A debate inside another one

Tim Maudlin debated Gerard 't Hooft about his cellular automaton interpretation of quantum mechanics in a series of Facebook posts, the fourth one being here https://www.facebook.com/tim.maudlin/posts/10155699914028398. Somewhere in the forest of comments I was engaged in a sort of sub-debate, with Tim, Hans, and others. Sabine was there too. The discussion was completely surrealistic, Tim and Hans completely misunderstood my point. This started by me intervening with a theoretical counterexample to a claim that all so called superdeterministic theories (in particular 't Hooft's) are not falsifiable, and of course it led to different topics. It is not known, but not a secret that the wavefunction collapse leads to violations of conservation laws, and that it is possible at least in principle to remove the collapse while remaining with a single world. But removing the collapse can be seen as superdeterministic (although I wouldn't call it like this, because it is based on spacetime, not on initial conditions), and I even proposed a principle to explain this, and experiments to test it. I paste here most of this *debate*, because there are some parts I am interested to keep. I skipped some parts in which I was not involved.

Maudlin's "(Information) Paradox Lost" paper

Tim Maudlin has an interesting paper in which he criticizes the importance given to the black hole information paradox, and even brings arguments that it is not even a problem: (Information) Paradox Lost. I agree that the importance of the problem is perhaps exaggerated, but at the same time many consider it to be a useful benchmark to test quantum gravity solutions. This led to decades of research made by many physicists, and to many controversies. I wrote a bit about some of the proposed solutions to the problem in some older posts, for example [1,2,3]. Maudlin's paper is discussed by Sabine here.

One of the central arguments in Maudlin's paper is that the well-known spacetime illustrating the information loss can be foliated into some 3D spaces (which are Cauchy hypersurfaces that are discontinuous at the singularity). These hypersurfaces have a part outside the black hole, and another one inside it, which are not connected to one another. Cauchy hypersurfaces contain the Cauchy data necessary to solve the partial differential equations, so the information should be preserved if we consider both their part inside and their part outside the black hole.

I illustrate this with this animated gif:

 

I made this gif back in 2010, when I independently had the same idea and wanted to write about it, but I don't think I made it public. Probably the idea is older. The reason I didn't write about it was that I was more attracted* to another solution I found, which led to an analytic extension of the black hole spacetime, and has Cauchy hypersurfaces but no discontinuities. I reproduce a picture of the Penrose diagram from an older post in which I say more about this:

 

A. The standard Penrose diagram of an evaporating black hole.

B The diagram from the analytic solution  I proposed.

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* The reason I preferred to work at the second solution is that it allows the information to become available after the evaporation to an external observer. The solution which relies on completing the Cauchy hypersurface with a part inside the black hole doesn't restore information and unitarity for an external observer. I don't know if this is a problem, but many physicists believe that information should be restored for an external observer, because otherwise we would observe violations of unitarity even in the most mundane cases, considering that micro black holes form and evaporate at very high energies. I don't think this argument, also given by Sabine, is very good, because there is no reason to believe that micro black  holes form at high energy under normal conditions. People arrive at high energies for normal situations because they use perturbative expansions, but this is just a method of approximation. And even so, I doubt anyone who sums over Feynman diagrams includes black holes. But nevertheless, I wouldn't like information to be lost for an outside observer after evaporation, but this is just personal taste, I don't claim that there is some experiment that proved this. And the solution I preferred to research allows recovery of information and unitarity for an external observer, and other things which I explained in the mentioned posts and my PhD thesis.

Are Single-World Interpretations of Quantum Theory Inconsistent?

A recent eprint caught my atention: Single-world interpretations of quantum theory cannot be self-consistent by Daniela Frauchiger and Renato Renner. In the abstract we read

We find that, in such a scenario, no single-world interpretation can be logically consistent. This conclusion extends to deterministic hidden-variable theories, such as Bohmian mechanics, for they impose a single-world interpretation. 

The article contains an experiment based on Wigner's friend thought experiment, from which is deduced in a Theorem that there cannot exist a theory T that satisfies the following conditions:

(QT) Compliance with quantum theory: T forbids all measurement results that are forbidden by standard [non-relativistic] quantum theory (and this condition holds even if the measured system is large enough to contain itself an experimenter).
(SW) Single-world: T rules out the occurrence of more than one single outcome if an
experimenter measures a system once.
(SC) Self-consistency: T's statements about measurement outcomes are logically consistent (even if they are obtained by considering the perspectives of different experimenters).

