Boddy, Carroll: trying to save physics by sacrificing the Universe

...but no saviors are needed: their irrational Boltzmann Brain alarmism misunderstands what a hypothesis includes...

External discussion: Jacques Distler will write a few critical sentences about the Boddy-Carroll paper tomorrow. I completely agree with Distler – as he will reproduce some ideas from the text below (and others). It's not possible for hypothetical future events to influence the present; and there is no particular framework of probability theory into which sentences of the kind "we're likely Boltzmann Brains" may be justifiably embedded. Each of these two bugs is enough to identify the paper as crap and the authors as nuts.

On his blog, the Preposterous Universe, Sean Carroll promoted a paper by himself and Kimberly Boddy:

The Higgs Boson vs. Boltzmann Brains (his blog)

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Can the Higgs Boson Save Us From the Menace of the Boltzmann Brains? (arXiv)

Last week, I was giving a popular physics talk in a planetarium in Northern Bohemia. It clearly turned out to be too complicated for the bulk of the audience (philosophers are sometimes annoying but a group of philosophers is more ready to listen to some almost real physics than a selected 1/1,000 fraction of the general public in a medium-size town!) but we've had some fun, anyway.

One of the longest discussions was dedicated to the phase transition that may destroy the Universe; the Higgs field instability is the most ordinary example of such a scenario. In a "seed of doom", the Higgs field (or, more generally, another usually scalar field) may penetrate to a new, lower energy state that is incompatible with life. This "seed of the new lifeless Universe" starts to expand almost by the speed of light and devour everything. You won't feel the pain because your nerves are slower than the inflating nothingness.



I wanted to calm the public. The Universe won't collapse anytime soon. At the end, however, I just couldn't tell them anything else than the truth. And the truth is that empirically, we only know that the approximate lifetime of the Universe after which the "seed of doom" starts to grow somewhere is unlikely to be much shorter than the current age of the Universe, 13.8 billion years. It may be comparable, it may be a bit shorter but it may also be much longer and infinite. If it is finite, it sounds sort of unlikely that it would be comparable to the current age of the Universe which means that it's probably much longer. Don't worry. But there's really no "solid" argument that would prove that the Universe won't start to disappear in the next 1 billion years.

You may find the "Higgs decay" scenario frightening. The Universe may die long before the Sun runs out of fuel in 7.5 billion AD and goes red giant. What a waste! It may be tomorrow. We're not able to present any solid enough proof that it won't happen. However, Boddy and Carroll are scared of something else: that the Universe won't die soon. So they claim that the unstable Higgs field is our savior from the genuine threat: the Boltzmann Brains. This fear is utterly irrational because the Boltzmann Brains aren't endangering us. They aren't endangering physics, either. The won't ever appear on the Earth (much like Category 6 hurricanes which are nothing else than another proof that Al Gore is a liar without any scruples). There's no reason to sacrifice the world (or billions of dollars).

Similar explanations have repeatedly occurred on this blog but here we go again.




Boddy and Carroll – and others – are scared of a competing theory that may "explain" all the observations we have had. It's a theory that doesn't require the usual prehistory that has led to our life – including the Big Bang, the formation of the galaxies, the Solar System, and the lengthy path of evolution, not to mention many less fundamental parts of our life story.

Instead, one may say that there will be infinitely many opportunities in our soon-to-be-almost-empty de Sitter space where brains locally indistinguishable from ours may be created out of pure thermal fluctuations. Carroll and others believe that because the number of such "freak brains" is infinite (when integrated over the whole infinite future of the Universe), they are predicted to be more likely to be "us" than anything else (i.e. the well-behaved brains that have evolved from the Big Bang and evolution).

But this just ain't the case.




If there is an infinite number of something, this infinite number doesn't mean that it "has to be us". For example, \(\pi\) has infinitely many digits in its decimal form but that doesn't mean that you are any of them. The probability that you are just a digit of \(\pi\) is zero which means that not even the infinite number of these digits may force you to become a digit. Even if the probability that you were a digit of \(\pi\) were nonzero, it may still decrease with the location in \(\pi\) so quickly that the overall probability that you are a digit will be tiny.

Let me hope that the previous paragraph sounds trivial to you but be sure that Sean Carroll and others don't understand this simple claim. They confuse the "number of some objects" with "their probability" which are completely different quantities (and in a general situation, completely uncorrelated quantities) because the probability that "you are something" is in no way uniform for all these "somethings".

That's one way to describe the fundamental mistakes in his reasoning.

Another, closely related way to describe the fallacy is to point out that a hypothesis in science is something that explains our observations. To do so, it must not only make assumptions about "how the world works" but also about "where and when we are located or living" i.e. essentially "who we are" (I mean primarily who we are relatively to the rest of the world, not who we are internally and structurally).

Different assumptions about "where we are" and "when we are living" obviously lead to different predictions of what we should be seeing (the world looks different if you manage to live inside the Sun, inside the Moon, or million of light years from the nearest galaxy). So these assumptions distinguish different scientific hypotheses and they may be tested and falsified separately from each other. As a hardcore Marxist, Sean Carroll clearly wants to confirm or falsify all these different hypotheses simultaneously, as a collective, but science just can't work like that.

This simple point has been discussed using many different words on this blog. Several years ago, Hartle and Srednicki introduced the catchy term "xerographic distribution" to emphasize that the assumptions about our location within the spacetime incorporated in a theory is a part of the hypothesis that is being tested i.e. validated or falsified.

Imagine that our Universe will converge to an empty de Sitter space – everything indicates it is so (the world is already dominated by the cosmological constant and each 11 billion years of the cosmic time or so, the linear distances will double which means that the particle-based mass density of the Universe will decrease by a factor of 8 or so). This de Sitter space has a certain Poincaré recurrence time comparable to \(\exp(S_{dS})R_{dS}\) after which it has to repeat up to arbitrarily small errors and people have said many things about the question whether the repetitions should be viewed as independent episodes (the ER-EPR correspondence is surely another conceptual reason to think that these repeated stories should be thought of as being "in the same region of the spacetime" i.e. not independent).

But I don't really think that such questions about the identification influence how science chooses the valid hypotheses.

Fine. The Universe will continue as a nearly empty de Sitter space which is still filled with the thermal radiation at the temperature which is tiny (the typical thermal wavelength is comparable to the curvature radius of the de Sitter space) but nonzero. And because it's nonzero, every state of matter has a nonzero probability and when it's given infinitely many opportunities to be realized, it will be realized. In particular, freaky Boltzmann Brains that perceive the same things as we do even though they haven't evolved through the nice scientific big-bang-evolution path are guaranteed to appear at some very distant moment in the future.

But that doesn't mean that our best theories (assuming that the de Sitter space won't collapse) actually predict that we are the Boltzmann Brains. Even though the Boltzmann Brains will be repeated infinitely many times, science can say – and actually does say – that we're not belonging to their transtemporal society. Again, the number of objects is a different thing than the probability that you are one of these things, stupid!

The usual physical theories with the Big Bang and an infinitely long-lived empty de Sitter space are compatible with our having evolved by the "almost straightforward history" involving the usual events after the Big Bang and evolution, among others, without some exponentially unlikely events. Why? Because these assumptions are a part of the standard physical theory combining cosmology and particle physics! Like most good theories in science, the standard cosmology says that life has evolved without a dependence on some super-unlikely fluctuations or events. As a good scientific theory, our Big Bang cosmology explicitly says that we are not Boltzmann Brains. This claim isn't incompatible with any other assumption of the theory just like the claim that you are not a digit of \(\pi\) (even though there are infinitely many such digits) is not incompatible with the biology of mammals.

So the standard cosmological theory is a different theory than any theory that claims that we are Boltzmann Brains. They are totally incompatible with each other because the standard cosmological theory says that everything we see is a result of a nearly inevitable evolution that was picking the most likely outcomes almost all the time – and that depended on no "super-unlikely" events or fluctuations.

Different hypotheses may be compared with each other. You may compare the standard cosmological theory with the Boltzmann Brain hypothesis of any kind. Needless to say, the standard cosmological theory wins because the Boltzmann Brain hypothesis predicts that whenever you look a bit further than before, you should almost certainly see a disorder that will leak the fact that your brain and its vicinity is just a giant thermal fluctuation. (By the Boltzmann Brain hypothesis, I mean the hypothesis that our brains/civilization etc. appeared from a thermal fluctuation that only began to resemble the usual evolution at times much shorter than the usual age of the Universe or in a region much smaller than the usual size of the visible Universe but is truly thermal elsewhere; if you include large fluctuations that have evolved "ordinarily" in the whole visible Universe for 13.8 billion years, then such a "generalized Boltzmann Brain" hypothesis isn't falsified and may in fact be a good description or philosophical incarnation of our observations.)

The standard cosmological theory predicts that the next galaxy you are going to observe with your next-generation telescopes will be similar to those you already know. And of course that the standard cosmological theory's predictions are pretty much right while the totally different predictions of the Boltzmann Brain hypothesis are falsified.

A scientific hypothesis working with the assumption that we are Boltzmann Brains is empirically falsified – by totally elementary observations, in fact. Simple observations (combined with simple logic) are the easiest ways to falsify a hypothesis in science. But it shouldn't be shocking that one needs at least some empirical data to falsify a hypothesis. That's how science has always worked. Science chooses the right and wrong hypotheses by looking at the empirical data. There's no reason to be ashamed of this fact. It is true and it has to be true, otherwise it wouldn't be science.

So we don't need to assume that our Universe will die in a few billion years if we want to protect our physical theories from the Boltzmann Brains' being us. The empirical evidence is overwhelming that we are not Boltzmann Brains. Because we know that we're not Boltzmann Brains, we may immediately eliminate every hypothesis or its part that would force us to believe that we are Boltzmann Brains. It's that simple. That's why we just don't have to be afraid of "being" Boltzmann Brains or postulate some "liberating doomsday" to protect the good feelings about our identity against some crazy ideas.

