[I]f you want a universe with certain very generic properties, you seem forced to one of three choices: (1) determinism, (2) classical probabilities, or (3) quantum mechanics. [My emphasis.]
Scott Aaronson, Quantum Computing since Democritus
Aaronson then proceeds by arguing that there are only two theories that are "like" probability theory: probability theory itself and quantum mechanics. Probability theory is based on the $1$-norm, whereas quantum mechanics is based on the $2$-norm.
Call $\{v_1,\ldots,v_N\}$ a unit vector in the $p$-norm if $|v_1|^{\ p}+\cdots+|v_N|^{\ p}=1.$
The slide below is from a presentation of his.
![[]](https://i.stack.imgur.com/NeOio.png)
Now, it seems that there is yet another candidate out there: the $0$-norm. For this purpose one would have to define $0^0=0$, and, if one does, one has a theory "like" probability theory that is determinism.
($0^0$ is indeterminate, which, I understand, means that you can do as you please, as long as you do it consistently.)
It was remarked that the $0$-norm can be seen as a special case of both the $1$ and $2$-norms, in fact, that it can even be defined as the intersection between the two.
From there on I'm sort of questioning the original trilemma posed by Aaronson. It seems one can choose one of three possible realities; not the three that were mentioned, but rather options 1, 2 and 3:
$$\begin{array}{|l|c|c|c|}
\hline
&\text{Determinism}&\text{Classical probabilities}&\text{Quantum mechanics}\\
\hline
\text{Option 1}&\color{green}{\text{True}}&\color{green}{\text{True}}&\color{green}{\text{True}}\\
\hline
\text{Option 2}&\color{red}{\text{False}}&\color{green}{\text{True}}&\color{red}{\text{False}}\\
\hline
\text{Option 3}&\color{red}{\text{False}}&\color{red}{\text{False}}&\color{green}{\text{True}}\\
\hline
\end{array}$$
Scientific evidence favours quantum mechanics. That seems to rule out Option 2 (by, e.g., Bell's theorem), but, mathematically, doesn't seem to point to Option 3 only (meaning, of course, that I don't see it ruling out Option 1). (Bell's theorem is sort of empty under determinism.)
How does one rule out that reality can be described by Option 1, notwithstanding the fact that it's certainly easier (and more practical) to describe it by Option 3? In what sense is Option 3 the scientific one, if it is indeed the scientific one? (Some might say that quantum mechanics is deterministic, but then, according to the table, it is also classically probabilistic.)
Taking it possibly further then necessary: If, on scientific grounds, we can't make a distinction between Option 1 and Option 3, isn't the whole thing (i.e., the statuses of determinism and classical probability) just exclusively philosophical?
Best Answer
It is certainly possible to scientifically distinguish between option 1 and 3 by either finding a deterministic hidden-variable theory underlying QM or proving that such a theory cannot be constructed. For example, if Bohmian interpretation can be extended to the relativistic regime (I know that there are proposals but I am not sure if they are accepted by the physics community as correct) option 1 is true and 3 is false.