Deriving the Lorentz transformation

My last few posts have been all about visualizing the Lorentz transformation, the coordinate transformation in special relativity. But where does this transformation come from? In this post, I’ll derive it from basic principles. I saw this derivation first probably a year ago, and have since tried unsuccessfully to re-find the source.  It isn’t the algebraically simplest derivation I’ve seen, but it is the conceptually simplest. The principles we’ll use to derive the transformation should all seem extremely obvious to you.

So let’s dive straight in!

The Lorentz transformation in full generality is a 4D matrix that tells you how to transform spacetime coordinates in one inertial reference frame to spacetime coordinates in another inertial reference frame. It turns out that once you’ve found the Lorentz transformation for one spatial dimension, it’s quite simple to generalize it to three spatial dimensions, so for simplicity we’ll just stick to the 1D case. The Lorentz transformation also allows you to transform to a coordinate system that is both translated some distance and rotated some angle. Both of these are pretty straightforward, and work the way we intuitively think rotation and translation should work. So I’ll not consider them either. The interesting part of the Lorentz transformation is what happens when we translate to reference frames that are co-moving (moving with respect to one another). Strictly speaking, this is called a Lorentz boost. That’s what I’ll be deriving for you: the 1D Lorentz boost.

So, we start by imagine some reference frame, in which an event is labeled by its temporal and spatial coordinates: t and x. Then we look at a new reference frame moving at velocity v with respect to the starting reference frame. We describe the temporal and spatial coordinates of the same event in the new coordinate system: t’ and x’. In general, these new coordinates can be any function whatsoever of the starting coordinates and the velocity v.

Screen Shot 2018-12-09 at 10.31.11 PM.png

To narrow down what these functions f and g might be, we need to postulate some general relationship between the primed and unprimed coordinate system.

So, our first postulate!

1. Straight lines stay straight.

Our first postulate is that all observers in inertial reference frames will agree about if an object is moving at a constant velocity. Since objects moving at constant velocities are straight lines on diagrams of position vs time, this is equivalent to saying that a straight path through spacetime in one reference frame is a straight path through spacetime in all reference frames.

More formally, if x is proportional to t, then x’ is proportional to t’ (though the constant of proportionality may differ).

Screen Shot 2018-12-09 at 10.41.03 PM.png

This postulate turns out to be immensely powerful. There is a special name for the types of transformations that keep straight lines straight: they are linear transformations. (Note, by the way, that the linearity is only in the coordinates t and x, since those are the things that retain straightness. There is no guarantee that the dependence on v will be linear, and in fact it will turn out not to be.)

 These transformations are extremely simple, and can be represented by a matrix. Let’s write out the matrix in full generality:

Screen Shot 2018-12-09 at 10.45.02 PM.png

We’ve gone from two functions (f and g) to four (A, B, C, and D). But in exchange, each of these four functions is now only a function of one variable: the velocity v. For ease of future reference, I’ve chosen to name the matrix T(v).

So, our first postulate gives us linearity. On to the second!

2. An object at rest in the starting reference frame is moving with velocity -v in the moving reference frame

This is more or less definitional. If somebody tells you that they had a function that transformed coordinates from one reference frame to a moving reference frame, then the most basic check you can do to see if they’re telling the truth is verify that objects at rest in the starting reference frame end up moving in the final reference frame. And again, it seems to follow from what it means for the reference frame to be moving right at 1 m/s that the initially stationary objects should end up moving left at 1 m/s.

Let’s consider an object sitting at rest at x = 0 in the starting frame of reference. Then we have:

Screen Shot 2018-12-09 at 10.52.06 PM.png

We can plug this into our matrix to get a constraint on the functions A and C:

Screen Shot 2018-12-09 at 10.54.59 PM.png

Great! We’ve gone from four functions to three!

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3. Moving to the left at velocity v and to the right at the same velocity is the same as not moving at all

More specifically: Start with any reference frame. Now consider a new reference frame that is moving at velocity v with respect to the starting reference frame. Now, from this new reference frame, consider a third reference frame that is moving at velocity -v. This third reference frame should be identical to the one we started with. Got it?

Formally, this is simply saying the following:

Screen Shot 2018-12-09 at 11.01.36 PM.png

(I is the identity matrix.)

To make this equation useful, we need to say more about T(-v). In particular, it would be best if we could express T(-v) in terms of our three functions A(v), B(v), and D(v). We do this with our next postulate:

4. Moving at velocity -v is the same as turning 180°, then moving at velocity v, then turning 180° again.

Again, this is quite self-explanatory. As a geometric fact, the reference frame you end up with by turning around, moving at velocity v, and then turning back has got to be the same as the reference frame you’d end up with by moving at velocity -v. All we need to formalize this postulate is the matrix corresponding to rotating 180°.

Screen Shot 2018-12-09 at 11.07.28 PM.png

There we go! Rotating by 180° is the same as taking every position in the starting reference frame and flipping its sign. Now we can write our postulate more precisely:

Screen Shot 2018-12-09 at 11.09.47 PM

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Now we can finally use Postulate 3!

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Doing a little algebra, we get…

Screen Shot 2018-12-09 at 11.12.42 PM.png

(You might notice that we can only conclude that A = D if we reject the possibility that A = B = 0. We are allowed to do this because allowing A = B = 0 gives us a trivial result in which a moving reference frame experiences no time. Prove this for yourself!)

Now we have managed to express all four of our starting functions in terms of just one!

Screen Shot 2018-12-09 at 11.18.23 PM.png

So far our assumptions have been grounded by almost entirely a priori considerations about what we mean by velocity. It’s pretty amazing how far we got with so little! But to progress, we need to include one final a posteriori postulate, that which motivated Einstein to develop special relativity in the first place: the invariance of the speed of light.

5. Light’s velocity is c in all reference frames.

The motivation for this postulate comes from mountains of empirical evidence, as well as good theoretical arguments from the nature of light as an electromagnetic phenomenon. We can write it quite simply as:

Screen Shot 2018-12-09 at 11.43.23 PM

Plugging in our transformation, we get:

Screen Shot 2018-12-09 at 11.43.28 PM

Multiplying the time coordinate by c must give us the space coordinate:

Screen Shot 2018-12-10 at 3.27.16 AM

And we’re done with the derivation!

