# Privileging the Hypothesis

Suppose that the police of Largeville, a town with a million inhabitants, are investigating a murder in which there are few or no clues—the victim was stabbed to death in an alley, and there are no fingerprints and no witnesses.

Then, one of the detectives says, “Well… we have no idea who did it… no particular evidence singling out any of the million people in this city… but let’s consider the hypothesis that this murder was committed by Mortimer Q. Snodgrass, who lives at 128 Ordinary Ln. It could have been him, after all.”

I’ll label this the fallacy of privileging the hypothesis. (Do let me know if it already has an official name—I can’t recall seeing it described.)

Now the detective may perhaps have some form of rational evidence that is not legal evidence admissible in court—hearsay from an informant, for example. But if the detective does not have some justification already in hand for promoting Mortimer to the police’s special attention—if the name is pulled entirely out of a hat—then Mortimer’s rights are being violated.

And this is true even if the detective is not claiming that Mortimer “did” do it, but only asking the police to spend time pondering that Mortimer might have done it—unjustifiably promoting that particular hypothesis to attention. It’s human nature to look for confirmation rather than disconfirmation. Suppose that three detectives each suggest their hated enemies, as names to be considered; and Mortimer is brown-haired, Frederick is black-haired, and Helen is blonde. Then a witness is found who says that the person leaving the scene was brown-haired. “Aha!” say the police. “We previously had no evidence to distinguish among the possibilities, but now we know that Mortimer did it!”

This is related to the principle I’ve started calling “locating the hypothesis,” which is that if you have a billion boxes only one of which contains a diamond (the truth), and your detectors only provide 1 bit of evidence apiece, then it takes much more evidence to promote the truth to your particular attention—to narrow it down to ten good possibilities, each deserving of our individual attention—than it does to figure out which of those ten possibilities is true. It takes 27 bits to narrow it down to ten, and just another 4 bits will give us better than even odds of having the right answer.

Thus the detective, in calling Mortimer to the particular attention of the police, for no reason out of a million other people, is skipping over most of the evidence that needs to be supplied against Mortimer.

And the detective ought to have this evidence in their possession, at the first moment when they bring Mortimer to the police’s attention at all. It may be mere rational evidence rather than legal evidence, but if there’s no evidence then the detective is harassing and persecuting poor Mortimer.

During my recent diavlog with Scott Aaronson on quantum mechanics, I did manage to corner Scott to the extent of getting Scott to admit that there was no concrete evidence whatsoever that favors a collapse postulate or single-world quantum mechanics. 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 possibilitywithout 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.

And yes, this is just the same fallacy committed, on a much more blatant scale, by the theist who points out that modern science does not offer an absolutely complete explanation of the entire universe, and takes this as evidence for the existence of Jehovah. Rather than Allah, the Flying Spaghetti Monster, or a trillion other gods no less complicated—never mind the space of naturalistic explanations!

To talk about “intelligent design” whenever you point to a purported flaw or open problem in evolutionary theory is, again, privileging the hypothesis—you must have evidence already in hand that points to intelligent design specifically in order to justify raising that particular idea to our attention, rather than a thousand others.

So that’s the sane rule. And the corresponding anti-epistemology is to talk endlessly of “possibility” and how you “can’t disprove” an idea, to hope that future evidence may confirm it without presenting past evidence already in hand, to dwell and dwell on possibilities without evaluating possibly unfavorable evidence, to draw glowing word-pictures of confirming observations that could happen but haven’t happened yet, or to try and show that piece after piece of negative evidence is “not conclusive.”

Just as Occam’s Razor says that more complicated propositions require more evidence to believe, more complicated propositions also ought to require more work to raise to attention. Just as the principle of burdensome details requires that each part of a belief be separately justified, it requires that each part be separately raised to attention.

As discussed in Perpetual Motion Beliefs, faith and type 2 perpetual motion machines (water ice cubes electricity) have in common that they purport to manufacture improbability from nowhere, whether the improbability of water forming ice cubes or the improbability of arriving at correct beliefs without observation. Sometimes most of the anti-work involved in manufacturing this improbability is getting us to pay attention to an unwarranted belief—thinking on it, dwelling on it. In large answer spaces, attention without evidence is more than halfway to belief without evidence.

Someone who spends all day thinking about whether the Trinity does or does not exist, rather than Allah or Thor or the Flying Spaghetti Monster, is more than halfway to Christianity. If leaving, they’re less than half departed; if arriving, they’re more than halfway there.

