Suppose you are given a box that lets you dial in any² number between 0 and 1 and returns some output value between 0 and 1 within some experimental error³. If after trying many values you derive a polynomial with 25 coefficients that lets you very closely approximate the average result⁴ for a given input you probably wouldn’t think you’d hit on anything deep about the operation of the box. In fact you’d probably guess that greater precision (averaging over more tests) would reveal subtle distinctions between your approximate function and the true value. On the other hand if after the same number of tests it appears that sin(x) is an equally good approximation you might think this was the true function and expect this to be born out by further experiments. You might even make hypothesises about the box’s mechanism on this basis.

My worry is that those theories we take to be simple and elegant really aren’t simple at all. For instance is it really the case that sin(x) is a simpler function than some 25 term polynomial with integer coefficients between 1 and 10? The obvious way to answer this is to ask how many symbols it takes to define each function but this answer depends on what we take to be our primitive terms. To put the point more formally the Kolmogorov complexity of a string depends on our choice of a universal prefix-free machine. However, it’s reasonable to think that so long as we pick one system to represent out theories in and stick with it then it will function as a useful measure of a theories complexity⁵.

However, in practice we never really fix one system and insist on writing all our theories in terms of it. When people discovered that the sin function was frequently useful in describing physical systems they stuck it into their toolkit. They didn’t stick with whatever previous system they had been using and include the definition of sin(x) in all of their theories. Yet if our idea of what a simple theory is changes in response to what seems to make good predictions we no longer have a good argument for the truth of our theories. If it had turned out that a parametrized solution to the equation y^3+x*y=x^2 had been widely useful in physical theories instead of solutions to x^2+y^2=1 then it would probably have been those functions rather than cos(x) and sin(x) that we regarded as elementary functions.

I don’t doubt that evolution has endowed us with a notion of simplicity that works well in everyday macroscopic scenarios. What I’m skeptical of is the claim that the abstract mathematical theories that underlie particle physics and cosmology are really especially simple. Certainly it’s true that they can be expressed in a form that strikes us as elegant and appears simple but they only do so by making use of many layers of abstraction. I’m not so sure that if we examined the mathematical framework for quantum mechanics written out as a formal statement in PA it would still strike us as particularly simple.

In short I’m worried that we underestimate the power of additional layers of abstraction. Sure, the mathematical concepts used in modern physics are the result of a series of definitions and abstractions each one of which strikes us as simple and elegant but the essential question is whether alternative theories giving similar agreement with the data would admit a similar chain of definitions. Given that no real work (to my knowledge) has been done about the additional complexity each layer of abstraction brings to a theory what reason do we really have to be confident about the simplicity of physics?

It started with someone bringing up the Fermi paradox (why haven’t advanced alien civilizations contacted us yet). The host then steered the question towards whether this was an argument against time travel as well (failure to see time travelers). Timothy Ferris replied that he didn’t find it very compelling because he expects time will be lack a deck of cards so that if you go back in time of forward in time you end up in one of many possible pasts or many possible futures. While he admitted it was just his expectation he clearly conveyed the sense that it was a possibility that experts would take seriously.

I happen to think the very idea of something being time travel requires that we go back into the past not merely enter some universe that looks like the past.¹ However, let’s set this point aside. I suspect the journalist was referencing some approaches to quantum mechanics that go by the name of sum over histories or multiple histories. Possibly he meant to refer to many minds or many worlds theories. The problem is that traveling to an ‘alternate’ past doesn’t even make sense in any of them. Supposing it was true and even meaningful that we have multiple histories in this QM sense we would have multiple presents as well. What the hell would it even mean for a person, who is really a superposition, to visit one component of a prior superposition? Pure many worlds theories only really make sense² if we understand them as collapsing down to a many mind’s theory and it certainly isn’t clear what it would mean for a mind that rides atop the superposition to time travel by itself, certainly not in the sense of some dude from the future appearing.

That’s confused and I was annoyed that he said it with such apparent authority but what really got my goat was when he talked about how interesting it would be if we ended up with quantum computers since we couldn’t explain their processing power with just one universe and would have to say that they use other universes to do their computations. This is just a lie that is being pushed on the public. The fundamental laws of nature could just offer us an oracle that computer anything we wanted as fast as we want. For all we know there is some special experiment we can do that reveals the true bits of 0′ (the set of the halting problem). Worse this is certainly not anything scientists have or can test. It is purely unjustified bad philosophical speculation that misleads the public.

