Friday, April 03, 2009

God and Boltzmann eggs: dogmas and infinite priors

At the end, I added some comments about the general proof of the second law of thermodynamics in quantum mechanics.

Apologies to readers who are believers but I found it funny and deep.

The discussions about the Boltzmann eggs have reminded me of the psychological power of dogmas, fundamentalism, and circular reasoning once again.

Imagine the following old-fashioned set of hypotheses:
  1. the world was created exactly as described in the Bible
  2. for example, the electromagnetic field was created after the Earth
  3. life was created within seven days
  4. and so on...
At some level, that's an internally self-consistent framework. That's why it could have been believed by decent and relatively smart people for long centuries. This framework can be reinforced in many ways:
  1. the infinite power of God is independently confirmed by many books of the Bible
  2. that's why He is really "infinitely^66" powerful, not just infinitely powerful
  3. because God is infinitely great, no competing hypothesis can ever have a chance and no experiment can be important enough to change the conclusions above; after all, thou shall only have one God
  4. and so on...
The Inquisition vs heretics

You can see that the framework is self-consistent and powerful, indeed. ;-) But some heretics have asked whether the framework was right. Well, if you use the internal logic of this framework, it is certainly right: any doubt about it is a heresy.

However, if you are open-minded at the beginning, if you formulate, read, or hear alternative explanations, and if you compare them with the creationist picture of the world, your conclusions may differ and will differ. The evidence will support a completely different theory about the origin of species - one that was written down by Charles Darwin - and a different theory about the origin of light that has been similar to the present light at least for 13.7 billion years.

This method to derive similar conclusions, some of which may turn out to be heretical, is called science. It requires the people to be open-minded at the beginning, to admit that any assumption can a priori be qualitatively wrong and any alternative assumption can a priori be qualitatively correct. And it requires the people to evaluate the evidence that gives us hints which hypotheses are more likely and which hypotheses are less likely.

Statistically antireligious bigots

Many people oppose religion because they think it is not scientific. But if you look what these people actually believe, you will find out that their reasons to consider religion unscientific are irrational - and that their own beliefs are equally unscientific as the beliefs in the religious example above. In fact, their beliefs are structurally isomorphic.

Consider the following set of postulates:
  1. there are infinitely many possible cosmological models
  2. many of them describe the past of our Universe as one that contains an infinite (or nearly infinite) spacetime volume with a nonzero density of matter
  3. in these infinite (or almost infinite) regions, all localized configurations of matter (microstates) such as those of eggs appear (nearly) infinitely many times
  4. on the other hand, the evolution from a tiny, low-entropy Universe appears only a few times
  5. it is thus infinitely (or almost infinitely) more likely that our life was born as a statistical fluctuation, from a Boltzmann egg, or our brain was directly created as a Boltzmann brain
  6. because of this (nearly) infinite discrepancy between the priors, no finite number of arguments or experiments can change the conclusion that we are just a statistical fluctuation that evolved completely randomly
  7. and so on...
Again, this set of ideas is internally self-consistent, as long as you don't care about too many observations about the reality such as the existence of irreversible phenomena, aging, evolution, natural selection, friction, memory, heat, or any other effect or thing in the physical Universe. ;-)

This set of ideas is so powerful that any hypothesis and any theory that includes the statement that the entropy in the past was low is excluded. It is permanently excluded, at least in the minds of the champions of this reasoning. Science as we know it is forever guaranteed to be in
compatible with this "statistical argument". Science as we know it sucks, these people think. The paradox is inevitable.

The blasphemy of thermodynamics

Now, is that right? You may guess what the right answer is. Of course, this set of ideas is not right. The people who believe it - such as Sean Carroll - are irrational bigots in the very same sense as Pat Robertson. The main difference is that Pat Robertson has an actual anti-aging protein shake (because he can leg-press 2,000 pounds) while Sean Carroll only offers a cartoon of an egg that defies aging. What is the mistake that the people like Sean Carroll are doing? The mistake is that they are simply unwilling to admit that their assumptions may be fundamentally wrong.

By assigning their assumptions with prior probabilities that are pretty much infinitely greater than the prior probabilities of all competing hypotheses, these people confine themselves into a dogmatic mode of reasoning. It is very clear that if you set the prior probabilities of all competing hypotheses to zero - or to exp(-10^{120}) which is effectively the same thing in any conceivable context - logical inference will never be able to "undo" this bad prior. Why?

