while also not believing in certain spooky things called "continuous random variables", which don't really fit into Cox's Theorem very well, if I understood Jaynes correctly.
I found a partial answer to the question I asked in the sibling comment. By chance I happened to need to generate random chords of a circle covering the circle uniformly. In searching on the net for Jaynes' solution I came across a few fragments of Jaynes' views on infinity. In short, he insists on always regarding continuous situations as limits of finite ones (e.g as when the binomial distribution tends to the normal), which is unproblematic for all the mathematics he wants to do. That is how the real numbers are traditionally formalised anyway. All of analysis is left unscathed. His wider philosophical objections to such things as Cantor's transfinite numbers can be ignored, since these play no role in statistics and probability anyway.
I don't know about the technicalities regarding Cox's Theorem, but I do notice a substantial number of papers arguing about exactly what hypotheses it requires or does not require, and other papers discussing counterexamples (even to the finite case). The Wikipedia article has a long list of references, and a general search shows more. 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 can google /"Cox's theorem" review/ but it is difficult for me to judge where the results sit within current understanding, or indeed what the current understanding is.
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 t...
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!