Summarizing our five postulates:

Screen Shot 2018-12-10 at 12.37.23 AM.png

And our final result:

Screen Shot 2018-12-10 at 3.29.09 AM.png

Pushing anti-anthropic intuitions

A stranger comes up to you and offers to play the following game with you: “I will roll a pair of dice. If they land snake eyes (i.e. they both land 1), you give me one dollar. Otherwise, if they land anything else, I give you a dollar.”

Do you play this game?

Here’s an intuitive response: Yes, of course you should! You have a 35/36 chance of gaining $1, and only a 1/36 chance of losing $1. You’d have to be quite risk averse to refuse those odds.

What if the stranger tells you that they are giving this same bet to many other people? Should that change your calculation?

Intuitively: No, of course not! It doesn’t matter what else the stranger is doing with other people.

What if they tell you that they’ve given this offer to people in the past, and might give the offer to others in the future? Should that change anything?

Once again, it seems intuitively not to matter. The offers given to others simply have nothing to do with you. What matters are your possible outcomes and the probabilities of each of these outcomes. And what other people are doing has nothing to do with either of these.

… Right?

Now imagine that the stranger is playing the game in the following way: First they find one person and offer to play the game with them. If the dice land snake eyes, then they collect a dollar and stop playing the game. Otherwise, they find ten new people and offer to play the game with them. Same as before: snake eyes, the stranger collects $1 from each and stops playing, otherwise he moves on to 100 new people. Et cetera forever.

We now ask the question: How does the average person given the offer do if they take the offer? Well, no matter how many rounds of offers the stranger gives, at least 90% of people end up in his last round. That means that at least 90% of people end up giving over $1 and at most 10% gain $1. This is clearly net negative for those that hand over money!

Think about it this way: Imagine a population of individuals who all take the offer, and compare them to a population that all reject the offer. Which population does better on average?

For the population who takes the offer, the average person loses money. An upper bound on how much they lose is 10% ($1) + 90% (-$1) = -$.80. For the population that reject the offer, nobody gains money or loses It either: the average case is exactly $0. $0 is better than -$.80, so the strategy of rejecting the offer is better, on average!

This thought experiment is very closely related to the dice killer thought experiment. I think of it as a variant that pushes our anti-anthropic-reasoning intuitions. It just seems really wrong to me that if somebody comes up to you and offers you this deal that has a 35/36 chance of paying out you should reject it. The details of who else is being offered the deal seem totally irrelevant.

But of course, all of the previous arguments I’ve made for anthropic reasoning apply here as well. And it is just true that the average person that rejects the offer does better than the average person that accepts it. Perhaps this is just another bullet that we have to bite in our attempt to formalize rationality!

Clarifying self-defeating beliefs

In a previous post, I mentioned self-defeating beliefs as a category that I am confused about. I wrote:

How should we reason about self defeating beliefs?

The classic self-defeating belief is “This statement is a lie.” If you believe it, then you are compelled to disbelieve it, eliminating the need to believe it in the first place. Broadly speaking, self-defeating beliefs are those that undermine the justifications for belief in them.

Here’s an example that might actually apply in the real world: Black holes glow. The process of emission is known as Hawking radiation. In principle, any configuration of particles with a mass less than the black hole can be emitted from it. Larger configurations are less likely to be emitted, but even configurations such as a human brain have a non-zero probability of being emitted. Henceforth, we will call such configurations black hole brains.

Now, imagine discovering some cosmological evidence that the era in which life can naturally arise on planets circling stars is finite, and that after this era there will be an infinite stretch of time during which all that exists are black holes and their radiation. In such a universe, the expected number of black hole brains produced is infinite (a tiny finite probability multiplied by an infinite stretch of time), while the expected number of “ordinary” brains produced is finite (assuming a finite spatial extent as well).

What this means is that discovering this cosmological evidence should give you an extremely strong boost in credence that you are a black hole brain. (Simply because most brains in your exact situation are black hole brains.) But most black hole brains have completely unreliable beliefs about their environment! They are produced by a stochastic process which cares nothing for producing brains with reliable beliefs. So if you believe that you are a black hole brain, then you should suddenly doubt all of your experiences and beliefs. In particular, you have no reason to think that the cosmological evidence you received was veridical at all!

I don’t know how to deal with this. It seems perfectly possible to find evidence for a scenario that suggests that we are black hole brains (I’d say that we havealready found such evidence, multiple times). But then it seems we have no way to rationally respond to this evidence! In fact, if we do a naive application of Bayes’ theorem here, we find that the probability of receiving any evidence in support of black hole brains to be 0!

So we have a few options. First, we could rule out any possible skeptical scenarios like black hole brains, as well as anything that could provide anyamount of evidence for them (no matter how tiny). Or we could accept the possibility of such scenarios but face paralysis upon actually encountering evidence for them! Both of these seem clearly wrong, but I don’t know what else to do.

A friend (whose blog Compassionate Equilibria you should definitely check out) left a comment in response, saying:

I think I feel somewhat less confused about self-defeating beliefs (at least when considering the black hole brain scenario maybe I would feel more confused about other cases).

It seems like the problem might be when you say “imagine discovering some cosmological evidence that the era in which life can naturally arise on planets circling stars is finite, and that after this era there will be an infinite stretch of time during which all that exists are black holes and their radiation.” Presumably, whatever experience you had that you are interpreting as this cosmological evidence is an experience that you would actually be very unlikely to have given that you exist in that universe and as a result shouldn’t be interpreted as evidence for existing in such a universe. Instead you would have to think about in what kind of universe would you be most likely to have those experiences that naively seemed to indicate living in a universe with an infinity of black hole brains.

This could be a very difficult question to answer but not totally intractable. This also doesn’t seem to rule out starting with a high prior in being a black hole brain and it seems like you might even be able to get evidence for being a black hole brain (although I’m not sure what this would be; maybe having a some crazy jumble of incoherent experiences while suddenly dying?).