Hypotheses must be given a chance

Well, Bayes' theorem leads you to use the evidence E and to adjust the prior probability of a hypothesis, P(H), to a new one, P(H|E), according to the formula
P(H|E) = P(H) P(E|H) / P(E)
You can see that if P(H) is zero, P(H|E) is also guaranteed to be zero regardless of the probabilities that the evidence occurs: P(E|H) is the probability that E occurs, as predicted by the hypothesis H, while P(E) is the probability that the evidence E emerges according to all competing hypotheses, weighted by their priors (which cannot be zero, otherwise all hypotheses would be incompatible with E).

In other words, if you assume that a hypothesis - such as a Godless origin of species or a low-entropy beginning of the Universe - is a priori impossible, no finite amount of observations or arguments will be able to change your mind. If that's the case, you are a closed-minded bigot. If the prior probability of a hypothesis is exp(-10^{120}), it is not quite zero but the practical result will be identical: no conceivable set of arguments or experiments may ever convince you that you are wrong.

In a rational, scientific approach, every qualitatively different hypothesis about the world must be given a chance to be supported by evidence and every hypothesis about the world must be allowed to fail. In science, there is simply no room for a priori judgements such as the "infinite God" or the "infinite dominance of high-entropy states in the past".

Although these systems of ideas involve (nearly) infinite objects that make certain statements (almost) infinitely certain, within the same system to calculate the probabilities, that doesn't mean that these statements are certain. This implication doesn't exist because of a simple reason, namely that these systems of ideas may be fundamentally wrong - which makes all their internal "infinite" statements worthless. And indeed, they are fundamentally wrong as the most basic scientific tests immediately show.

A reader wanted me to prove that by collecting additional evidence, we can fix the wrong priors for the low-entropy initial state and find out that the probability of a low-entropy initial state of the Universe is actually high. However, if the prior probability for a low-entropy initial state is zero or exp(-10^{120}), such a task is impossible. It is impossible for the very same reason why you can never defeat God in His own disciplines. :-)

If you accept the assumption that God and all the statements about Him are infinitely greater than anything you can ever do and say, it is simply impossible to show that evolutionary biology is superior relatively to creationism. If you accept the infinite truth of Genesis as your zeroth approximation about reality, an (almost) infinite struggle against the wind mills is the only possible fate awaiting the heretics. ;-)

In the very same sense, if you accept the assumption that the high-entropy initial states are exp(10^{120}) times more likely than the low-entropy initial states, all the people who disagree with you, the heretics who believe thermodynamics up to its second law or even beyond ;-), are obliged to find an effectively infinite miracle that beats your effectively infinitely powerful argument.

Except that you're wrong in the very same sense as your Christian colleagues. The nearly infinite factors that you like to talk about do not measure the actual validity of your framework. They only determine how much blinded you are in your defense of your wrong dogmas.

Past and future: symmetries

The hypothesis that the Universe in the distant past should have a higher entropy than the current entropy is simply fundamentally wrong. It contradicts the second law of thermodynamics which holds according to all observations. It contradicts the proper theoretical derivations that are relevant for statistical physics and thermodynamics. Because the entropy increases with time, the entropy in the distant past had to be lower than today, not higher than today. Any retrodiction about the past that ends up with a qualitatively different conclusion is wrong - in the scientific sense.

What's exactly wrong with this framework? Some time ago, I wrote 7 postulates of this "high-entropy" ideology. They said that one can imagine models with infinite spacetime volumes - sure, one can do so. One can imagine that certain events are repeated infinitely many times - sure, you can think about such models. Another postulate number 4 said that the Big Bang evolution only occurs a few times. That's not really true because the evolution from the Big Bang (and Darwin's evolution) can also be repeated infinitely many times across your big multiverse.

So the point 4 is really incorrect. The evolution of mammals from more primitive organisms will still occur much more often than a spontaneous emergence of a mammal from a chaotic piece of matter.

But the most incorrect postulate is the postulate number 5. This postulate claimed that the infinite spacetime volume in some cosmological models implied that life has emerged as a statistical fluctuation. That's simply wrong. There exists no logical procedure that would allow one to derive such a conclusion simply because there exists no universal law that probabilities are proportional to the number of possibilities.

Even if you can write down infinitely many possibilities describing our Universe in a certain way, it doesn't mean that one of these possibilities has to be right. These possibilities can still be wrong - and a completely different possibility that you didn't include into your ensemble, even if it is just one possibility, can be correct. The number of microstates is not the same thing as the probability. These are completely different concepts.

I used to think that people must have already understood what was wrong with Zeno's paradoxes. For example, the trajectory of Achilles trying to catch up with the tortoise can be divided to infinitely many pieces. But that doesn't mean that he can't ever overtake the animal, does it? ;-) Sums of infinitely many numbers may converge!

