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?