Friday, July 26, 2019

Beauty of physical laws doesn't have the purpose of pleasing humans

Under the previous "baryon number R-symmetry" blog post, Santa Claus wrote several sentences that have increased my blood pressure. The most effective sentence was the following one:
But beauty in the dark sector would be wasted beauty.
Wow, you are just joking, Santa, aren't you?

Paul Dirac wrote this important observation on a blackboard in Moscow in the early 1950s. The sentence is still there as of 2019. Vladimir continues in the nice tradition introduced by his predecessor Joseph – to send every janitor to the Gulag if he indicates a tendency to erase the principle above.

Clearly, Santa Claus implicitly assumes that it's important whether some beauty involved in the laws of physics impresses many humans with some aesthetic sense. Humans don't usually look at the dark matter – it's hard because dark matter is dark – which is why throwing the beauty to the dark sector is similar to throwing it into the trash bin, he thinks! It almost seems as if he believed that Mother Nature would deserve to be criticized for such a wasteful behavior. ;-)

In the college, I used to have lots of intellectual polemics with the Christians. It just seems to me that I haven't met an intellectually self-confident Christian for some 20 years so virtually all my opponents have been some sort of leftists, usually of the cheapest possible anti-Trump (and now perhaps even anti-Boris) neo-Marxist type. They're increasingly uniform, increasingly stupid, and increasingly boring – and I no longer enjoy the interactions with these cloned NPCs.

But in the college, it was normal for would-be smart Christians to defend Creationism in the student hostels – and to mock Darwin's theory of evolution. While the atmosphere was basically friendly in the big picture, the arguments looked like this:
Look at a bird sitting on a spruce. Do you really believe that it may suddenly become a pig? It's so ludicrous, Darwin is ludicrous, QED!
Well, this said something about the quality of their argumentation rather than the solidity of Darwin's theory. Two basic facts about Darwin's theory that should be comprehensible to 5-year-old kids are that it takes millions or billions of years so you can't see it ongoing while looking at a tree; and the point isn't the change of an organism during its one life but the variation from one organism to its offspring. OK, the would-be argument above was clearly fighting a straw man.

These days, we are surrounded by people who dislike theoretical physics to a similar extent to which the fundamentalist Christians disliked Darwin's theory. Many of them offer remarkable conspiracy theories to the public and to each other – e.g. the theory that physicists shouldn't pay attention to arguments that look like aesthetic ones – and one may run into real trouble when he criticizes such breathtakingly scientifically illiterate inkspillers and hecklers.

And now, Santa Claus apparently believes that the beauty of the physical laws is only relevant if it shows up, if some humans are clearly seeing its effects, if and when it pleases some humans. I can't quite figure out why Santa Claus wrote such an amazing stupidity. Was it a part of his propaganda efforts to mock or delegitimize aesthetic considerations in theoretical physics? Or does he really believe in the anthropocentric belief that the pleasure of the humans matters in Nature and determines which theories are more likely? Can a generally intelligent person believe such a thing?

No, the statement really is completely ludicrous. The physical laws should have mathematical beauty, as Dirac pointed out, and it's obviously true for all physical laws, not just some physical laws that create objects that many people see so that they could enjoy the aesthetic pleasure. Well, humans' aesthetic pleasure is correlated with the actual beauty of the physical laws. But correlation isn't causation and even if there's a causation, you must be careful about the direction of the causal relationship.

Physicists with a good intuition feel some pleasure associated with beauty when they are exposed to beautiful laws of physics. But it's because their brains have evolved – or been trained – to feel happy when they see the beauty of more conventional types and the special kind of beauty that some physical laws have may have a similar impact on these humans' psychology. But it is not true that the pleasure of these humans or other humans is the purpose of the beauty of physics. Physical laws really don't have any purpose, after all. Physicists may just exploit their intuitive or subconscious thinking and "feelings" when judging certain aspects of the proposed laws of physics because these methods have good reasons to be good guides.

But the pleasure isn't the cause of the beauty of the actual or rationally preferred laws of physics. It clearly cannot be the cause – instead, human feelings and humans themselves are just some of the distant implications of the laws of physics that had been chosen for billions of years or "outside the spacetime". Instead, the "beauty" is just a poetic description of some rather mathematical traits that may be more accurately described by some calm, rational language.