At the end, I really think that Carroll's totally wrong reasoning is tightly linked to an ideology that blinds his eyes. As a hardcore leftist (or at least a person pretending to be one in order to improve his social status in a hard left-wing environment), he believes in various forms of egalitarianism. Every "object" has the same probability. Also, much like climate "scientists" (and I am only talking about "scientists" in the quotation marks here, not about genuine scientists), he wants to "collectively test" (and "collectively trust") models (e.g. climate models) whether they are right. But none of these things is scientifically true. Objects, people, and their categories are created unequal, probabilities aren't proportional to the numbers of objects in any reasonable sense, and hypotheses must be validated individually and not "in collectives" because at most one of the inequivalent theories or models may be right at the very end and it's just wrong to clump a right theory with the wrong ones because the very purpose of science is to be disentangling the right ones from the wrong ones.

Let me offer you an analogy that should hopefully clarify why Boddy's and Carroll's way of thinking is totally silly.

Imagine that we discover a stone that looks like a display and it displays one decimal digit every hour. Such a stone looks like a result of Intelligent Design but it doesn't matter whether it's man-made, UFO-made (OK, I meant ET-made), or natural. Assume it's natural but your task is to predict what the object will do. Once people begin to watch the digits and remember them, they record the following sequence:

4,1,5,9,2,6,5,3,5,8,9,7,9,...

It looks like a random sequence of digits. However, someone realizes that they look like digits (starting from the third one) of \(\pi\):\[

\pi\approx 3.1415926535897932384626\dots

\] This person will predict that the next digits will be 3,2,3,8 and the prediction is confirmed. It's great. Note that by now, 17 hours after the records began, people have recorded 17 digits from the stone so far.

But someone will start to claim that there is no reason why the digits should be taken from the beginning of \(\pi\). The same sequence of 17 digits appears roughly once in a sequence of \(10^{17}\) digits of \(\pi\) and because \(\pi\) has infinitely many digits, the same 17-digit sequence is bound to appear infinitely many times somewhere.

In fact, someone else will change the statement and say that they will appear somewhere in\[

e\approx 2.718281828459045235360\dots\qquad\\
\qquad \dots 28747135266249775724709369995\dots

\] or somewhere in its powers \(e^n\) where \(n\) is a nonzero integer. Because there are infinitely many numbers of the form \(e^n\) and just one number \(\pi\), someone else may even claim that it's more likely that the stone emits random digits from a number of the form \(e^n\) and not from \(\pi\).

Needless to say, such a claim is unjustified because there's no reason why the \(e^n\)-based explanation should be "equally likely" as an explanation based on \(\pi\). And indeed, the empirical evidence will keep on accumulating (new digits are coming every hour!) that the \(\pi\)-based explanation is the right one while the other hypotheses are just wrong.

The guy or babe who invented the \(\pi\) theory of the stone used \(\pi\) and not \(e^n\) and he or she did claim that the digits are taken almost from the beginning of \(\pi\), too. He or she isn't "obliged" to consider some faraway sequences in \(\pi\) (or even in other numbers) to be "equally justified" predictions of his or her theory because the theory includes the statement that the digits are taken almost from the beginning. The place in \(\pi\) from which the digits are being taken isn't "obliged" to be "typical" – on the contrary, it's a point of the explanation that it is a very special point, the beginning. So every inequivalent statement is a competing hypothesis and it will finally lose. Too specific theories based on specific enough locations in other numbers will be strictly falsified; theories explicitly or effectively claiming that the digits are random will be "fuzzily" (but increasingly robustly) falsified because they predict that (very/extremely) long \(\pi\)-like patterns are (very/extremely) unlikely. But they're being observed which makes these random explanations increasingly falsified.

In this analogy, Boddy's and Carroll's claim about the "desirable" Higgs decay of the Universe is analogous to the statement that \(e\) is a rational number. If the digits of \(e\) start to get repeated, then the same is true for \(e^n\) as well and the numbers \(e^n\) won't have the sufficient infinite diversity of sequences of digits to match the observed digits emitted by the stone. In this way, the analogous Boddy and Carroll will argue, the \(\pi\) theory is protected against the Boltzmann Brain explanation – the explanation assuming that the digits are being taken from a random faraway place of one of the numbers \(e^n\).

Indeed, academically speaking, the Boltzmann Brains-like \(e^n\) theory of the stone could be falsified in this way: if \(e\) were rational, it just couldn't generate an aperiodic sequence of digits. But what Boddy and Carroll (and others) don't understand is that it is not neccessary for \(e\) to be a rational number if we want to scientifically establish that the digits are actually being taken from \(\pi\). The empirical evidence is enough. And indeed, \(e^n\) for \(n\neq 0\) aren't rational numbers which means that the strategy to disprove the \(e^n\) theory of the stone is hopeless because it depends on propositions that are wrong. And indeed, the evidence from the stone keeps on arriving and confirming the \(\pi\) theory while falsifying any other simple enough hypothesis.

Also, I want to mention that we may know a reason or we may not know any reason why the stone prefers \(\pi\) over \(e^n\). But even if we don't know any such deeper reason, it doesn't mean that the \(\pi\) and \(e^n\) explanations are equally likely. Instead, the empirical data heavily break this symmetry and imply that \(\pi\) is vastly preferred. That observation really means that deeper theories about the inner workings of the stone are either "encouraged" or "totally required" to prefer \(\pi\) over \(e^n\). You may believe that \(\pi\) and \(e^n\) are equally good for the stone but much like any belief in science that has observable consequences, your belief may be proved to be wrong and indeed, it is proved to be wrong in this case, too. There's nothing "holy" or "infallible" about egalitarianism; in fact, it's one of the crappiest ideologies around.

We know that we aren't Boltzmann Brains and we don't need to assume a "doomsday scenario" – a Higgs vacuum decay or any other doomsday scenario one could talk about (the very fact that Boddy and Carroll single out the Higgs vacuum decay is a piece of demagogy or a hint that they're unable to localize the actual reasons that lead to certain conclusions) – to be sure that we aren't Boltzmann Brains because the observations we have already made are enough to be absolutely sure. The state-of-the-art scientific theories claim that we are results of a nearly inevitable evolution involving a very dense and hot Universe after the Big Bang, structure formation, and evolution of species and these scientific theories are explicitly stating that we are not random thermal fluctuations that would have to be super-exponentially unlikely.

Someone may think he has reasons to think that we should be Boltzmann Brains or it should be likely that we are Boltzmann Brains. But "where we are" represent a part of the scientific hypotheses – the xerographic distribution – that needs to be tested much like any other part of a scientific theory. The tests are very easy, have been done long before we became homo sapiens, and the result is that the Boltzmann Brain xerographic distribution is safely falsified. So why do people keep on talking about it? It's as safely falsified a scientific hypothesis as any other falsified scientific hypothesis. In fact, more so. It's one of the key principles of the scientific method that we are gradually ceasing to discuss scientific hypotheses and paradigms that have been falsified.

So Boddies and Carrolls of the world, please stop emitting this crap and attempting to raise the stakes by incorporating ever more irrational and ever more megalomaniac "requirements" concerning a doomsday. No doomsday is necessary for the science we have learned to work.

And that's the memo.

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Aaronson's anthropic dilemmas

This text has been expanded and covers the rest of the book... Originally posted on May 1st

If you read my previous observations on Scott Aaronson's book including all the comments, you will see my remarks about all the chapters up to Chapter 15 about the quantum computation skeptics – where I agree with almost everything Aaronson writes although he seems to focus on the dumb criticisms and writes too little about the more intelligent ones (and e.g. about the error-correcting codes).

Chapter 16 is about learning; perhaps too much formalism if we compare it with the relatively modest implications for our understanding of the process of learning.

Chapter 17 is the most hardcore "computational complexity" part of the book and hopefully the last one that is intensely focusing on the complexity classes. It's about interactive proof systems. Aaronson often wants to present all of computer science as a "fundamental scientific discipline" so he tries to apply these superlatives to aforementioned "interactive issues", too.

I have a lot of trouble to get excited about these problems.




In the interactive proof system, two beings – a verifier and a prover – are exchanging messages whose goal is to ascertain whether a given string belongs to a language or not. The prover cannot be trusted while the verifier only has finite resources. It looks like an immensely contrived game – from game theory – to me. Detailed questions about such a game seem about as non-fundamental to me as the question whether chess is a draw.

The only true reason why I would want to prove \(P=NP\) or its negation (or even the numerous less important results of this sort) would be to get a million of dollars.




Needless to say, I think that Scott Aaronson is a world's top professional in the computational complexity theory – and I think that the quantum aspect is an optional cherry on a pie for him, an extra X-factor that he adopted to feel rather special about the computational complexity theorists themselves.

But for me, this is a portion of mathematics that is completely disconnected from fundamental problems of natural sciences. I like to think about important scientific problems. But the complexity papers aren't really about the beef, about particular problems. They are thinking about thinking about problems – and they don't really care what are the "ultimate" problems and whether they're true (e.g. in Nature). In this sense, suggesting that this is a fundamental layer of knowledge about the world or the existence is as silly as the proclamations of anthropologists who study dances of wild tribes in the Pacific but who also try to study the interactions among scientists. These anthropologists are trying to put themselves "above" the physicists, for example, even though in reality, they are inferior stupid animals in comparison – people who completely miss the beef of physics and who may only focus on the irrelevant, superficial, sociological makeup on the surface. In some sense, Scott as a computational complexity theorist is doing the same thing as the anthropologists but with more mathematical rigor. ;-)

Moreover, the computational complexity theory seems to be all about a particular "practical" quantity I don't really care about much – namely computational complexity. I am probably too simple a guy but I primarily care about the truth, especially the truth about essential things, and I don't really care how hard it is to find or establish the truth. So the whole categorization of problems to polynomially or otherwise easy ones – and Aaronson defines dozens of complexity classes and discusses their relationships – is just something orthogonal to the things I find most important.

But let me stop with these negatively sounding remarks about the discipline. Computer science is surely a legitimate portion of maths and Aaronson is talking about it nicely.

Chapter 18 is about "fun with the anthropic principle".

This part of the book doesn't need any physics background – because this principle used by some physicists isn't about any scientific results, either. It's about their emotional prejudices and unsubstantiated beliefs in proportionality laws between probabilities and souls (which boils down to the fanatical egalitarianism of many of these folks).

The chapter is at least as wittily written as the rest of the book. The end of the chapter talks too much about complexity again but let's focus on the defining dilemmas in the early parts of the chapter. After a sensible introduction to Bayes' formula and its simple proof, Aaronson talks about some characteristic problems in which people's attitudes to the anthropic reasoning dramatically differ.