I think this is a really good point that clears up a lot of my confusion on the topic. My response ended up being quite long, so I’ve decided to make it its own post.

 

*** My response starts here ***

 

The key point that I was stuck on before reading this comment was the notion that this argument puts a strong a priori constraint on the types of experiences we can expect to have. This is because P(E) is near zero when E strongly implies a theory and that theory undermines E.

Your point, which seems right, is: It’s not that it’s impossible or near impossible to observe certain things that appear to strongly suggest a cosmology with an infinity of black hole brains. It’s that we can observe these things, and they aren’t actually evidence for these cosmologies (for just the reasons you laid out).

That is, there just aren’t observations that provide evidence for radical skeptical scenarios. Observations that appear to provide such evidence, prove to not do so upon closer examination. It’s about the fact that the belief that you are a black hole brain is by construction unmotivateable: this is what it means to say P(E) ~ 0. (More precisely, the types of observations that actually provide evidence for black hole brains are those that are not undermined by the belief in black hole brains. Your “crazy jumble of incoherent experiences” might be a good example of this. And importantly, basically any scientific evidence of the sort that we think could adjudicate between different cosmological theories will be undermined.)

One more thing as I digest this: Previously I had been really disturbed by the idea that I’d heard mentioned by Sean Carroll and others that one criterion for a feasible cosmology is that it doesn’t end up making it highly likely that we are black hole brains. This seemed like a bizarrely strong a priori constraint on the types of theories we allow ourselves to consider. But this actually makes a lot of sense if conceived of not as an a priori constraint but as a combination of two things: (1) updating on the strong experiential evidence that we are not black hole brains (the extremely structured and self-consistent nature of our experiences) and (2) noticing that these theories are very difficult to motivate, as most pieces of evidence that intuitively seem to support them actually don’t upon closer examination.

So (1) the condition that P(E) is near zero is not necessarily a constraint on your possible experiences, and (2) it makes sense to treat cosmologies that imply that we are black hole brains as empirically unsound and nearly unmotivateable.

Now, I’m almost all the way there, but still have a few remaining hesitations.

One thing is that things get more confusing when you break an argument for black hole brains down into its component parts and try to figure out where exactly you went wrong. Like, say you already have a whole lot of evidence that after a finite length of time, the universe will be black holes forever, but don’t yet know about Hawking radiation. So far everything is fine. But now scientists observe Hawking radiation. From this they conclude that black holes radiate, though they don’t have a theory of the stochastic nature of the process that entails that it can in principle produce brains. They then notice that Hawking radiation is actually predicted by combining aspects of QM and GR, and see that this entails that black holes can produce brains. Now they have all the pieces that together imply that they are black hole brains, but at which step did they go wrong? And what should they conclude now? They appear to have developed a mountain of solid evidence that when put together (and combined with some anthropic reasoning) straightforwardly imply that they are black hole brains. But this can’t be the case, since this would undermine the evidence they started with.

We can frame this as a multilemma. The general reasoning process that leads to the conclusion that we are black hole brains might look like:

  1. We observe nature.
  2. We generate laws of physics from these observations.
  3. We predict from the laws of physics that there is a greater abundance of black hole brains than normal brains.
  4. We infer from (3) that we are black hole brains (via anthropic reasoning).

Either this process fails at some point, or we should believe that we are black hole brains. Our multilemma (five propositions, at least one of which must be accepted) is thus:

  1. Our observations of nature were invalid.
  2. Our observations were valid, but our inference of laws of physics from them was invalid.
  3. Our inference of laws of physics from our observations were valid, but our inference from these laws of there being a greater abundance of black hole brains than normal brains was invalid.
  4.  Our inference from the laws of there being a greater abundance of black hole brains from normal brains was valid, but the anthropic step was invalid.
  5. We are black hole brains.

Clearly we want to deny (5). I also would want to deny (3) and (4) – I’m imagining them to be fairly straightforward deductive steps. (1) is just some form of skepticism about our access to nature, which I also want to deny. The best choice, it looks like, is (2): our inductive inference of laws of physics from observations of nature is flawed in some way. But even this is a hard bullet to bite. It’s not sufficient to just say that other laws of physics might equally well or better explain the data. What is required is to say that in fact our observations don’t really provide compelling evidence for QM, GR, and so on.

So the end result is that I pretty much want to deny every possible way the process could have failed, while also denying the conclusion. But we have to deny something! This is clearly not okay!

Summing up: The remaining disturbing thing to me is that it seems totally possible to accidentally run into a situation where your best theories of physics inevitably imply (by a process of reasoning each step of which you accept is valid) that you are a black hole brain, and I’m not sure what to do next at that point.

The Anthropic Dice Killer

Today we discuss anthropic reasoning.

The Problem

Imagine the following scenario:

A mad killer has locked you in a room. You are trapped and alone, with only your knowledge of your situation to help you out.

One piece of information that you have is that you are aware of the maniacal schemes of your captor. His plans began by capturing one random person. He then rolled a pair of dice to determine their fate. If the dice landed snake eyes (both 1), then the captive would be killed. If not, then they would be let free.

But if they are let free, the killer will search for new victims, and this time bring back ten new people and lock them alone in rooms. He will then determine their fate just as before, with a pair of dice. Snake eyes means they die, otherwise they will be let free and he will search for new victims.

His murder spree will continue until the first time he rolls snake eyes. Then he will kill the group that he currently has imprisoned and retire from the serial-killer life.

Now. You become aware of a risky way out of the room you are locked in and to freedom. The chances of surviving this escape route are only 50%. Your choices are thus either (1) to traverse the escape route with a 50% chance of survival or (2) to just wait for the killer to roll his dice, and hope that it doesn’t land snake eyes.

What should you do?

 

 

 

(Think about it before reading on)

 

 

 

A plausible-sounding answer

Your chance of dying if you stay and wait is just the chance that the dice lands snake eyes. The probability of snake eyes is just 1/36 (1/6 for each dice landing 1).