A proof of the inconsistency of Bohmian mechanics (discovered by de Broglie and rediscovered and further developed by David Bohm) would already be a  big deal, because despite being rejected with enthusiasm by many quantum theorists, it was never actually refuted, neither by reasoning, nor by experiment. Bohmian mechanics is based on two objects: the pilot-wave, which is very similar to the standard wavefunction and evolves according to the Schrödinger equation, and the Bohmian trajectory, which is an integral curve of the current associated to the Schrödinger equation. While one would expect the Bohmian trajectory to be the trajectory of a physical particle, all observables and physical properties, including mass, charge, spin, properties like non-locality and contextuality, are attributes of the wave, and not of the Bohmian particle. This explains in part why BM is able to satisfy (QT). The pilot-wave itself evolves unitarily, not being subject to the collapse. Decoherence (first discovered by Bohm when developing this theory) plays a major role. The only role played by the Bohmian trajectory seems (to me at least) to be to point which outcome was obtained during an experiment. In other words, the pilot-wave behaves just like in the Many-Worlds Interpretation, and the Bohmian trajectory is used only to select a single-world. But the other single-worlds are equally justified, once we accepted all branches of the pilot-wave to be equally real, and the Bohmian trajectory really plays no role. I will come back later with a more detailed argumentation of what I said here about Bohmian mechanics, but I repeat, this is not a refutation of BM, rather some arguments coming from my personal taste and expectations of what a theory of QT should do. Anyway, if the result of the Frauchiger-Renner paper is correct, this will show not only that the Bohmian trajectory is not necessary, but also that it is impossible in the proposed experiment. This would be really strange, given that the Bohmian trajectory is just an integral curve of a vector field in the configuration space, and it is perfectly well defined for almost all initial configurations. This would be a counterexample given by Bohmian mechanics itself to the Frauchiger-Renner theorem. Or is the opposite true?

But when you read their paper you realize that any theory compatible with standard quantum theory (which satisfies QT and SW) has to be inconsistent, including therefore standard QT itself. Despite the fact that the paper analyzes all three options obtained by negating each of the three conditions, it is pretty transparent that the only alternative has to be Many-Worlds. In fact, even MW, where each world is interpreted as a single-world, seems to be ruled out. If correct, this may be the most important result in the foundations of QT in decades.

Recall that the Many-Worlds Interpretation is considered by most of its supporters as being the logical consequence of the Schrödinger equation, without needing to assume the wavefunction collapse. The reason is that the unitary evolution prescribed by the Schrödinger equation contains in it all possible results of the measurement of a quantum system, in superposition. And since each possible result lies in a branch of the wavefunction that can no longer interfere with the other branches, there will be independent branches behaving as separate worlds. Although there are some important open questions in the MWI, the official point of view is that the most important ones are already solved without assuming more than the Schrödinger equation. So perhaps for them this result would add nothing. But for the rest of us, it would really be important.

My first impulse was that there is a circularity in the proof of the Frauchiger-Renner theorem: they consider that it is possible to perform an experiment resulting in the superposition of two different classical states of a system. Here by "classical state" I understand of course still a quantum state, but one which effectively looks classical, as a measurement device is expected to be before and after the measurement. In other words, their experiment is designed so that an observer sees a superposition of a dead cat and an alive one. Their experiment is cleverly designed so that two such observations of "Schrödinger cats" lead to inconsistencies, if (SW) is assumed to be true. So my first thought was that this means they already assume MWI, by allowing an observer to observe a superposition between a classical state that "happened" and one that "didn't happen".

But the things are not that simple, because even if a quantum state looks classical, it is still quantum. And there seem to be no absolute rule to forbid the superposition of two classical states. Einselection (environment-induced superselection) is a potential answer, but so far it is still an open problem, and at any rate, unlike the usual superselection rules, it is not an exact rule, but again an effective one (even if it would be proven to resolve the problem). So the standard formulation of QT doesn't actually forbid superpositions of classical states. Well, in Bohr's interpretation there are quantum and there are classical objects, and the distinction is unbreakable, so for him the extended Wigner's friend experiment proposed by Frauchiger and Renner would not make sense. But if we want to include the classical level in the quantum description, it seems that there is nothing to prevent the possibility, in principle, of this experiment.