I’m not sure whether to be mad at the people who promote this crap or applaud the physicists for great PR. Maybe we should just adopt this for math. Push the whole confusion about Godel’s theorem a bit more and try selling the Banach-Tarski paradox as a proof that “space is an illusion.”

Bit about quantum computer

However, it appears some recent research suggests that different brain areas are implicated in making risky decisions and ambiguous decisions. In fact it even appears that risk-aversion and ambiguity-aversion can vary separately between individuals. Does this doom Bayesianism?

I don’t have a clue. This just begs the question of what the hell theories like Bayesianism are supposed to accomplish. If it is a claim about rationality then this result is irrelevant. If it is about a ‘good-enough’ approximation to actual scientific decision making then this could just be pushing the goodness of the approximation. If it aims to be some kind of idealization of actual decision making given arbitrary time and generally agreed upon errors ruled out then yes this conclusion could be very damning.

It is very difficult to underestimate the importance of precision and naming. Each of these different notions of Bayesianism really should be given differing names.

The point I want to illustrate is that whether or not this is the actual world or a matrix style simulation is an instance of the sleeping beauty problem. In particular the same reasoning that supports the two-thirds solution to the sleeping beauty problems seems to guarantee we are in a repetitive simulation. I don’t know if this puzzle has appeared in the literature yet (it seems kinda obvious to me so I have a hard time believing it hasn’t) but it is new to me (no I haven’t read all the papers I link so it could be there). Since I find the reasoning supporting the two-thirds solution compelling I expect some other solution will ultimately be forthcoming but I expect that solution will be more interesting than the problem itself.

More concretely it certainly seems possible that we are actually in a virtual world like the matrix. It remains possible even if we suppose that our lives are in fact arbitrarily long (the ‘real’ world has conquered aging) and every time we die in the virtual world our memories are wiped and the simulation is restarted. While we might think ‘a priori’ such a world is unlikely since it seems possible we should assign it some non-zero probability. However, since this matrix world repeats the experience we are having infinitely many times it would seem that the same reasoning which allows us to support the two-thirds solution in the sleeping beauty problem requires that we find the matrix world arbitrarily more probable than our own world, i.e., the probability of the matrix world conditionalized on matrix world or real world is one (or arbitrarily close to one). Of course the virtual world aspect isn’t central to the problem. We could repeat the same argument in favor of Nietchzie’s world of eternal recurrence or any situation which involves making us experience our lives so far arbitrarily many times.

The Sleeping Beauty Problem

In the sleeping beauty paradox we drug the hapless heroine into a deep slumber on sunday. At this point we flip a coin. If the coin lands heads we wake her up on monday tell only giving her time to realize she is awake but not informing her of the day or any other information before we render her unconscious and wipe her memory of the event. In this case we do not wake her again till wednesday. If instead the coin lands tails we do the same thing on monday, wake her wipe her memory and put her back to sleep, but instead of letting her sleep will wednesday we do the same again on tuesday. The puzzle is when woken is it rational for sleeping beauty to believe the probability of tails is still 1/2 or should she now believe it is 2/3.

If you are interested in the reasons for one side or the other you can read the philosophical arguments about the sleeping beauty problem. I may write up my thoughts on the problem in more detail later but rather than get into the details here I will just assume Elga’s argument for the two-thirds answer is correct.

In effect Elga’s solution seems to mandate the following means of calculating the credence we should assign to the coin being tails upon waking up. Consider all the ways you could have the experience you just had (being woken up by the experimenter) where the coin lands tails and weight each of those by its probability of occurring. Divide this number by the sum of all the ways you could have this experience weighted by probability and divide. Since their are two ways you could have the experience of waking up when the coin is tails (it is monday or it is tuesday) and each of these has probability 1/2 (if the coin lands tails you are guaranteed to have both) the numerator is one. The only other way you could have the experience of waking up is if the coin landed heads and it is monday the denominator is 3/2. Hence giving the ‘right’ answer of 2/3.

This solution has great intrinsic appeal. If we want to calculate the probability of T given E we usually sum up all the ways T and E could happen weighted by their unconditional probability and divide by the sum of all the ways E could happen weighted by unconditional probability. Though in this case we may not technically be calculating the probability of tails given that you had the experience of waking up since this is the normal means of updating on new information it seems appealing to do the same in the sleeping beauty case.

Sleeping Beauty and Repeating Simulations

However, let us consider what happens if we apply this reasoning to the matrix example. In this case instead of the coin flip we have the choice of world (real or virtual) and instead of being woken up twice our simulation is repeated n times. So instead of H₁, T₁ and T₂ as Elga had we have the following outcomes.

