3 ms·

Sure, half of the math (if not more) was done in the same way e.g. by using tools or concepts informally before they were defined rigorously.

kgwgk 2y ago
To the extent that one doesn’t need measure theory to do statistical mechanics there is no need to teach measure theory in a statistical mechanics course. “If anyone wants to concentrate his attention on infinite sets, measure theory, and mathematical pathology in general, he has every right to do so. And he need not justify this by pointing to useful applications or apologize for the lack of them; as was noted long ago, abstract mathematics is worth knowing for its own sake. But others in turn have equal rights. If we choose to concentrate on those aspects of mathematics which are useful in real problems and which enable us to carry out the important substantive calculations correctly – but which the mathematical pathologists never get around to – we feel free to do so without apology.” [Jaynes 2003, p. 673] By the way, rigour and generality are different things.
- 331c8c71 2y ago
  > To the extent that one doesn’t need measure theory to do statistical mechanics there is no need to teach measure theory in a statistical mechanics course. I never assumed otherwise. That being said, I couldn't be bothered to memorize and reproduce the handwavy exposition that made sense to a particular physicist. And it's not that I was there to prove a point - I was asked about the basic constructions like probability distributions, densities etc. > infinite sets, measure theory, and mathematical pathology > By the way, rigour and generality are different things. Measure theory and Kolmogorov axiomatization addressed very real problems like Bertrand's paradox plaguing classical probabilit to the point that it was often seen as a black art rather than science. Moreover, stochastic processes in continuous time are not very intuitive and a sound theory would be next to impossible to develop without solid foundations.
  - kgwgk 2y ago
    > Measure theory and Kolmogorov axiomatization addressed very real problems like Bertrand's paradox. They don’t solve the “paradox” - and it can be “solved” without them. https://bayes.wustl.edu/etj/articles/well.pdf https://bayes.wustl.edu/etj/articles/well.pdf

kgwgk 2y ago

To the extent that one doesn’t need measure theory to do statistical mechanics there is no need to teach measure theory in a statistical mechanics course. “If anyone wants to concentrate his attention on infinite sets, measure theory, and mathematical pathology in general, he has every right to do so. And he need not justify this by pointing to useful applications or apologize for the lack of them; as was noted long ago, abstract mathematics is worth knowing for its own sake. But others in turn have equal rights. If we choose to concentrate on those aspects of mathematics which are useful in real problems and which enable us to carry out the important substantive calculations correctly – but which the mathematical pathologists never get around to – we feel free to do so without apology.” [Jaynes 2003, p. 673] By the way, rigour and generality are different things.

331c8c71 2y ago
> To the extent that one doesn’t need measure theory to do statistical mechanics there is no need to teach measure theory in a statistical mechanics course. I never assumed otherwise. That being said, I couldn't be bothered to memorize and reproduce the handwavy exposition that made sense to a particular physicist. And it's not that I was there to prove a point - I was asked about the basic constructions like probability distributions, densities etc. > infinite sets, measure theory, and mathematical pathology > By the way, rigour and generality are different things. Measure theory and Kolmogorov axiomatization addressed very real problems like Bertrand's paradox plaguing classical probabilit to the point that it was often seen as a black art rather than science. Moreover, stochastic processes in continuous time are not very intuitive and a sound theory would be next to impossible to develop without solid foundations.
- kgwgk 2y ago
  > Measure theory and Kolmogorov axiomatization addressed very real problems like Bertrand's paradox. They don’t solve the “paradox” - and it can be “solved” without them. https://bayes.wustl.edu/etj/articles/well.pdf https://bayes.wustl.edu/etj/articles/well.pdf