Is there a Mathematical system which is stochastic at its core? Ok so I'm a programmer, I'm not a mathematician, I've got a minor in math but I didn't even do particularly well at it so please bear with some possibly really stupid thoughts.  Just please try to explain to me why what I'm asking is stupid so that I don't keep making a fool of myself.
I recently found myself thinking along the following lines:
If you take the stand that it's solipsism to talk about anything that cannot be put to use in  predicting an outcome then cannot you apply the same axiom to basic mathematical operations?  For example does addition really have any meaning outside of the context of placing two objects in a box and predicting how many are inside?  Yeah, I realize that's a dangerous question to ask a board full of mathematicians but I imagine you can ask this question about any axiom.
The problem is that in reality if you put one thing and then another into a box it does not mean that there will be two things in the box.  There will likely be two things of course but if the objects are point particles like electrons there's a chance there will be one or three, or a million electrons there.  Heck, the same is even possible with apples, just incredibly unlikely.
I don't have the mathematical wherewithal to think through what this would mean but intuitively it would seem that this might not have many implications for addition, but might for subtraction (unless you could somehow have negative amounts of particles which I won't rule out), and certainly for things like integration and derivation.
I would assume that someone's done work along these lines before?  Has anyone actually created a system around it?  Was it useful?  Are there any accessible books or articles about it?
Just interested
 A: Your question is undoubtedly philosophical in nature: it is predicated on the premise "it's solipsism to talk about anything that cannot be put to use in predicting an outcome".  But solipsism is a philosophical term (and more fundamentally, an inherently philosophical -- as opposed to mathematical or scientific -- idea)!
The philosophical view you advocate seems to be, roughly, that the truths of arithmetic are (only) empirical in nature.  As with many basic philosophical ideas, this has been propounded before and has gotten a lot of discussion.  Famously, it was the view of arithmetic espoused by John Stuart Mill.  For a pretty good quick summary of his positions, see
http://plato.stanford.edu/entries/mill/#GeoAri
Mill's ideas on arithmetic and geometry have generally been poorly received by later philosophers.  Notably, much of the work of Immanuel Kant is a reaction against Mill's ideas, especially his notion of the "synthetic a priori".  
So far as I know, essentially no very few mathematicians or philosophers of mathematics have accepted Mill's ideas.  Rather, we have followed the path of Frege and viewed logic as being the foundation for mathematics.  (Later work by Russell, Godel and others have showed that there are some pitfalls and limitations of the logicist approach, but it has nevertheless been the one mathematicians have adopted for the last hundred years.)
[As Qiaochu says, if you really want to talk mathematics you should remove the overtly philosophical lead to your question.  Instead, you should say more about the mathematical aspects of your question: what mathematical problem are you trying to solve?]
A: You are looking at the problem of mathematics without knowing mathematics. Most of what you write applies to Chinese, provided you do not understand Chinese. 
For one thing: A question like «does addition really have any meaning outside of the context of placing two objects in a box and predicting how many are inside?» is meaningless to anyone who understands arithmetic and, better, "higher arithmetic": it is obvious to him/her that addition is meaningful!
More generally, your "axiom" regarding solipsism is quite against observation.
A: The first thing that comes to mind is Quantum Electrodynamics. I don't really know much more than the name, but the first chapters of Richard Feynmans's Six Easy Pieces (I think that was the one) are pretty accessible. They raise the problems of trying to keep track of single photons, and how they behave completely differently from regular solid objects. With a bunch of photons you can only keep track of the distribution of photons (more or less). Look for Feynmann vidoes on the web. He was entertaining and you won't regret the time.