Reading the Frauchiger-Renner paper made me think that there is an important open problem in QT, because it doesn't seem to prescribe how to deal with classical states:

Does QT allow quantum measurements of classical (macroscopic) systems, so that the resulting states are non-classical superpositions of their classical states?

I am not convinced that we are allowed to do this even in principle (in practice seems pretty clear it is impossible), but also I am not convinced why we are forbidden. To me, this is a big open problem. Can the answer to this question be derived logically from the principles of standard QT, or should it be added as an independent, new principle?

My guess is that we don't have a definitive solution yet. It is therefore a matter of choice: those accepting that we are allowed to perform any quantum measurements on classical states, perhaps already accept MWI, and consider that it is a logical consequence of the Schrödinger equation. Those who think that one can't perform on classical states quantum measurements that result in Schrödinger cats, will of course object to the result of the paper of Frauchiger and Renner and consider its proof circular.

I will not rush with the verdict about the Frauchiger-Renner paper. But I think at least the open problem I mentioned deserves more attention. Nevertheless, if their result is true, it will pose a big problem not only to Bohmian mechanics, but also to standard QT. And also to my own proposed interpretation, which is based on the possibility of a single-world unitary solution of the Schrödinger equation (see my recent paper On the Wavefunction Collapse and the references therein).

An attempt to refute my Big-Bang singularity solution

I learned recently about a paper which attempts to refute one of my papers. While being sure about my proofs, I confess that I was a bit worried, you never know when you made a mistake, a silly assumption that you overlooked. But as I was reading the refutation paper, my worries dissipated, and were replaced by amusement and I actually had a lot of fun. Because that so-called refutation was something like: "I will refute Pythagoras's Theorem by showing that it doesn't apply to triangles that are not right."

My paper in cause about Big-Bang singularities is arXiv:1112.4508 (The Friedmann-Lemaitre-Robertson-Walker Big Bang singularities are well behaved). As it is known, the main mathematical tool used in General Relativity is semi-Riemannian geometry, and this works only as long as the metric is regular. The metric ceases to be regular at singularities, but I developed the extension of semi-Riemannian geometry at some degenerate metrics, so it applies to a large class of singularities, in arxiv:1105.0201. And this allowed me to find descriptions of such singularities in terms of quantities that are still invariant, but as opposed to the usual ones, they remain finite at singularities. More about this can be found in my PhD thesis arxiv:1301.2231. In the paper arXiv:1112.4508, I give a theorem that shows that, if the scaling function of the FLRW universe is smooth at the Big-Bang singularity, then I can apply the tools I developed previously, and get a finite description of both the geometry, and the physical quantities involved.

The paper attempting to refute my result is arxiv:1603.02837 (Behavior of Friedmann-Lemaitre-Robertson-Walker Singularities, by L. Fernández-Jambrina). Both my paper and this one appeared this year in International Journal of Theoretical Physics. I think F-J is a good researcher and expert in singularities. But for some reason, he didn't like my paper, and he "refuted" it. The "refutation" simply takes the case that was explicitly not covered in my theorem, namely when the scaling function of the FLRW solution is not derivable at the singularity, and checks that indeed my tools don't work in this case. Now, while my result is much more humble than Pythagoras's Theorem, I will use it for comparison, since it is well-known by everybody. You can't refute Pythagoras's Theorem by taking triangles that are not right, and proving that the sum of squares of two sides is different than the square of the third. Simply because the Theorem makes clear in its hypothesis that it refers only to right triangles. My theorem also states clearly that the result doesn't refer to FLRW models whose scaling function is not derivable at the singularity. And F-J even copies the Theorem's enounce in his paper, so how could he miss this? So what F-J said is that my theorem can't be applied to some cases, which I made clear that I leave out (I don't claim my theorem solves everything, neither that it cures cancer). Now, is the case when the scaling function is not derivable important? Yes, at least historically, because some classical solutions fit here. But the cases covered by my theorem include what we know today about inflation. So I think that my result is not only correct, but also significant. In addition to this, F-J says that I actually don't remove the Big-Bang singularity. This is also true, and stated in my paper from the beginning. I don't remove the singularities, I just try to understand them to describe them in terms of finite quantities that make sense both geometrically and physically. But he wrote it as if I claim that I try to remove them and he proves that I don't, not that I accept them and provide a finite-quantities description of them.