Hair colors in the Universe

At the beginning, God flips a fair coin. If the coin lands heads, He creates two rooms – one with a red-haired person and one with a green-haired person. If it lands tails, He creates just one room with a red-haired person.

You find yourself in a room with mirrors and your task is to find the probability that the coin landed heads. Well, you look into the mirror that's a part of each such room. If you see you are green-haired, the probability is 100% that the coin landed heads because the other result is incompatible with the existence of a green-haired person.

What about if you see you are a redhead?

A natural (and right!) solution, one mentioned at the beginning, is that the probability is 50% that the coin landed heads. The existence of a redhead is compatible with both theories (heads/tails) so you are learning nothing if you see a redhead in the mirror. You should therefore return to the prior probabilities and both theories, heads and tails, have 50% odds by assumption.

In my opinion (LM), this is really the most correct calculation and justification one may get. I tried to "improve" Aaronson's justification a bit.

Now, one may also (incorrectly!) argue that the probability of heads is just 1/3 instead of 1/2 if we see a redhead. The tails hypothesis is twice as likely, 2/3, than the heads hypothesis because – and again, this is an explanation using my language – it makes a more nontrivial, yet correct, prediction of the observed hair color. The heads hypothesis allows both colors so the probability that "you" will be the person with the red hair color is just 1/2.

But I believe this argument is just wrong. It doesn't matter how predictive the hypotheses are! By assumption, the prior probability of heads and tails were 50% vs 50%. The tails hypothesis is more predictive because it allows you to unambiguously predict your hair color – it has to be red because you're the only human in that Universe. But we know that this doesn't increase the probability of heads above 50%.

For that reason, we also don't need an additional "adjustment" of the argument – and this adjustment is wrong by itself as well – that returns the value 1/3 back to 1/2. We may return from 1/3 to 1/2 if we give the Universes with larger numbers of people – in this case, the heads Universe – a higher "weight". There is no reason to adjust these weights. The point is that the prior probabilities of heads and tails are completely determined here by an assumption so any inequivalent "calculation" of these prior probabilities based on the number of people in the Universe is wrong. We just know it to be wrong. We're told it is wrong!

Aaronson "calculates" the value 1/3 of the probability by Bayes' formula. But the calculation is just conceptually wrong because the prior probabilities of heads/tails are given as 1/2 vs 1/2 at the very beginning and the observation of a redhead provides us with no new data and no room to update the probabilities of hypotheses. The observation of a greenhead does represent new data. The arguably invalid update in the case of the observation of a redhead plays one role: to counteract the update from the green observation so that the probability of heads weighted-averaged over the people in the Universe will remain equal to the probability of tails. But it's not the redhead's "duty" to balance things in this way. By seeing his red color, he just learns much less information about the Universe than the greenhead (namely nothing) so he has no reasons to update.

Using slightly different words, I may point to a very specific error in the Bayesian calculation leading to the result 1/3, too. Aaronson says that the probability \(P({\rm redhead}|{\rm heads})\) is equal to 1/2 – probably because in the two-colored heads Universe, there are two folks and they have "the same probability". But that's a completely wrong interpretation of the quantity that should enter this place of Bayes' formula. The factor \(P(E|H)\) that appears in the formula should represent the probability with which the hypothesis \(H\) predicts some property of the Universe we have actually observed, \(E\) i.e. the evidence. And what we have observed isn't that a random person in the Universe is a redhead. Instead, we have observed that our Universe contains at least one redhead; in particular, the predicted probabilities \(P({\rm redhead}|{\rm heads})+P({\rm greenhead}|{\rm heads})\) don't have to add to one because both "redhead" and "greenhead" refer to the observation of at least one human of the given hair color so these two colorful observations are not mutually exclusive. (You should better avoid propositions with the word "I" because this word is clearly ill-defined across the Universes; there's no accurate "you" or "I" in a completely different Universe than ours because the identification of the right Universe around you is a part of the precise specification of what "I" or "you" means; you should treat yourself as just another object in the Universe that may be observed, otherwise you may be driven to spiritually motivated logical traps.) The probability of this actual observation – evidence – is predicted by the heads hypothesis to be 1, not 1/2. With the correct value 1, we get the correct final value 1/2 that the heads scenario is right!

I must mention the joke about the engineer, physicist, and mathematician who see a brown cow outside the train. The first two guys say some sloppy things – cows are brown here (engineer); at least one cow is brown here (physicist) – but the mathematician says that there's at least one cow that's brown at least from one side in Switzerland. This is the correct interpretation of the evidence! The situation in the previous paragraph is completely analogous. (There's a difference: people are less afraid to say unjustifiable and/or wrong propositions that are probabilistic in character, e.g. "I am generic", than Yes/No statements about facts that are "sharply wrong" if they're wrong. But probabilistic arguments and conclusions are often wrong, too!) I am surprised that even Scott Aaronson either fails to distinguish the different statements; or deliberately picks one of those that actually don't follow from the observations! This is the kind of the elementary schoolkid's mathematical sloppiness that powers most of the anthropic reasoning.

At the end, the error of the Bayesian calculation may also be rephrased as its acausality. It effectively assumes that the probabilities of different initial states are completely adjustable by some backward-in-time notions of randomness even though they may be determined by the laws of physics – and by the very formulation of this problem, they are indeed determined by the laws of physics in this scenario!

Madman

A madman kidnaps 10 people, puts them in a room, throws 2 dice, and if he gets 1-1 (snake-eyes), he kills everyone. If he gets something else, he releases everyone, kidnaps 100 other people, confines them, and throws again. Again, 1-1 means death for everyone, another result means that 100 people are released and 1,000 new people are kidnapped. And so on, and so on.

You know the rough situation and you know that you're kidnapped and confined in the potentially lethal room (but you don't know whether some people have already been released). What's the probability that you will die now?

Obviously, you know the whole mechanism of what will happen. He will throw dice. The probability to get 1-1 is obviously 1/36. That's the chance you will die.

Aaronson presents a different, "anthropic" calculation telling you that the chances to die are vastly higher, essentially 8/9. Why? Well, the madman almost certainly releases the first 10 people and then probably the 100 people as well etc. but at some moment, he sees snake-eyes so, for example, he kills 100,000 people and releases the previous 10,000+1,000+100+10 = 11,110 people. Among the folks who have ever been confined to the scary room, about 100,000/111,110 = 8/9 of them die. So this could be your chance to die; the ratio doesn't seriously depend on the number of people who die as long as it is high enough.

Which result is correct? Aaronson remains ambiguous, with some mild support for 8/9. I think that the only acceptable answer is 1/36. The argument behind 8/9 is completely flawed. It effectively assumes that you're a "generic" person among those who are kidnapped on that day – there's a uniform distribution over those people. But that's not only wrong; it's mathematically impossible.

The average number of people who will die is\[

\sum_{n=1}^\infty 10^n \zav{\frac{35}{36}}^n \frac{1}{36}

\] but this is divergent because \(q=350/36\geq 1\). Chances are nonzero that the madman will run out of people on Earth and won't be able to follow the recipe. At any rate, the reasoning behind \(p=8/9\) strongly assumes that the geometric character of the sequence remains undisturbed even when the number of the hostages is arbitrarily large. It effectively forces us to deal with an infinite average number of people and there's no uniform measure on infinite sets because there exists no \(P\) such that \(\infty\times P = 1\).

I think that this is not just some aesthetic counter-argument. It's an indisputable flaw in the calculation behind \(p=8/9\) and the latter result must simply be abandoned. In this case, we know very well it's wrong. If the madman tries to causally stick to his recipe as long as it's possible, the probability for each kidnapped person to die is manifestly \(p=1/36\).

The wrong, anthropic results often make unjustified calculations based on the "genericity" of the people – assumptions that some probability measures are uniform even though there is absolutely no basis for such an assumption and in our scenario, this uniformity assumption explicitly contradicted some assumptions that were actually given to us! And the anthropic arguments also tend to make acausal considerations.

Doom Soon and Doom Late

This is also the case of the "doomsday is probably coming" argument. Imagine that there are two possible worlds. In one of them, the doom arrives when the human population is just somewhat higher than 7 billion (Doom Soon). In the other one (Doom Late), the population reaches many quintillions (billions of times larger than the current population).

Again, just like in the hair color case, if we have reasons to expect that the prior probability of both worlds are equally or comparably large, then we have no justification to "correct" or "update" these probabilities. The existence of 7 billion people is compatible both with Doom Soon and with Doom Late. So both possible scenarios remain equally or comparably likely!

The totally irrational anthropic argument says that Doom Soon is 1 billion times more likely because it would be very unlikely for us to be among the first 7 billion – one billionth of the overall human population throughout the history. This totally wrong argument says that we're observing something that is unlikely according to the Doom Late scenario – only 1/1,000,000,000 of the overall history's people have already lived – and our belief that we live in the Doom Late world must be reduced by the factor of one billion, too.

That's wrong and based on all the mistakes we have mentioned above and more. The main mistake is the acausality of this would-be argument. The argument says that we are "observing" quintillions of people. But we are not observing quintillions of people. We are observing just 7 billion people. If the Doom Late hypothesis is true, one may derive that the mankind will grow by another factor of one billion. But if we can derive it, then it's not unlikely at all that the current population is just 1/1,000,000,000 of the overall history's population. Instead, it is inevitable: \(p=1\). So the suppression by the factor of 1 billion is completely irrational, wrong, idiotic, and stupid.

The only theory in which it makes sense to talk about quintillions of people – the Doom Late theory – makes it inevitable that the people aren't distributed uniformly over time. Instead, they live in an exponentially growing tree. So there's manifestly no "intertemporal democracy" between them that could imply that we're equally likely to be one of the early humans or one of the later ones. We're clearly not. It is inevitable that in most moments of such Universe's history, the number of people who have already lived is a tiny fraction of the cumulative number of the people in the history (including the future).

Aaronson offers another idiotic argument that may sometimes be heard. A valid objection to the Doom Soon conclusion is that it could have been done by the people in the world when the population was just 1 million or another small number – e.g. by the ancient Greek philosophers. And they would have been wrong: the doom wasn't imminent. Aaronson says that it doesn't matter because "most" of the people who make the argument are right.

