David, To my mind there are three presentations of a "theory" of probability. Two arrive at essentially the same theory by somewhat different means; these are frequentist and Bayesian presentations of "standard" probability theory. The third comes from a completely different direction: quantum mechanics. i remember when i first encountered the Dirac presentation of QM and the interpretation of <a| M |b> as a probability amplitude. My first thought was -- hang on, doesn't that come with an obligation to prove that this aligns with (satisfies the axioms of) a theory of probability. In attempting to work that out for myself, i realized that it didn't; discovered a whole cottage industry of people who had made a similar observation; and argued to myself that of the various notions of probability put forward, this one enjoyed being rigourously employed in physical calculations verified to many decimal places. Best wishes, --greg On Tue, May 12, 2009 at 10:52 AM, Jeff Egger <jeffegger@yahoo.ca> wrote:
When I took a graduate course in probability, my lecturer began with a rather fine speech about the relationship between probability and (finite) measure theory; in it, he discouraged identifying the two. His point was that, insofar as probabilistic phenomena occur in the real world, no mathematical theory can aspire to do more than model probability---and that, while (finite) measure theory has been very successful at modelling probability, it also has shortcomings.
Intrigued, I sought him out later for more thoughts on the subject. In the ensuing conversation, I gathered two tidbits of information which readers of the list may appreciate: that Gromov believes that the future of probability theory lies in bicategory theory; and that discontent with measure theory stems, at least in part, from its failure to adequately handle conditional probabilities.
To be honest, the latter point heartened me even more than the first. From a purely aesthetic point of view, it has always irked me that one can meaningfully assign probabilities to things which are not events; I interpret this as meaning that the (standard) notion of event is too narrow. Of course, it is also the case that the (standard) formula for a conditional probability may result in the indeterminate 0/0, so it would seem that [0,1] is also too small a codomain for the map "probability", even classically understood (i.e., not getting into the "free probability" of Voiculescu).
Cheers, Jeff.