Over at Asymptotia, Len Adleman (the A in RSA, founder of DNA computation (but not the A in DNA!), and a discoverer of the APR primality testing algorithm) has a guest post about the foundations of quantum theory. Len suggests, if I understand him correctly, that one should attempt to understand the uncertainty arising in quantum theory as being of the same nature as the fact that there exists statements which cannot be proven true or false within a fixed set of powerful enough axioms.
First of all, I know I’ve heard a similar argument before, but can’t seem to find the reference! Any foundations (or other) people want to supply those as I’m sure they would be welcomed in the comment section of Asymptotia. Second, I find it interesting that Len seems most troubled by the uncertainty arising in quantum theory and not by, for example, Bell inequalities. I’m no so sure many of us are troubled by this aspect (that it is probabilistic and not deterministic) of quantum theory, in and of itself. That is to say if the world had a probabilistic local hidden variable theory, would we be arguing about the foundations of quantum theory? Third, this of course brings to mind the Kochen-Specker theorem which shows that there is no non-contextual hidden variable theory of quantum mechanics. Indeed contextuality reminds one a lot of a “choice of axiomatic system.” It would indeed be neat if one could make this into a more established result. But in particular one would need to argue why Hilbert space best captures the idea of a set of axioms. Finally, because I think Bell inequalities are essential for understanding what makes quantum theory truely unique (yes I’m biased), I’m curious as to whether mathematicians have ever considered the notion of “local axioms”, i.e. axioms which live in spacetime?