But again, this is completely irrelevant. Whether most people say something is an entirely different question from the question whether it's right. And indeed, in this particular case, we may show that the probability is very high that the "majority" that uses the anthropic arguments is wrong! What's important is that the methodology or logic leading to the "doomsday is coming" conclusion is invalid as a matter of principle. It doesn't matter how many people use it! One can't or shouldn't invent excuses why these arguments are flawed by saying that some quintillions of completely different (and much less historically important, per capita) people at a different time would reach a valid conclusion. I don't care. I want to reach a correct conclusion myself and I don't give a damn whether some totally different people are right. Of course that they're mostly wrong.

Anthropic principle and a loss of predictivity: what is the real problem?

At the end, it's mentioned that the anthropic principle is often criticized for its inability to predict things. It's indeed unfortunate if a theory makes no prediction. But it's not a valid logical argument against a theory. The correct theory may make much fewer or much less accurate or unambiguous predictions than some people might hope!

The actual problem – one that may be used as an argument against the anthropic principle – is sort of the opposite one. A valid argument is that the alternative explanations that are more accurate, tangible, and predictive have not been excluded. There may be an old-fashioned calculation of the value of the cosmological constant, \(\Lambda\sim 10^{-123}\). And science proceeds by falsification of the wrong theories, not by "proofs" of correct theories.

We know that the anthropic explanation would have been wrong as an explanation of – now "materialistically" understood – features of Nature simply because we have better explanations that we know to be much more likely to be true than the anthropic one. And the same thing may happen – and, I think, it is likely to happen – in the future, too. If you can't really show that this expectation is wrong, you shouldn't pretend that you have proved it!

Perhaps, science will be forced to switch to anthropic arguments because beyond a certain point, there just won't be any old-fashioned explanations. Maybe quintillions of people will live in that future world and the claim that the "open problems are explained anthropically" will therefore be true for a "majority" of the mankind that will have lived throughout the history. But that won't change the much more important fact that the anthropic principle will have been wrong throughout the whole previous history of physics.

Aaronson is clearly close to all the anthropic misconceptions discussed above – which may be correlated with herd instincts, mass hysterias, "consensus science", and other pathologies. This is also manifest in his humiliating comments about the role of Adam and Eve. Well, I don't want to discuss the literal interpretation of the Bible which I don't believe, of course. But he wants to suggest that there is some uniform measure that makes it less likely to "feel that I am an early human".

But this is just totally wrong. There is absolutely no justification for such a uniform measure and because the population was growing pretty much exponentially (demonstrably so), this fact indeed pretty much allows us to prove that each early human was exponentially more important than the current ones and we're more important than the future ones.

In recent years, I got sort of interested in the history, e.g. the local history, and I studied the villages etc. that existed on the territory of Pilsen and in its vicinity. There were just hundreds of people and a few lousy houses and the folks didn't have almost anything but they were clearly very important because the hundreds of thousands of people who live here today have arisen from the small number of ancestors. So each of the ancestors is just much more important than an average contemporary human in the overall historical scheme of things. "Adam and Eve" were clearly even more important, if I express it in this way.

If we divide some consciousness or soul or something based on the spiritual importance, it's totally plausible to say that Adam and Eve (plus Jesus and His close relatives, or whoever counts) have 50% of it and the rest is divided among later humans, if you allow me to express the point more concretely than what is really possible. The argument "I can't be special or early because it is unlikely due to some uniform measure on the history's humans" is completely wrong. It is acausal, it uses mathematically non-existent measures, and it uses uniform measures that have no justification and that sometimes contradict legitimately calculable measures.

So I agree with Scott Aaronson that the anthropic reasoning may be defined as some part of probability theory that is more about feelings and opinions than about solid results. Well, most of the people – including Aaronson himself – clearly end up with completely wrong arguments and results which is just another way to disprove the anthropic principle. ;-) I can't be generic because almost all people seem to be morons. In fact, even these would-be generic people may use the same arguments because almost all of these stupid folks are still much smarter than generic insects and bacteria that are far more numerous. The whole idea of "considering oneself generic in a set" is just a way to contaminate a correct or rational argument or result by an incorrect or irrational one that is believed by the inferior life forms.

Chapter 19 is about the free will.

I found it amusing and agreed with what he had to say. He starts by pointing out some errors of free will supporters as well as foes – the "absence of free will means that all criminals have to be liberated" (silly: we sometimes punish toxic machines even though they don't have a free will!) and "undetermined implies random" (not the same thing).

Then he discusses childish examples with a Predictor who knows how you will act (that shouldn't be possible if the free will really exists) – Robert Nozick has played with such things. And finally, he gets to the Conway-Kochen free-will theorem which I was waiting for. There's an elementary, useful explanation what it says and how it guarantees quantum-certified random numbers.

Chapter 20 is on time travel.

Time machines are primarily wanted because they could speed up computation, something that isn't too important for your humble correspondent, especially because time machines and (macroscopic) closed time-like curves are impossible. The complexity chatter is legitimate maths but it doesn't turn me on. I still see that discipline as a conglomerate of many largely disconnected results with connections that are at most ad hoc. It must be easy to get lost in that jungle.

I must say: it's crazy that people like Peter Shor listen to these talks, how you compute something quickly by constructing a time machine and allowing your grandfather to have sex with your grandmother. Then he leaves the MIT seminar room and criticizes string theorists for not being sufficiently down-to-Earth and connected with the observations. Holy cow! Who is disconnected here?

The closed time-like curves are impossible for various deep reasons and one could discuss it. But Scott Aaronson chooses a different attitude, one of a spoiled frat who screams "I want them I want them I want them because I want fast computers!" The links to physics – which are an important motivating theme of the book – seem mostly bogus to me because he's ready to ignore the physics insights whenever it's appropriate to study abstract problems about "computation in the worlds with totally different laws of physics than ours".

Chapter 21: cosmology

This chapter shows that Aaronson has a pretty good background in cosmology. In most cases, it's pretty manifest that he got this knowledge from conversations with physicists and cosmologists (in many cases folks I know rather well) but that doesn't change my feeling that his presentation of the energy density in the universe, expansion, entropy of the Universe, and the holographic principle including things like Bousso's light sheet is more accurate, complete, and meaningful than what most actual experts would be able to write down.

In this chapter I noticed, perhaps even more strongly than in the previous chapters, that the switching between these physics topics and the omnipresent topic of complexity classes is somewhat unnatural – that Aaronson must also realize that even this chapter is constructed out of two largely disconnected topics. Physics and cosmology impose certain laws and limitations on all the objects inside – humans, computers, cucumbers, and everything else. Everyone must respect them and someone who proposes new computers or cucumbers should better learn about the laws. But one can't learn a sufficient amount about the laws just by discussing what kind of a computer we would like. The limitations imposed by the mathematical insights summarized in computer science belong among the limitations but they don't exhaust the full list because there's also physics that imposes constraints on what the mathematicians and computer scientists label as indisputable (and pretty much arbitrary) axioms.

As long as we talk about computers in the real world, the laws of physics/Nature are always primary and fundamental. Aaronson seems to implicitly ignore this fact at many places.

Chapter 22: answering all students' questions

The last lecture in 2006 – when he began to write the book – had the same format as the last lectures in courses by Feynman: the instructor could have been asked any question by students and was turned into an oracle. There are fun questions in the list. In some of them, Aaronson just reiterates his opinions about unsettled conjectures on the complexity classes (will they be proved or disproved in the future?). But there are also speculations on laws transcending quantum computers and their limits and so on.

In his answer to the last question, Aaronson suggests that computer scientists could be working in physics departments. It's just a historical accident, we hear, that they're not there. Well, I don't think so. It's applied maths. They're not really learning mechanics, field theory, and so on, because it's not needed. And they don't really care much whether some axioms they build upon may be realized in the Universe around us. So it's not physics. QCD or string theory are very far from mechanics but people doing it still start by learning and then are building on foundations that do include the characteristic subdisciplines in physics. They have to because the subdisciplines of physics are tightly connected to a compact whole. Computer scientists are doing something else – not trying to find out the ultimate underlying laws ("axioms" in the language of mathematicians) but choosing arbitrary axioms, regardless of their agreement with the empirical data, and getting interesting results and conclusions out of them. So it's maths, not physics.

Of course that with some very inclusive definition, all quantitative thinkers or all scholars or perhaps all employed people are doing some "generalized physics". But I don't think it's right to promote this inflation and degradation of the word "physics".

My (undergraduate) Alma Mater, the Faculty of Mathematics and Physics of the Charles University in Prague, is vaguely divided to the sections Physics – Mathematics – Computer Science – Teaching of M/Ph/CS. So computer science is de iure put on par both with mathematics and physics as an independent branch. Still, it shares the floors and buildings with some kind of mathematicians (and especially philosophers of mathematics and set theorists), not with physicists, and for good reasons.

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Greene et al.: too large landscapes are unstable

For years, I've been intrigued by the general idea that the usual KKLT arguments – supporting the view that the number of vacuum (dS or AdS) solutions to string theory is googol-like huge, most of those are stable, and they realize the anthropic principle (including the anthropic explanation of a tiny cosmological constant) within string theory – fail because there is an overlooked mechanism that tries to harm the classes of vacua that are too numerous and prefer the more unique vacua (or their small families).



In the landscape category of this blog, you will find numerous articles discussing various additional instabilities that may appear on configuration spaces of too high dimensions.

See e.g. Resonance tunneling and landscape percolation, Landscape decay channels, Disorder on the landscape, Locally predictive landscape, among others. I have personally spent some time with new ways in which the compactification manifolds could decay – even to several disconnected, simpler pieces – and various new factors that could enhance the decay rates. Now, there's a new addition to the collection with some famous author names.




Brian Greene, David Kagan, Ali Masoumi, Erick J. Weinberg, and Xiao Xiao released the following hep-th preprint:

Tumbling through a landscape: Evidence of instabilities in high-dimensional moduli spaces

Incidentally, I think it is a mistake for Brian et al. not to cite several of the papers discussed in the aforementioned TRF blog entries.

At any rate, the main message of the new paper is clear: if you look at portions of the stringy or similar configuration space with too many scalars (those that result from "too complicated" compactification manifolds with too high Hodge numbers etc. which are typically necessary to obtain the anthropically huge sets of compactifications), the naive calculations of the rate of the lethal bubble nucleation has to be modified and one actually finds many more instabilities and more abrupt instabilities on that sub-landscape. This dramatically reduces the percentage of the viable vacua in these large classes.