The idea that all the microstates are equally likely is valid in certain contexts but one must be very careful what the contexts are. The conclusion is valid for the configurations of matter after a long enough time when "thermalization" could have occurred. If a closed, non-integrable system with many degrees of freedom evolves randomly, it eventually ends up in a "random" or "generic" point of its phase space or Hilbert space (more precisely, its subspace where the conservation laws are respected).

But if you omit the assumption about the thermalization time, the statement that all microstates are equally likely is not correct. In general, it is wrong. And if you study the state of the Universe in the distant past, it is fundamentally wrong. The truth is diametrically different. While the assumption about "generic" states is getting increasingly valid in the future, it is becoming increasing invalid if you go into the past, essentially for the same reason. A few basic observations - observations of an increasing entropy - are enough to show that the idea that the entropy decreases with time is incorrect.

If you insist that people should give you (almost) "infinite arguments" or "miracles" to defeat your assumption that all the microstates in the past are equally likely, the only thing you are showing is that your bigotry and your unwillingness to objectively look at the evidence is (almost) infinitely large.

No cosmological model that includes the well-established laws of local physics can ever predict that the eggs unbreak as often as they break. The reason is that the breaking of eggs is a local phenomenon. It means that the breaking of eggs only depends on the local behavior of your model of physics. It follows that it can't possibly be influenced by any changes of the cosmological features of your model of the Universe.

The idea that the entropy should increase both into the future as well into the past implies that the entropy should be minimized at the present. But the present is not a scientific term. The present means something else than what it meant when you were reading the previous sentence. No objective law of physics can assign the "present" with a privileged role. The idea that the present is special brutally contradicts the time-translational symmetry of the laws of Nature. The latter symmetry (linked to the energy conservation by Noether's theorem) has been verified with a huge accuracy - unlike the time-reversal symmetry of the macroscopic phenomena that has been (easily and repeatedly) ruled out.

OK: which symmetry wins?

Again, there are two possible symmetries of the macroscopic, effective laws of physics that you may consider and appreciate in this context. One of them is the time-translational symmetry and the other one is the time-reversal symmetry. However, if you have ever looked outside the window of your office or anywhere in the real world, you must know that the macroscopic phenomena look pretty much identical if you shift them in time, but they look dramatically different if you revert them in time.

For example, you can watch a movie after it was shot - and it looks OK. On the other hand, if you watch a time-reverted movie, you get something like this:

Some people find this video as sensible as the original one - yet they still have the guts to call me "reactionary". ;-)

Do you know why the people in the normal world look in front of them, in the direction of their motion, and not in the opposite direction? It may sound surprising for Sean Carroll but it's because the eyes exist in order to help the people to plan for the future. They can't plan the past because the past is already gone. ;-)

Also, to do their present job, the eyes collect the information from the past light cone. Again, they can't collect any information from the future because it is not yet decided i.e. not yet available at the present. :-) Because of quantum mechanics, the future developments will depend on random, probabilistic outcomes of events that can't be known now (see e.g. the free will theorem that makes this point really sharp). That differs from the past which is already settled. But that's not just quantum mechanics: in any realistic sense, the future is "unknown" even in classical physics.

The correct arrow of time is also crucial for the relatively low concentration of traffic accidents: the video above wouldn't work in this way because the pedestrians or drivers walking or driving backwards would be colliding all the time.

There exists no corner of the Universe - and no other Universe - where the people are making plans for the past. The difference between the past and the future is huge and logical in character. An arrow of time must always exist in every Universe whose basic architecture resembles ours. And by choosing a proper terminology, the past is what cannot be changed while the future is what you may plan for. This logical arrow of time inevitably agrees with the thermodynamic arrow of time and the direction of decoherence.

So the observational data speak in a clear language. The macroscopic features of the phenomena in the real world are compatible with the time-translational symmetry but they are completely incompatible with the time-reversal symmetry at the macroscopic scale. Theories that effectively describe macroscopic phenomena therefore cannot be time-reversal symmetric. And indeed, they are not. Friction, diffusion, viscosity, aging, decoherence, and other phenomena introduce an arrow of time into the realm of macroscopic phenomena.

Not only the arrow of time for these phenomena is compatible with the (approximately, but almost exactly) time-reversal-symmetric (or exactly CPT-invariant) underlying microscopic processes. In fact, the coefficients of the time-reversal-asymmetric phenomena - such as the coefficients of friction or the speed of decoherence - may be quantitatively calculated from the first principles, too. Boltzmann's H-theorem was the first pioneering calculation showing how to do so. This calculation and its followups agree with all the observations while the dramatically different philosophical frameworks don't agree with any observations.