• R₁: The world we experience is real and we are living our one and only life (after death we just cease to exist or go on to some other type of existence).
• V₁: The world we experience is just a simulation and we are on the 1st repetition of the simulation.
• V₂: The world we experience is just a simulation and we are on the 2nd repetition of the simulation.
• V_n: The world we experience is just a simulation and we are on the nth repetition of the simulation.

• V: The world we experience is just a simulation and we are on some repetition of the simulation, i.e., V=V₁v V₂…v V_n.

Since we only want to compare the probabilities of this world being real and being in this particular simulation we will conditionalize our probabilities on VvR, i.e., P(X) denotes the probability of X given ‘V or R’ and hence P(V)+P(R)=1. Importantly P(V) and P(R) do not give our current probabilities for V and R but the ‘a priori’ probabilities for these worlds before it is conditionalized on any experience. As we are simply postulating some possible world where we are thrust into simulators we can assume that all n repetitions of the simulation are guaranteed, i.e., there is no chance the simulation will break or we will die before all repetitions are complete.

Applying the same solution Elga does to this situation we calculate the current credence we should hold for being in situation V to be P(V)*n/(P(R)+n*P(V)). Note that if we hold P(V) and P(R) fixed and allow n to go to infinity our rational credence in V goes to 1. Of course one might have a lower probability for being in a situation where the simulation is repeated n+1 times than one where it is repeated n times but we can avoid this problem by considering a world where the simulation is repeated infinitely (omega) many times. Countable additivity guarantees the credence we should have in this case is just the limit of our prior answer as n goes to infinity. Since we seem happy to consider the possibility that our universe will keep expanding forever it seems perfectly possible that such a situation could exist. Heck one could just pick some system of particles and dynamical laws which repeat at regular intervals forever and postulate psycho-physical laws which makes the motion of these particles create the experiences of your life. So long as we assign non-zero ‘a priori’ probability to such a situation the above argument shows we should believe it is infinitely more likely we are in this repeating simulation than the real world, i.e., the credence we should give V given V or R is 1.

Potential Solutions

Obviously this result seems absurd. It seems totally unreasonable that we should be required to believe the world is infinitely more likely to be a simulation or even that we will repeat our current life after we die just because we are having some experience. So what moves can we make to avoid this result? Our first implication might be to discard the most abstract and non-intuitive principle we used, countable additivity. However, this really does no good as just using finite additivity we should be able to prove the credence we should assign to being in V is greater than any number below 1. Alright so what about giving up the assumption that P(V) isn’t 0? Remember P(V) refers to the conditional probability of V given V or R so this would require believing we have grounds to believe that the real non-recurring world was infinitely more likely than this world filled with simulations. Even if you are willing say this about a world running indefinite simulations of this one are you really prepared to (essentially) rule out a world of eternal recurrence?

A more substantive move might be to reject the idea that ‘a priori’ probabilities exist. After all most philosophers are skeptical of objective prior probabilities so don’t we get rid of this paradox for free when we give up objective priors? Even if true it would still be pretty cool to show that reasonable assumptions actually excluded objective priors. However, nothing in this argument required that these probabilities be objective only that we have some notion of likelihood about various ways the world might have turned out. I have been careful to put ‘a priori’ in scare quotes everywhere because this probability function can be the result of any model we adopt and need not be a consequence of logic alone. If the physicists come up with some theory which gives probabilities for various types of initial conditions (and at least some of these allow for a universe which lasts forever) it seems the problem reoccurs. Do we really want to be philosophically committed to the absence of such a physical theory?

Moreover, denying the existence of this sort of ‘a priori’ probability function leaves us at a loss about whether or not to believe we are in some sort of simulation. If I observed certain very bizarre or intuitively absurd events it would be perfectly rational of me to conclude I was in some type of simulation. For instance if a giant black unix terminal opened up in the sky and what appeared to be computer commands appeared inside with corresponding effects out in the world it seems reasonable I should accept that I’m in a simulation. Yet despite never having seen (or at least remembered) the real world I would judge certain ways the real world might be more likely than others, e.g., fundamental physical objects are particles not toasters or starfish. Moreover, if I was trying to predict what the real world was like I would first judge how probable that sort of world seemed and then modify that judgement by how likely that sort of world was to cause simulated experiences like mine. These judgments may not be a priori in the sense of being a priori truths but if does seem like our basic intuitions of probability give us the sort of probability which lead to all this trouble.

