Yesterday's post concluded:
To form accurate beliefs about something, you really do have to observe it. It's a very physical, very real process: any rational mind does "work" in the thermodynamic sense, not just the sense of mental effort... So unless you can tell me which specific step in your argument violates the laws of physics by giving you true knowledge of the unseen, don't expect me to believe that a big, elaborate clever argument can do it either.
One of the chief morals of the mathematical analogy between thermodynamics and cognition is that the constraints of probability are inescapable; probability may be a "subjective state of belief", but the laws of probability are harder than steel.
People learn under the traditional school regimen that the teacher tells you certain things, and you must believe them and recite them back; but if a mere student suggests a belief, you do not have to obey it. They map the domain of belief onto the domain of authority, and think that a certain belief is like an order that must be obeyed, but a probabilistic belief is like a mere suggestion.
They look at a lottery ticket, and say, "But you can't prove I won't win, right?" Meaning: "You may have calculated a low probability of winning, but since it is a probability, it's just a suggestion, and I am allowed to believe what I want."
Here's a little experiment: Smash an egg on the floor. The rule that says that the egg won't spontaneously reform and leap back into your hand is merely probabilistic. A suggestion, if you will. The laws of thermodynamics are probabilistic, so they can't really be laws, the way that "Thou shalt not murder" is a law... right?
So why not just ignore the suggestion? Then the egg will unscramble itself... right?
It may help to think of it this way - if you still have some lingering intuition that uncertain beliefs are not authoritative:
In reality, there may be a very small chance that the egg spontaneously reforms. But you cannot expect it to reform. You must expect it to smash. Your mandatory belief is that the egg's probability of spontaneous reformation is ~0. Probabilities are not certainties, but the laws of probability are theorems.
If you doubt this, try dropping an egg on the floor a few decillion times, ignoring the thermodynamic suggestion and expecting it to spontaneously reassemble, and see what happens. Probabilities may be subjective states of belief, but the laws governing them are stronger by far than steel.
I once knew a fellow who was convinced that his system of wheels and gears would produce reactionless thrust, and he had an Excel spreadsheet that would prove this - which of course he couldn't show us because he was still developing the system. In classical mechanics, violating Conservation of Momentum is provably impossible. So any Excel spreadsheet calculated according to the rules of classical mechanics must necessarily show that no reactionless thrust exists - unless your machine is complicated enough that you have made a mistake in the calculations.
And similarly, when half-trained or tenth-trained rationalists abandon their art and try to believe without evidence just this once, they often build vast edifices of justification, confusing themselves just enough to conceal the magical steps.
It can be quite a pain to nail down where the magic occurs - their structure of argument tends to morph and squirm away as you interrogate them. But there's always some step where a tiny probability turns into a large one - where they try to believe without evidence - where they step into the unknown, thinking, "No one can prove me wrong".
Their foot naturally lands on thin air, for there is far more thin air than ground in the realms of Possibility. Ah, but there is an (exponentially tiny) amount of ground in Possibility, and you do have an (exponentially tiny) probability of hitting it by luck, so maybe this time, your foot will land in the right place! It is merely a probability, so it must be merely a suggestion.
The exact state of a glass of boiling-hot water may be unknown to you - indeed, your ignorance of its exact state is what makes the molecules' kinetic energy "heat", rather than work waiting to be extracted like the momentum of a spinning flywheel. So the water might cool down your hand instead of heating it up, with probability ~0.
Decide to ignore the laws of thermodynamics and stick your hand in anyway, and you'll get burned.
"But you don't know that!"
I don't know it with certainty, but it is mandatory that I expect it to happen. Probabilities are not logical truths, but the laws of probability are.
"But what if I guess the state of the boiling water, and I happen to guess correctly?"
Your chance of guessing correctly by luck, is even less than the chance of the boiling water cooling your hand by luck.
"But you can't prove I won't guess correctly."
I can (indeed, must) assign extremely low probability to it.
"That's not the same as certainty, though."
Hey, maybe if you add enough wheels and gears to your argument, it'll turn warm water into electricity and ice cubes! Or, rather, you will no longer see why this couldn't be the case.
"Right! I can't see why couldn't be the case! So maybe it is!"
Another gear? That just makes your machine even less efficient. It wasn't a perpetual motion machine before, and each extra gear you add makes it even less efficient than that.
Each extra detail in your argument necessarily decreases the joint probability. The probability that you've violated the Second Law of Thermodynamics without knowing exactly how, by guessing the exact state of boiling water without evidence, so that you can stick your finger in without getting burned, is, necessarily, even less than the probability of sticking in your finger into boiling water without getting burned.