So far, I haven't been able to "compress" the reason why this is claimed to occur into several words – their arguments are a mixture with some numerical calculations that make the main idea somewhat less penetrable to me; but I haven't spent enough time with the paper so far so this may change. At any rate, I do guess that the increased instabilities have something to do with some new freedom to construct instantons – or new (more complicated) instantons – that don't exist if you consider with one scalar or several scalars only.

Instead, let me offer you a few approximate formulae that summarize their main conclusions.

The bubble nucleation rate has the form\[

\Gamma = A e^{-B}

\] where the exponent \(B\), normally related to the instanton action, has a more important impact on the stability of the minimum in the landscape. They look at the percentage of the minima in the sub-landscape with \(N\) for which \(B\geq 1\) – well, one surely wants \(B\) to be much greater than one for the Universe to be stable enough and to be a candidate to match ours, so let's look at a more general separation of the minima to stable and unstable ones.

They find out that the fraction of the minima in the landscape for which the exponent \(B\) is greater than \(B'\) i.e. that are more stable than a certain bound goes like\[

\frac{\#({\rm stable})_N}{\#(\rm all)_N} \sim \exp[-0.001\cdot\lambda\cdot B'\cdot \exp(b N)]

\] This formula results from some partly uncontrollable set of assumptions and some numerical calculations – at least I can't prove that every piece is robust now. But they suggest that \(\lambda\) could be of order \(0.01\). If that's so, the double exponential decrease produces, for \(B\sim 1\), the fraction \(10^{-20}\) or so for \(N=18\) and \(10^{-1,300}\) for \(N=23\). For higher values, the double exponential drop of the percentage becomes insane.

At that moment, for \(N\geq 23\) or so, the ratio is pretty much zero and one sort of expects (well, if he assumes that all the vacua are generic and the distributions are quasi-uniform, something I have a big problem to believe, but let me omit this complaint in this blog entry) that you won't find a single stable enough minimum on the landscape. The number of vacua in those classes, even though it's often claimed to be a "very high" number such as \(10^{500}\), is simply not high enough to beat the "instability disease" that cripples most of the candidates.

I think that if these formulae were right and if one would still try to follow the KKLT-like multiverse thinking, but with the corrected maths, the mechanism could still prefer the vacua with \(N\sim 18\) or so, using the example above, and there could perhaps be still enough minima on such a sub-landscape to anthropically explain the tiny cosmological constant, assuming that the potential would be a function of a high enough order (or complexity) in these fields. So it seems to me that their conclusion that they debunk the anthropic explanation of the cosmological constant may be premature or incorrect.

Your humble correspondent would prefer a principle that favors vacua with the smallest values of \(N\), the number of scalar fields, such as \(N=1\), and some completely different, not-naively-statistical and anthropic, explanations of the small C.C., but my preferences don't matter, of course. It is perfectly plausible that Nature favors some intermediate values of \(N\). There could be \(10^{200}\) vacua for \(N=17\) and \(10^{-80}\) of them could be stable enough. That could still produce vacua with the minimum achievable positive cosmological constant around \(10^{-123}\) in the Planck units by some natural naive anthropic estimates.

Again, to repeat some points I have done many times in this category, I feel that too much intuition from quantum field theory has been blindly imported to string theory. However, string theory may modify many of the conclusions and even in quantum field theory with very many scalar fields, the events may simply proceed differently.

Just a trivial example. Consider the \(N\)-dimensional unit ball. What is its volume (in the units of the volume of a unit \(N\)-dimensional cube)? This is a fun exercise I numerically "derived" with the help of a Commodore 64 when I was 10 years old and the answer is\[

V(B^N) = \frac{1}{(N/2)!} \pi^{N/2}

\] where \(n!=n\cdot (n-1)!\), \(0!=1\), \((-1/2)!=\sqrt{\pi}\). Note that for very large \(N\), the factorial ultimately grows faster than any simple exponential (or power law). Indeed, Stirling's formula says\[

X! \sim \sqrt{2\pi X}\zav{\frac Xe}^X, \quad X\to \infty.

\] The most important factor over here is \(X^X\). This ultimately beats \(E^X\) for any fixed base \(E\) such as \(E=e\). Incidentally, you may calculate the volume of the ball analytically by computing the integral \(\int d^n x\,\exp(-|x|^2)\) in two different ways, by a decomposition of the integral into \(N\) one-dimensional factors yielding \(\sqrt{\pi}\) from the Cartesian coordinates or by a calculation in spherical coordinates whose angular part produces the volume of the sphere and the radial part generates a version of the Euler integral for the factorial.

Now, this volume of the \(N\)-ball or a similar \(N\)-sphere (or something similar) naturally appears at various points of the calculation. So you may get something like \(10^N\) vacua but the rates and other quantities may be modified by coefficients that include things of the sort \(N^N\) – and the latter inevitably wins if \(N\) is really large.

Many papers have been written and what really matters on the landscape of highly complicated compactification manifolds hasn't been settled yet. But one must surely be careful because much of our intuition has been trained in field theories with small values of \(N\sim{\mathcal O}(1)\) and they may break down when we switch to complicated landscapes.

Tomorrow, Keith Copsey will release a related but inequivalent paper that will argue that orientifold planes – pretty much inevitable in all semi-realistic F-theory vacua – suffer from a perturbative instability (deformations allowed when you lift the orientifold planes to M-theory) that may destroy the stringy landscape as we know it. I find this claim confusingly far-reaching because for large enough manifolds in Planck units, the instabilities from some non-local states or transitions must be hugely suppressed and essentially described by the effective low-energy field theory, am I wrong?

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There are 921,497 CICY four-folds

The constructive part of the "landscape" is finite and under control

James Gray, Alexander S. Haupt, and Andre Lukas posted a highly impressive yet amusing preprint on maths of string theory,

All Complete Intersection Calabi-Yau Four-Folds (arXiv)

---

Text with results, Mathematica results, C+Mathematica code (supplementary website)

They looked for all possible eight-dimensional Calabi-Yau manifolds – we call them Calabi-Yau four-folds because it's sensible to count the "complex dimension" which is just four – of a certain constructive type, namely the complete intersections in products of projective spaces (CICYs).




At least in principle, this subclass is easily accessible because almost everyone knows what a projective space is, everyone can construct their Cartesian products, and everyone can write a set of complex polynomial equations that define submanifolds of these products of projective spaces. In total, 660 of these products ("ambient manifolds") harbor some CICYs.

Nevertheless, the work is formidable and it was done for the first time. The result is that there exist 921,497 topologically distinct complete intersection Calabi-Yaus. Each of them is given by certain matrices. You may download all these matrices: the compressed files have a few megabytes but they uncompress to hundreds of megabytes.




While Calabi-Yau three-folds may have both positive and negative values of the Euler characteristic \(\chi\) – in fact, mirror symmetry universally pairs manifolds with opposite values \(\pm \chi\) – the CICY Calabi-Yau four-folds have a non-negative Euler characteristic. In the complete list of CICYs, \(\chi\) is in between \(0\) and \(2610\).

Almost all of these Calabi-Yau four-folds are elliptic fibrations. It means that these 8-real-dimensional manifolds may be imagined as 6-dimensional real manifolds ("the base") with a 2-dimensional torus attached at each point. Physically, it means that they may be used as the hidden dimensions of generalized, nonperturbative type IIB string theory compactifications envisioned by Cumrun Vafa – i.e. as the F-theory compactifications. Four spacetime dimensions remain large; 6 dimensions of the base are compactified; and the 2 dimensions of the torus ("elliptic fiber", a fiber that is an "elliptic curve") are used in the F-theoretical way. Their complex structure (shape) \(\tau\) determines the type IIB dilaton-RR_scalar complex field which may depend on the location in the 6-dimensional based and which may undergo nontrivial \(SL(2,\ZZ)\) monodromies if you make a round trip around singular fibers.

Whether these CICY topologies form a substantial part of all Calabi-Yau four-folds is unknown but it's plausible that the answer is Yes. Note that F-theory on four-folds is the scenario in which the huge landscapes with \(10^{500}\) vacua is often being discussed. the number 921,497 is so much smaller – it could be analyzed by the world's grad students, one student for each topology, if you were able to teach some maths to grad students in the humanities as well – because it only counts different topologies. The vacua carry some extra decoration for each topology, namely the generalized electromagnetic fluxes, and the number of the integers that determined these fluxes is a power of a googol for a complicated enough topology.

Much of the irrational disgust by string theory is caused by people's widespread mathphobia. People think that if the number of possibilities or solutions to certain conditions is large, the topic ceases to be a science and it can't be analyzed. But as this paper and others show, it may often be analyzed and the possibilities may be, in fact, fully listed and classified. In principle, one may also find the right vacuum that describes the Universe around us if it exists in a given set.

By the way, there's another math-oriented stringy paper constructing a non-linear realization of \(E_8\) and connecting it with the bosonic fields in M-theory, including a dual gravitational potential of a sort.

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We don't live in a simulation

In 2011, I already wrote a text about the theories that our universe is a computer simulation:

Rebooting the Cosmos

Needless to say, a regular reader has seen lots of other criticisms of discrete physics, a more general concept. Let me return to this topic – and change the focus somewhat – in the wake of a sensible text at Backreaction,

The simulation hypothesis and other things I don’t believe

As David Tong explained using different words in his Silver Prize Winning Essay written for a crackpot foundation whose basic mission does include the promotion of the "Simulation Hypothesis", there is strong scientific evidence today that the world isn't discrete (and it isn't simulated).




We do encounter integers and discrete mathematical structures in physics but in all the cases, we may see that they're derived or emergent. They're just limited discrete aspects of a more general and more fundamental underlying continuous structure, or they're a rewriting of a continuous structure into discrete variables (eigenstates in a discrete spectrum) which makes it impossible to understand the value of certain parameters.

Quite generally, if the Universe were fundamentally discontinuous, it couldn't have continuous symmetries such as the rotational symmetry, the Lorentz symmetry, and even descriptions in terms of gauge symmetries (which aren't real full-fledged symmetries but redundancies) would be impossible. In a fundamentally discrete world, many (or infinitely many) continuous parameters would have to be precisely fine-tuned for the product to "look" invariant under the continuous transformations.




