Aaronson goes on:

Look, obviously the physicists had their reasons for teaching quantum mechanics that way, and it works great for a certain kind of student. But the “historical” approach also has disadvantages, which in the quantum information age are becoming increasingly apparent. For example, I’ve had experts in quantum field theory – people who’ve spent years calculating path integrals of mind-boggling complexity – ask me to explain the Bell inequality to them, or other simple conceptual things like Grover’s algorithm. I felt as if Andrew Wiles had asked me to explain the Pythagorean Theorem.

And then, did anyone clap?

This is a false analogy. I don't think it's a surprise, I am not convinced that it's an actual problem, and if it is, I don't think Aaronson makes any progress to a solution.

The Pythagorean theorem is part of the common heritage of all mathematics education. Moreover, it's the direct ancestor to the problem that Wiles famously solved. It's going to be within his wheelhouse. But a quantum field theorist who's been deep into that corner of physics might well not have had to think about Bell inequalities since they were in school. It's like asking an expert on the voyages of Zheng He about how Charlemagne became Holy Roman Emperor. There are multiple aspects of Bell inequalities that someone from a different specialization could want "explained", even if they remember the gist. First, there are plenty of questions about how to get a clean Bell test in the laboratory. How does one handle noise, how do we avoid subtly flawed statistics, what are these "loopholes" that experimentalists keep trying to close by doing better and better tests, etc. Aaronson has nothing to say about this, because he's not an experiment guy. And again, that's entirely fair; some of us are best as theorists. Second, there are more conceptual (dare I say "philosophical"?) questions about what exactly are the assumptions that go into deriving Bell-type inequalities, how to divide those assumptions up, and what the violation of those inequalities in nature says about the physical world. Relatedly, there are questions about who proved what and when, what specifically Bell said in each of his papers, who built on his work and why, etc. Aaronson says very little about all of this. Nothing leaps out at me as wrong, but it's rather "101". The third broad category of questions are about mathematical specifics. What particular combination of variables appears in which inequality, what are the bounds that combination is supposed to satisfy, etc. The expressions that appear in these formulae tend to look like rabbits pulled out of a hat. Sometimes there are minus signs and factors of root-2 and such floating around, and it's hard to remember where exactly they go. Even people who know the import of Bell's theorem could well ask to have it "explained", i.e., to have some account given of where all those arbitrary-looking bits came from. I don't think Aaronson does particularly well on this front. He pulls a rabbit out of his hat (a two-player game with Alice and Bob trying to take the XOR of two bits), he quotes a number with a root-2 in it, and he refers to some other lecture notes for the details, which include lots of fractional multiples of pi and which themselves leave some of the details to the interested reader.

Aaronson leads into this rather unsatisfying discussion thusly:

So what is Bell’s Inequality? Well, if you look for an answer in almost any popular book or website, you’ll find page after page about entangled photon sources, Stern–Gerlach apparatuses, etc., all of it helpfully illustrated with detailed experimental diagrams. This is necessary, of course, since if you took all the complications away, people might actually grasp the conceptual point!

However, since I’m not a member of the Physics Popularizers’ Guild, I’m now going to break that profession’s time-honored bylaws, and just tell you the conceptual point directly.

The tone strikes me, personally, as smarmy. But there's also an organizational issue. After saying he'll "just tell you the conceptual point directly", he then goes through the XOR rigmarole, which takes more than a page, before he gets to "the conceptual point" (that quantum mechanics is inconsistent with local hidden variables). It's less direct than advertised, for sure. I have not systematically surveyed pop-science explanations of Bell's theorem prior to 2013, but the "page after page of entangled photon sources..." rings false to me.

Aaronson tries to back up his perspective in chapter 9, where he makes the following contention:

Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let’s try to generalize it so that the numbers we used to call “probabilities” can be negative numbers.

This is a bait-and-switch, or more charitably, poor organization. Later he will admit that he needs to introduce not just negative numbers, but complex numbers too. What arguments does he give to justify bringing complex numbers into the picture? Why prefer ordinary quantum theory over what we might call "real-amplitude" quantum theory? He provides three suggestions. The first is based on a continuity argument ("if it makes sense to apply an operation for one second, then it ought to make sense to apply that same operation for only half a second"). He argues that this can only be made to work if the amplitudes are complex rather than only real. But this does not hold. We can simply say that in real-amplitude quantum theory, the time evolution operators belong to the subgroup of the orthogonal group that is continuously connected to the identity. This is actually what would be analogous to regular quantum theory, where we make unitary operators by taking the exponential of -iHt, where H is a Hamiltonian and t is an amount of time. In the real-amplitude theory, we just use an antisymmetric matrix as a generator instead of an anti-Hermitian one.