So your chance of death is only 1/36 (≈ 3%) if you wait, and it’s 50% if you try to run for it. Clearly, you are better off waiting!

But…

You guessed it, things aren’t that easy. You have extra information about your situation besides just how the dice works, and you should use it. In particular, the killing pattern of your captor turns out to be very useful information.

Ask the following question: Out of all of the people that have been captured or will be captured at some point by this madman, how many of them will end up dying? This is just the very last group, which, incidentally, is the largest group.

Consider: if the dice land snake eyes the first time they are rolled, then only one person is ever captured, and this person dies. So the fraction of those captured that die is 100%.

If they lands snake eyes the second time they are rolled, then 11 people total are captured, 10 of whom die. So the fraction of those captured that die is 10/11, or ≈ 91%.

If it’s the third time, then 111 people total are captured, 100 of whom die. Now the fraction is just over 90%.

In general, no matter how many times the dice rolls before landing snake eyes, it always ends up that over 90% of those captured end up being in the last round, and thus end up dying.

So! This looks like bad news for you… you’ve been captured, and over 90% of those that are captured always die. Thus, your chance of death is guaranteed to be greater than 90%.

The escape route with a 50% survival chance is looking nicer now, right?

Wtf is this kind of reasoning??

What we just did is called anthropic reasoning. Anthropic reasoning really just means updating on all of the information available to you, including indexical information (information about your existence, age, location, and so on). In this case, the initial argument neglected the very crucial information that you are one of the people that were captured by the killer. When updating on this information, we get an answer that is very very different from what we started with. And in this life-or-death scenario, this is an important difference!

You might still feel hesitant about the answer we got. After all, if you expect a 90% chance of death, this means that you expect a 90% chance for the dice to land snake eyes. But it’s not that you think the dice are biased or anything… Isn’t this just blatantly contradictory?

This is a convincing-sounding rebuttal, but it’s subtly wrong. The key point is that even though the dice are fair, there is a selection bias in the results you are seeing. This selection bias amounts to the fact that when the dice inevitably lands snake-eyes, there are more people around to see it. The fact that you are more likely than 1/36 to see snake-eyes is kind of like the fact that if you are given the ticket of a random concert-goer, you have a higher chance of ending seeing a really popular band than if you just looked at the current proportion of shows performed by really popular bands.

It’s kind of like the fact that in your life you will spend more time waiting in long lines than short lines, and that on average your friends have more friends than you. This all seems counterintuitive and wrong until you think closely about the selection biases involved.

Anyway, I want to impress upon you that 90% really is the right answer, so I’ll throw some math at you. Let’s calculate in full detail what fraction of the group ends up surviving on average.

Screen Shot 2018-08-02 at 1.16.15 AM

By the way, the discrepancy between the baseline chance of death (1/36) and the anthropic chance of death (90%) can be made as large as you like by manipulating the starting problem. Suppose that instead of 1/36, the chance of the group dying was 1/100, and instead of the group multiplying by 10 in size each round, it grew by a factor of 100. Then the baseline chance of death would be 1%, and the anthropic probability would be 99%.

We can find the general formula for any such scenario:

Screen Shot 2018-08-02 at 4.54.30 AM.png

IF ANYBODY CAN SOLVE THIS, PLEASE TELL ME! I’ve been trying for too long now and would really like an analytic general solution. 🙂

There is a lot more to be said about this thought experiment, but I’ll leave it there for now. In the next post, I’ll present a slight variant on this thought experiment that appears to give us a way to get direct Bayesian evidence for different theories of consciousness! Stay tuned.

What do I find conceptually puzzling?

There are lots of things that I don’t know, like, say, what the birth rate in Sweden is or what the effect of poverty on IQ is. There are also lots of things that I find really confusing and hard to understand, like quantum field theory and monetary policy. There’s also a special category of things that I find conceptually puzzling. These things aren’t difficult to grasp because the facts about them are difficult to understand or require learning complicated jargon. Instead, they’re difficult to grasp because I suspect that I’m confused about the concepts in use.

This is a much deeper level of confusion. It can’t be adjudicated by just reading lots of facts about the subject matter. It requires philosophical reflection on the nature of these concepts, which can sometimes leave me totally confused about everything and grasping for the solid ground of mere factual ignorance.

As such, it feels like a big deal when something I’ve been conceptually puzzled about becomes clear. I want to compile a list for future reference of things that I’m currently conceptually puzzled about and things that I’ve become un-puzzled about. (This is not a complete list, but I believe it touches on the major themes.)

Things I’m conceptually puzzled about

What is the relationship between consciousness and physics?

I’ve written about this here.

Essentially, at this point every available viewpoint on consciousness seems wrong to me.

Eliminativism amounts to a denial of pretty much the only thing that we can be sure can’t be denied – that we are having conscious experiences. Physicalism entails the claim that facts about conscious experience can be derived from laws of physics, which is wrong as a matter of logic.

Dualism entails that the laws of physics by themselves cannot account for the behavior of the matter in our brains, which is wrong. And epiphenomenalism entails that our beliefs about our own conscious experience are almost certainly wrong, and are no better representations of our actual conscious experiences than random chance.

How do we make sense of decision theory if we deny libertarian free will?

Written about this here and here.

Decision theory is ultimately about finding the decision D that maximizes expected utility EU(D). But to do this calculation, we have to decide what the set of possible decisions we are searching is.

EU confusion

Make this set too large, and you end up getting fantastical and impossible results (like that the optimal decision is to snap your fingers and make the world into a utopia). Make it too small, and you end up getting underwhelming results (in the extreme case, you just get that the optimal decision is to do exactly what you are going to do, since this is the only thing you can do in a strictly deterministic world).

We want to find a nice middle ground between these two – a boundary where we can say “inside here the things that are actually possible for us to do, and outside are those that are not.” But any principled distinction between what’s in the set and what’s not must be based on some conception of some actions being “truly possible” to us, and others being truly impossible. I don’t know how to make this distinction in the absence of a robust conception of libertarian free will.

Are there objectively right choices of priors?