I am not saying that it is "impossible" for all these parameters to be fine-tuned for the product to look e.g. Lorentz-invariant; I just say it is very unlikely. The scientific theory that requires this huge unexplained fine-tuning is less likely than scientific theories that work with no fine-tuning (or less fine-tuning). It's less likely simply because given any sensible, quasi-uniform a priori distribution of the parameters, it is insanely unlikely that they will have the "right" values if there are so many wrong values.

EPR-style experiments show that local realism is incompatible with the observations. Locality must be fundamentally respected because it may only follow from the Lorentz invariance, the Lorentz invariance is observed to hold as well, and making it accidentally hold in a fundamentally Lorentz-violating theory would require the nearly infinite amount of luck and fine-tuning, as explained in the previous paragraph. So the reality is local but non-realist. It is quantum, stupid.

It means that the proposal that our Universe is a simulation run on a classical computer is ruled out immediately. This is a very naive idea, indeed. Again, I am not saying that it is "impossible" to write a computer program that behaves in a way that resembles the reality. It may have a wave function treated as a collection of classical degrees of freedom; it can make the wave function "objectively collapse" at various points (although this guarantees that some disagreement with the observations is inevitable, as I have often emphasized). But it may look "more or less OK".



But when a simulation may "look OK", it's a different statement than the statement that the simulation may "be" the reality. These are totally different questions. The simulation may do a good job in emulating Nature around us. However, we may still prove by the scientific method that it must be a simulation, not the real deal!

I wrote a lot about the flawed idea of discrete physics. But in this blog entry, I want to write a few more words about a more ambitious aspect of this reasoning, the idea of an "Overlord". We are a simulation that a more intelligent "Overlord" is playing with. This is such a childish and mathless idea that we are really leaving proper physics. The first thing I would say – in agreement with Sabine Hossenfelder – is that this idea is just religion in "modernized clothes" and its details reveal unusually naive anthropomorphism that we usually associate with very primitive forms of religions.

But some people apparently think that if they replace the old Gentleman floating on the cloud by a modern (or slightly futuristic) computer with its data floating in the cloud, it's not only much cooler but also much more scientific. Well, it's not. The decoration and makeup may look modern, technological, and geeky but the central idea in this scenario is still the same primitive religion.

(Some IT companies may think that their cloud technology is the newest state-of-the-art invention but clouds have been a basic tool used by the Overlords for thousands of years.)

Brian Greene promoted these ideas in the last 2 chapters of his latest major popular book, The Hidden Reality. It's a fun reading that tries to suggest that it's connecting physics with the ideas of science-fiction movies and books. However, it is not. These ideas are pure science-fiction. They don't contain any "physical added value" whatsoever. They show a physicist trying to make an excursion to the world of science-fiction.

Don't get me wrong. I do like many of these books and movies. I think that some of them are clever in the ways how the rules of the game are sketched. I have watched some of these movies many times, thought about them, and I have perhaps tried to write my own science-fiction novels, too. Those activities may require some thinking. However, it is important to realize that this thinking is certainly not scientific thinking.

The Ultragirl who plays with the simulation is usually visualized in a totally anthropomorphic way. She is a girl so she just wants to have some fun. She may be bored. She doesn't want to be bored. She may have bought the device with the simulation in the mall. She may even be sexy and tease you. She may even look like a human babe.

Needless to say, all these ideas are preposterous. If there were a totally new fundamental layer beneath our Universe, and I have argued it can't exist, it would not only have no reason to include beings that are physically similar to humans with the same kind of lust. Even broader feelings such as fun and boredom would almost certainly be irrelevant. They're as silly as they are in the case of Moses and the Lord who apparently created the Solar System within a week some 6,000 years ago but who needed to relax on the sixth or seventh day (sorry, I am really uncertain at this point!) because all the workers want their weekends to be relaxing.

But the anthropocentric delusions don't stop with fun and boredom. Some people, including Brian Greene, are even promoting a scenario in which the simulation runs on a limited device that only simulates something or uses rougher, approximate algorithms to simulate things that aren't too important because the simulated beings aren't observing these aspects too accurately (or they don't observe them at all).

It may be a good idea for you to buy a cheaper computer if you don't need the most powerful one. You may have good reasons to save some electricity and run simpler programs if you don't need the state-of-the-art ones. You may pay smaller bills and you may save some money for other things. But to imagine that the underlying Overlords are trying to save the money in the same sense as we do is the ultimate anthropomorphic naivite. It's not just anthropomorphic: it's an idea that envisions a layer underlying the whole observable Universe to resemble the early 21st century culture of the U.S. urbanite teenagers.

Holy Christ and Saint Simulator, we must ask: Why? All these assumptions about the Overlords are completely irrational. There is absolutely no reason to assume that the beings underlying our existence would resemble the U.S. urbanite teenagers of 2007. In fact, if we have some experience and if we can imagine a reasonable "landscape of possible ideas", we must agree that the probability that the underlying reality respects these principles is virtually zero.

Brian Greene faithfully yet uncritically sketched many other arguments in favor of the "Simulation Hypothesis" that I find utterly idiotic. One of them is a sort of the anthropic propaganda. He says that in the asymptotic future (of our particular Universe or perhaps the whole multiverse), there will probably be so many computers with so much power that the number of simulated TRF readers will be vastly greater than the number of biological TRF reader. For this reason, Brian thinks, it's far more likely that you are a simulated being, not a physical or biological one.

There are several reasons why this argument is dumb. One of them is that there is no such a law that the probability that you belong to a subset S of the whole set W is equal to the ratio of the number of elements N(S)/N(W). In most cases, the ratio isn't even well-defined because both the numerator and denominator are infinite. But even when they're finite, there's absolutely no reason why the probability should be equal to the ratio and in almost all cases, it is not equal. Only if all elements of W are equally likely – and that pretty much occurs only when there is a reason why they are equally likely (thermalization balancing all the microstates, for example) – the probabilities may be represented by the ratio. But it is a negligible fraction of the situations. There isn't any symmetry or democracy between biological and simulated beings because they're qualitatively different objects so there can't be any justification of the "equal odds" assumptions.

Regardless of the number of simulated beings "somewhere", we may still present scientific evidence that there is no "simulated layer" in our reality. The continuous symmetries are such an example. If you want to deny this argument, you may deny it by saying that your meme that "the number of simulated beings is vastly greater than the number of genuine beings" is so powerful (because the number of simulated beings is so high) that it may render the smallness of the "we are simulated beings" prior probabilities irrelevant.

But such a method to render rational arguments and calculations irrelevant is just a sleight-of-hand. It is exactly equivalent to the statement that all scientists and atheists are deluded a*s*oles because God is infinite and by His infinite powers, He may neutralize any scientist, his arguments, and inflate an arbitrarily unlikely possibility into the most likely outcome. To argue in this way means to totally abandon rational discourse and replace it by a kindergarten boys' pissing contest trying to find out whose God or Father is more infinite. You're just showing that your faith is infinite and won't be affected by any finite evidence, however extensive. But this only proves your bigotry. It doesn't prove anything about Nature.

Moreover, one may conjecture equally convincingly that in the future world, the number of biological beings will be vastly dominant; yes, I am just saying that the other side could be active in the pissing contest, too. I don't want to spend too much time with these anthropic arguments because they're just totally unjustifiable. In fact, there aren't any infinite numbers in physics that could ever "strengthen an argument". All physically meaningful numbers are always finite (some renormalization etc. is sometimes needed to see it) and if you think about possible infinite numbers such as the volume of our not-only-observable Universe, they never affect probabilities of observable propositions. The latter are only affected by intensive or local, and therefore finite, quantities. Someone's infinite size as the "ultimate argument" belongs to religion, not to physics.

Another would-be scientific yet unscientific line of defense that the champions of the Simulated Universe Hypothesis often employ are the demands of loyalty towards our Overlord. Some of them are as dead serious about this line as the most hopeless religious fundamentalists. They say that the Overlord may be deliberately trying to mislead us and we must just stay humble and respect Her great power because we're such tiny insects.

Nice.

Well, let me say that I know that this Overlord doesn't exist but even if I had doubts about Her existence, as a scientist, I don't have and can't have any respect towards Her "authority" that would affect my reasoning. Such an influence of fear on reasoning is just not kosher for a scientist – and I would surely extend this moral principle to North Korean scientists and simulated scientists, too. It's just about the most important, defining features of science. So even if the Overlord exists, she may politely sc*ew Her own asshole, or whatever f*cking parts of the body She is expected to have. Give me a break with this stuff. Worshiping a being, especially one that doesn't exist according to all the rational reasoning, is just plain stupid.

When She is presented as omnipotent in this way, the champions of Her dominion over the world have no problem to say that She may fine-tune things to values I find unlikely because it's Her right. She has the right to confuse me, too. Because these folks live in a mental world that assumes that She exists as if it were the most reliable and universal axiom of science – in a complete analogy with religious bigots in Christianity, Islam, and perhaps other old-fashioned religions – they consider their uncritical faith to be a sign of their being scientific.

But it's exactly the opposite. In religion, one can make assumptions (such as the Simulator's omnipotence) and require that everyone else worships these assumptions and treats them as facts. All other facts must be distorted, adjusted, or completely overlooked so that they become compatible with the basic religious axioms and of course that if you're willing to say awkward things, it's possible to a large extent. However, this approach is unacceptable in science.

In science, we always question the assumptions. We're comparing competing theories whose assumptions differ. That's really another defining feature of science. So feel free to create a personality cult around the Female Simulator but a person who is thinking scientifically will always question it and will always try to estimate the probability (and probable truth values of various assumptions) using the most rational, indirectly empirically based, arguments that are available.

Let me phrase this point about "Her right to confuse" differently. Some champions of the Simulation Hypothesis (and religions) think that if they postulate that "She" can arrange things in such a way that scientific strategies to investigate the foundations of the Universe become misleading or impossible, they make their Simulation Hypothesis more likely because they "hurt" the other hypotheses. But the answer is the opposite one. They only hurt their own, Simulation Hypothesis because they are adding additional unnatural assumptions to it: they are making it less likely. The probabilities of the competing (scientific) explanations can't be lowered by axioms done within the Simulation Hypothesis because – with apologies to the Overlord – the competing scientific explanations lie beyond the Overlord's power. So if the Simulation Hypothesis contains additional axioms that "hurt the Overlord-free scientific explanations" and if someone believes that the scientific explanations really become less likely because of that, he is a victim of circular reasoning. He can only prove what he wants (scientific explanations are disfavored or impossible) if he assumes it from the beginning, anyway.

