Agreed re: the bashing of mainstream math in PT:TLOS. AFAIK, his claims that mainstream math leads to paradoxes are all false; of course trying to act as though various items of mainstream math meant what an uneducated first glance says they mean can make them look bad. (e.g. the Banach-Tarski paradox means either “omg, mathematicians think they can violate conservation of mass!” or “OK, so I guess non-measurable things are crazy and should be avoided”) It’s not only unnecessary and annoying, but also I think that using usual measure theory would clarify things sometimes. For instance the fact that MaxEnt depends on what kind of distribution you start with, because a probability distribution doesn’t actually have an entropy, but only a relative entropy relative to a reference measure, which is of course not necessarily uniform, even for a discrete variable. Jaynes seems to strongly deemphasize this, which is unfortunate: from PT:TLOS it seems as though MaxEnt gives you a prior given only some constraints, when really you also need a “prior prior”.
Agreed re: the bashing of mainstream math in PT:TLOS. AFAIK, his claims that mainstream math leads to paradoxes are all false; of course trying to act as though various items of mainstream math meant what an uneducated first glance says they mean can make them look bad. (e.g. the Banach-Tarski paradox means either “omg, mathematicians think they can violate conservation of mass!” or “OK, so I guess non-measurable things are crazy and should be avoided”) It’s not only unnecessary and annoying, but also I think that using usual measure theory would clarify things sometimes. For instance the fact that MaxEnt depends on what kind of distribution you start with, because a probability distribution doesn’t actually have an entropy, but only a relative entropy relative to a reference measure, which is of course not necessarily uniform, even for a discrete variable. Jaynes seems to strongly deemphasize this, which is unfortunate: from PT:TLOS it seems as though MaxEnt gives you a prior given only some constraints, when really you also need a “prior prior”.