I say all this, because people really do construct these huge edifices of argument in the course of believing without evidence. One must learn to see this as analogous to all the wheels and gears that fellow added onto his reactionless drive, until he finally collected enough complications to make a mistake in his Excel spreadsheet.
there are so many beautiful things about this post. i have only a smallish idea what you are talking about a good deal of the time, but i can easily understand this: "To form accurate beliefs about something, you really do have to observe it."
your posts here have helped solidify this for me. this is true, i feel strongly, for all great art, for reliable science, and for joy in living. observation is everything.
this sentence: "One of the chief morals of the mathematical analogy between thermodynamics and cognition is that the constraints of probability are inescapable; probability may be a "subjective state of belief", but the laws of probability are harder than steel."
man. this is one of those sentences i'll be meditating on for a while. god that's a good one. just simply as a piece of writing, that is nice work. i'm going to put it on a t-shirt or something. maybe a bumper sticker, see how many accidents i can cause on the way home...
Reading today's fare is a bit like eating unflavored oatmeal. :-)
It seems to me that the person who can read this and understand it, already knows it.
But the person who does not know it, cannot understand it and will be frustrated by reading it.
I'm not sure what your intention is with the whole series of posts, but if you'd like to enligthen the muggles, the trick is to explain it in a concise, striking, unusual, easily understood, entertaining manner.
Of course, that takes genius. :-)
But otherwise you are writing primarily for people who already know it.
In yet other words: some of your posts, I will forward to my wife. Others, I won't. This one is one of the latter.
Interesting stuff about the preservation of phase space volume, though. I appreciate it, I previously knew nothing about that.
I would be interested in seeing you talk about belief and probability in cases where the deck is not quite so stacked as it is in your thermodynamic examples. The principles should be the same, but I imagine your argument will be less tractable. No?
I thought this post was actually very easy to follow. It actually gave me the vocab I needed to settle scores of arguments with dumbarse religious types who say daft things like "Evolution is only a religion" and "You can't PROVE that blah blah blah magic fairy dust blah blah blah blah".
I love that. "I don't know it with certainty - that's because of your ridiculous understanding of what 'certainty' means - but it is mandatory that I expect it to happen."
hang on - they don't say "evolution is only a religion" they say "Evolution is only a theory".
....dumbarse atheistic types who don't proofread their own rantings....
Well the incarcerated* Kent Hovind did used to say that evolution is a religion. But I never heard him saying it was "just" a religion.
*HAD to include that
I have to point out that Kent Hovind is in prison for tax evasion, not for being wrong about evolution (though I'm sure he'd like you to think that!).
I've actually met the man, I went to a Christian school as a child and he was a guest speaker at more than one assembly. It's only years later that I realize how ignorant and intentionally blind to the facts he is, and that his arguments rested entirely on straw men and a misunderstanding of the evidence. It slowed my understanding of science by at least five years, maybe more, and I'm more than just a little pissed about that.
It wasn't until I saw japanese mud-skippers in a nature show a few years ago that it really clicked - one of his slams against evolution was "why don't you see fish crawling out of the water?" Well, there you go. 100% fish, living on land - they crawl into mud periodically to breath, pretty cool really. Their front fins are even elongated and act more like fingerless arms than paddles. I wouldn't be surprised if there were a hundred other similar species of fish out there.
When you realize that the theory of evolution has been accurately guiding biologists' expectations for life in bizarre places, and archaeologists' and paleontologists' expectations for artifacts and fossils, for many decades, it becomes clear who was right and who was wrong, just as a matter of practical application. It can be hard to see this when you've been taught to dismiss evolution out of hand, though, and Hovind's intellectually bankrupt arguments give you a fake foundation on which to base such dismissal, so long as you don't question it too much.
Walking catfish-- they actually just sort of slither, but they can locomote.
I would be interested in seeing you talk about belief and probability in cases where the deck is not quite so stacked as it is in your thermodynamic examples.
Okay: It's less stacked for lottery tickets than thermodynamics, and it's less stacked for roulette wheels than lottery tickets.
If you stick your fingers in the game anyway, it is mandatory to expect your fingers to get toasted, with not quite as extreme a probability.
The principles should be the same, but I imagine your argument will be less tractable. No?
The principles are exactly the same, and the real force of the argument is exactly unchanged. But to the unenlightened ones the argument seems to have less emotional force, and they do indeed try to ignore the laws of probability, and the expected proportion of them burn their hands.
What do you think the relation between the mental category of "certainty" and probability is?
For the primitive it is not true that "the sun will rise with 100% certainty" - it is simply "certain that the sun will rise." What's more, I think these statements are -not equivalent-.
For the "educated westerner" it is true that "the sun will rise with certainty very close to 100%, given some assumptions about the nature of the universe in earth's neighborhood." Certainty is not a necessity any longer.