For the question of fine-tuning, a rational argument is that the fine-tuning of 40+ parameters of the Lorentz-breaking Standard Model to the Lorentz-preserving values is almost infinitely unlikely, so a fundamental theory that doesn't respect the Lorentz symmetry is almost fully excluded. That's true for the Sexy Simulator Hypothesis, Loop Quantum Gravity, or any other silly fairy-tale of this sort. It doesn't matter that you may imagine that such an unlikely scenario is true and you may collect a billion of gullible simpletons around the religion that it is actually true. Science is still saying what it is saying and it is saying that the probability that such an assumption about the Simulator holds is basically zero. It's the strongest way how science may disfavor a hypothesis. So science is against you in the most violent way that the cautious and peaceful structure of science admits.

To summarize, I think that if you admit that there is a difference between science and religion, the proponents of the Simulated Universe Hypothesis are squarely on the religious side and all the feelings that their theories are modern, geeky, technological, and advanced are just about the makeup, not about the essence of the hypothesis that is the same primitive religion that the mankind has played with at least for thousands of years.

And that's the memo.

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Nicolaus Copernicus: 540th birthday

Off-topic, Higgs: Fox News, BBC, and others are suddenly excited by the possibility suggested by the Higgs boson mass that our Universe is intrinsically unstable. See some 4-month or years old TRF blog entries.

Mikołaj Kopernik was born on February 19th, 1473 – half a millennium and 9.5 months before your humble correspondent – into a rich family in Toruń (thorn) in Royal Prussia, a part of the Crown of the Kingdom of Poland.

At that time, the nationality of the people was more associated with the territory and not with ethnic groups: modern European 19th century nation states weren't born yet (American readers will forgive me but they still haven't invented the concept as of early 21st century).

However, Nicolaus spoke Polish, German, and Latin very well and equally well; he also spoke Greek and Italian. At some school, he was pretty much led to register himself as a German which doesn't mean much. His father was a successful copper trader at the Wall Street, selling the commodity to Danzig. His mother died when Nicolaus was a small boy but she was a member of a very rich dynasty.




Some people claim that the name "Koperník" is actually related to copper that his father traded. I find it unlikely. Such an influence of the job on the name seems too fast to me. It's more likely that it's related to the dill plant – koperek or kopernik in Polish, kopr in Czech, a Slavic word – that was abundant in Prussia.

He studied not only the mathematical-astronomical disciplines but also some theological ones (canon law) and some kind of humanities, too. I don't want to go into the boring list of institutions he was affiliated with. More interestingly, he organized defense against the attacks by the Teutonic Order (in Czech, we call it The Order of Germanic Knights); he worked as the canon in Frauenberg; drew some maps; and wrote macroeconomic analyses mostly focusing on inflation such as the Treatise on the Coin (he was a clear monetary hawk).

Some people could imagine that Copernicus was a life-long heretic of a sort but that's completely wrong. Theology was earning him a very convenient life. For example, in 1503, your humble correspondent's Kingdom of Bohemia gave him sinecure at the Collegiate Church of the Holy Cross in Wrocław, Silesia, Bohemia (yes, our territory was a bit bigger than today). Sinecure means that he didn't have to do anything. It was handy.




In 1514, he began to write blog entries about the heliocentric theory. And he wrote a book about it etc. Again, it would be a mistake to think that the Church was disgusted. For example, Pope Clement VII – and two cardinals – were told what the heliocentric theory had taught us by Johann Widmanstetter, the Holy Father's secretary. The Pope was so happy about the new insights that he gave the secretary a valuable gift.

The heliocentric system wasn't really new; some Greeks such as Aristarchus of Samos were discussing these models already 300 years before Christ. However, Copernicus was at least a major figure who revived the heliocentric idea in the Middle Ages and incorporated the state-of-the-art knowledge of astronomy into it so that it could have been considered a viable alternative to the prevailing Ptolemy's model.



This diagram of the Solar System looks really, really simple. It's remarkable why people weren't attracted by the simple model much more strongly than they actually were. This preference for "simpler models" is something we take for granted but it apparently wasn't always obvious. Copernicus was among the folks who helped us to prioritize the ideas in this way. We should be more precise when we say what is simple. The ultimate application of the model and calculations of observed phenomena may be rather complex; what's important is that the primary starting ideas such as the diagram above is simple.

Copernicus hasn't really faced any problems with the fundamentalist Christian ideology during his lifetime. His main problem came in 1538 when Johannes Dantiscus, a new bishop who had still been friend of Copernicus' at that time, accused Copernicus of coitus with Anna Schilling, a housekeeper (an act incompatible with his celibate). In 1538, Copernicus relinquished the Breslau sinecure (one of the jobs where he had no duties) and this fact is probably related to the accusation. Nevertheless, Copernicus never married and never had children.

The real controversy about the Copernican system only began to be built after Copernicus' death. Various Dominicans and Luther's collaborators and others were gradually finding inconvenient truths in Copernicus' writing and they were building fan clubs of haters who would ultimately transform Copernicus' theory into a magnificent heresy. Note that Giordano Bruno was burned at the stake in 1600, 57 years after Copernicus' death. (Bruno's valid claims went beyond Copernicus; he also recognized that the Sun was a star and he even claimed that there are "infinitely many" inhabited planets orbiting various other stars in the Universe.) The trial against Galileo took place in 1633.

Dozens of blog entries at this blog mention Copernicus. I am going to single out the myths about epicycles where I try to explain that the Copernican model, while having a more modern and simpler "core", didn't away with the need for epicycles. Epicycles were a clever phenomenological theory to improve the precision of the orbits – which are not exactly circular in the real world – some kind of higher harmonic corrections to the leading harmonic orbits in a Fourier expansion. Most contemporary people who feel very clever by slinging mud on epicycles understand astronomy much less than the 15th and 16th century astronomers.



There's really nothing wrong with the geocentric frame; it is just another coordinate system in which the motion of the celestial bodies (yellow Sun, blue Earth, red Mars) may be described. In general relativity, we're allowed to use both frames but it's still true that the metric tensor (gravitational) field in the heliocentric system is "simpler", more stationary, closer to the flat Minkowski space, and that's why the heliocentric frame was much straighter a path towards the Newtonian theory, a non-relativistic limit of GR.

Also, I have to mention that Copernicus is often cited as the "role model" for the proponents of the anthropic principle and the multiverse. They say that the world is always greater than previously thought and this process has been underway since Copernicus, we hear. It's the Copernican Revolution according to them. Well, I think they are rewriting the history a bit. Copernicus hasn't really enlarged the Universe and the revolution was about the change of the celestial body at the center of the Universe. (As I mentioned above, Giordano Bruno surely did propose to extend the world. Maybe the anthropic folks should talk about the Brunian Revolution.)

But OK, the role of the Earth was downgraded – and in the same way, the anthropic folks want to downgrade the whole observable Universe. The problem is that if the former step were legitimate, it doesn't follow that the latter step is legitimate. These are two different questions and attempts to identify them in between the lines is an ideology, not a proper logical reasoning. There's another key difference between the two situations: the worlds outside the Earth may be observed (so their existence is really indisputable today) while the worlds outside the observable Universe cannot be observed (which is a big reason why their existence is questionable). ;-)

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A proof of Intelligent Design?

I hope that the title isn't too provocative. ;-)

Bill Z. has brought my attention to a December 2012 nuclear physics paper that was updated 3 days ago,

The fate of carbon-based life as a function of the light quark mass

They (Evgeny Epelbaum, Hermann Krebs, Timo A. Lähde, Dean Lee, Ulf-G. Meißner) try to determine the precision with which God or non-God had to fine-tune the average light quark mass – a parameter defined as \((m_{\rm up}+m_{\rm down})/2\) – in order to guarantee that there would be enough carbon, oxygen, and other elements that are crucial for the type of life that is recommended by 4 of 5 dentists.

The detailed calculations are concerned with the Hoyle state. What is it and what did the authors of the new paper conclude?




In 1954, Fred Hoyle noticed that we were lucky about a seemingly technical coincidence in nuclear physics that was apparently needed for us to exist. Note that in the baryonic (proton- and neutron-based i.e. visible) matter in the Universe around us, hydrogen and helium are the dominant elements.

It's no coincidence. It was mostly hydrogen (\(Z=1\)) and helium (\(Z=2\)) – and just some lithium (\(Z=3\)), aside from negligible trace amounts of heavier elements (\(Z\geq 4\)), that was directly produced during the Big Bang nucleosynthesis in the first three minutes after the Big Bang. One may reconstruct the temperature in the Universe during these early formative stages of our Cosmos and calculate, using the statistical methods, the ratios of the concentrations of these three light elements. The results seem to match the observations of hydrogen, helium, lithium rather impressively – and this agreement is one of the important pieces of evidence supporting the Big Bang paradigm.

Looking from a practical perspective, are the three lightest elements enough to get everything we need to be happy? Well, lithium may be helpful for some laptop and cell phone batteries and helium is useful at most for funny tricks to change your voice into the voice of the Smurfs (fine, it's also great as the gas in balloons, either for kids or adults, and as the coolant in NMR and the LHC). We could also describe helium as the main waste product of the thermonuclear reactions in the Sun and other stars if it weren't too disrespectful.

Hydrogen is useful for a huge fraction of compounds we need and love. But where is the rest? It's obvious that the three elements aren't enough to build life and the civilization as we know it. In particular, two other major heavier elements behind the miraculous project of life – carbon and oxygen – seem to be absent. Yes, these are the same two elements found in the gas that they call a pollution but we call it life.

Aside from hydrogen, carbon, and oxygen, the three major elements, nitrogen, phosphorus, and sulfur are three more elements that are paramount for life we know. Other elements such as silicon or calcium or fluorine (because I mentioned the dentists) may be helpful to shape our bodies at various moments and/or to create various intelligent gadgets but they're no longer a universal "must". Where do all these elements come from?



