Quantum mechanics is natural, not plagued by problems; maths relevant for physics worships the rules that are natural for continuous, not discrete, structures
I received my copy of Scott Aaronson's book – thanks to him – and I sometimes find some time to read a chapter or two. So let me post several comments on my impressions – and generalized comments about thinkers like him.
First, the book is very witty, narcisistically witty. Its language is very informal. It's totally OK with me because at least one-half of my parental background is extremely detached from any world resembling the Academia (and even rural if one returns by one more generation). However, the average-man-on-the-street language sometimes sounds really bizarre and astroturfal if you realize that the author is a archetypical model of a left-wing Academic carefully parroting all these people's collective delusions – including "gems" such as the very global warming hysteria – and confined to a snobby ivory tower. ;-)
The preface tells you a funny story how Scott Aaronson didn't sue the authors of a commercial who quoted a few sentences from his talk or whatever it was. But it already presents some "basic scientific philosophy". The only problem is that most of the basic theses are wrong and even when it comes to the correct ones, Scott talks the talk but doesn't walk the walk.
Now, after I have finished reading Chapters 1-3 as well, the preface looks much less agreeable to me than it did before, probably because I see what was actually the "grander goal" of various sentences and I can evaluate in a more "moral" way than just by their literal validity.
Right after I finished the preface, I thought that the comments about the author's "suspicion about everything that is continuous" were the only major problem. But right now, it seems that most of the basic philosophic tenets are deeply flawed.
For example, the counter-intuitive nature of quantum mechanics is already discussed in the preface. And if I realize what he actually wrote, it seems so fundamentally flawed to me! First, Scott wants to make as "modest" claims about the interpretations as possible. So he announces he won't pick any preferred one but he will agree with anyone who says that there is a problem with quantum mechanics and disagree with every interpretation that presents quantum mechanics as a complete and consistent framework to do physics.
In most respects, I would also prefer to be independent of "tastes" and ill-defined words and philosophies that don't produce in principle observable differences from others but the main message that goes beyond these things would be exactly the opposite one's than Aaronson's. There isn't any problem with quantum mechanics. If you want to find a problem with a theory within the scientific method, you must actually show what it is – its incorrect prediction of an observation or an internal inconsistency. These people's vague feeling that "something must be wrong with a theory" even though they know very well that there doesn't exist any sensible observation that would actually back this criticism only shows that these people's thinking is unscientific. They prefer emotions, feelings, animal instincts, dogmas, and stupidity over cold reasoning and empirical evidence.
The problem gets even more manifest when the author says what we should do when we find out that our intuition was wrong. We should fix our intuition to be sure that next time, we will have more sensible expectations and what Nature is telling us won't look crazy or extremely unlikely to us again. I originally thought that this was what he was writing and I would agree with that.
However, now I think that this "right reaction" was just the beginning of his explanation that ended up with a totally different conclusion. He thinks that he was told that his physical intuition was wrong while no one gave him a path to correct his intuition. It's like flunking him on an exam without providing any hint how he could have done better, Aaronson wrote. He thinks that it's sensible for him to switch to other courses where he can earn As.
But whether someone told him how to correct his intuition, the most important thing is still that he has failed the exam! His intuition didn't agree with Nature's actual behavior so he and his musings had to be flunked. The purpose of science isn't to provide help to children who were left behind. The purpose of science is to find the truth by eliminating the ideas that are wrong. In Aaronson's analogy, flunking the bad students isn't a negative side effect of science; it is its very purpose! And I think that much like the Islamic terrorists, he is personifying the wrong ideas about quantum mechanics – they become a student in his metaphor – in order to convert such a student to a human shield whose destruction should be considered inhuman. But it's still totally necessary to destroy what is behind the student if we want to study these things scientifically!
Now, it's sensible for a flunked student to try something else (falsified ideas in science usually get no additional opportunities: that's a difference showing why the personification is misleading). But it's still important that if he hasn't found a way to correct his knowledge or intuition, he has not mastered the subject. In these circumstances, it's kind of painful for him to write a book about the subject he has not mastered! Claiming that the exam was hard, the examiners were cruel, and they didn't provide the poor whining student with any hope how to get better shouldn't be used as an excuse to write rubbish about things he doesn't understand.