The unreasonable beauty of mathematics in the natural sciences*

Imagine a man and a woman, seeing and liking each other at a party or club or so. They start talking, the mutual attraction is obvious, but they want to be casual for two minutes. So they exchange informal formalities about doesn't matter what. Then he asks her: "so, what do you do?", and she replies "I'm a poet". What if the guy would say something like "I hate poetry!", or even declare proudly "I never knew how to use letters to write words and stuff, and I don't care!". Or imagine she's a musician, and he says "I hate music!". There are two things we can say about that kind of guy. First, he is very rude, he never ever deserves a second chance with that girl or any other human being for that matter. He should be isolated, kept outside society. Second, or maybe this should be first, how on earth can he be proud for being illiterate!

You probably guessed that this story is true. OK, In my case it was about math instead of poetry, and the genders are reversed. This happened to me or to anyone in the same situation quite often. There is no political correctness when it comes about math, maybe because one tends to believe that if you like math, you have no feelings, and such a remark wouldn't hurt you. And I actually was never offended when a girl said such outrageous things like that she hates math. Because whenever a girl told me she hates math, I knew she calls math something that really is boring and ugly, and not what I actually call math. Because math as I know it is poetry, is music, and is a wonderful goddess.

The story continues, years later. You talk about physics, with people interested in physics, or even with physicists. And you say something about this being just a mathematical consequence of that, or that certain phenomenon can be better understood if we consider it as certain mathematical object. It happens sometimes that your interlocutor becomes impatient and says that this is only math, and you were discussing physics, that math has no power there, and so on. Or that math is at best just a tool, and it actually obscures the real picture, or even that it limits our power of understanding.

People got the wrong picture that math is about numbers, or letters that stand for unknown numbers, or being extremely precise and calculating a huge number of decimals, or being very rigid and limited. In fact, math is just the study of relations. You will be surprised, but this is actually the mathematical definition of math. Numbers come into math only incidentally, as they come into music, when you indicate the duration or the tempo. Math is just a qualitative description of relations, and by relations we can understand a wide rainbow of things. I will detail this another time.

Imagine you wake up and you don't remember where you are, or who you are, like you were just born. You are surrounded by noise, which hurts your ears and your brain, meaningless random violent noise. You run desperately, trying to avoid it, but it is everywhere. And you finally find a spot where everything becomes suddenly wonderful: the noise becomes music, a celestial, beautiful music, and everything starts making sense. You are in a wonderful Cathedral, and you are tempted to call what you are listening "music of the spheres". The same music played earlier, but you were in the wrong place, where the acoustics was bad, or the sounds reached your ear in the wrong order, because of the relative positions of the instruments. Or maybe your ears were not yet tuned to the music. The point is that what seemed to be ugly noise, suddenly became so wonderful.

So, when someone says "I hate math!", all I hear is "I am in the Cathedral you call wonderful, but in the wrong place, where the celestial music becomes ugly violent noise!".

If you are interested in physics, you entered the Cathedral. But if you hate math, you will not last here, and maybe it is better to get out immediately! And if you are still interested in physics, come inside slowly, carefully choosing your steps, to avoid being assaulted by the music of the spheres, to allow it gently to enter in your mind, and to open your eyes. Choose carefully what you read, what lectures you watch, and ask questions. Don't be shy, any question you will ask is the right question for your current position, and for your next step.

There are some places in the Cathedral where the music is really beautiful. If you meet people there, to share the music, to dance, you will feel wonderful. If not, you will feel lonely. So you will want to share that place, you will want to invite your friends to join you.

The reason I love physics, is that I want to find these places. The reason I read blogs and papers, is that I want them to help me find such places. The reason I write papers, and I blog about this, is that I would like to  share my places with others. I attend conferences (four so far this year) because they are like concerts, where you get the chance to listen some wonderful music, and to play your own.

But these are just words. I would like to write more posts in which I show the unreasonable beauty of math in physics, with concrete examples. Judging by the statistics, I have a few readers; judging by the number of comments, I don't really touch many of them. I know sometimes I am too serious, or too brief when I should explain more, especially when mathematical subtleties are involved. I am not very good at explaining abstract things to non-specialists, but I want to learn. I would like to write better, to be more useful, so, I would like to encourage comments and suggestions. Ask me to clarify, to explain, to detail, to simplify. Tell me what you would like to understand.