I’ve written about this here.

If you say no, then there are no objectively right answers to questions like “What should I believe given the evidence I have?” And if you say yes, then you have to deal with thought experiments like the cube problem, where any choice of priors looks arbitrary and unjustifiable.

(If you are going to be handed a cube, and all you know is that it has a volume less than 1 cm3, then setting maximum entropy priors over volumes gives different answers than setting maximum entropy priors over side areas or side lengths. This means that what qualifies as “maximally uncertain” depends on whether we frame our reasoning in terms of side length, areas, or cube volume. Other approaches besides MaxEnt have similar problems of concept dependence.)

How should we deal with infinities in decision theory?

I wrote about this here, here, here, and here.

The basic problem is that expected utility theory does great at delivering reasonable answers when the rewards are finite, but becomes wacky when the rewards become infinite. There are a huge amount of examples of this. For instance, in the St. Petersburg paradox, you are given the option to play a game with an infinite expected payout, suggesting that you should buy in to the game no matter how high the cost. You end up making obviously irrational choices, such as spending $1,000,000 on the hope that a fair coin will land heads 20 times in a row. Variants of this involve the inability of EU theory to distinguish between obviously better and worse bets that have infinite expected value.

And Pascal’s mugging is an even worse case. Roughly speaking, a person comes up to you and threatens you with infinite torture if you don’t submit to them and give them 20 dollars. Now, the probability that this threat is credible is surely tiny. But it is non-zero! (as long as you don’t think it is literally logically impossible for this threat to come true)

An infinite penalty times a finite probability is still an infinite expected penalty. So we stand to gain an infinite expected utility by just handing over the 20 dollars. This seems ridiculous, but I don’t know any reasonable formalization of decision theory that allows me to refute it.

Is causality fundamental?

Causality has been nicely formalized by Pearl’s probabilistic graphical models. This is a simple extension of probability theory, out of which naturally falls causality and counterfactuals.

One can use this framework to represent the states of fundamental particles and how they change over time and interact with one another. What I’m confused about is that in some ways of looking at it, the causal relations appear to be useful but un-fundamental constructs for the sake of easing calculations. In other ways of looking at it, causal relations are necessarily built into the structure of the world, and we can go out and empirically discover them. I don’t know which is right. (Sorry for the vagueness in this one – it’s confusing enough to me that I have trouble even precisely phrasing the dilemma).

How should we deal with the apparent dependence of inductive reasoning upon our choices of concepts?

I’ve written about this here. Beyond just the problem of concept-dependence in our choices of priors, there’s also the problem presented by the grue/bleen thought experiment.

This thought experiment proposes two new concepts: grue (= the set of things that are either green before 2100 or blue after 2100) and bleen (the inverse of grue). It then shows that if we reasoned in terms of grue and bleen, standard induction would have us concluding that all emeralds will suddenly turn blue after 2100. (We repeatedly observed them being grue before 2100, so we should conclude that they will be grue after 2100.)

In other words, choose the wrong concepts and induction breaks down. This is really disturbing – choices of concepts should be merely pragmatic matters! They shouldn’t function as fatal epistemic handicaps. And given that they appear to, we need to develop some criterion we can use to determine what concepts are good and what concepts are bad.

The trouble with this is that the only proposals I’ve seen for such a criterion reference the idea of concepts that “carve reality at its joints”; in other words, the world is composed of green and blue things, not grue and bleen things, so we should use the former rather than the latter. But this relies on the outcome of our inductive process to draw conclusions about the starting step on which this outcome depends!

I don’t know how to cash out “good choices of concepts” without ultimately reasoning circularly. I also don’t even know how to make sense of the idea of concepts being better or worse for more than merely pragmatic reasons.

How should we reason about self defeating beliefs?

The classic self-defeating belief is “This statement is a lie.” If you believe it, then you are compelled to disbelieve it, eliminating the need to believe it in the first place. Broadly speaking, self-defeating beliefs are those that undermine the justifications for belief in them.

Here’s an example that might actually apply in the real world: Black holes glow. The process of emission is known as Hawking radiation. In principle, any configuration of particles with a mass less than the black hole can be emitted from it. Larger configurations are less likely to be emitted, but even configurations such as a human brain have a non-zero probability of being emitted. Henceforth, we will call such configurations black hole brains.

Now, imagine discovering some cosmological evidence that the era in which life can naturally arise on planets circling stars is finite, and that after this era there will be an infinite stretch of time during which all that exists are black holes and their radiation. In such a universe, the expected number of black hole brains produced is infinite (a tiny finite probability multiplied by an infinite stretch of time), while the expected number of “ordinary” brains produced is finite (assuming a finite spatial extent as well).

What this means is that discovering this cosmological evidence should give you an extremely strong boost in credence that you are a black hole brain. (Simply because most brains in your exact situation are black hole brains.) But most black hole brains have completely unreliable beliefs about their environment! They are produced by a stochastic process which cares nothing for producing brains with reliable beliefs. So if you believe that you are a black hole brain, then you should suddenly doubt all of your experiences and beliefs. In particular, you have no reason to think that the cosmological evidence you received was veridical at all!

I don’t know how to deal with this. It seems perfectly possible to find evidence for a scenario that suggests that we are black hole brains (I’d say that we have already found such evidence, multiple times). But then it seems we have no way to rationally respond to this evidence! In fact, if we do a naive application of Bayes’ theorem here, we find that the probability of receiving any evidence in support of black hole brains to be 0!

So we have a few options. First, we could rule out any possible skeptical scenarios like black hole brains, as well as anything that could provide any amount of evidence for them (no matter how tiny). Or we could accept the possibility of such scenarios but face paralysis upon actually encountering evidence for them! Both of these seem clearly wrong, but I don’t know what else to do.

How should we reason about our own existence and indexical statements in general?

This is called anthropic reasoning. I haven’t written about it on this blog, but expect future posts on it.