The second argument is that the number of parameters needed to specify a mixed state scales better for complex amplitudes than for real. This is a style of argument that has a considerable cachet among aspiring reconstructors of the quantum formalism, but it too has shortcomings. Aaronson invokes the principle that states for independent quantum systems combine via the tensor product. He asserts that this is true, and then argues that this makes the parameter counting work out nicely for complex but not real amplitudes. Plainly, then, this case for complex amplitudes can't be better than the case for the tensor product. It replaces one mathematical "brute fact" with another. People who go into more depth about this invoke a premise they call "tomographic locality". The conceptual challenge is then, if tomographic locality failed to hold true, would that actually be so bad? Would we find it stranger than, for example, quantum entanglement? See Hardy and Wootters (2010) and Centeno et al. (2024).

The third argument is given almost in passing. It's a "well, I guess that's nice" property which holds for the complex-amplitude theory and fails for the real-amplitude version. Bill Wootters noticed it. Of course, he also found something that works out nice only when the amplitudes are real instead. See Wootters (2013) for a more recent explanation of the latter, which he first published in 1980.

What Aaronson calls starting "directly from the conceptual core" strikes me instead as merely discarding some old prefatory material, like the Bohr model of hydrogen, and replacing it with new, like some chatter about classical computation. His "conceptual core" is the same old postulate. He just applies it in somewhat different settings, so he ends up doing matrix algebra instead of differential equations. I once thought that would be easier on students, but then I actually had to teach a QM class, and then I ended up "reviewing" a lot of matrix algebra.

A physicist who learned quantum mechanics the old-fashioned way, and who now sees "quantum" being hyped as the next Bitcoin, might well have some questions at this point. "So, you're telling me that these highly idealized models of hypothetical, engineered systems bring us closer to the secrets of the Old One than studying natural phenomena will? I'm sure you have your own good reasons for wanting to know if QURP is contained in PFUNK, but I want to understand why ice floats on water, why both iron and charcoal glow the same kind of red when they get hot, why a magnet will pick up a steel paperclip but not a shiny soda can." And: "I get the desire for a 'conceptual core' to quantum physics. But have you actually isolated such a thing? From where I stand, it looks like you've just picked one of the important equations and called it the important equation. Shouldn't your 'conceptual core' be a statement with some punch to it, like the big drama premise of special relativity? What's your counterpart to each observer who feels herself motionless will measure the same speed of light?"

Here's how Aaronson begins chapter 9:

There are two ways to teach quantum mechanics. The first way – which for most physicists today is still the only way – follows the historical order in which the ideas were discovered. So, you start with classical mechanics and electrodynamics, solving lots of grueling differential equations at every step. Then, you learn about the “blackbody paradox” and various strange experimental results, and the great crisis these things posed for physics. Next, you learn a complicated patchwork of ideas that physicists invented between 1900 and 1926 to try to make the crisis go away. Then, if you’re lucky, after years of study, you finally get around to the central conceptual point: that nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex.

This is wrong in a few ways. First, that "years of study"? Yeah, I saw complex probability amplitudes in my first term of college. Before they showed us all the blobby/cloudy pictures of electron orbitals, they took two minutes to explain what was being plotted. Our first full-blown quantum mechanics course was at the advanced age of ... sophomore year. And we're not talking about something squeezed in on the last day before summer vacation. See above regarding how it's the third equation in the first chapter of the ubiquitous standard undergrad QM textbook. This is not an idea sequestered in the inner sanctum of knowledge; it's babby's first wavefunction.

Second, the orthodox method is not really "historical". It can't be. The physicists who did all that work from 1900 through 1925--27 knew much more physics than college kids do today. They were professionals! Pick up the Dover reprint of the Sources of Quantum Mechanics collection, and see how many of the papers in it make sense using only first-year physics. Dirac was thinking about Poisson brackets, not a block on an inclined plane. The capsule "histories" in QM textbooks are caricatures, and sometimes quite poor ones at that.