In Tuesday’s post I started discussing this essay (PDF format), by mathematician Doron Zeilberger. I wholeheartedly seconded the sentiments from the first part of the essay, in which he lamented the generally poor state of mathematical communication.
But I’m a little skeptical of this part:
The purpose of mathematical research should be the increase of mathematical knowledge, broadly defined. We should not be tied up with the antiquated notions of alleged “rigor”. A new philosophy of and attitude toward mathematics is developing, called “experimental math” (though it is derided by most of my colleagues; I often hear the phrase, “It’s only experimental math”). Experimental math should trickle down to all levels of education, from professional math meetings, via grad school, all the way to kindergarten. Should that happen, Wigner’s “unreasonable effectiveness of math in science” would be all the more effective!
Let’s start right now! A modest beginning would be to have every math major undergrad take a course in experimental mathematics.
As I will discuss further in a future post, I share Zeilberger’s distaste for excessive rigor in mathematics education. But I don’t think an emphasis on rigor is antiquated. Someone has to worry about the logical details. Otherwise it’s too easy to delude ourselves that we have found the truth when really we have not.
As for experimental mathematics, I’m not sure what such an undergraduate course would look like. I’d like to see Zeilberger’s proposed syllabus. The experiments have to be about something, after all, and you need some understanding of the underlying abstractions before you know what experiments to carry out.
To a certain extent, experimental mathematics is already part of virtually every course in mathematics. It is the rare math professor who does not emphasize the importance of working out concrete examples in solving problems, and that is really all experimental mathematics is. In the past twenty years or so, the availability of extraordinary computing power has made it possible to carry out experiments of a thoroughness and complexity that was previously unthinkable. That is why it has almost become a discipline unto itself. But it does not render obsolete our usual standards of logical rigor, and it does not introduce any new notion of mathematical truth. It is a useful tool for mathematicians to use in their research, but it does not change the discipline in any fundamental way.
Here’s Zeilberger’s closing paragraph:
Please don’t misunderstand me. Personally, I love (quite a few) rigorous proofs, and it’s okay for anyone who loves them to look for them in his or her spare time. However, for the research and teaching that we get paid for, we should adopt a much more open-minded attitude
to mathematical truth similar to the standards of the “hard” physical sciences. We need to abandon our fanatical insistence on “rigorous” proofs.
If we are talking about education, then I agree with that closing sentence. But for researchers I do not see how it is workable. If we eliminate an insistence on rigorous proofs, then with what do we replace it? Does everyone get to decide for himself when the experimental data is sufficient to establish truth? Research would be a lot easier if that were the case!
Or maybe Zeilberger’s idea is that a particular mathematical model can be tested against experiments conducted in the real world. For example, Isaac Newton was able to obtain demonstrably correct results, even though his work on calculus was riddled with logical infelicities. Euclidean geometry is undeniably useful, even though Euclid’s Elements fall short of modern standards of rigor. Perhaps practical usefulness should replace logical rigor as our communal standard.
But where does that leave pure mathematics? We pure types put so much emphasis on logical rigor because the objects we study are entirely abstract. They have no real-world existence. So how can we say anything about them beyond what logic can tell us?
In short, if Zeilberger had presented his argument with greater rigor, I would be better able to decide what I think of it!