In science, the previous sentence makes a difference. You may find some features of Boltzmann's derivations "ugly" but they are physically deep, true, quantitatively accurate (in the thermodynamic limit), and robust (i.e. invariant under detailed modifications of the assumptions).

You know, the very concept of entropy is slightly ill-defined because the clustering of states into macroscopically indistinguishable ensembles depends on many choices. Correspondingly, the laws (e.g. the second law) can sometimes be violated for a little while, with a small probability. Nevertheless, the impact of this clustering of microstates on macroscopic physics is very real and pretty much universal. This impact is studied by a science called thermodynamics and even children can feel that something is hot and something else is cool!

Anthropic brainwashing

The fallacy of the infinite priors - i.e. the people's inability to realize that they are blinded by some dogmas that can be wrong (and that often are wrong) appears at many places in the world. The champions of the anthropic reasoning suffer from similar fallacies all the time, too. Their defining assumption is the assumption of "typicality". It says that all the different Universes should be equally likely.

Now, some of them agree that it is an arbitrary "egalitarian" assumption that has no reason to hold exactly. However, if they're anthropic believers, they will always tell you that the assumption about the equal probabilities of all vacua must be the correct zeroth approximation. All other effects must be small corrections.

If you ask them about the evidence supporting their assumption, you won't get any. The reason is simple: there is no evidence. The assumption is nothing else than a dogma. Those people only differ in their degree of fanaticism i.e. in their tolerance for "corrections" that are allowed to modify their egalitarian priors. But none of the anthropic people is able to understand that their assumption may be qualitatively, fundamentally wrong.

And be sure that it is fundamentally wrong. The number of all vacua in quantum gravity is really infinite - it's enough to consider the AdS5 x S5 vacua to see why - and there is no uniform probability distribution on an infinite countable set. The weight of the "numerous" classes of vacua that make the total number so infinite must be dramatically reduced in order for the total probability to converge. It is plausible and, in fact, extremely likely that the probability of the excessively large classes is reduced so that even these whole classes are less likely than the small classes.

This answer is suggested e.g. by the analogies with thermodynamics above. Not only the high-entropy microstates are less likely than the low-entropy microstates. But even the sum of the probabilities of the huge number of high-entropy microstates - i.e. the probability of a high-entropy macrostate in the past - is smaller than the probability of a low-entropy macrostate in the past. That's how the world works, whether you like it or not. The probabilistic distribution on the space of vacua in string theory, assuming that there is a distribution that waits to be clarified by science, is likely to obey the same rule. It seems necessary for the total probability to converge.

Moral of the story

If someone tries to throw away an alternative argument by claiming that the a priori probability of his argument is (almost) infinitely higher, you should always be careful. Is there some evidence that it is (almost) infinitely higher? And if such claimed evidence is based on the claim that there are many possibilities that are equally likely, is there actually some evidence that the numerous possibilities are equally likely?

If Lee Smolin tells you that you may ignore string theory because there are roughly 12 competing theories (a high number!) with various kinds of swimming octopi that are equally likely as string theory, does he actually have some evidence that the crackpot fantasies are equally likely as string theory?

To make the inconsistency and irrationality of his whining more obvious, Smolin says that the "alternative" theories should be supported because there are roughly 12 of them which is (impressively) many while string theory should be suppressed because it has 12 or so scenarios which is (too) many. I still can't believe that some people may be so dumb that they are ready to buy this incoherent propaganda as a package.

The number of hypotheses and their probabilities are completely different animals. Unless there exists some evidence, these quantities ("N" and "Prob") cannot be assumed to be negatively correlated and they cannot be assumed to be positively correlated.

In the thermodynamic and anthropic cases, there is no evidence. In fact, there exists counter-evidence suggesting that these statements are wrong. In the thermodynamic case and in the Smolin case, this counter-evidence is overwhelming. But there are many other situations where both the evidence and the counter-evidence is absent. The people who try to strengthen their propositions by talking about (nearly) infinite numbers often have no rational justification for their statements and the (nearly) infinite quantities only determine their fanaticism, i.e. their willingness to replace rational arguments by shrill irrational arguments.

The (nearly) infinite catastrophes hypothetically caused by global warming are just another example. There exists no rational evidence that global warming will cause any problems - and this missing rational evidence for finite problems cannot be replaced by irrational evidence that global warming will cause infinite problems. ;-) But you can surely find many other examples outside the field of global warming pseudoscience, too.

