On 11/21/2022 5:54 PM, Bruce Kellett wrote:
On Tue, Nov 22, 2022 at 12:44 PM Brent Meeker <[email protected]> wrote:On 11/21/2022 4:38 PM, Bruce Kellett wrote:On Tue, Nov 22, 2022 at 11:35 AM Brent Meeker <[email protected]> wrote: But frequencies are how we test probabilistic theories. Testing is not a theoretical grounding of the theory.It's not the axiomatic ground of Kolmogorov's theory. But so what? We tested Euclid's theory of geometry by making measurements which weren't in his axioms. That doesn't mean Euclid's wasn't a good theory of geometry. I can see Deutsch crossing off Pythagora's theorem saying, "No matter how precise our instruments they only yield rational quantities!" Physics is not mathematics and it's never going to have data to infinitely many decimal places. That frequencies only yield rational number approximations to Born rule predictions doesn't seem like a big deal to me.What is a probability? We can't define it as a limiting frequency, since repeats of a sequence of measurements of a spin are going to give a range of answers for the frequency of spin-up, and this sequence converges to some limit only in probability. That is then circular -- probability is defined in terms of probability.
But in fact we will reject a theory that predicts that the observed data is very improbable.
Perhaps 'probability' is a primitive concept -- not definable in terms of anything physical. Nevertheless, like language, it is essential for our understanding of our experience of the world.
I think it is term that refers to different things in different contexts, like "energy" or "distance". It has a definition is mathematics, but that's not the same as it's operational definition in physics or law or finance. It can be defined in terms of symmetry in physics, as a die is fair if the probability of each face is independent and = 1/6. Then this can be tested under frequentist definitions. How many throws yielded :: out of 10,000 trials? If you're playing craps then you adopt a degree of belief that :: will appear 1/6 of the time and being a decision theorist you bet accordingly.
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