OK, so why are the more beautiful proposed theories of physics more likely to be true? We could discuss various kinds of beauty but here I will pick the symmetry, the most famous well-defined type of beauty of the physical laws. The argument below will also apply to (string) dualities; I won't discuss it again but the same argument indicates that a theory with many dual descriptions is preferred. Consider any system in Nature that you want to describe by a theory.

OK, it's a four leaf clover. It has a \(D_4\) symmetry which is a semidirect product of \(\ZZ_4\) and a \(\ZZ_2\). You may rotate it by multiples of 90 degrees and it stays the same; and it stays the same in the mirror. The group has 8 elements. More powerful cases have continuous symmetries which really have an infinite number of elements.

Fine. A good theory of a four leaf clover explains this \(D_4\) symmetry. It should better do so because we may observe the four leaf clover and see that the actual symmetry is present. Well, it's slightly violated but the degree of violation is small. In the corresponding situations with the true laws of physics, experiments could only impose upper bounds on the values of the parameters that break the symmetry.

But even without seeing that the symmetry is preferred, we have a very good reason to prefer a symmetric theory over an asymmetric one. Why is it so? Imagine that you try to describe the shape – or birth – of the four leaf clover by a class of theories \(T(\lambda_i)\) that depend on some parameters. Let's assume that you're just fitting the shape. The theories for generic values of the parameters \(\lambda_i\) will disrespect the \(D_4\) symmetry.

Well, we need to describe roughly four leaves and the given theory \(T(\lambda_i)\) must also make a choice which of the leaves from the observations should correspond to the "first leaf of the theory itself" – because those aren't equivalent, due to the general lack of symmetries. It means that when we're comparing the theory \(T(\lambda_i)\) with the actual observed plant, we also need to identify the "leaves and directions from the observations" with "those in the theory". In effect, we are really comparing the observations with\[

T(\lambda_i; \gamma,\epsilon)

\] where \(\epsilon=\pm 1\) tells you whether you should take a mirror image before you compare the theory with the experiments and \(\gamma\) is a multiple of 90 degrees, one of the four choices, that tells you how you should rotate the leaf before you compare it to the theory. The funny thing is that if the symmetry doesn't hold, only one of the eight choices \((\gamma,\epsilon)\) will work correctly. It means that you must really divide the pie of the prior probability of the theory \(T(\lambda_i)\) into eight equal slices. This reduces the probability of each.

On the other hand, there may be points in the \(\lambda_i\) parameter space for which the \(D_4\) symmetry is restored, so the theory \(T(\lambda_i; \gamma,\epsilon)\) becomes independent of \(\epsilon\) and \(\gamma\). More precisely, for the good special values of \(\lambda_i\), the theories with different choices of \(\epsilon\) and \(\gamma\) are equivalent to each other.

If you don't care about the symmetry, and you don't care about it a priori, it's a coincidence. But if the prior probability distribution for the different theories is uniform on the \(\lambda_i\)-space, in the vicinity of the symmetry-restoring point (or locus), it's simply a fact that "all the eight theories including the parameters \(\epsilon\) and \(\gamma\)" will be valid simultaneously if they're valid at all.

Think about it but it really means that the theory with the restored symmetry is 8 times more likely than another theory at a seemingly similarly large region of the parameters \(\lambda_i\). Equivalently, if you make an observation that indicates that the symmetry indeed holds, the theories where it really fundamentally holds become 8 times more likely than the (previously equally likely) theories where the symmetry doesn't fundamentally hold – simply because the symmetry-breaking theories see their probabilities suppressed by another factor of \(1/8\) for the low probability that you also pick \(\epsilon\) and \(\gamma\) correctly.

When you have infinite symmetries, e.g. the continuous Lie groups, the preference for the symmetric theories naively becomes infinite. However, in practice, the preference is just finite because the experiments will never make you quite certain that the symmetry is preserved precisely; and – e.g. if you deal with non-compact groups – some elements of the group that are very far from the identity (large boosts etc.) should be dismissed or suppressed a priori.