And that's the memo.

Off-topic: Fermi (GLAST) essentially kills all Lorentz-breaking theories (which includes all existing alternatives to string theory): the paper has appeared in the regular Science Magazine.
Bonus: quantum mechanical H-theorem

I urge all the interested readers to look at this proof of the increasing entropy using the tools of quantum mechanics. This quantum-mechanical "upgrade" of Boltzmann's own proof is very transparent. (I also recommend you a similar cute, solid, quantum-field-theory-friendly proof by Steven Weinberg, Quantum Field Theory vol. 1, pages 150-151.)

The only assumption that is being made is that one can choose a basis of the Hilbert space in which the off-diagonal elements quickly decohere, i.e. can be set to zero after each step "dt". Of course, if this assumption were not adopted, the entropy defined by the sum of "-p_i ln(p_i)" would have to be interpreted in terms of eigenvalues "p_i" of the density matrix and the sum wouldn't change at all (because of unitarity).

If we do accept the assumption/approximation, the evolution of the nonzero, diagonal entries in the density matrix, i.e. the probabilities, follows T-symmetric microscopic laws that can be replaced by Fermi's golden rule, by another assumption that the "p_i" are probabilities that combine a couple of practically indistinguishable states (the usual assumptions of Fermi's golden rule), and one gets a very explicit, quantitative formula for the manifestly positive increase of entropy of any system with many degrees of freedom (i.e. any system where the entropy makes sense).

While the proof is a moral explanation of the origin of the entropy increase, the most accurate (and highest) formula for the entropy growth is actually obtained if we distinguish all microstates (a basis of the Hilbert space) and "p_i" are the individual diagonal entries of the density matrix - even though the assumptions of Fermi's golden rule don't strictly hold with this choice.

In this quantum mechanical proof, it is decoherence that kills the information and spreads the probabilities over many microstates. But any kind of rounding, coarse graining etc. will have the same qualitative effect. The entropy defined by the formula may sometimes decrease for a while but only by steps (and for time intervals) that are small enough for one of the assumptions above to become invalid.

All this stuff is very serious, very well understood, very universal, very quantitative, and very accurate. The origin of the entropy increase is very local and very microscopic. Attempts to create the illusion that the arrow of time is not understood even at a vague level - and that it should be getting connected with some mysterious or global ideas about religion or cosmology - are attempts to propagate pure ignorance and stupidity.

1 comment:

  1. I'll be honest, I didn't read your whole article but I think I should comment on something about the book of Genesis that I've been thinking about for awhile.

    Let's say there's a God for this example. Now, had God conducted/inspired Moses, a guy before him, or as a lot of people think, a bunch of lying, manipulative scribes in the 8th century who pulled Israel's pre-exilic history out of their asses and no one said "Hey! We don't remember any of this happening. We have a long tradition of oral history and this isn't part of it." to write the beginning of the book of Genesis as purely a science book, do you really think it would have gone over well with a bunch of goat farmers 3000 years ago?

    It would be like them trying to read the physics portion your blog. I'm a modern, rather intelligent human, 3000 years later and I don't understand nearly any of the mentions of physcis (I can't even spell it) on your blog or really anything about the creation of the universe or evolution (okay, I understand it much better than the other one) other than what I've tried to understand from watching the Discovery channel. If I go any deeper into it, I run into quantum flucations, D-branes, and things I can't even look up because the definitions include words I don't understand either!

    Basically my point is, there's a damn good reason why Genesis isn't a science book and why it shouldn't be read like one either by the beleiver or the non-believer. The people would have said "Gravity? Fuck you Moses! You have heatstroke!" "Evolution? My family has had the same flock of sheep from my grandfathers time and none of them look any different." Really, it wouldn't have gone over well back then. If anything, it's to let the people know that it was YHWH who did all that creating and not some other God (Don't even try pulling "But it says 'elohim'!"). I remember them having a problem with other gods a few chapters later. It had to be digestable for people back then.

    I'm not actually sure why I wrote this rant now. It wasn't directed at you but I had thought this awhile back and had nowhere to write it. Something in your post must have sparked my imagination In summation, reading Genesis as allegorical isn't a cop-out and I don't see it being at such a war with science.

    P.S. I tried rewriting Genesis in modern scientific language and had you seen it, you would understand my point. It would be a lot of " And the LORD said "Let there be the force of X which caused Y, (which you'll discover in about 2800 years. Look just trust me. That's how it works.)