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

If you're really interested in the topic, let me first recommend David Gries' "The Science of Programming", a classic introductory work on the topic.

It actually is possible to prove programs correct to some extent. You can write preconditions and postconditions and then prove that given a state that meets the preconditions, the resulting state after execution will meet the postconditions.

Where it gets tricky, however, is loops. For these, you additionally need to find a loop invariant and to show correct termination you need to find an upper bound function on the maximum possible number of iterations remaining after each loop. You also have to be able to show that this decreases by at least one after each iteration through the loop.

Once you have all this for a program, the proof is mechanical. But unfortunately, there's no way to automatically derive the invariant and bound functions for loops. Human intuition suffices for trivial cases with small loops, but realistically, complex programs quickly become intractable.