Paul,

I don't think I ever mentioned it to you but there is a whole area of physics concerned with finite element calculations. It was started back when computers were first introduced to perform calculational results. There was a lot of good stuff available back when I was a graduate student and I would presume the issue has continued to be developed. In other words, there exists a body of work dedicated to a mathematics equivalent to denying the axiom of choice and most all mathematical physicists should be aware of it.

On the other hand, there have been some advances in handling mathematatical representations on computers (essentially dealing with named variables and defined functions) which can be used to represent the same concepts we use in continuous mathematics. So, as long as the final results are representable in finite numbers, the transformations can progress through continuous representations (which of course can all be proved through limiting processes).

What I am getting at is that the question, "do we actually use continuous things in our thoughts about real things?" is a difficult question to answer.

Have fun -- Dick