To start, I would like to write about vectors. They are so fundamentals in all areas of physics and mathematics, so I think it's a good idea to start with them. You may think they are too simple, and that you know all about them from high school, but you don't know the whole story. Later, when I will say something about quantum mechanics and relativity, they will be necessary (after all, according to quantum mechanics, the state of the universe is a vector). On the other hand, if you will understand them well, you will be around half of the way to understand some modern physics.

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* You surely guessed that the title is a reference to Wigner's brilliant and insightful lecture, The unreasonable effectiveness of mathematics in the natural sciences.

Update, October 14, 2014

I just watched an episode of the Colbert Report, where the mathematician Edward Frenkel was invited in April this year. It was about Frenkel's new book and about his movie. He discusses at some point precisely the fact that it is so acceptable to hate math, as opposed to hating music or painting. Here is what he says for The Wall Street Journal:

It's like teaching an art class where they only tell you how to paint a fence but they never show you Picasso. People say 'I'm bad at math,' but what they're really saying is 'I was bad at painting the fence.'

Also see this video:

Will science end after the last experiment will be performed?

Science is supposed to work like this: you make a theory which explains the experimental data collected up to this point, but also proposes new experiments, and predicts the results. If the experiment doesn't reject your theory, you are allowed to keep it (for a while).

I agree with this. On the other hand, much of the progress in science is not done like this, and we can look back in history and see.

Now, to be fair, making testable predictions is something really excellent, without which there would be no science. To paraphrase Churchill,

Scientific method is the worst form of conducting science, except for all the others.

I am completely for experiments, and I think we should never stop testing our theories. On the other hand, we should not be extremists about making predictions. Science advances in the absence of new experiments too.

For example, Newton had access to a lot of data already collected by his predecessors, and sorted by Kepler, Galileo, and others. Newton came with the law of universal attraction, which applies to how planets move, in conformity with Kepler's laws, but also to how bodies fall on earth. His equation allowed him to calculate from one case the gravitational constant, but then, this applied to all other data. Of course, later experiments were performed, and they confirmed Newton's law. But his theory was already science, before these experiments were performed. Why? Because his single formula gave the quantitative and qualitative descriptions of a huge amount of data, like the movements of planets and earth gravity.

Once Newton guessed the inverse square law, and checked its validity (on paper) on the data about the motion of a planet and on the data about several projectiles, he was sure that it will work for other planets, comets, etc. And he was right (up to a point, of course, corrected by general relativity, but that's a different story). For him, checking his formula for a new planet was like a new experiment, only that the data was already collected by Tycho Brahe, and already analyzed by Kepler.

Assuming that this data was not available, and it was only later collected, would this mean that Newton's theory would have been more justified? I don't really think so. From his viewpoint, just checking the new cases, already known, was a corroboration of his law. Because he could not come up with his formula from all the data available. He started with one or two cases, then guessed it, then checked with the others. The data for the other cases was already available, but it could very well be obtained later, by new observations or experiments.

New experiments and observations that were performed after that were just redundant.

Now, think at special relativity. By the work of Lorentz, Poincaré, Einstein and others, the incompatibility between the way electromagnetic fields and waves transform when one changes the reference  frame, and how were they expected to transform by the formulae known from classical mechanics, was resolved. The old transformations of Galileo were replaced by the new ones of Lorentz and Poincaré. As a bonus, mass, energy and momentum became unified, electric and magnetic fields became unified, and several known phenomena gained a better and simpler explanation. Of course, new predictions were also made, and they served as new reasons to prefer special relativity over classical mechanics. But assuming these predictions were not made, or not verified, or were already known, how would this make special relativity less scientific? This theory already explained in a unified way various apparently disconnected phenomena which were already known.

