Maybe, with the caveat that Borel-Cantelli involves non-finitary notions like "infinitely often." But, I suspect, one can make it work using the ideas of Vovk et al.
" probabilities are not objective properties of physical systems, but rather numerical assessments attached to propositions about these systems."
It seems like it wouldn't be unreasonable to make the jump from probabilities to mathematical entities more broadly(?)
Say, geometry is actually the right set of rules to use when reasoning about appropriate physical systems, but there is nothing deeply geometric *about* the physical system itself. (something something about nominalism/vagueness / complex adaptive systems on the question of what an "appropriate" system is)
I suppose the natural question then is to ask why these structures should obtain at all? If there’s some idea of correspondence then it feels like we’re just sneaking platonism in through the back door
I think this revolves around the exact nature of the correspondence relation. In general, the passage from object language to metalanguage involves both abstraction, i.e., selective loss of detail, and idealization, i.e., introduction of limiting processes and such. One could view it either as Platonism (additional detail removed via abstraction is actually worldly imperfections) or as pragmatism (which would also go nicely with the idea of choosing the _right_ level of abstraction for the task at hand). This is a "choose your own adventure" kind of thing.
this was great :)
Thank you! I'm surprised you were not put off by my pragmatist leanings here.
I have pragmatist sympathies ! I understand the pull towards it
"role of the probability theory is to identify events of probability 0 or 1"
IOW: Everything comes down to the Borel-Cantelli lemmas* ... when the aggregated randomness yields certainty in either direction.
*and other results with a family resemblance to B-C
Maybe, with the caveat that Borel-Cantelli involves non-finitary notions like "infinitely often." But, I suspect, one can make it work using the ideas of Vovk et al.
Excellent post! I really appreciate all the references.
Thank you!
Great post. What did you think of Burdzy's newer book?
I haven't had a chance to look at it carefully yet.
" probabilities are not objective properties of physical systems, but rather numerical assessments attached to propositions about these systems."
It seems like it wouldn't be unreasonable to make the jump from probabilities to mathematical entities more broadly(?)
Say, geometry is actually the right set of rules to use when reasoning about appropriate physical systems, but there is nothing deeply geometric *about* the physical system itself. (something something about nominalism/vagueness / complex adaptive systems on the question of what an "appropriate" system is)
Yes, that's exactly it. It's the distinction between object language and metalanguage, and its operative range is very broad.
I suppose the natural question then is to ask why these structures should obtain at all? If there’s some idea of correspondence then it feels like we’re just sneaking platonism in through the back door
I think this revolves around the exact nature of the correspondence relation. In general, the passage from object language to metalanguage involves both abstraction, i.e., selective loss of detail, and idealization, i.e., introduction of limiting processes and such. One could view it either as Platonism (additional detail removed via abstraction is actually worldly imperfections) or as pragmatism (which would also go nicely with the idea of choosing the _right_ level of abstraction for the task at hand). This is a "choose your own adventure" kind of thing.