A thought experiment: imagine a murderous psychopath who has decided to go on an unusual rampage. He will start by abducting one random person. He rolls a pair of dice, and kills the person if they land snake eyes (1, 1). If not, he lets them free and hunts down ten new people. Once again, he rolls his pair of die. If he gets snake eyes he kills all ten. Otherwise he frees them and kidnaps 100 new people. On and on until he eventually gets snake eyes, at which point his murder spree ends.

Now, you wake up and find that you have been abducted. You don’t know how many others have been abducted alongside you. The murderer is about to roll the dice. What is your chance of survival?

Your first thought might be that your chance of death is just the chance of both dice landing 1: 1/36. But think instead about the proportion of all people that are ever abducted by him that end up dying. This value ends up being roughly 90%! So once you condition upon the information that you have been captured, you end up being much more worried about your survival chance.

But at the same time, it seems really wrong to be watching the two dice tumble and internally thinking that there is a 90% chance that they land snake eyes. It’s as if you’re imagining that there’s some weird anthropic “force” pushing the dice towards snake eyes. There’s way more to say about this, but I’ll leave it for future posts.

Things I’ve become un-puzzled about

Newcomb’s problem – one box or two box?

To almost everyone, it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly.

– Nozick, 1969

I’ve spent months and months being hopelessly puzzled about Newcomb’s problem. I now am convinced that there’s an unambiguous right answer, which is to take the one box. I wrote up a dialogue here explaining the justification for this choice.

In a few words, you should one-box because one-boxing makes it nearly certain that the simulation of you run by the predictor also one-boxed, thus making it nearly certain that you will get 1 million dollars. The dependence between your action and the simulation is not an ordinary causal dependence, nor even a spurious correlation – it is a logical dependence arising from the shared input-output structure. It is the same type of dependence that exists in the clone prisoner dilemma, where you can defect or cooperate with an individual you are assured is identical to you in every single way. When you take into account this logical dependence (also called subjunctive dependence), the answer is unambiguous: one-boxing is the way to go.

Summing up:

Things I remain conceptually confused about:

  • Consciousness
  • Decision theory & free will
  • Objective priors
  • Infinities in decision theory
  • Fundamentality of causality
  • Dependence of induction on concept choice
  • Self-defeating beliefs
  • Anthropic reasoning

More on quantum entanglement and irreducibility

A few posts ago, I talked about how quantum mechanics entails the existence of irreducible states – states of particles that in principle cannot be described as the product of their individual components. The classic example of such an entangled state is the two qubit state

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This state describes a system which is in an equal-probability superposition of both particles being |0 and both particles being |1. As it turns out, this state cannot be expressed as the product of two single-qubit states.

A friend of mine asked me a question about this that was good enough to deserve its own post in response. Start by imagining that Alice and Bob each have a coin. They each put their quarter inside a small box with heads facing up. Now they close their respective boxes, and shake them up in the exact same way. This is important! (as well as unrealistic) We suppose that whatever happens to the coin in Alice’s box, also happens to the coin in Bob’s box.

Now we have two boxes, each of which contains a coin, and these coins are guaranteed to be facing the same way. We just don’t know what way they are facing.

Alice and Bob pick up their boxes, being very careful to not disturb the states of their respective coins, and travel to opposite ends of the galaxy. The Milky Way is 100,000 light years across, so any communication between the two now would take a minimum of 100,000 years. But if Alice now opens her box, she instantly knows the state of Bob’s coin!

So while Alice and Bob cannot send messages about the state of their boxes any faster than 100,000 years, they can instantly receive information about each others’ boxes by just observing their own! Is this a contradiction?

No, of course not. While Alice does learn something about Bob’s box, this is not because of any message passed between the two. It is the result of the fact that in the past the configurations of their coins were carefully designed to be identical. So what seemed on its face to be special and interesting turns out to be no paradox at all.

Finally, we get to the question my friend asked. How is this any different from the case of entangled particles in quantum mechanics??

Both systems would be found to be in the states |00 and |11⟩ with equal probability (where |0⟩ is heads and |1⟩ is tails). And both have the property that learning the state of one instantly tells you the state of the other. Indeed, the coins-in-boxes system also has the property of irreducibility that we talked about before! Try as we might, we cannot coherently treat the system of both coins as the product of two independent coins, as doing so will ignore the statistical dependence between the two coins.

(Which, by the way, is exactly the sort of statistical dependence that justifies timeless decision theory and makes it a necessary update to decision theory.)

I love this question. The premise of the question is that we can construct a classical system that behaves in just the same supposedly weird ways that quantum systems behave, and thus make sense of all this mystery. And answering it requires that we get to the root of why quantum mechanics is a fundamentally different description of reality than anything classical.

So! I’ll describe the two primary disanalogies between entangled particles and “entangled” coins.

Epistemic Uncertainty vs Fundamental Indeterminacy

First disanalogy. With the coins, either they are both heads or they are both tails. There is an actual fact in the world about which of these two is true, and the probabilities we reference when we talk about the chance of HH or TT represent epistemic uncertainty. There is a true determinate state of the coins, and probability only arises as a way to deal with our imperfect knowledge.

On the other hand, according to the mainstream interpretation of quantum mechanics, the state of the two particles is fundamentally indeterminate. There isn’t a true fact out there waiting to be discovered about whether the state is |00⟩ or |11⟩. The actual state of the system is this unusual thing called a superposition of |00⟩ and |11⟩. When we observe it to be |00⟩, the state has now actually changed from the superposition to the determinate state.

We can phrase this in terms of counterfactuals: If when we look at the coins, we see that they are HH, then we know that they were HH all along. In particular, we know that if we had observed them a moment later or earlier, we would have gotten H with 100% certainty. Give that we actually observed HH, the probability that we would have observed HH is 100%.

But if we observe the state of the particles to be |00⟩, this does not mean that had we observed it a moment before, we would be guaranteed to get the same answer. Given that we actually observed |00⟩, the probability that we would have observed |00⟩ is still 50%.

(A project for some enterprising reader: see what the truths of these counterfactuals imply for an interpretation of quantum mechanics in terms of Pearl-style causal diagrams. Is it even possible to do?)

