Re: Axioms of elementary probability

[Jeff Egger <jeffegger@yahoo.ca>]

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.

----- Original Message ----
> From: Ross Street <street@ics.mq.edu.au>
> To: David Espinosa <david@davidespinosa.net>; Categories <categories@mta.ca>
> Sent: Tuesday, May 12, 2009 2:53:13 AM
> Subject: Re: categories: Axioms of elementary probability
> 
> A couple of years ago, Voevodsky gave an interesting talk at the
> Australian Math Soc
> Annual Meeting (at RMIT. Melbourne) about a categorical approach to
> probability theory.
> Google told me about:
> 
>    http://www.math.miami.edu/anno/voevodsky.htm
> and
>    http://golem.ph.utexas.edu/category/2007/02/
> category_theoretic_probability_1.html
> 
> Ross
> 
> On 09/05/2009, at 4:02 PM, David Espinosa wrote:
> 
> > Here's a question about elementary (naive, finitist) probability.
> > The proper, self-dual axioms for elementary probability are presumably
> >
> >  P(0) = 0
> >  P(X) = 1
> >  P(A u B) + P(A n B) = P(A) + P(B)