Incidentally, this particular opposition to science – "science must be corrected so that no poor child is left behind" – is a predominantly left-wing assaults against the dignity of science (I count spoiled whining frats who flunked an exam among the leftists). The association of the phrase with George W. Bush is an exception that confirms the rule.
May we help the students to understand how to fix their intuition (about quantum mechanics and, more generally, about other things)? I have tried to do so for years and the results are mixed. But these are mere pedagogical efforts, not a part of the scientific research. Even if almost no one can be explained how to correct his intuition, it means absolutely nothing for the validity of quantum mechanics. The validity of a scientific theory doesn't boil down to the science's being accessible to most people. Indeed, most of the modern science – and not just quantum mechanics – is pretty much inaccessible to most people in practice. But that doesn't mean that there's anything wrong about the science! To show that there is something wrong with a scientific theory, one must find valid evidence of an internal contradiction or a contradiction with the experiments according to the proper professional rules of science, not just to whine that these things are hard to understand!
At the end, the scientific claims on quantum mechanics in the book (so far) are bad but they aren't that bad. He wants the information to become the central player. It's just fine with me. It's an OK sketch of the philosophy predestined as a theme for a book. However, if such a meme is promoted to a key dogma that decides about the validity of all other detailed scientific questions, you shouldn't be shocked that one ends up with wrong answers to most questions. Science can't be guaranteed to obey predetermined dogmas. No one can guarantee that if you try to write a book where a whole scientific discipline is presented as a consequence of a short slogan, it will be a valid book. Indeed, most likely, it will be garbage.
Chapter 1 focuses on Democritus, atomists, and some very general philosophical questions about experience vs abstract scientific theories. But they aren't really brought to the level of the state-of-the-art science. So in some sense, it remains a superficial chapter about the ancient Greek philosophy that is disconnected with the rest.
Chapter 2 talks about sets, their elements, axioms, quantifiers, axioms of Peano Arithmetics, cardinals, ordinals, Axiom of Choice, continuum hypothesis, models for axioms, and general observations about the consistency of axiomatic systems. It's nicely written but given the fact that the book is supposed to be primarily a book about a discipline of physics, I would say that this chapter is pretty much disconnected from the primary topic of the book. The questions related to set theory aren't "remolded" to become useful for or valid from the viewpoint of physics. The most likely reason is that the author just hasn't understood what the relationship is. So he's only eclectically combining things he has heard from different sources.
What's the relationship?
Axioms and set theory may be used as a starting point to define the mathematical frameworks used in physics. But axiomatic systems and physics aren't the same thing. The most important difference between mathematics and natural sciences is that in mathematics, one may study pretty much any sets of axioms he finds interesting; in natural sciences, we want to learn the true axioms that are respected by Nature, at least within an approximation.
I am afraid that Scott Aaronson – and pretty much everyone who is trying to reshape physics as a discipline of computer science – just fails to understand this basic fact about the scientific method. He apparently thinks that if he writes down some facts about the integers or discrete sets or Peano algorithms or algorithms, they – because they look "elementary" or "fundamental" from some viewpoint (as starting points, they contain no reducible beef in them) – must be fundamental things behind physics, too.
But that's now how natural sciences work. Peano Axioms or algorithms or Turing machines may be the elementary concepts in various branches of maths and computer science but they're not guaranteed to be – and they actually aren't – the key ideas upon which the structure of physics is built. In science, we must be open-minded what are the right axioms – even what is the "general spirit" of the axioms, the philosophy of the right theories, and the character of the mathematical structures that are used to describe the world. Those things simply aren't up to us.
Scott Aaronson and pretty much all the computer scientists trying to conquer physics, "discrete physicists", and related sets of people just can't accept this basic point.
They pompously think that they're in charge, they may order Nature to follow their dogmas, and it's Nature's duty to find a way to follow them (and to pay income taxes, carbon taxes, and 50 other types of taxes as well). But Nature won't accept this role. Nature is a harsh examiner who evaluates Scott Aaronson's predictions and – more generally – intuition. She finds out that he fails in Test 1 and Test 2, among others (he thinks that Nature should be discrete etc.), so she flunks him. When he starts to whine that Nature is obliged to help him to get a chance to get an A, She just tells him to sh*t up, f*ck off, and, if needed, hang himself. He has no f*cking business to tell Nature how She should behave or even to promote the culture of entitlement or other left-wing diseases and corruption.