My claim would be that, for most, heuristic descriptions of possibility/probability and an understanding of the mathematical laws or probability are absolutely disjoint. The reason that you can even think about low probability events is not mere knowledge - you must actually switch the context in which you are framing the problem - you must "step back" and examine the lottery in the context of theory in which you (rightly) believe.
What I'm saying here is that arguing against -heuristic descriptions- with -actual probabilities- (even if just approximations) is like arguing against a shaman's perception of the weather with modern supercomputer-driven approximations. You have to consider that people have an investment in their heuristic descriptions - to leave them would be like to leave a nice warm place which makes you happy (most of the time) but might have some nagging problems (e.g. playing the lottery).
the ep. dealmaker said: "I would be interested in seeing you talk about belief and probability in cases where the deck is not quite so stacked as it is in your thermodynamic examples."
It seems reasonable for a financial analyst to understand that the lottery and coin flipping aren't "stacked".
Hmm, what else could he mean...
Perhaps he means something like the weather.
New rule for the lottery: every time you play but don't win, stick your hand in a pan of hot water!
I favor taking energy from earth rotation. Put a horizontal gyroscope across a circular track around the North Pole, and let the earth rotate under it. Take energy from the relative motion.
I suspect the real problem with the lottery is that people are familiar with the amounts of money necessary to purchase a ticket and recognize that it won't bring them much happiness or satisfaction... but they're not familiar with the amounts of money typically given away in a lottery jackpot, and they imagine that it will make them much, much happier than it is actually likely too.
Even people who know that the lottery has a negative expected financial value buy tickets. They'll accept a resource loss if it leads to greater utility, and people tend to perceive the slim chance of winning lots of money to be sufficiently valuable that they'll take an expected loss to have the chance.
The irrationality comes not from dismissing the statistical loss of money, but from believing that winning offers much greater utility than it will.
I suspect the people who suspect a real problem with the lottery have never played it.
I don't play regularly, or at all anymore. I can actually count on one hand the number of times I have, but in all those occasions the primary joy from that was not the possibility that I might become more wealthy. It was because it was fun to engage with my peers in a group discussion of "What If."
From what I have witnessed, this seemed to be a popular activity: the discussion of fantasy. This didn't mean that anyone had any illusions about the possibility of winning. I can do that math.
Simply viewing it as a probability game ignores a motivation: it's fun to dream. And it's fun to do so together. "What would you get?" "Who would you give money to?" "Would you quit right away or give two-weeks' notice?" and so on.
Of course, because I only bought lottery tickets with people who bought lottery tickets with me means that my sample is biased towards those who bought them with me. And that I bought lottery tickets.
Edit: Just a note that the "What If" game need not be a social activity. Obviously.
The "what if" game can be played even if you don't buy the ticket.
What's more, there's another "what if" game that you're neglecting... that's the "what if I invest this money in something actually achievable here and now?"
This is the game that investors and entrepreneurs play, and if you actually put money into the end-result of that game you have a higher expected payoff than that with the "lottery ticket what if" game
You may find a waste of hope interesting. Like taryneast suggests, everyone plays the "what if?" game- what matters is what you play it about. "What if Brad Pitt leaves Angelina Jolie for me?" is a less profitable question to think about than "What if I talk to the cute guy at the coffee shop?". And since you only think about one or two of those questions at a time, there is a real trade-off involved with planning for the first instead of the second.
"I favor taking energy from earth rotation. Put a horizontal gyroscope across a circular track around the North Pole, and let the earth rotate under it. Take energy from the relative motion."
The energy to put it there will come out of your allowance.
A more practical way to take energy from the rotations of the earth-moon system is to exploit the tides...
That was actually a joke. Though people would be hard-pressed to guess what happens if you try it.
Gyroscopes are very unintuitive, because people intuitively but incorrectly think that pushing on something changes its position, a mistake that gyroscopes bring out.
How do you distinguish between principles as solid as the conservation laws vs. the commonly held belief (recently established as wrong) that adults don't have significant neuroplasticity?
Which ones seem to have stupidly huge amounts of evidence, lower complexity, deeper ties to the rest of our theories/models of reality, etc?
ie, the usual way: downgrade based on complexity (more complex assumptions = lower probability), upgrade based on huge amounts of evidence, etc.
Or do I misunderstand your question?
No, you understood me correctly.
The problem is a result of confusing consensus with knowledge.
And that's a really easy mistake to make-- it isn't as though there's a handy index to how much evidence there is for commonly held beliefs.
I hope there will be handy indexes once we've accumulated enough accurate beliefs, widely.
It doesn't help that our most accurate beliefs (e.g. the standard model of physics) are some of the most difficult to understand, or that beliefs with lots of evidence (e.g. evolution, the age of Earth) are not widely held.
This is gold.
"The rule that says that the egg won't spontaneously reform and leap back into your hand is merely probabilistic."