One said that Maxwell unified the electric and magnetic fields with his equations. While I agree with this, the unification became even better understood in the context of special relativity. There, it became clear that the electric and magnetic fields are just part of a four-dimensional tensor $F$. The magnetic field corresponds to the spatial components $F_{xy}$, $F_{yz}$, $F_{zx}$, and the electric field to the mixed, spatial and temporal, components $F_{tx}$, $F_{ty}$, $F_{tz}$ of that tensor. Scalar and vector potentials turned out to be unified in a four-dimensional vector potential. Moreover, the unification became clearer when the differential form of Maxwell's equations was found, and even clearer when the gauge theory formulation was discovered. These are simple conceptual jumps, but they are science. And if they were also accompanied by empirical predictions which were confirmed, even better.

Suppose for a moment that we live in an Euclidean world. Say that we performed experiments and tested the axioms of Euclid. Then, we keep performing experiments to test various propositions that result from these axioms. Would this make any sense? Yes, but not as much as it is usually implied. They already are bound to be true by logic, because they are deduced from the axioms, which are already tested. So, why bother to make more and more experiments, to test various theorems in Euclidean geometry? This would be silly. Unless we want to check by this that the theorems were correctly proven.

On the other hand, in physics, a lot of experiments are performed, to test various predictions of quantum mechanics or special relativity, or of the standard model of particle physics, which follow logically and necessarily from the postulates which are already tested decades ago. This should be done, one should never say "no more tests". But on the other hand, this gives us the feeling that we are doing new science, because we are told that science without experiment is not science. And we are just checking the same principles over and over again.

Imagine a world where all possibly conceivable experiments were done. Suppose we even know some formulae that tell us what experimental data we would obtain, if we would do again any of these experiments. Would this mean that science reached its end, and there is nothing more to be done?

Obviously it doesn't mean this. We can systematize the data. Tycho Brahe's tables were not the final word in the astronomy of our solar system. They could be systematize by Kepler, and then, Kepler's laws could be obtained as corollaries by Newton. Of course, Kepler's laws have more content that Brahe's tables, because they would apply also to new planets, and new planetary systems. Newton's theory of gravity does more than Kepler's laws, and Einstein's general relativity does more than Newton's gravity. But, such predictions were out of our reach at that time. Even assuming that Tycho Brahe had the means to make tables for all planets in the universe, this would not make Kepler's laws less scientific.

Assuming that we have all the data about the universe, science can continue to advance, to systematize, to compress this data in more general laws. To compress the data better, the laws have to be as universal as possible, as unified as possible. And this is still science. Understanding that Maxwell's four equations (two scalar and two vectorial) can be written as only two, $d F = 0$ and $\delta F = J$ (or even one, $(d + \delta)F=J$), is scientific progress, because it tells us more than we previously knew about this.

But there is also another reason not to consider that science without experiments is dead. The idea that any theory should offer the means to be tested is misguided. Of course, it is preferred, but why would Nature give us the mean to check any truth about Her? Isn't this belief a bit anthropocentric?

Another reason to not be extremist about predictions is the following. Researchers try to find better explanation of known phenomena. But because they don't want they claims to appear unscientific, they try to come up with experiments, even if it is not the case. For example, you may want to find a better interpretation of quantum mechanics, but how would you test it? Hidden variables stay hidden, alternative worlds remain alternative, if you believe measurement changes the past, you can't go back in time and see it changed without actually measuring it etc. It is like quantum mechanics is protected by a spell against various interpretations. But, should we reject an alternative explanation of quantum phenomena, because it doesn't make predictions that are different from the standard quantum formalism? No, so instead of calling them "alternative theories", we call them "interpretations". If there is no testable difference, they are just interpretations or reconstructions.

A couple of months ago, the physics blogosphere debated about post-empirical science. This debate was ignited by a book by Richard Dawid, named String Theory and the Scientific Method, and an interview. His position seemed to be that, although there are no accessible means to test string theory, it still is science. Well, I did not write this blog to defend string theory. I think it has, at this time, bigger problems that the absence of means to test what happens at Plank scale. It predicts things that were not found, like supersymmetric particles, non-positive cosmological constant, huge masses for particles, and it fails to reproduce the standard model of particle physics. Maybe these will be solved, but I am not interested about string theory here. I am just interested in post-empirical science. And while string theory may be a good example that post-empirical science is useful, I don't want to take advantage of the trouble in which this theory is now.