An oft-encountered mode of privilege is to try to make uncertainty within a space, slop outside of that space onto the privileged hypothesis. For example, a creationist seizes on some (allegedly) debated aspect of contemporary theory, argues that scientists are uncertain about evolution, and then says, “We don’t really know which theory is right, so maybe intelligent design is right.” But the uncertainty is uncertainty within the realm of naturalistic theories of evolution—we have no reason to believe that we’ll need to leave that realm to deal with our uncertainty, still less that we would jump out of the realm of standard science and land on Jehovah in particular. That is privileging the hypothesis—taking doubt within a normal space, and trying to slop doubt out of the normal space, onto a privileged (and usually discredited) extremely abnormal target.

Similarly, our 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.

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John Wilkes Booth: You know you really ought to do something about that stomach.

Zangara: I do everything about the stomach!

Booth: Oh yes?

Zangara: I give up wine. No good! I give up smokes. No good! I quit my work. No good! I move Miami. No good! I TAKE APPENDIX OUT! No good! Nothing no good! Nothing! Nothing! Nothing!

Booth: Have you considered shooting Franklin Roosevelt?

Zangara: You think that help?

Booth: It couldn't hurt...

-- Assassins shows up how privileging hypotheses is done.

"Is it cold out in space, Bowie?"

"You can borrow my jumper if you like, Bowie!"

"Does the cold of deep space make your nipples get pointy, Bowie?"

"Do you use your pointy nipples as telescopic antennae to transmit data back to earth?"

"I bet you do, you freaky old bastard you!"

[...]

"Receiving transmission...from David Bowie's nipple antennae!"

--Flight of the Conchords helps

That's pretty far out!

I don't know, this "Mortimer Q. Snodgrass" fellow seems pretty suspicious to me. I mean, a weird name like that is probably an alias. And "Ordinary Lane"? At a power of 2 no less? Tell me he plays tennis and I'll be convinced he did it.

And even if he wasn't the murderer, he's probably guilty of something. Check his computer for pirated music! ;)

Mortimer Snodgrass? Or maybe it was Homer Dalrymple. :) Either way, it sounds like a mad scientist's name.

(Six year late reply, I know, but I just heard the relevant chapter of HPMOR and was curious to see if anyone else caught the reference!)

Well, I think that John Q. Wiffleheim of 1234 Norkle Rd is a more likely candidate.

I agree that privileging a hypothesis is a common error. I don't agree that it applies in the example used, though.

If you have a tradition thousands of years old saying that a particular spot was the site of Nazareth in 4BC, or of Troy in 1200BC, it isn't irrational to privilege the hypothesis that that spot was indeed the site of Nazareth, or of Troy.

Similarly, when the entire world has used the single-world hypothesis almost exclusively until the recent past, it isn't unfairly privileging it to still consider it a major contender.

You might think this is more like evolution vs. creationism. I don't mean that we should keep teaching creationism in school as an alternative today. But we haven't got as strong an argument for many-worlds as we do for evolution.

Also, AFAIK there's just these 2 competing hypotheses: One-world, many-world. We don't have the 7-worlds hypothesis and the 23-worlds hypothesis and the pi-worlds hypothesis. We could have the countable-worlds hypothesis and the uncountable-worlds hypothesis, but AFAIK we don't even have those. How can you say it's irrational to consider 1 of the only 2 hypotheses available?

Also, AFAIK there's just these 2 competing hypotheses: One-world, many-world.

Reminiscent of the guy who was asked what were the odds he would win the lottery, and replied, "Fifty-fifty, either I win or I don't." The corresponding heuristic-and-bias is I think known as "packing and unpacking" or something along those lines.

I remember the Daily Show had a funny example of this in action. They were interviewing people about the possibility of the Large Hadron Collider destroying the earth, and they talked to a physicist and a crazy survivalist. The former said it was impossible for the LHC to destroy the earth, while the latter used basically that argument: "There are two possibilities: it can destroy us, or not. So, that's about a 50/50 chance."

Then later the interviewer followed the survivalist to his bunker and asked him: if everyone died but them, don't they have an obligation to mate to repopulate the earth? (They were both men.) The survivalist said, "Um, no, because that doesn't work. It's impossible." And then the interviewer came back with, "well, there's two possibilities: we'll produce a baby, or we won't, so that's 50/50 -- pretty good odds."

I'm sure someone would love to dig up the clip...

I remember the Daily Show had a funny example of this in action. They were interviewing people about the possibility of the Large Hadron Collider destroying the earth, and they talked to a physicist and a crazy survivalist. [...] I'm sure someone would love to dig up the clip...

Sure! Didn't take more than three years for someone to do that, either!

Though apparently your mind edited out how the interviewee's "there's a 0% chance it [them reproducing] will work" makes a great parallel with how John Ellis, who's otherwise amazing in this video, earlier explains that "there is 0% chance", "zero", of the LHC destroying the world[*]. Sigh.

