Why can't programs be proven?

Question

Why can\'t a computer program be proven just as a mathematical statement can? A mathematical proof is built up on other proofs, which are built up from yet more proofs and

Answer 1

As others pointed out, (some) programs can indeed be proven.

One problem in practice however is that you first need something (i.e. an assumption or theorem) that you want to prove. So to prove something about a program you first need a formal description of what it should do (e.g. pre- and post-conditions).

In other words, you need a formal specification of the program. But getting even a reasonable (much less a rigorous formal) specification is already one of the hardest things in software development. Therefore it is generally very difficult to prove interesting things about a (real-world) program.

There are however some things which can be (and have been) more easily formalized (and proven). If you can at least prove that your program will not crash, that's already something :-).

BTW, some compiler warnings/errors are essentially (simple) proofs about a program. E.g., the Java compiler will prove that you never access an uninitialized variable in your code (otherwise it will give you a compiler error).