The idea that science will continue to exist after we will exhaust all experiments, which I am not sure describes fairly the real position of Richard Dawid, was severely criticized, for example in Backreaction: Post-empirical science is an oxymoron. And the author of that article, Bee, is indeed serious about experiment. For example, she entertains a superdeterministic interpretation of quantum mechanics. I think this is fine, given that my own view can be seen as superdeterministic. In fact, if you want to reject faster-than-light communication, you have to accept superdeterminism, but this is another story. The point is that you can't make an experiment to distinguish between standard quantum mechanics, and a superdeterministic interpretation, because that interpretation came from the same data as the standard one. Well, you can't in general, but for a particular type of superdeterministic theory, you can. So Bee has an experiment, which is relevant only if the superdeterministic theory is such that making a measurement A, then another one B, and then repeating A, will give the same result whenever you measure A, even if A and B are incompatible. Now, any quantum mechanics book which discusses sequences of spin measurements claims the opposite. So this is a strong prediction, indeed. But how could we test superdeterminism, if it is not like this? Why would Nature choose a superdeterministic mechanism behind quantum mechanics, in this very special way, only to be testable? As if Nature tries to be nice with us, and gives us only puzzles that we can solve.

Are sciences and arts perversions?

According to Wikipedia, perversion is

a type of human behavior that deviates from that which is understood to be orthodox or normal.

Now consider the human mind. We evolved so that we find food, make children, avoid predators etc. All these are just means that serve to our survival in a universe that is trying to kill us. Or, even better, they serve to the selfish gene, to its replication.

So, the human mind shouldn't care about things that don't serve this purpose. What evolutionary purpose can be in doing math and physics? Indulging yourself in such activities doesn't serve the purpose of your survival and replication. One may say that, at least for some, science is their job, they earn money, and they survive. But researchers know better that jobs in the industry are safer and better paid, and with better success rate. And better success at ladies (although artists are doing even better). But anyway, sciences and arts are recent, so they can't be the product of mutation and selection. So, science and arts are perversions of the original purpose of the brain.

While they are not the product of evolution, they may be a byproduct. In order to survive, our ancestors had to identify patterns around them, use these patterns to make predictions. To anticipate when a wild animal will attack them, to recognize comestible fruits, to identify a sexual partner with good potential, all these require pattern recognition and the ability to make predictions. And this is why we became intelligent. So, even if we are using our intelligence to other purposes like sciences, this is a byproduct of evolution.

Nature has a way to reward you when you do something good for your genes. This is why we like to eat and to have sex. This is why we feel proud and happy when our children accomplish tasks or acquire new skills. But the blind gene doesn't know the future, so she can't reward us for actually doing something good for her. Instead, she rewards us for guessing patterns. We feel happy when we guess a pattern, and especially when a long anticipated prediction is confirmed. We identify patterns in sounds and drawings too, so this is why we like music and other arts. Even literature, builds on our predictions and anticipation. During anticipation, the brain produces the drugs that will make us happy. Building anticipation and suspense is the craft of accumulating this happiness in the consumer of one's art.

We are surprised when we make predictions which we consider safe, and turn out to be wrong. Sometimes, the anticipation accumulates the feel-good drug, and the surprise makes it explode. So this is how we laugh. Jokes are just clever ways to manipulate us into making predictions that turn out to be wrong in an unexpected and usually harmless way.

OK, so evolution explains all these as perversions of our mind, as byproducts. However, we know that science helped us to survive better. As a result of science and of its progeny, technology, we live longer, in better conditions, we find and produce food easier, we can take care better of our children and it also helps the children of our children.

So, science really helps the replication of the selfish gene.

I can't help but asking, did the selfish gene have a secret plan all this time?

A confused sleeping beauty 2

This post contains a small twist of the original experiment discussed in the previous post, A confused sleeping beauty. The new version doesn't require putting anyone to sleep and removing her memories, because we replace memory removal with lack of information.

Confusing Sleeping Beauty without erasing her memory

Sleeping Beauty is no longer required to sleep, but she may still need to sleep, to remain beautiful.

Consider the following settings:

- We toss a fair coin.

- If it lands heads, we will ask once Sleeping Beauty her belief for the proposition that the question landed heads.

- If the coin lands tail, we ask her twice.

This is similar to the original experiment, but instead of erasing her memory, we just do the following:

- Before asking her any question, we toss the coin a large number of times.