(The clip is great from start to finish, but IMHO the funniest part is what John Oliver says in answer to "This place is perfectly safe" towards the end of the video. I was going to say that the only people it could be said to make fun of are annoying nitpickers, but on reflection, it's actually feels like a really great dig at people who make terrible arguments and want you to take them seriously, even though they really should realize the flaw themselves.)

[*] Technically, the video only suggests that it is world-destroying that Ellis claims to have "0% chance", and this is the Daily Show, but I think we can safely assume that ths is not selective editing to make him look like a bad Bayesian.

1 vs. many is a very natural divide, not at all a good example of the packing and unpacking fallacy.

Once you accept that there exists something isomorphic to a wave function, it's more like:

many worlds vs. many worlds and an orang-utan vs. many worlds and an apple tree vs. many worlds and a television vs. many worlds and a blue castle vs. (...) vs. many worlds and a character-of-natural-law-violating process that constantly kills all the worlds except one.

All cases except the last case contain many worlds, but Phil packed them together. I think that's the intuition Eliezer was getting at.

We shouldn't be afraid here to sound Orwellian. Copenhagen people believe in the many worldeaters interpretation. We believe in the no worldeaters interpretation.

So true - My "8 worlds and an orang-utan" hypothesis never got the respect it deserved.

"Should array indices start at 0 or 1? My compromise of 0.5 was rejected without, I thought, proper consideration."

--Stan Kelly-Bootle

Props for the perseverance, man. Props ;-)

That is exactly and perfectly right and I should use this example henceforth.

I think you are demonstrating a dramatic failure to update by saying that a hypothesis held by 99.99+% of humanity, and even by most people who have thought about the issues, is not worth considering.

I'd like to know what the distribution of opinions of quantum physicists and cosmologists is.

There have been polls, with a dramatic range of support. Wikipedia leads me to the most MWI-friendly poll. I think the low-water mark is about 10% of some other group of quantum theorists. I suspect that the variation is due to wording issues and local social pressure (by "local" I mean the conference), but the page suggests different communities:

The findings of this poll are in accord with other polls, that many- worlds is most popular amongst scientists who may rather loosely be described as string theorists or quantum gravitists/cosmologists. It is less popular amongst the wider scientific community who mostly remain in ignorance of it.

Antia Lamas saw MWI win a poll for least favorite interpretation. On that page, Michael Nielsen mentions a poll where MWI came 3rd, after Copenhagen and decoherence...but if decoherence is an interpretation, it sure sounds like MWI to me.

Why you insist on being dogmatic on this is beyond me. In your writings on the subject, you admit you don't understand the math behind quantum mechanics, which is in fact the model. Why be so sure you are right about the interpretation of the model you don't understand?

People look kindly on those who are humble when commenting on things outside of their expertise. People that go around making bold claims about things about which they are not that knowledgeable are labeled cranks, and rightfully so.

Similarly, when the entire world has used the single-world hypothesis almost exclusively until the recent past, it isn't unfairly privileging it to still consider it a major contender.

It's not a major contender because of hearsay of powerful evidence like we have with legends. It's a major because it's been unfairly privileged ever since someone thought of it. It's far more complicated than the hypotheses that they haven't thought of, so by Occam's razor, it's far more likely to be a hypothesis that nobody's thought of than that one.

It's not like a legend about the city of Nazareth. It's not even like a legend about the birth of a god. It's like concluding that there's a god because life has clearly been optimized, and you haven't thought of any alternative hypotheses yet. Once Many-Worlds has been suggested, it's like concluding there's a good chance of there being a god, because you would have thought there was one before you thought of the alternative hypothesis.

Just because you haven't thought of an alternative hypothesis doesn't mean there isn't one. It does mean that you have to discount it on the, rather high, chance that it has already been disproven. Most have. But if there's enough alternatives, if your hypothesis is complicated enough from the beginning, there's bound to be an alternative hypothesis that actually explains it.

If you have a tradition thousands of years old saying that a particular spot was the site of Nazareth in 4BC, or of Troy in 1200BC, it isn't irrational to privilege the hypothesis that that spot was indeed the site of Nazareth, or of Troy.

I agree. It ought to have taken the first people to grasp the math at least 30 seconds to discard the Nazareth concept.

I don't know if there's another name for privileging the hypothesis, but it seems closely related to anchoring, in that it involves establishing an unjustifiable "starting point" from which to search.

It seems like it's a special case of anchoring to me.