This seems to leave us only two possible options: bite the bullet or abandon the ‘two-thirds’ solution to the Sleeping Beauty problem. As both the limiting frequency and a number of compelling arguments favor the ‘2/3′ option I am loathe to give it up. Additionally biting the bullet may not be quite as unpalatable as it seems. It seems to be a valid scientific possibility that the universe will collapse and be recreated in a new big bang. If this happens indefinitely our world effectively is a recurring world and the world of repeated simulation seems to lose any apparent advantage. However, this draws the paradoxes of infinity deeply into our judgments about the world. In particular the sleeping beauty problem asks us to calculate the probability of being in a world where the coin lands heads versus one where it lands tails. Once we assume the world is recurring both possibilities present the same number of experiences of waking, countably many in both cases, nullifying the ‘two-thirds’ solution. Alternatively we might postulate that our world actually consists of countably many (or more) copies (as in many worlds interpretations of quantum mechanics). However, this still leaves the above problem and adds the additional puzzle of what it even means for the same experience to be happening countably many times.

Ultimately I’m pretty much at a loss about this problem. Maybe someone reading this will have some ideas.

Anyway I was having a conversation with Sharon today when she said (more or less), “If my dad had been home schooled that would explain his personality.” While intuitively I immediately knew what she meant, though a precise analysis is somewhat complicated, the sentence itself seemed deeply troublesome. In particular it seemed to come out necessarily false, or at least self-defeating like a Moorean sentence, on a possible world’s account of counterfactuals.

Why? Well I’m not entirely sure. My intuition seems to be based on a principle like: an explanation of some effect must necessarily cite things which ’cause’ the explanadum. For example the fact that Mary threw a rock towards my bedroom window at 3pm does not explain why my window glass is broken if it was actually smashed at noon by John. This holds despite the fact that throwing a rock towards a window is quite definitely the type of act which causes window glass to break. For a more detailed discussion read below. Admittedly the idea that an explanation must somehow cite causes of the explanadum is clearly incomplete. Types of mathematical explanation certainly don’t satisfy this condition and there are probably other exceptions. Still it seems something is right about this idea. An explanation must cite things which actually account for the truth or probability of the explanadum not merely things which might have done so but are ruled out by other facts.

As another example of this intuition suppose that Iraq would have publicly assassinated president Bush if we hadn’t invaded over WMDs. This would certainly have guaranteed a war with Iraq but an explanation citing this fact wouldn’t be valid. An obvious analysis might suggest that this is merely a case of an explanation citing false explanans. However, the explanans in this case seems only to be the conditional ‘If no war over WMD then Bush would have been assassinated.’ Moreover, this explanation is still invalid even if the prior probability of the US going to war over WMDs was arbitrarily small. As we often accept explanations which only show why the explanadum is probable we can’t discredit this explanation by saying it relies on false or missing explanans.

Frankly I’m baffled as to what criteria of explanation the above example violates. Whatever it may be I think it is the same feature which makes me uncomfortable with the sentence about Sharon’s dad. On a possible world’s analysis this sentence asks us to consider the closest possible world where her dad was home schooled. Since this should be the smallest modification of this world possible presumably whatever factors did cause her dad’s personality should still be present. For this discussion we will assume the cause wasn’t a consequence of not being home schooled (though this does raise the interesting question of whether A can ever explain B if not A also causes A or if the explanation must account for both cases). Yet this seems to guarantee the home schooling explanation is superseded by the real explanation as in the John and Mary example.

Now it seems clear to me this sort of possible world’s account is not really what is intended by the sentence. In asserting this claim Sharon did not mean to say that in the closest possible world where her dad was home schooled this was the explanation of his personality. Rather it seems she meant to say something more like, “Home schooling is the sort of thing which would explain this sort of personality.” Though whether this should be cached out in terms of some kind of inductive confidence relation, measure over many possible worlds or what is unclear. Still it seems something of a hack to stipulate a special translation of what seems to be a fairly normal sort of counterfactual.

While I am left in a state of some confusion it certainly raises interesting issues. It seems an interesting question as to whether a possible world’s analysis of this can be saved by an appropriate specification of what facts should be considered in the closeness relation in the same sort of manner we only consider facts before 1945 when evaluating conditionals about the US not dropping the bomb. Also these considerations totally destroy my ignorant intuition that an explanation was something which made the explanadum likely or expected. If anyone knows any good literature on this I would love to hear a reference.