- Then we ask Beauty, but not in the same order in which we tossed. For example, when we toss a coin, if it landed heads, we write down a question and don't ask it yet. If it landed tails, we write down two questions, and don't ask them yet. Then we shuffle the questions and we ask Beauty one at a time. We take care to keep track for each question to which toss is connected.

To prevent the possibility that she adjust her estimates by counting counting the number of heads and tails about which she was already asked, we don't tell her whether she guessed or not, until the end of the experiment.

We see that the most rational answer she can give is 1/3. On the other hand, of course she knows that the probability that when the coin was tossed it landed heads is 1/2.

A confused sleeping beauty

The Sleeping Beauty problem

 A recent post by Sean Carroll reignited a debate about the "Sleeping beauty problem".

This is a simple problem of probabilities, involving tossing a coin. But for some reason, it seems to be no agreement about its solution.

Consider the following experiment:

- On Sunday, put Sleeping Beauty to sleep.

- Toss a fair coin.

- We are interested to ask Sleeping Beauty the question 

Q. What is your belief now for the proposition that the coin landed heads?

- If the coin comes up heads, wake up Sleeping Beauty on Monday and ask her the question. Then drug her to forget that awakening.

- If the coin comes up tails, wake up Sleeping Beauty both on Monday and Tuesday and ask her the question. Each time drug her to forget that awakening.

- In both cases, don't forget to wake her on Wednesday and end the experiment.

If you have trouble convincing a Beauty to let you put her to sleep and drug her, you can try your luck with people who already have very short memory, like Lucy Whitmore from "50 First Dates", Leonard from "Memento", Allie from The Notebook, or Dory from "Finding Nemo". Or you can make the experiment with Dory from "Finding Nemo". 

You can also perform the experiment with Dory from "Finding Nemo".

Those thinking they know the answer are mainly in one of two camps: halfers, who think she should answer 1/2, and thirdirs, who think she should answer 1/3. Thirdirs say that when Beauty is waken and interviewed, she thinks she can be in one of three situations. Since only in one of the cases the coin turned up heads, the answer must be 1/3. Halfers say that this answer is wrong, being probably caused by drug abuse, and since the coin is fair, the answer should be 1/2. There is nothing that can provide new information to Sleeping Beauty, so this answer should remain 1/2.

I will not detail here the debates still ongoing on the net, and the articles which are written about this. I just want to explain why I think that this debate is based on different understandings of the question.

Another experiment

Consider the following experiment.

- Prepare a large box, in which you can put apples and oranges, without seeing its content.

- Toss a fair coin.

- If the coin comes up heads, put one orange in the box.

- If the coin comes up tails, put two apples in the box.

- Repeat this many times.

- At the end, randomly extract a fruit from the box. Unless the experiment took too long, the fruits are not yet rotten, so you can extract a fruit.

- Then answer the following questions:

1. What is the probability that the fruit you will extract was introduced after the coin landed heads?

2. What fraction of the total times the coin was tossed, it landed heads?

The answer to question 1 is of course 1/3, because 1/3 of the fruits are oranges, and oranges were placed in the box when the coin landed heads.

The answer to question 2 is of course 1/2, because the coin is supposed to be fair.

My claim is that thirdirs were actually answering question 1, and halfers were answering question 2.

The question Sleeping Beauty was asked can be seen as equivalent to both question 1 and question 2.

To see how it can be seen as equivalent to question 1, consider a combination of the two experiments. Say that Sleeping Beauty is not only asked the question, but also it is given a fruit to put in the box. If the coin landed heads, she will receive an orange, and if it landed tails, she will receive an apple. She will put them in the box, then she will be put to sleep and forget about the awakening. Say the experiment is ran a large number of times. At the end, she can just count the fruits, and she will find that 1/3 of them are indeed oranges, so she will know that indeed the answer to the question is 1/3. Asking her about her belief that the coin landed heads that time is the same as asking her about her belief that she will receive an orange.

It is true that for every time the coin landed tails, she gets two apples, while every time it landed heads, she gets only one orange. This is why some tend to understand the question as being actually question 2.

Removing the confusion

So the dispute between thirdirs and halfers is due to the fact that they interpreted the question differently, and consequently answered different questions.

Instead of asking Sleeping Beauty the question as originally stated, we could just ask her two questions:

Q1. What is your belief that this awakening occurred following an event in which the coin landed heads?

Q2. What is your belief that when the coin was tossed, it landed heads?