Chapter 3 reformulates Gödel's Completeness and Incompleteness Theorems in terms of (hypothetical) algorithms that either stop or not or decide whether another algorithm stops or not. By a trivial "I am a liar" argument, such an algorithm can't exist which proves the Incompleteness Theorem. I totally agree that the computer scientist's way of thinking about such things simplifies much of Gödel's thinking tremendously. Turing gets some credit here. And Aaronson talks about the consistency of axioms supplemented with additional assumptions about their (in)consistency. It's all nice. But again, these things haven't really passed the tests of "validity from the viewpoint of physics".
A physics-oriented question appears at the end of Chapter 3: Are we really talking about continuous objects themselves or about finite sequences of symbols that talk about continuum?
That's a good question and I am inclined to say the latter. If we talk about specific things, these specific things are always countable because they must be describable by a finite sequence of symbols. Even when we talk about intervals of real numbers that arguably contain an uncountably infinite number of real numbers, we must still specify the endpoints of the interval by a finite sequence of words or symbols – and such sequences are "discrete" i.e. "countable".
This is why I tend to consider the very fact that real numbers are uncountable to be nothing else than a linguistic curiosity: the actual, well-defined real numbers you may ever encounter form a countable set! This is why the uncountability of the real numbers – and the whole discipline of maths based on this formally provable claim and similar claims – doesn't have implications for "talking about physics".
On the other hand, the fact – let me now fully assume that I do take this position, and I mostly do – that we only talk about finite sequences of symbols does not mean at all that discrete maths and its axioms should be the foundation of physics. Even when I say that we are talking about finite sequences from a discrete set, it's still important what we're saying about them. And to do physics, we should organize these sequences and "discrete objects" in such a way that they talk about properties of continuous objects because those are ultimately fundamental in physics, as we know because of many general reasons.
So talking uses finite sequences from a countable set but in physics, it still matters what we are saying, and if we're not saying things about intrinsically continuous structures and properties that continuous structures may have, it will be either invalid or non-fundamental as chatter about physics!
I hope that the dear TRF reader crisply understands the difference between the statements "all language and communication uses discrete symbols" (true) and "Nature is fundamentally based on axioms describing discrete mathematical structures" (false). But I do feel that Scott Aaronson must be overlooking this not-so-subtle difference. It reminds me of Amy Farrah Fowler's silly comments to Sheldon Cooper when she suggested that neuroscience was more fundamental than theoretical physics because she may understand the functioning of a brain that is discovering an important theory in physics. The problem is that while she may, she doesn't. To find a right theory is a task that can't be reduced to the understanding of the functioning of the brain; after all, she hasn't completed the task. Similarly, theoretical physics can't be reduced to grammar, linguistics, and the history of languages because it still matters what the physicist is saying and a linguist has no idea about that – that's why linguists don't have any credentials to claim to master theoretical physics, either. I could add a few similar examples with additional occupations. Secretaries and bakers also provide something important for the physicists but they are not physicists themselves. My broader point is that it is comparably silly to start with Peano Arithmetics as the foundation of physics as it would be to start with some basic rules of grammar or linguistics. They may be the natural starting points in a discipline but the discipline just isn't physics. What is true and fundamental in physics is decided according to the rules of physics, not according to the rules of bakeries, bureaucracy, linguistics, or set theory.
At the end, pretty much all the silly "conceptual" things that people (not only Scott Aaronson) are saying about the foundations of science boil down to the same problem: these people just refuse the scientific method as a method to verify the basic tenets of their belief systems. From this viewpoint, some people's belief that alternative medicine or homeopathy must be much more potent than just the narrow-minded modern Western medicine is based on the same "intellectual mistake" as Scott Aaronson's belief that continuous objects or quantum mechanical postulates are "suspect". It's a belief. They aren't willing to put this belief at risk and test it. They aren't willing to find out whether it is right. They aren't willing to learn that their belief systems are actually based on tons of stinky feces. They prefer to believe rubbish and search for allies and excuses explaining why flunked spoiled frats should still write lots of things about things they haven't mastered.
I may continue with Chapter 4-??? sometime in the future.