The heavier elements are abundantly enough produced by helium burning in stars that have gone red giants. Our carbon, oxygen, and other elements arose from the production in these red giant stars that once existed but they are no more. In 7.5 billion years or so, our beloved Sun will become a red giant, too. It will devour the Earth and other planets – that's the less dramatic part of the story – but it will also produce completely new carbon, oxygen, and other elements that may be incorporated into the bodies of a future extraterrestrial civilization.

Fine. How do the red giants produce carbon (\(Z=6\))?

Note that six is a multiple of two: we call these numbers "even". So it has the right number of protons to arise from several, namely three, nuclei of helium which seem abundant. Moreover, the ordinary carbon nuclei we need have 6 protons and 6 neutrons, the same number, so it seems appropriate to combine three helium-4 nuclei to create carbon-12:\[

\Large {}^4_2{\rm He} + {}^4_2{\rm He} + {}^4_2{\rm He} \rightarrow {}^{12}_6{\rm C}

\] It's somewhat unlikely for three helium-4 nuclei – which I will call the alpha-particles just like everyone else – to hit each other and produce the carbon nucleus directly. So the reaction actually proceeds in two steps, with an unstable level of beryllium-8 or \({}^8_4{\rm Be}\) in between. This beryllium-8 nucleus combines with another alpha-particle to get the desired carbon but this second step has too low a rate.

We wouldn't get enough carbon in this way (if it were just a generic fusion of these nuclei) and it's actually known that something special is going on. There is a resonance, a \(0^+\) state of carbon-12 known as the Hoyle state. Fred Hoyle actually predicted – using the apparent abundance of carbon as the only input – that it should be somewhere over there and indeed, the prediction was soon experimentally confirmed.

There exists a state of carbon-12 whose mass/energy is equal to the mass/energy of three free alpha-particles plus \(\varepsilon=397.47(18)\keV\); we say that the state is \(\varepsilon\) above the three-alpha threshold (a threshold, in general, is the minimum mass/energy of an unstable/composite object that allows the corresponding state to decay to particular products without violating the energy conservation law).

In the relevant region of the parameter space, the reaction rate for the carbon-12 production – via the Hoyle resonance – may be approximated by \[

r_{3\alpha} = \Gamma_\gamma (N_\alpha/k_B T)^3 \exp(-\varepsilon/k_B T).

\] You see that up to an overall normalization constant, this is equal to the third power of the number (density) of alpha-particles per unit volume (because three of them have to meet) and a Boltzmannian factor that exponentially decreases with energy. This \(\varepsilon\) shouldn't be too high because the exponential suppression could be severe. It shouldn't be too small, either, for other reasons. In the past, it was argued that a 15% deviation of \(\varepsilon\) from the known value could still allow enough carbon etc. for life.

Now, what have they found about the dependence of this accident on the light quark mass?

They use a novel numerical method, nuclear lattice simulations, to calculate the dependence. It's something like lattice QCD except that it seems to work with some composite pion fields and the low-energy emergent nuclear physics mess instead of the fundamental QCD fields. The light quark mass is translated to the mass of the pion \(M_\pi\) and they discuss the dependence of the energies of several energy levels on either the light quark mass or the pion mass which is almost the same dependence.

I don't want to bore you with all the details – you may read the original paper, it's just 4 pages long. Instead, let me repost a graph summarizing some partial results of their analysis:



On the \(x\)-axis, they depicted the relative change of the binding energy of the alpha-particle; on the \(y\)-axis, you see the corresponding (much larger) relative change of the three energies related to the helium-4, beryllium-8, and carbon-18 nuclei, namely of\[

\eq{
\Delta E_h &= E_{12}^* - E_8 - E_4\\
\Delta E_b &= E_8 - 2E_4\\
\varepsilon &= E_{12}^* - 3E_4 = \Delta E_h + \Delta E_b
}

\] Because of the final relationship for \(\varepsilon\), it's not surprising that the yellow curve is in between the other two. But what may be surprising is that these two curves – and therefore all three curves – have pretty much the same slope. What does it mean? It means that the several fine-tunings are actually not independent from each other.

You could think that for the nuclear factory to work and produce the elements, you may need several miracles – several anthropic conditions, in this case three – and therefore God has to be even greater than the size He would adopt if there were just one miracle. God's omnipotence seems like the third power of a generic god's power (or three times? It depends whether His omnipotence is quantified on the log scale).

However, the new paper shows that this ain't the case. The three coincidences aren't independent from each other. Pretty much because of mathematical identities, they're more or less equivalent to one coincidence only. If one identity for the nuclear level energies holds, the other two will probably hold as well, with a highly acceptable accuracy. It means that one miracle is enough and life is much more likely than what you would expect if you thought that the three conditions are independent of each other.

Now, you could claim that we still need meta-God to explain the mathematical "metamiracle" that the three conditions are actually almost equivalent to each other. If you did so, you would cover these questions by lots of exciting religious fog. But the matter of fact is that they can actually explain this "metamiracle" – at least in a preliminary way – in terms of completely non-mysterious, irreligious arguments, too. The same slopes kind of follow from the alpha-cluster structure of beryllium-8 and carbon-12 nuclei. I won't present this derivation in its full glory but the alpha-particle-based compositeness of the two nuclei sort of rationally explains why the two slopes are almost the same.

Even though several levels and energy differences are involved, the authors de facto show that there is only one "miracle" we need for a sufficient production of the heavier elements behind life. Moreover, the tolerated error for \(\alpha_{\rm elmg}\) as well as \(m_q\) could be around 2 percent or so: the fine-tuning isn't extreme.

Although this very topic may make you "wish" that there is some evidence for the Intelligent Design and/or a stunningly convincing role for the anthropic principle, and the very fact that a paper about this metaphysical and mysterious question was written could manipulate you into a more spiritual thinking, I would say that the actual results of their analysis imply exactly the opposite conclusion. Different "miracles" aren't really independent from each other and they're not "terribly unlikely miracles", anyway. Good luck at the 1-in-50 level seems to be enough for the amount of carbon to be just fine. At most, you may need two such 1-in-50 fine-tunings – one for the fine-structure constant and one for the light quark mass – except that I think that only some combination of them will have a high enough impact on the essential processes needed for the elements of life to arise.

Now, you could still argue that 1-in-50 is a low chance. The probability \(p=0.02\) or so is pretty small, some of you could say, and this strengthens some arguments in favor of God, Intelligent Design, the anthropic principle, or something along these lines. Well, perhaps. I don't think it's a right way to think about this coincidence. Why?

First, \(p=0.02\) is equivalent to a "bump just a little bit larger than a 2-sigma bump". To make extraordinary claims about God or the anthropic principle – and one really doesn't know which of these (or other metaphysical) explanations "follows" from the "miracle" – and justify them by not-so-extraordinary evidence such as 2-sigma bumps seems to betray the lack of evidence. Extraordinary claims require extraordinary evidence and this ain't one.

Second, this not-so-extreme probability \(p=0.02\) is the \(p\)-value before the look-elsewhere effect of a particular type is included. What I want to say is that we're computing the probability that a particular system of nuclear furnaces will be able to produce a particular type of life (determined by its elements etc.). However, there could very well be other types of life – perhaps \({\mathcal O}(50)\) types of life – that may arise in the same parameter space which means that the probability that at least one of these types of life will be allowed for a "random" choice of the values of parameters may approach 100 percent.

These observations of coincidences that are needed for life are intriguing but we shouldn't get carried away for two basic reasons. First, as argued above, the probabilities that we get a tolerable value of the parameters (values compatible with life) aren't extremely tiny and we should treat these "modestly suggestive" low probability just like any other 2-sigma bumps in physics. They're not enough to settle a question, they're not enough for a paradigm shift.

Second, it's pretty much guaranteed that if we calculate the odds that some "conditions constraining parameters that makes the theory friendly to life as we know it" are obeyed, it's unsurprising that the answer will probably be Yes because our type of life does exist, after all. The proposition that "conditions apparently needed for this life to arise with a significant probability were satisfied" is pretty much tautologicaly true. These conditions simply aren't independent of some empirical known facts. We're just measuring the answer to the question "Does life exist?" using a different, perhaps more contrived, procedure. But the fact that many such questions have "Yes" answers isn't a miracle; it's pretty much tautologically guaranteed because these questions were cherry-picked by their equivalence to the existence of life (or some of the aspects of this existence).

If the probabilities arising in similar anthropic coincidences were much tinier, i.e. much more extreme than 2-sigma bumps, I could be impressed. The tiny cosmological constant could be an indication of this sort. However, we may only argue that the "probability that the cosmological constant is below \(10^{-120}m_{\rm Planck}^4\) is of order \(10^{-120}\)" if we adopt a uniform probability distribution for cosmological constants in the interval comparable to \((0,m_{\rm Planck}^4)\).

While some plausible models that make the uniform distribution look natural exist, they're not "inevitably true" and it's still easy to imagine that this uniform probability distribution is a completely naive, wrong expectation. If we replace it by another one – one that follows from a slightly sophisticated mechanism and one that gives tiny values of the cosmological constant with far higher probabilities – the "unavoidably impressive" miracle goes away once again. If you wanted to convince me that there is a miracle that harbors strong evidence in favor of the anthropic reasoning or God or anything like that, you would have to show me a coincidence that has a tiny probability according to the right calculation of probabilities (a calculation which takes all mathematically guaranteed correlations such as those above into account) and a nicely justifiable probability distribution for the parameters.

If you used a quasi-uniform one, you would have to convince me that it's reasonable to expect that the distribution is quasi-uniform for that situation. It would have to be so reasonable – almost inevitable – that, in fact, I would find your anthropic principle or God more likely as an explanation than the mundane possibility that there simply exists a better argument or "better theory" telling you that a non-uniform distribution is actually a much more sensible (and likely) one. Or a better theory that simply allows you to calculate the observed value. How strong evidence is needed to prefer God over the "better theory" depends on subjective prior probabilities but be sure that 2-sigma or even 3-sigma bumps are way too small for people like me to pick God or His best pal, the anthropic anti-God, instead of a "better, so far unknown, theory".

If you can't show me such a thing, I would keep on insisting that there doesn't exist any tangible evidence to believe the anthropic/religious paradigm and because these things aren't mathematically elegant or explaining any true pre-existing mysteries in physics, they don't really deserve to become a part of physics, at least at this moment.

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