Computer-free math is obsolete
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 café 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.
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Would that mean that mathematics is nothing but number crunching. Mind-boggeling complex number crunching, but number-crunching nonetheless? Is there any place for creativity in this process? How does that fit in?
(I left most of my mathematics behind with my BSc over ten years ago, so it is not surprising that I would have more questions then answers.)
Comment by Ed Bilodeau — 14/6/2006 @ 7:47
Zeilberger is most certainly not a number cruncher. He says quite the opposite, stating that mathematicians are a superior race.
What he comments on, is the fact that if you are not using computers to do math, you are exploring the world using your own two feet instead of using a jeep and a GPS.
Notice that most, if not all, mathematician use computers these days to assist, at least, with the algebra. But it would be akin to the jogger who takes his car to drive to the gym: mathematicians still, very much, do the same type of math, with the lemma/theorem/proof routine. However, Zeilberger goes a step further and says we have to move to a new type of mathematics, just like going from the marathon to the rally.
This is partially motivated by the fact that journals won’t publish his work, and he claims that it is because he is ahead of his time.
(Don’t get mixed up and read that Zeilberger is not a published mathematician, this is not what I’m saying!)
Comment by Daniel Lemire — 14/6/2006 @ 9:03