Also related to availability - the very fact that it enters your conscious mind, even if you immediately discount it, is going to wear down that thought groove, making it more likely to be consciously (or subconsciously) accessed.

Just to confirm:

Another way of explaining the 'locating the hypothesis' concept would be to say: "When answering a question with a large number of possible answers, it takes more work to narrow down the possibilities (generate the reasonable hypotheses) than it does to test those hypotheses for correctness."

Is that right?

That is correct, and even more importantly "When answering a question with a large enough number of possible answers, any single possible answer will have a bigger chance of being a false positive than true positive if tested"

You are asserting a false duality. Either many-worlds, OR a collapse postulate. You use evidence AGAINST a collapse as evidence FOR many worlds, which is very weak evidence. Here is a third alternative- the wavefunction is not real- merely a mathematical formalism used to calculate probability distributions (this map doesn't have to be the territory). Here is a fourth- collapse is an approximation to a small, non-linear self-coupling in the equation that governs time evolution. Here is a fifth- evolution is governed by both the advanced and retarded Green function solution to the Schroedinger equation, and what appears to be collapse is a sort of beat-resonance between the two. Here is a sixth- there are(non-local) degrees of freedom apart from the wavefunction and 'collapse' occurs because our existing theory is confused about what devices actually measure. I could keep going.

Every one of the above has a huge advantage over many worlds- there is positive evidence to update in their favor. Because they accurately reproduce most of quantum mechanics, all that evidence that we can use to push us to "quantum mechanics is probably right" CAN lead us to any of the above theories.

Many worlds does NOT have Born probabilities, and so IT DOES NOT MAKE PREDICTIONS. No one knows how to use many worlds to do anything at all. So you are doing a very weird sort of Bayesian process- you use Copenhagen or one of the above theories' predictions to update your belief to "quantum mechanics is probably right." Now starting from this new belief, you use other facts to update to "many worlds is probably right." Unfortunately, you didn't notice that in switching to many worlds, all of that evidence that pointed to quantum mechanics is gone.

If you start from an agnostic prior, many world's has no predictions to push you in the direction of "this is the right theory."

You use evidence AGAINST a collapse as evidence FOR many worlds, which is nonsense.

That actually isn't nonsense, even if (or rather, even though) there are not only two hypotheses. Given that collapse outright excludes many worlds, evidence against collapse is evidence in favor of many worlds. It is evidence that merely becomes weaker the more additional probability mass there is for the additional hypotheses.

I retract the overly-strong word "non-sense", I'm not sure how to markup a strike out so I merely edited the above post.

EHeller: what if the decision-theoretic approach by Wallace et al. turns out to work? Would you consider MWI "heavily" favoured then?

grand total of zero

I have perceived exactly one world all my life. Isn't that evidence that exactly one world exists?

But that's exactly what you'd perceive if many worlds was true.

So, Many Worlds is a garage dragon?

If many worlds wasn't favored by the evidence that distinguishes between the two explanations and was the more complex explanation from reality's point of view, then yes, but I don't think it qualifies for dragonhood under this criteria.

Surely spontaneous collapse is the garage dragon here. Zero evidence, highly unlikely.

OK, on the one hand we have many-worlds. As you say, no direct subjective corroborating evidence (it’s what we’d see either way). What’s more, it’s the simplest explanation of what we see around us.

On the other hand, we have one-world. Again, ‘it’s what we’d see either way’. However, we now have to postulate an extra mechanism that causes the ‘collapse’.

I know which of these feels more like a privileged complex hypothesis pulled out of thin air, like a dragon.

Could whomever downvoted me above let me know where I’m going wrong here?

How is postulating entire worlds simpler than collapse?

Because that's not what's actually being postulated. What's being postulated is "you know the basic math of QM? well... Just take that math really seriously and avoid adding too many extra rules. a CONSEQUENCE of that is many worlds."

ie, "take the quantum amplitudes over configuration space and the linear update rule. Also keep the whole born statistics thing for now. Hopefully we'll be able to derive it from the rest. And that's it. Don't add any rules about the rest of the amplitude field going to zero or any other such nonsense. Just have all QM all the time"

Voted this down, then changed my mind and undid it. This is a genuine question, the answer to which was graciously accepted. Downvoting people who need guidance to understand a concept and are ready to learn is exactly what we don't want to do.

I tend to think that the Bible and the Koran are sufficient evidence to draw our attention to the Jehovah and Allah hypotheses, respectively. Each is a substantial work of literature, claiming to have been inspired by direct communication from a higher power, and each has millions of adherents claiming that its teachings have made them better people. That isn't absolute proof, of course, but it sounds to me like enough to privilege the hypotheses.

