Just read Doron Zeilberger’s 72nd Opinion. The man is quite a bit pretentious (“mathematician is really another species, higher than homo sapiens”) though he doesn’t lose one bit of respect from me.
Back in 2002, I had a discussion, in a cafÃ© in the old town of Saint Malo, where I basically tried to convey the following message, but not as well put:
(…) most of human mathematics is completely useless. It was developed by humans for human consumption. In order for humans to understand it, it had to proceed in tiny steps, each comprehensible to a human. But if we take the “mesh size” of each step, dA, to be larger, one can do potentially much bigger and better things, and the computer’s dA is much larger, so we can (potentially) reach a mountain-top much faster, and conquer new mountain-tops where no humans will ever tread with their naked brains.
So this activity of computer-generated mathematics is the future. Unfortunately, many human mathematicians still don’t realize the importance of this activity, and dismiss it as “just a computer program” and “no new mathematics”.
At the time, the mathematician I was talking to, a man I greatly respect, objected that nobody could predict what could be useful, so to claim that non-computable math. is worth less than computable math., was just foolish. It managed to silence me. Indeed, while I believe that algorithms are a higher form of mathematics, I cannot prove that it is, and neither can Zeilberger, but he makes a great case for it:
[computable math] is a methodology that will make all computer-free math obsolete very soon.