My unusual hobby
&gt; What&#x27;s really amazing to me is that Stephen Kleene probably proved this without the help of a computer, but I was able to verify it with the highest possible scrutiny. It&#x27;s as if he wrote an impressive program without ever having run it, and it turned out not to have any bugs!<p>This offhand comment (which we can all forgive) makes it seem like mathematics happens in a vacuum. I can understand the temptation to think that way, and I hope I can make it clear that this is not the case.<p>General theorems typically grow out of a set of representative examples of interesting behavior (unit tests, if you will), are judged by their relationship to other mathematical objects and results (integration tests), and are reviewed by other mathematicians (user acceptance tests).<p>Mathematicians don&#x27;t just write down a proof and cross their fingers. The proof is often one of the last steps. There is a high-level intuition that explains why a theorem or proof sketch is probably true&#x2F;correct before a formal proof is found. These explanations are usually compact enough to help one re-derive proof (&quot;X doesn&#x27;t have enough &#x27;space&#x27; to fill Y, and Z tracks that gap&quot;, or something like that), but aren&#x27;t formal enough to put into any formal logic system I know of.