This is in fact the general problem here. If there is a large group of people claiming that some religion is true, that is quite enough evidence to call your attention to the hypothesis. That is in fact why people's attention is called to the hypothesis: paying attention to what large groups of people say is not remotely close to inventing a random idea.

I think that "privileging the hypothesis" is an example of special pleading (http://en.wikipedia.org/wiki/Special_pleading) being applied to the selection of a hypothesis, as opposed to the evaluation of the hypothesis.

Eliezer, speaking of "privileging the hypothesis," what do you think about the proscription in statistics against "data dredging," or using past data to support post hoc hypotheses suggested by the data? What do you think about the view of descriptive science being inferior to hypothesis-driven science?

Based on your analysis, it would indeed seem that a hypothesis that could be located prior to an experiment might be more probable than a hypothesis that could only be located after an experiment.

Yet is there an over-emphasis placed on the state of mind of the particular experimenter prior to data collection? What if another scientist on the other side of the world had a hypothesis, which our original experimenter only came to after doing a study looking at something else. Can the second scientist say that the study confirms his hypothesis (because he held it in advance), while the first scientist cannot (because he only came to his hypothesis after doing the study)?

What if the post-hoc-hypothesized effect is very strong and related to plausible mechanisms in the field? What if it showed up in lots of previous studies that were looking at other things?

EDIT: Non-Eliezer people are invited to reply to this comment also.

There's nothing inherently wrong with data dredging. Considering all possible hypotheses and keeping the ones suggested by the data is just Solomonoff induction. It only becomes problematic if you don't have a consistent prior, e.g. if you keep the hypothesis with the greatest likelihood ratio rather than the greatest posterior.

Hypothesis-driven has its place in the human practice of science, because humans have a hard time computing a prior after having seen the data. But that's a problem with the humans, not with the math.

i.e. if you keep the hypothesis with the greatest likelihood ratio rather than the greatest posterior

...or if you believe everything that has p<.05.

It only becomes problematic if you don't have a consistent prior, i.e. if you keep the hypothesis with the greatest likelihood ratio rather than the greatest posterior.

If that were true, you would never need to hold out a validation set.

And the corresponding anti-epistemology is to talk endlessly of "possibility" and how you "can't disprove" an idea

Something to point out to someone who says this is that 'possibility' is not a constraint - EVERYTHING is possible. As far as I know (and correct me if i'm wrong cause that would be a major fuckup), you can't assign a probability of zero to anything. You can't seperate the possible things from the impossible ones, and then focus only on the possible. 'Possible', by itself, applies to everything, so you don't say anything by declaring something 'possible'. It's only when you start talking about degrees of possibility that the word has any meaning.

Exceptions to "everything is possible" include logical contradictions (such as mathematical falsehoods).

I voted you up, but I'm genuinely confused here - does the concept of probability/possibility apply to a strict, axiomatic, isolated (yet human created and thus fallible) system like mathematics?

Not all logical contradictions have to do with mathematics. For instance, it's impossible (barring childish equivocation etc.) that a given pancake be both buttered and not buttered. Pancakes tend to have a very poor grasp of math.

Your confusion may have to do with epistemic v. metaphysical possibility. I can imagine that I am so deeply, profoundly confused about the universe that I could be mistaken about arithmetic; therefore, it's sort of epistemically possible that two and two be five. However, as it happens, it's not actually possible that two and two be five. Because I am part of a philosophy department in which David Lewis is practically worshiped, I'll put it this way: I can think about two and two making five, but there's no possible world in which that thought is reality.

Not all logical contradictions have to do with mathematics. For instance, it's impossible (barring childish equivocation etc.) that a given pancake be both buttered and not buttered.

It does have to do with mathematics, though. A buttered pancake is a pattern, an unbuttered pancake is a different pattern. Each pattern can be expressed as a series of bits, and the first series will not equal the second.

but there's no possible world in which that thought is reality.

Are you 100% certain about that?

Are you 100% certain about that?

No. Do I need to explain epistemic possibility versus metaphysical possibility again?

I find your distinction between "epistemic" and "metaphysical" possibility to be fairly useless. Are you using a different brain when you consider "epistemic" possibility than when you consider "metaphysical" possibility? Does your concept of "metaphysical" possibility somehow not reside within your own mind, but rather "out there" somewhere in the realm of Platonic logic? That seems suspicious. And even if it were somehow "out there", how do you know you're not mistaken about what it is?

More to the point: can you name a single "metaphysical" possibility that does not reduce to an "epistemic" possibility when considered from the context of your own brain?

It is a metaphysical possibility that the universe is actually entirely random and only seems to us as if it were lawful. But it is not epistemically possible for me to ever be 100% certain that the universe is random, or that it is lawful.