This example requires a level of education that doesn't match my belief of the expected audience of this post.
The low importance in the distinction between mathematical certainty and realistic likelihood is valid, but involving quantum probability kills the post for me.
The example doesn't require quantum physics. Just ordinary classical mechanics.
My point still holds. Most people, myself included, don't have a belief that an egg will spontaneously reform according any laws of physics. To use it as an example of the difference between certainty and likelihood is ineffective.
If it were something too open to debate, it would take away from the point.
The point is as stated. There is a non-zero probability it will happen, so you shouldn't use "certain", but any reasonable person will act on the belief it isn't going to happen.
If he used religion, which is also extremely unlikely to be correct, it would distract from the point.
This looks like a valuable special case of scope insensitivity.
A time to turn off advocatus diaboli and really *be* a clever (hopefully) skeptic... One could say that there is still a difference between probabilities so high/low that you can use ~1/~0 writings and probable but not THAT probable situations such as 98:2 (there is the obvious question of threshold but please bear with me here). You suggest about the same course of actions for, say, a scientific theory of the first and the second type while I believe that the first type is to be in practice equated to the 1/0 and thus called (quasi-)deterministic whereas the second is probably wrong unless you can find a (quasi-)deterministic explanation for the rogue 2 (and thus change the theory) - so in that sense there is no longer place for intrinsically non-deterministic theories like quantum theory.
One may respond that some intrinsically non-deterministic theories do *work* (whether we are speaking of statistics or quantum theory) - but it is a difficult question whether that means they are true or that they are close to an (unknown) deterministic theory. Do we have actual reasons *besides* "working" to believe world is non-(quasi-)deterministic? Threshold uncertainty may be one - but then the unknown deterministic theory may derive the threshold itself so that the uncertainty is only a property of our wrong map.
Addendum: and besides Bell's theorem. Every time I'm convinced by it (and it happened three times in my life, including one three days ago) an hour or a day later I notice I'm confused - confused enough to deconvince myself even though I still don't know what is wrong.
From what I understand, the Many Worlds interpretation of quantum physics is deterministic - everything possible does happen. It only seems probabilistic from inside one of the worlds. You can't predict the outcome of a quantum event, since different instances of you will observe all possible outcomes. Take with a grain of salt, since Many Words is unproven (possibly unprovable?) and my understanding is surface level at best.
On the macro level, a coin toss of a fair coin becomes predictable if you have perfect knowledge and enough computational power. The point of probabilities and statistics is that they give us the rules for mapmaking with imperfect knowledge and limited computational power.
In short, a deterministic universe doesn't lead to certainty in our maps - hence "probability may be a "subjective state of belief", but the laws of probability are harder than steel."
A common reaction to QM is that it doesn't matter since quantum randomness will never manifest itself at the macroscopic level -- that is, in the world of sticks and stones we can see with the naked eye. An appeal is usually made to the "law of large numbers", according to which random fluctuations at the atomic (or lower level) will cancel each other out in a macroscopic object, so that what is seen is an averaged-out behaviour that is fairly predictable.
Something like this must be happening in some cases, assuming QM is a correct description of the micro-world, or there would not even be an appearance of a deterministic macro-world. Since deterministic classical physics is partially correct, there must be a mechanism that makes the QM micro-world at least approximate to the classical description.
However, it it were the case that all macroscopic objects behaved in a 100% deterministic fashion, there would be no evidence for QM in the first place -- since all scientific apparatus is in the macro-world ! A geiger-counter is able to amplify the impact of a single particle into an audible click. Richard Feynman suggested that if that wasn't macroscopic enough, you could always amplify the signal further and use it to set off a stick of dynamite! It could be objected that these are artificial situations. However, because there is a well-known natural mechanism that could do the same job: critical dependence on initial conditions, or classical chaos.
Oops! Beautiful. Your comment described my implicit assumptions probably better than I could, before showing me the error in my thinking. I will have to try and accept the consequences of QM on a deeper level. "It all adds up to normality" is a weak consolation, if you happen to be far from the median after all. It's also becoming blindingly obvious I should finally just sit down and read Feynman.
Huh. The universe is non-deterministic after all. Like, for real. I knew May was going way too peacefully.
Edit: Forgot to say: Thank you for that!
I don't think that Eliezer would disagree with this.
As I understand it, he generally argues for following the numbers and in this post he tries to bind the reader's emotions to reality: He gives examples that make it emotionally clear that it already is in our interest to follow the numbers ('hot water need not *necessarily* burn you, but you correctly do not count on this. Getting burned is bad') and forces one to contrast this realisation with examples where common intuition/behaviour doesn't follow the numbers ('you do not *necessarily* loose money in a lottery, but you are mistaken to count on this. Loosing money is bad').