly the same predictions?
Another intriguing answer came from JGWeissman. Apparently, as we learn new physics, we tend to discard inconvenient versions of old formalisms. So electromagnetic potentials turn out to be "more true" than electromagnetic fields because they carry over to quantum mechanics much better. I like this answer because it seems to be very well-informed! But what shall we do after we discover all of physics, and still have multiple equivalent formalisms - do we have any reason to believe simplicity will still work as a deciding factor? And the question remains, which definition of real numbers is "correct" after all?
Eliezer, bless him, decided to take a more naive view. He merely pointed out that our intuitive concept of "truth" does seem to distinguish between "physics" and "God created physics", so if our current formalization of "truth" fails to tell them apart, the flaw lies with the formalism rather than with us. I have a lot of sympathy for this answer as well, but it looks rather like a mystery to be solved. I never expected to become entangled in a controversy over the notion of truth on LW, of all places!
A final and most intriguing answer of all came from saturn, who alluded to a position held by Eliezer and sharpened by Nesov. After thinking it over for awhile, I generated a good contender for the most confused argument ever expressed on LW. Namely, I'm going to completely ignore the is-ought distinction and use morality to prove the "strong" version of Occam's Razor - that shorter theories are more "likely" than equivalent longer versions. You ready? Here goes:
Imagine you have the option to put a human being in a sealed box where they will be tortured for 50 years and then incinerated. No observational evidence will ever leave the box. (For added certainty, fling the box away at near lightspeed and let the expansion of the universe ensure that you can never reach it.) Now consider the following physical theory: as soon as you seal the box, our laws of physics will make a localized exception and the victim will spontaneously vanish from the box. This theory makes exactly the same observational predictions as your current best theory of physics, so it lies in the same equivalence class and you should give it the same credence. If you're still reluctant to push the button, it looks like you already are a believer in the "strong Occam's Razor" saying simpler theories without local exceptions are "more true". QED.
It's not clear what, if anything, the above argument proves. It probably has no consequences in reality, because no matter how seductive it sounds, skipping over the is-ought distinction is not permitted. But it makes for a nice koan to meditate on weird matters like "probability as preference" (due to Nesov and Wei Dai) and other mysteries we haven't solved yet.
ETA: Hal Finney pointed out that the UDT approach - assuming that you live in many branches of the "Solomonoff multiverse" at once, weighted by simplicity, and reducing everything to decision problems in the obvious way - dissolves our mystery nicely and logically, at the cost of abandoning approximate concepts like "truth" and "degree of belief". It agrees with our intuition in advising you to avoid torturing people in closed boxes, and more generally in all questions about moral consequences of the "implied invisible". And it nicely skips over all the tangled issues of "actual" vs "potential" predictions, etc. I'm a little embarrassed at not having noticed the connection earlier. Now can we find any other good solutions, or is Wei's idea the only game in town?