I think quantum mechanics really is different. Consider a fundamental aspect of the quantum world, familiar to anybody reading Scott Aaronson's blog: the 2-norm.
People are familiar with how probability works in the classical world. For a given event with several different possible outcomes, like flipping a coin, the probability of those outcomes sums to 1 (here, 1/2 + 1/2 = 1).
In the quantum world, probability doesn't work like that. Rather, the sum of the absolute value of the probability squared equals one. A quantum coin flip has probability (actually called amplitude) 1/sqrt(2) for heads and 1/sqrt(2) for tails. |1/sqrt(2)|^2 + |1/sqrt(2)|^2 = 1.
This has huge implications. It means probabilities can be negative, since |-1/sqrt(2)|^2 + |-1/sqrt(2)|^2 = 1 (actually it goes further - the probabilities can be complex numbers!). It means negative probabilities can interfere and cancel out positive probabilities. And a whole bunch of other wild stuff. All of quantum mechanics (or so Scott claims) follows from this.
I submit that a world where something as basic as the probabilities of all outcomes summing to 1 not holding is a world so alien that attempts to explain it in terms of our own experience will only end in confusion.
Sure, but we don't need to explain everything in terms of our world to still try to explain concepts. We can use contradictions, or games[0], or experiments, or any number of mechanisms.
Infinity doesn't really map to anything else that I understand. Some of the better tutorials I've gotten on infinity are literally just saying, "hey, look at this scenario. Do you think it works the same way as with really big numbers? Well it DOESN'T! Now watch me shove even more guests into your hotel! :)"
I'm not pushing back against the idea that quantum mechanics is a completely new paradigm, I'm pushing back against the idea that math, by itself, is a good enough mechanism for teaching new paradigms.
The math/pop-science dichotomy actually reminds me a bit of my early physics lessons on electricity. I'd either get a really awful analogy (electricity is just like water flowing through a pipe), or I'd get the formula to add up the resistors and calculate voltage and then get told, "well, this is all you're going to need to know for the test."
Neither approach was concerned with understanding how electricity worked; one just said "think of it as something else", and the other said "don't think about it."
The problem is that our best way to describe QM is using math, and the experiments can only verify that the predictions made with math agree with the results.
There are many interpretations of what the math mean, like many words or wave function collapse. But the interpretations are equivalent, in any experiment they use the same math to get the same results.
So you may like one interpretation or the other, but the only sure part is the math.
Games, analogies, think experiments are nice and are useful to understand how to apply the math and get an intuitive idea of how the math work and how some approximations may simplify the calculations.
I still sometimes move my finger in circles to think and do calculations about spin, in spite the spin of a particle is not exactly equal to the spin of a ball. It's a nice analogy and (for me) it's useful, but the real particles don't behave like balls or moving finger.
Another important part is to understand how to translate an actual lab setup to math and how to relate the result of the calculation and the experiments.
I totally agree, the poster understands the calculation rules, but calls the wave function or amplitudes the probabilities, where practically everyone calls their norm squared the probability (or probability density)
It was for pedagogical purposes. I mentioned they were actually called amplitudes. For the intended audience, the difference doesn't matter. Scott Aaronson uses a similar tactic on p110 of Quantum Computing Since Democritus.
People are familiar with how probability works in the classical world. For a given event with several different possible outcomes, like flipping a coin, the probability of those outcomes sums to 1 (here, 1/2 + 1/2 = 1).
In the quantum world, probability doesn't work like that. Rather, the sum of the absolute value of the probability squared equals one. A quantum coin flip has probability (actually called amplitude) 1/sqrt(2) for heads and 1/sqrt(2) for tails. |1/sqrt(2)|^2 + |1/sqrt(2)|^2 = 1.
This has huge implications. It means probabilities can be negative, since |-1/sqrt(2)|^2 + |-1/sqrt(2)|^2 = 1 (actually it goes further - the probabilities can be complex numbers!). It means negative probabilities can interfere and cancel out positive probabilities. And a whole bunch of other wild stuff. All of quantum mechanics (or so Scott claims) follows from this.
I submit that a world where something as basic as the probabilities of all outcomes summing to 1 not holding is a world so alien that attempts to explain it in terms of our own experience will only end in confusion.