So it appears to me that an epistemic possibility is something which exists only in my map, whereas a metaphysical possibility is something which exists in the territory and may be represented in my map. The fact that I represent both concepts within my own brain doesn't seem to change this.

So it appears to me that an epistemic possibility is something which exists only in my map, whereas a metaphysical possibility is something which exists in the territory and may be represented in my map. The fact that I represent both concepts within my own brain doesn't seem to change this.

This would seem to make it useless to talk of metaphysical possibilities, seeing as there is no way to directly access the territory.

I don't think so. I cannot directly access reality, but it still seems very useful to me to speak of it. Even if only to have something for my beliefs to try to correspond to.

In that case, you can refer directly to your map. Instead of saying, "The sky is blue," you can say, "My map of the territory contains a blue sky." (Naturally, this is only necessary when context requires; if you're in an ordinary conversation, there's no need to go that far.) To me, it seems that the only time you need to really refer to the territory is when you're talking directly about the map-territory relationship, e.g. "As research continues, our understanding of quantum physics will hopefully increase." But to speak descriptively of the territory is to commit the Mind Projection Fallacy. After all, there's no difference between saying, "I believe X," versus just "X"; the two statements convey exactly the same information, and this information pertains only to the speaker's map, not the territory. In my view, then, all possibilities are of the "epistemic" sort. To add a second type, "metaphysical", seems wholly unnecessary.

I am in complete agreement with what you said that I need only talk about reality to have something to check my map against.

But I believe we may be using the term "epistemic possibility" differently.

It appears to me that when you say that X is epistemically possible, that it is possible that your map can contain X (and your map is correct in this aspect). I.e. that one can have a true belief that X.

What I mean when I say that X is epistemically possible is that I have a justified true belief that X. In that sense epistemic possibility is stronger than metaphysical possibility for me in that I require the justification of the belief as well as its truth.

Does the concept of probability/possibility apply to mathematics?

That's a very interesting question. Philosophers have been arguing about the concept of possibility (and its dual, necessity) for some time.

There's a sense of "necessary randomness", that Chaitin has written very extensively about.

http://www.umcs.maine.edu/~chaitin/

Corresponding to this notion, there are stochastic models of (generally-agreed) necessary truths. The best known is probably the "probability of n being prime":

But there are plenty others - e.g. the 3n+1 problem

http://en.wikipedia.org/wiki/Collatz_conjecture

or the question of who wins (first or second player) an integer-parametrized family of combinatorial games.

More exotically, Neal Stephenson's "Anathem" and Greg Egan's short stories "Luminous" and "Dark Integers" explore the possibility that what we think of as "necessary truths" are in fact contingent truths, frozen at some point in the distant past, and exerting a pervasive influence. (Note: I think this might sound ridiculous to a logician, but moderately reasonable to a cosmologist.) It is quite difficult to tell the difference between a necessary truth and a contingent truth which has always been true.

More prosaically, we do make errors and (given things like cosmic rays and other low-level stochastic processes) it seems unlikely that any physical process could be absolutely free of errors. We might believe something to be impossible, but erroneously. Your answer to the question "Are there any necessary truths?" probably depends on your degree of Platonism.

Definitional contradictions are impossible. For example, I can say that I will encounter a married bachelor, or a non-female vixen, with P=0. This doesn't actually say anything about the world; I could figure out that there are no married bachelors without leaving in my room, simply by knowing that bachelor is defined as "man who is not married." Mathematical truths (like 2+2=4) and non-contradiction (as mentioned deeper in this thread with the buttering of pancakes) are specific instances of definitional contradiction.

You're generally right, though. Truths that actually involve looking at the world, i.e. ones that are not inherently about language, cannot have P=0 or P=1.

Actually, certain truths about the self and "subjective" experience may also hit P=0 and P=1. It seems I can be certain of my existence, even if I can't be certain of what I am or what causes my existence. I can also be certain that there exists some X such that X exists. I also think I can be certain that it looks like there's a computer screen in front of me, that my knee feels slightly uncomfortable, and that I am literate. All of this seems to have P=1; the implied causes may be false - I may not actually have knees, for example - but I certainly do seem to be having the sense-experience. I don't think this contradicts your point in any practical sense, though.

I don't know if what I'm about to say is a nitpick, but I think it's relevant to the issue, so I'll say it anyway:

Definitional contradictions are impossible. For example, I can say that I will encounter a married bachelor, ... with P=0. This doesn't actually say anything about the world; I could figure out that there are no married bachelors without leaving in my room, simply by knowing that bachelor is defined as "man who is not married."

