Thursday, May 03, 2007

Popper's Probabilities

Commenting on Basic Sets, I replied to Torbjörn’s comment (#49) that in most parts of theoretical physics they use frequentist probabilities (in connection with my new argument against standard set theory), as follows—I repeat myself in order to refer to this post in my defence of Popper (sections linked to below).

50 years ago, I noted (in comment #52), Popper noticed that many scientists were not really frequentists, for all that they might call themselves that. While they took certain real numbers in the equations of QM to be probabilities (i.e. they took QM, quantim mechanics, to be saying something that could be tested against observed frequencies) they were not frequentists because, basically, frequentism is either finite or infinite. If it is finite then probabilities cannot be real numbers (see §2 for more reasons). But if it is infinite (as it is usually) then such probabilities actually say nothing about our finite observations—they cannot be tested against observed frequencies because each value for the limit frequency is compatible with any initial values (cf. §3 and §5).

So, most of those scientists who regarded themselves as frequentists, but who were also realists about QM (unlike
Lewis for example), actually believed in single-case propensities (see §4). The propensity bit is the idea that the QM equations model something real, something chancy that is tested via frequencies. The single-case bit is the idea that even a single particle behaves probabilistically, e.g. in 2-slit experiment, and that widely separated things might be causally independent. Note that by QM I mean nothing too technical, just the QM of chemistry, and of 2-slit experiments. Such elementary experiments do appear to be telling us that the underlying stuff of reality is well modelled by Schrödinger's equation. And I take it for granted that there is some such stuff. But of course, I hope that there are some coherent objections to some of that, for me to think about...

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