probability theory
1. Anyone interested in probability theory who does not read at least the early chapters of http://www.amazon.com/Probability-Theory-The-Logic-Science/dp/0521592712 is not seriously interested, is some kind of ideologue (common concerning probability), is for whatever reason unable to read it, or is somewhere on a spectrum.
2. Recently I bought a book whose title says it is about ‘rigorous’ probability theory. By this is meant Kolmogorov theory, which amounts to an elaborate mathematics of Venn diagrams. It is what we used in my grad school engineering course on ‘random’ processes. There are serious reasons to be dissatisfied with Venn diagrams as a basis for probability theory; what, for instance, is the correspondence between a proposition (or predicate) and a set of points? But what I really want to take issue with is the notion of ‘rigor’ assumed in the text. The proofs almost immediately call upon the ‘Axiom of Choice’. If you know what that means, let me ask you something about it. Suppose someone designed a bridge and they said it was a new type of bridge whose structure is proven sound by means of the ‘Axiom of Choice’; would you trust this bridge to stay up?
I sure wouldn’t.
(Jaynes OTOH, for the sake of meaningful rigor, sticks with finite sets and, where necessary, sets arrived at by a well-defined limiting process involving finite sets.)
3. If Karl Popper really said the sorts of things about probabilities that Jaynes says he did (and I’m sure Popper did), then Popper is way overrated in the popular opinion.
4. For heaven’s sake, read what you can of http://arxiv.org/abs/0712.3781
2. Recently I bought a book whose title says it is about ‘rigorous’ probability theory. By this is meant Kolmogorov theory, which amounts to an elaborate mathematics of Venn diagrams. It is what we used in my grad school engineering course on ‘random’ processes. There are serious reasons to be dissatisfied with Venn diagrams as a basis for probability theory; what, for instance, is the correspondence between a proposition (or predicate) and a set of points? But what I really want to take issue with is the notion of ‘rigor’ assumed in the text. The proofs almost immediately call upon the ‘Axiom of Choice’. If you know what that means, let me ask you something about it. Suppose someone designed a bridge and they said it was a new type of bridge whose structure is proven sound by means of the ‘Axiom of Choice’; would you trust this bridge to stay up?
I sure wouldn’t.
(Jaynes OTOH, for the sake of meaningful rigor, sticks with finite sets and, where necessary, sets arrived at by a well-defined limiting process involving finite sets.)
3. If Karl Popper really said the sorts of things about probabilities that Jaynes says he did (and I’m sure Popper did), then Popper is way overrated in the popular opinion.
4. For heaven’s sake, read what you can of http://arxiv.org/abs/0712.3781