Has anyone written an up to date review of what Cox-style theorems are known to be sound and how well they suffice to found the mathematics of probability theory?
I don't know. But I will say this: I am distrustful of a foundation which takes "propositions" to be primitive objects. If the Cox's Theorem foundation for probability requires that we assume a first-order logic foundation of mathematics in general, in which propositions cannot be considered as instances of some larger class of things (as they can in, for personal favoritism, type theory), then I'm suspicious.
I'm also suspicious of how Cox's Theorem is supposed to map up to continuous and non-finitary applications of probability -- even discrete probability theory, as when dealing with probabilistic programming or the Solomonoff measure. In these circumstances we seem to need the measure-theoretic approach.
Further: if "the extension of classical logic to continuous degrees of plausibility" and "rational propensities to bet" and "measure theory in spaces of normed measure" and "sampling frequencies in randomized conditional simulations of the world" all yield the same mathematical structure, then I think we're looking at something deeper and more significant than any one of these presentations admits.
In fact, I'd go so far as to say there isn't really a "Bayesian/Frequentist dichotomy" so much as a "Bayesian-Frequentist Isomorphism", in the style of the Curry-Howard Isomorphism. Several things we thought were different are actually the same.
Among my friends interested in rationality, effective altruism, and existential risk reduction, I often hear: "If you want to have a real positive impact on the world, grad school is a waste of time. It's better to use deliberate practice to learn whatever you need instead of working within the confines of an institution."
While I'd agree that grad school will not make you do good for the world, if you're a self-driven person who can spend time in a PhD program deliberately acquiring skills and connections for making a positive difference, I think you can make grad school a highly productive path, perhaps more so than many alternatives. In this post, I want to share some advice that I've been repeating a lot lately for how to do this:
That's all I have for now. The main sentiment behind most of this, I think, is that you have to be deliberate to get the most out of a PhD program, rather than passively expecting it to make you into anything in particular. Grad school still isn't for everyone, and far from it. But if you were seriously considering it at some point, and "do something more useful" felt like a compelling reason not to go, be sure to first consider the most useful version of grad that you could reliably make for yourself... and then decide whether or not to do it.
Please email me (lastname@thisdomain.com) if you have more ideas for getting the most out of grad school!