But words have histories behind them, and there is a reason why the term "bachelor" exists. The term "bachelor" carries connotations that go beyond simply "union(male,~married)". To borrow from an example from Hubert Dreyfus (and do forgive me for reading him), if I told you I was having a party and I wanted you to bring bachelors, would you consider bringing priests or gay men?

What's actually happening is that we believe "bachelor" has one meaning, while expecting people to imagine a different clump of conceptspace ("connotation") when we actually use it.

Only when you confine the issue into being a purely logcal one, with "bachelor", "man", etc. as suggestively-named LISP tokens can you identify purely logical (P = 0 or 1) truths. But at that point, you've destroyed the mutual information between those words and the outside world, including the usage of those terms in the outside world. And in that case, your statement is no longer about bachelors, but rather, about abstract logical relationships in Platonic space.

Another failure mode of arguing from a definition is that you could be wrong about the definition.

'Possible', by itself, applies to everything, so you don't say anything by declaring something 'possible'. It's only when you start talking about degrees of possibility that the word has any meaning.

Part of the problem is that "possibility" has different meanings depending on context. In everyday speech, it seems to be used to indicate degrees of probability. When people declare a certain event "possible" in everyday speech, they usually mean that it has a low but nontrivial probability, given the everyday state of the world. In this sense, I might say that it's "impossible" for me to become a NFL player, even though in a philosophical discussion we would recognize that the probability that I could become a NFL player is greater than 0.

Problems occur people equivocate between different meanings of "possibility," or introduce a certain meaning into a type of discussion where it doesn't belong. For instance, it's "possible" that the Flying Spaghetti Monster exists and created the world, but this is not the kind of "possibility" that people deal with in everyday life.

It may be that privileging the hypothesis -- or, more specifically, unjustifiably promoting a hypothesis about the goodness of a particular product or service to people's attention -- is the business end of TV advertising.

I think this is a great follow up to: http://lesswrong.com/lw/o3/superexponential_conceptspace_and_simple_words/ (scroll to the very end of that post)

Good point. Also, Yvain follows up that point with his parable that shows the hypothetical example of teaching reformed Nazis that "Untermenschen are people too", and how that's nearly as distorting as Nazi ideology itself.

Wait, did I just Godwin?

ideological organisations like think tanks, political parties and religions do this

Eliezer: privileging the hypothesis is known as the Prosecutor's fallacy. I like your name better though.

I see how this applies to different deities of the same complexity. What about the Maimonidean-type "negative theology" - http://en.wikipedia.org/wiki/Negative_theology#In_the_Jewish_tradition ? Basically this implies a perfectly simple diety "of reference class size 1". It seems harder to say that the hypothesis is arbitrary in this case.

a perfectly simple diety

No, that's why I'm asking ;). The reference you point to certainly provides no immediate answer (pretty much a placeholder); I agree that simplicity can be fake, but if you define something as

"God's existence is absolute and it includes no composition and we comprehend only the fact that He exists, not His essence. Consequently it is a false assumption to hold that He has any positive attribute... still less has He accidents (מקרה), which could be described by an attribute. Hence it is clear that He has no positive attribute whatever. The negative attributes are necessary to direct the mind to the truths which we must believe... When we say of this being, that it exists, we mean that its non-existence is impossible; it is living — it is not dead; ...it is the first — its existence is not due to any cause; it has power, wisdom, and will — it is not feeble or ignorant; He is One — there are not more Gods than one… Every attribute predicated of God denotes either the quality of an action, or, when the attribute is intended to convey some idea of the Divine Being itself — and not of His actions — the negation of the opposite"

it sounds like it's perfectly simple, by definition. BTW, even if wrong, Maimonides should get credit for recognizing the virtue of simplicity ;). This was 14th century.

Anyway, my intellectual toolkit is not sufficient to figure this out from the moment, so I am asking for help regurgitating this a little, if anyone wants to take this up as an exercise. The question does have persona significance to me.

"Perfectly simple" means "the mathematics is simple", not "the explanation does not have many apparent details". This so-called definition is a classic example of a mysterious answer - an actually simple deity could be described by positive attributes.

Indeed, I think we could end up calling the simple deity by his holiest of names: Math.

What properties does "Math" have that would justify calling it a "deity"?

Actually, back up: since when is mathematics (the human endeavor) simple from a mathematical perspective?

What properties does "Math" have that would justify calling it a "deity"?

It contains the almighty hammer Mjölnir. It is omniscient. (By volume - sure, it knows all wrong things that can possibly be represented too but hey, every other deity I have studied is defined as something outright logically incoherent so they can't talk.) So in conclusion... not much justification at all until you worship it a bit and it starts to get personified.