Predictive differences

The second difference between the two cases is a straightforward experimental difference. Suppose that Alice and Bob identically prepare thousands of coins as we described before, and also identically prepare thousands of entangled particles. They ensure that the coins are treated exactly the same way, so that they are guaranteed to all be in the same state, and similarly for the entangled pairs.

If they now just observe all of their entangled pairs and coins, they will get similar results – roughly half of the coins will be HH and roughly half of the entangled pairs will be |00⟩. But there are other experiments they could run on the entangled pairs that would give difference answers than 

The conclusion of this is that even if you tried to model the entangled pair as a simple probability distribution similar to the coins, you will get the wrong answer in some experiments. I described what these experiments could be in this earlier post – essentially they involve applying an operation that takes qubits in and out of superposition.

So we have both a theoretical argument and a practical argument for the difference between these two cases. They key take-away is the following:

According to quantum mechanics an entangled pair is in a state that is fundamentally indeterminate. When we describe it with probabilities, we are not saying “This probabilistic description is an account of my imperfect knowledge of the state of the system”. We’re saying that nature herself is undecided on what we will observe when we look at the state. (Side note: there is actually a way to describe epistemic uncertainty in quantum mechanics. It is called the density matrix, and is completely different from the description of superpositions.)

In addition, the most fundamental and accurate probability description for the state of the two particles is one that cannot be described as the product of two independent particles. This is not the case with the coins! The most fundamental and accurate probability description for the state of the two coins is either 100% HH or 100% TT (whichever turns out to be the case). What this means is that in the quantum case, not only is the state indeterminate, but the two particles are fundamentally interdependent – entangled. There is no independent description of the individual components of the system, there is only the system as a whole.

Quantum mechanics, reductionism, and irreducibility

Take a look at the following two qubit state:

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Is it possible to describe this two-qubit system in terms of the states of the two individual particles that compose it? The principle of reductionism suggests to us that yes it should be possible; after all, of course we can always take any larger system and describe it perfectly fine in terms of its components.

But this turns out to not be the case! The above state is a perfectly allowable physical configuration of two particles, but there is no accurate description of the state of the individual particles composing the system!

Multi-particle systems cannot in general be reduced to their parts. This is one of the shocking features of quantum mechanics that is extremely easy to prove, but is rarely emphasized in proportion to its importance. We’ll prove it now.

Suppose we have a system composed of two qubits in states |Ψ1⟩ and |Ψ2⟩. In general, we may write:

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Now, as we’ve seen in previous posts, we can describe the state of the two qubits as a whole by simply smushing them together as follows:

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So the set of all two-qubit states that can be split in their component parts is the set of all states arising from all possible values of α1, α2, β1, and β2 such that all states are normalized. I.e.

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However, there’s also a theorem that says that if any two states are physical possible, then all normalized linear combinations are physically possible as well. Because the states |00⟩, |01⟩, |10⟩, and |11⟩ are all physically possible, and because they form a basis for the set of two-qubit states, we can write out the set of all possible states:

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Now the philosophical question of whether or not there exist states that are irreducible can be formulated as a precise mathematical question: Does A = R?

And the answer is no! It turns out that A is much much larger than R.

The proof of this is very simple. R and A are both sets defined by a set of four complex numbers, and they share a constraint. But R also has two other constraints, independent of the shared constraint. That is, the two additional constraints cannot be derived from the first (try to derive it yourself! Or better, show that it cannot be derived). So the set of states that satisfy the conditions necessary to be in R must be smaller than the set of states that satisfy the conditions necessary to be in A. This is basically just the statement that when you take a set, and then impose a constraint on it, you get a smaller set.

An even simpler proof of the irreducibility of some states is to just give an example. Let’s return to our earlier example of a two-qubit state that cannot be decomposed into its parts:

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Suppose that ⟩ is reducible. Then for both |00⟩ and |11⟩ to have a nonzero amplitude, there must be a nonzero amplitude for the first qubit to be in the state |0⟩ and for the second to be in the state |1⟩. But then there can’t be zero amplitude for the state |01⟩. Q.E.D.!

More precisely:

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So here we have a two-qubit state that is fundamentally irreducible. There is literally no possible description of the individual qubits on their own. We can go through all possible states that each qubit might be in, and rule them out one.

Let’s pause for a minute to reflect on how totally insane this is. It is a definitive proof that according to quantum mechanics, reality cannot necessarily be described in terms of its smallest components. This is a serious challenge to the idea of reductionism, and I’m still trying to figure out how to adjust my worldview in response. While the notion of reductionism as “higher-level laws can be derived as approximations of the laws of physics” isn’t challenged by this, the notion that “the whole is always reducible to its parts” has to go.

In fact, I’ll show in the next section that if you try to make predictions about an system but analyze it in terms of its smallest components, you will not in general get the right answer.

Predictive accuracy requires holism

So suppose that we have two qubits in the state we already introduced:

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You might think the following: “Look, the two qubits are either both in the state |0, or both in the state |1⟩. There’s a 50% chance of either one happening. Let’s suppose that we are only interested in the first qubit, and don’t care what happens with the second one. Can’t we just say that the first qubit is in a state with amplitudes 1/√2 in both states |0⟩ and |1⟩? After all, this will match the experimental results when we measure the qubit (50% of the time it is |0⟩ and 50% of the time it is |1⟩.”

Okay, but there are two big problems with this. First of all, while it’s true that each particle has a 50% chance of being observed in the state |0⟩, if you model these probabilities as independent of one another, then you will end up concluding that there is a 25% chance of the first particle being in the state |0⟩ and the second being in the state |1⟩. Whereas in fact, this will never happen!

You may reply that this is only a problem if you’re interested in making predictions about the state of the second qubit. If you are solely looking at your single qubit, you can still succeed at predicting what will happen when you measure it.

Well, fine. But the second, more important point is that even if you are able to accurately describe what happens when you measure your single qubit, you can always construct a different experiment you could perform that this same description will give the wrong answer for.

What this comes down to is the observation that quantum gates don’t operate the same way on 1/√2 (|00⟩ + |11⟩) as on 1/√2 (|0⟩ + |1⟩).

