From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4834 Path: news.gmane.org!not-for-mail From: Greg Meredith Newsgroups: gmane.science.mathematics.categories Subject: Re: Axioms of elementary probability Date: Wed, 13 May 2009 12:59:05 -0700 Message-ID: Reply-To: Greg Meredith NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: multipart/alternative; boundary=00504502c74fc0420b0469d0a42f Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1242303736 11007 80.91.229.12 (14 May 2009 12:22:16 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 14 May 2009 12:22:16 +0000 (UTC) To: Jeff Egger , Original-X-From: categories@mta.ca Thu May 14 14:22:04 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1M4Zwo-0005XB-RU for gsmc-categories@m.gmane.org; Thu, 14 May 2009 14:22:02 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1M4ZGW-0005Yr-Gl for categories-list@mta.ca; Thu, 14 May 2009 08:38:20 -0300 Original-Content-Type: text/plain; charset=ISO-8859-1 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4834 Archived-At: 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 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 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.