Actually, back up: since when is mathematics (the human endeavor) simple

If you left it at this I'd say never...

from a mathematical perspective?

... but I'll never cease to be amazed at what a mathematian will describe as "simple" or even "trivial" when he is in his mathematical perspective groove!

So a math professor is going through the proof of a theorem on the blackboard in front of his class. Partway through, a student stops him to ask about the justification for a particular step. The professor furrows his brow, stares at the chalkboard for a moment, then walks briskly from the room. Twenty minutes later he returns, his chalk worn down to a nub, and announces triumphantly, "it's obvious".

I know this one as: professor walks into a class, scrawls an equation on the blackboard, and announces "I'm sure you'll all agree that this is obvious." Then he stops, stares at it, walks away, comes back 20 minutes later and says "Yes, that's right, it is obvious."

Twenty minutes later he returns, his chalk worn down to a nub, and announces triumphantly, "it's obvious".

Exactly. I wonder if anyone has a good link to a particularly witty or authoritative expression of this parody. I find it warrants reference rather frequently.

It's one of the standard famous anecdotes about Norbert Wiener.

(Oddly enough, the reason I know so much about Wiener is because Dan Simmons in Hyperion & Fall of Hyperion based Sad King Billy on him.)

You mean the mythical Welsh hero/king Math? :-)

"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"

I think there's more to it than historical accident. After all, it was a historical accident, of sorts, that people believed one could sail directly west from Europe to arrive in Asia, but once a continent was found in between it was no trouble at all to overturn that belief. Historical accident is not the only reason, or necessarily the major reason, that we are still struggling with single-world QM today. I think if we had started with Everett, and someone had later come up with Copenhagen, the latter would gain a nontrivial number of adherents. Note: this should not be taken as justifying Copenhagen in any way. I am merely saying that this particular analysis of the egregious error that is Copenhagen is not complete.

Im just reading Thomas Schelling's Theory of Conflict and one of his key tenets is that providing an identifiable point around which the discussion can be centered will tend to lead the discussion to be centered around that (classical anchoring). However, he brings out that in many cases, having a "line in the sand" brings benefits to all sides by allowing intermediate deals to be struck when only extremes were possible before.

This article, however, clearly demonstrates that having a line in the sand can be just as bad as it can be good, as it is with all of biases. However, I really recommend Schelling hit on "what is good" (in the evolutionary sense) about this phenomenon.

The following formula is difficult to read at a first glance because of the unfortunate line break: (water -> ice cubes + electricity)

I was first trying to parse it as: water "minus" "is greater than" ice cubes...

I've heard the foo->bar construct in C pronounced something like that at least once.

The wikipedia article for Abductive Reasoning claims this sort of privileging the hypothesis can be seen as an instance of affirming the consequent.

Good post.

I'm not sure that 'privileging the hypothesis' deserves to be called a fallacy, though. It's only a bad idea because of the biases that humans happen to have. It can lead to misconceptions for us primates, but it's not a logical error in itself, is it?

It may not be a completely generic bias or fallacy, but it certainly can affect more than just human decision processes. There are a number of primitive systems that exhibit pathologies similar to what Eliezer is describing, speech recognition systems, for example, have a huge issue almost exactly isomorphic to this. Once some interpretation of a audio wave is a hypothesis, it is chosen in great excess to it's real probability or confidence. This is the primary weakness of rule-based voice grammars, that their pre-determined possible interpretations lead to unexpected inputs being slotted into the nearest pre-existing hypothesis, rather than leading to a novel interpretation. The use of statistical grammars to try to pound interpretations to their 'natural' probabilistic initial weight is an attempt to avoid this issue.

This problem is also hidden in a great many AI decision systems within the 'hypothesis generation' system, or equivalent. However elegant the ranking and updating system, if your initial possible list is weak, you distort your whole decisions process.

This problem is also hidden in a great many AI decision systems within the 'hypothesis generation' system

At that point we're dealing with a full-fledged artificial heuristic and bias - the generation system is the heuristic, and the bias is the overly limited collection of hypotheses it manages to formulate for explicit attention at a given point.

I'd reserve "fallacy" for motivated or egregious cases, the sort that humans try to get away with.

Is then the ability to explicitly (at a high, abstract level) reach down to the initial hypothesis generation and include, raise, or add hypotheses for consideration always a pathology?

I can imagine a system where extremely low probability hypotheses, by virtue of complexity or special evidence required, might need to be formulated or added by high level processes, but you could simply view that as another failure of the generation system, and require that even extremely rare or novel structures of hypotheses must go through channels to avoid this kind of disturbance of natural frequencies, as it were.