At any rate, it is completely rational to pick a special theory – a theory with a perfect symmetry – if it is possible. The symmetry and the beauty may be labeled aesthetic or even emotional criteria because that's how ordinary people often look at the symmetry and the beauty. But it doesn't mean that this is why the physicists are actually preferring symmetric and beautiful theories. They prefer them because Bayesian inference generalizing the arguments above implies that such theories – and special points in the parameter space of the theories – simply are more likely to be true. Even if a physicist wouldn't be capable of a Bayesian argumentation like mine above, he has learned from experience that theories with certain "aesthetic qualities" simply tend to be more promising.

More generally, if two theories explain the data and one of them doesn't need to be fine-tuned – doesn't need any other or extra parameters such as \(\epsilon\) and \(\gamma\) but not even the counterparts of some or all \(\lambda_i\) – the more predictive theory is preferred if both of them are compatible with the empirical data. Again, this may be said to be an aesthetic criterion but the reason why it works is absolutely rational and doesn't depend on any human emotions let alone the human pleasure.

If you invent a possible stupid reason why another person may do or believe XY, it doesn't mean that it's the actual reason why he does XY! And it doesn't mean that there's something wrong for the person to do or believe XY. To assume that it has to be the reason (and XY's behavior or beliefs are therefore bad) is a clear fallacy – or a demagogic technique. It's incredible how many people are easily fooled by such fallacies or demagogy.

In string theory, to actually compare the theory with experiments, we need to specify the value of a parameter \(V\) which determines which compactification or which vacuum should be picked from the landscape. You may view \(V\) to be a counterpart of \(\epsilon\) or \(\gamma\); or a counterpart of a \(\lambda_i\). From a practical perspective, you could choose the latter but from a theoretical perspective, the former is more accurate because different values of \(V\) (different string vacua) aren't really different theories. There is just one string/M-theory and the choice of the vacuum is analogous to the right rotation of the four leaf clover.

If you had a fundamental theory that would explain all the things as string theory but didn't have the parameter \(V\), it would be preferred over the string explanation. But you don't really have it and it seems likely that such a theory – a consistent non-stringy theory of quantum gravity – cannot exist for mathematical reasons. The available evidence indicates that the fundamental theory simply has very many vacuum-like solutions and we're surrounded by one of them. You may only challenge this conclusion if you construct your own theory – which you probably wish to have a unique vacuum or a small number of vacuum-like solutions – which also allows us to calculate the graviton scattering amplitudes at all energies.

Just wishing that such a competitor of string theory "should" exist isn't good enough, however. And if you prefer a non-existent theory with a small number of solutions – over string theory, a damn explicit and real theory – just because you want the number of vacuum-like solutions to be low, you only show your prejudice. The number of values of \(V\) (the number of vacuum-like solutions) of the right physical theory isn't something you may choose. Mother Nature has chosen it with her half-sister, Auntie Mathematics. If you don't like the number, choose a different multiverse where the laws of mathematics are more pleasing for you and become a refugee.

The large multiplicity of the stringy vacua is analogous, in our four leaf clover analogy, to the observation of plants with many leaves that don't have a symmetry. The underlying string/M-theory still has an immense and pretty much perfect beauty, however. The beauty is only violated – along with the corresponding suppression of the probabilities according to the Bayesian argumentation – once you actually do a comparison with the experiments. But up to the moment when you actually do such a meaningful comparison to the experiments – which would favor some value or values of \(V\), a discrete parameter from a large or countable set – the suppression of the probabilities by the number of the vacua doesn't occur.

As I have sketched, these things must be discussed calmly, rationally, and as a form of Bayesian inference. But the essence of these arguments is similar to – or captured by – the people's aesthetic thinking or perception. But just because these criteria "sound like some arts or emotions" doesn't make them invalid! When someone spins something as an irrational thinking, it doesn't mean that it's really irrational. Physicists themselves have deliberately compared themselves to artists. Well, we may say that it has backfired because these efforts to "look cultural" are now being demagogically interpreted by some evil, dishonest individuals as a failure of rationality of theoretical physics – even though there is none.

It seems to me that Santa Claus is confused about every single elementary issue above. He has probably never thought about the relationship between the truth and the symmetry or the beauty rationally. So he preferred to mindlessly buy the low-brow liars' suggestions that there's something wrong about the aesthetic arguments in physics in general – that they're just some irrational criteria based on nothing else than feelings. And that's why Santa Claus actually believes that it matters whether the beauty is hidden in a dark sector because it won't please the humans over there. It's both stupid and sad.

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