I had difficulty with what I think was that chapter too. I asked about one part on math.stackexchange, it looks like someone answered my immediate question but judging by my followup there might have been more that I didn't understand. (I no longer remember enough math to really understand the question or answer.)
Yeah I read about 1/3d of the proof of Cox's theorem until I realized even if I followed every step I wouldn't gain any intuition from it, then I skipped the rest
I believe that what Jaynes does is quite standard: start with a minimalistic set of axioms (or principles, or whatever) and work your way to the intiuitive results later on. Euclid geometry is just like that!
I just skimmed over the details of the proofs (and I am a mathematician by training!). I did not read Jaynes for such details. I just guess that if they were wrong, somebody would have already reported them. The meaty part is elsewhere.
I first started reading Jaynes Probability Theory around 2 years ago, I did not last long, I wasn't able to follow the derivations in Chapter 2, so I gave up, assuming it would only get harder from there what was the point in continuing?
This was a big mistake! For several reasons
As I'm finally rectifying my mistake I figured I'd write an explanation for what Chapter 2 is actually about
The reasoning in Chapter 2 shows that any rules of plausible-inference must match our own theory of probability after a suitable change of units. The convoluted functional-equation argument is needed to construct a function which translates the alien's plausibility into our probability .
This is very different to how is usually defined, for Jaynes is a translation between the plausibilities of the alien and our "nicer" probabilities that obey the sum and product rules. Contrast this with standard probability where is a function from subsets of the sample space to real numbers obeying certain axioms.
Any alien civilization's concept of "probability the dice lands five" must be a monotonic function of our probability . That's amazing when you think about it, It shows that probability is discovered not invented. [1]
Furthermore, if you want the sum and product rule to take their natural forms you have to pick our units, this is analogous to how degrees Kelvin are defined to make the laws of thermodynamics look natural. [2]
Near the end of Chapter 2 (after we've shown uniqueness) Jaynes switches to a more traditional function where is some logical proposition like "the dice lands five".
Why Jaynes doesn't mention the intuitive explanation until after dragging you through a hard-to-follow argument involving tons of functional equations and Calculus I have no idea, I guess Thousand-year old vampires are bad at pedagogy? [3]
Hopefully I've convinced you Jaynes isn't that scary, and motivated you to start reading. A few quick tips
I feel a popularization of these ideas should be possible, they feel more fundamental (read: philosophically interesting) than most popularized science. Cantor's theorem pales in comparison! Somebody get Numberphile to in on this! ↩︎
Or, so I'm told. I don't know thermodynamics lol ↩︎
This has also made me think the world deserves a "Second edition" that corrects some of his egregious teaching, If you know of something like this (other books which take his approach to probability) please let me know ↩︎