It's not clear to me whether you're talking about the "mathematics of programming" or the "mathematics of computer programs", which I think is an important distinction to make. Programming is a creative process: you start from a specification and construct a program to achieve it. A computer program is a static object, unambiguous, with formally defined operational semantics. The computer program is easily amenable to mathematical analysis (in principle), but the creative process of producing programs is not as easily amenable to automation.

The entire field of Formal methods is devoted to the automated, mathematical analysis of computer programs. It is a very active research area.

Of course, many people (including some respondents in this thread) would object to the fact that your mathematical models can never include all of the real world concerns that might be relevant. Of course it's true, and it's a very legitimate concern. Einstein said it best:

"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

Some people consider this a shortcoming of trying to model things mathematically. I prefer to think of it as also a shortcoming of the real world ;)

Of course, you could have an interesting epistemological discussion about whether logic and mathematics are necessary to say anything about external reality with any degree of certainty, but I'm not really equipped to have such a discussion...