The computer is a tool I use to help me solve problems. Part of the process of solving problems, for me at least, is to write code that executes, and spend some time in the debugger to see if my assumptions were right. The computer is a tool for solving problems, dammit! It is ridiculous to think that you should solve the problem using pencil and paper (b/c computers no good for problem solving!?!), with various mathematical proofs (b/c we can and should prove everything?!?) and the computer is just an annoying end point for getting solutions out.
Programming is not just about implementing solutions, but also about finding solutions.