Suppose you take your qubit and pretend that the other one doesn’t exist. Then you apply a Hadamard gate to just your qubit and measure it. If you thought that the state was initially 1/√2 (|0⟩ + |1⟩), you will now think that your qubit is in the state |0⟩. You will predict with 100% confidence that if you measure it now, you will observe |0⟩.

But in fact when you measure it, you will find that 50% of the time it is |0⟩ and 50% of the time it is |1⟩! Where did you go wrong? You went wrong by trying to describe the particle as an individual entity.

Let’s prove this. First we’ll figure out what it looks like when we apply a Hadamard gate to only the first qubit, in the two-qubit representation:

Screen Shot 2018-07-17 at 10.02.27 PMScreen Shot 2018-07-17 at 10.13.22 PM

So we have a ​25% chance of observing each of |00⟩|10⟩|01⟩, and|11⟩. Looking at just your own qubit, then, you have a 50% chance of observing |0⟩ and a 50% chance of observing |1⟩.

While your single-qubit description told you to predict a 100% chance of observing |0⟩, you actually would get a 50% chance of |0⟩ and a 50% chance of |1⟩.

Okay, but maybe the problem was that we were just using the wrong amplitude distribution for our single qubit. There are many choices we could have made for the amplitudes besides 1/√2 that would have kept the probabilities 50/50. Maybe one of these correctly simulates the behavior of the qubit in response to a quantum gate?

But no. It turns out that even though it is correct that there is a 50/50 chance of observing the qubit to be |0⟩ or |1⟩, there is no amplitude distribution matching this probability distribution that will correctly predict the results of all possible experiments.

Quick proof: We can describe a general two-qubit state with a 50/50 probability of being observed in |0⟩ and |1⟩ as follows:

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For any ⟩, we can construct a specially designed quantum gate U that transforms ⟩ into |0⟩:

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Applying U to our single qubit, we now expect to observe |0⟩ with 100% probability. But now let’s look at what happens if we consider the state of the combined system. The operation of applying U to only the first qubit is represented by taking the tensor product of U with the identity matrix I: U ⊗ I.

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What we see is that the two-qubit state ends up with a 25% chance of being observed as each of |00⟩, |01⟩, |10⟩, and |11⟩. This means that there is still a 50% chance of the first qubit being observed as |0⟩ and |1⟩.

This means that for every possible single qubit description of the first qubit, we can construct an experiment that will give different results than the model predicts. And the only model that always gives the right experimental predictions is a model that considers the two qubits as a single unit, irreducible and impossible to describe independently. 

To recap: The lesson here is that for some quantum systems, if you describe them in terms of their parts instead of as a whole, you will necessarily make the wrong predictions about experimental results. And if you describe them as a whole, you will get the predictions spot on.

So how many states are irreducible?

Said another way, how much larger is A (the set of all states) than R (the set of reducible states)? Well, they’re both infinite sets with the same cardinality (they each have the cardinality of the continuum, |ℝ|). So in this sense, they’re the same size of infinity. But we can think about this by considering the dimensionality of these various spaces.

Let’s take another look at the definitions of A and R:

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Each set is defined by four complex numbers, or 8 real numbers. If we ignored all constraints, then, our sets would be isomorphic to ℝ8.

Now, each share the same first constraint, which says that the overall state must be normalized. This constraint cuts one dimension off of the space of solutions, making it isomorphic to ℝ7.

That’s the only constraint for A, so we can say that A ~ ℝ7. But R involves two further constraints (the normalization conditions for each individual qubit). So we have three total constraints. However, it turns out that one of them can be derived from the others – two normalized qubits, when smushed together, always produce a normalized state. This gives us a net two constraints, meaning that the space of reducible states is isomorphic to ℝ6.

The space of irreducible states is what’s left when we subtract all elements of R from A. The dimensionality of this is just the same as the dimensionality of A. (A 3D volume minus a plane is still a 3D volume, a plane minus a curve is still two dimensional, a curve minus a point is still one dimensional, and so on.)

So both the space of total states and the space of irreducible states are 7-real-dimensional, while the space of reducible states is 6-real dimensional.

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You can visualize this as the space of all states being a volume, through which cuts a plane that composes all reducible states. The entire rest of the volume is the set of irreducible states. Clearly there are a lot more irreducible states than reducible states.

What about if we consider totally reducible three-qubit states? Now things are slightly different.

The set of all possible three qubit states (which we’ll denote A3) is a set of 8 complex numbers (16 real numbers) with one normalization constraint. So A3 ~ ℝ15.

The set of all totally reducible three qubit states (which we’ll denote R3) is a set of only six complex numbers. Why? Because we only need to specify two complex numbers for each of the three individual qubits that will be smushed together. So we start off with only 12 real numbers. Then we have three constraints, one for the normalization of each individual qubit. And the final normalization constraint (of the entire system) follows from the previous three constraints. In the end, we see that R3 ~ ℝ9.

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Now the space of reducible states is six-dimensions less than the space of all states.

How does this scale for larger quantum systems? Let’s look in general at a system of N qubits.

AN is a set of 2N complex amplitudes (2N+1 real numbers), one for each N qubit state. There is just one normalization constraint. Thus we have a space with 2N+1 – 1 real dimensions.

On the other hand, RN is a set of only 2N complex amplitudes (4N real numbers), two for each of the N individual qubits. And there are N independent constraints ensuring that all states are normalized. So we have:

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The point of all of this is that as you consider larger and larger quantum systems, the dimensionality of the space of irreducible states grows exponentially, while the dimensionality of the space of reducible states only grows linearly. If we were to imagine randomly selecting a 20-qubit state from the space of all possibilities, we would be exponentially more likely to ending up with a space that cannot be described as a product of each of its parts.

What this means is that irreducibility is not a strange exotic phenomenon that we shouldn’t expect to see in the real world. Instead, we should expect that basically all systems we’re surrounded by are irreducible. And therefore, we should expect that the world as a whole is almost certainly not describable as the sum of individual parts.