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!
In a high school physics class, we were required to find the value of \pi by measuring the diameter and circumference of a cylindrical block. To me, this made sense as physics, because it taught us that we really could do experiments and measure things. But if that counts as experimental math, then count me out. That was not what I want in a math class.
On the other hand, if experimental math is just applied math -- solving real world problems by modeling them mathematically -- then I'm for it.
On rigor -- I agree that it is important. But it is not everything. I want students to develop some good mathematical intuition. But I also want them to know the importance of rigorously testing the ideas that come out of that intuition. What bothers me about a lot of textbooks, is that they don't have a good balance.
I'm a bit confused about how you would view proofs in a less "rigorous" situation in the K-12 (or more specifically the grade 9-12) ages. One issue is that we don't necessarily know "practical usefulness" until after we've gone through the process of learning logical rigor - you have to learn the tool *first* before you can reconsider situations where that tool isn't the most applicable.
To remove rigor from the later grades, and worse-still, to introduce "experimental" processes to kids at an age where they don't understand WHY it is experimental, is to invite the very same doubt in what they are taught that the creationists want to instill when they require "critical analysis of the evidence for and against". Kids would be taught an edge case without understanding why it is on the edge, and have no idea that it is. The result would be that it, even if false, might be accepted in their minds as being true. It would weaken their defenses against pseudo-education.
Intuition, even if *right*, must still be tested. The only test is to then be rigorous.
this is extremely true in computer programming, which is the most practical application of mathematics that is still accessible to people with a mere bachelors or lower.
As such, I would worry that a de-emphisis on rigor would result in a student body unprepared for the discipline required for proper programming, testing, and debugging - a reliance on intuition leads to missing a key detail in implementation (often written by someone else) that leads to program-crashing bugs.
As an example of an experimental approach, "How to Fold a Julia Fractal" introducing complex numbers:
(and other articles / videos on that site)
I don't see this sort of material as a substitute, I see it as the introductory chapter that is currently missing from curricula. It sketches out the playing field and gives you intuitive exposure to concepts that can later be formalized.
I think many people, the author of this article included, dismiss the idea of experimental mathematics without really understanding what it is.
As I understand it, experimental math is the practice of applying inductive reasoning, in the form of the scientific method, to mathematics.
For example, before Wile's proof of Fermat's last theorem, we could claim that it's truth is a theory (not theorem) with very strong experimental support since a counterexample was not found after checking many natural numbers. Scientific theories are not infallible in the same way as theorems, but that doesn't mean they're useless.
I've seen Zeilberger speak on the subject and he seems to believe that several very bright mathematicians are wasting their time in the pursuit of rigorous proofs for theorems that everybody is already 99% certain are true.
He used the example of Tom Hales' FlySpeck project to formalize his previous proof of the Kepler Conjecture. Zeilberger argued that Tom's brilliance is being wasted proving something that everyone already knows to be true. His time would be better off spent discovering new results. If Tom and other mathematicians used their talents to discover new mathematical results instead of in the pursuit of rigorously proving things we already know, then perhaps our mathematical knowledge would advance a lot faster than it is now.
It should be noted that Kurt Goedel himself was a proponent of experimental mathematics (at least according to my friend who is also a proponent of experimental math, and a reader of Goedel).
In case you are unaware, Zeilberger is a pretty awful source for opinions about mathematics. In particular, he's a finitist and not a very well thought-out one at that. He thinks that the vast majority of modern mathematics is worthless nonsense. Take, for example, this opinion of his where he speaks favorably of Wolfgang Mueckenheim, a well-known crank: http://www.math.rutgers.edu/~zeilberg/Opinion68.html
Or consider his viewpoint that we should replace education in calculus with education in "discrete calculus" and "difference equations": http://www.math.rutgers.edu/~zeilberg/Opinion115.html
It seems useful to me to separate between education and research. The latter obviously should be rigorous, the first much less. I show my kids (14, 15 years old) the Pythagoras Theorem by means of a concrete example:
Second picture with a = 3 and b = 4. Then I let the kids calculate the surface area of square c.
This is pure induction; it could hardly be less rigorous. Still it's completely convincing.
I certainly don't talk about how the Pythagoras Theorem relates to Euclides' axioms.
So I am certainly not averse to experimental mathematics. But my big question is always how we can concretize mathematical subjects. It's a pedagogical fact that kids up to 16 generally aren't capable of performing abstract operations. I refer to Piaget for instance.
Last year, Doron Zeilberger was one of the featured speakers at the undergraduate research conference JMU hosts every fall. During the talk, he mentioned, favorably, my book on the Monty Hall problem. So I'm not inclined to be too critical of him!
I don't think many people here understand what's experimental math. It's not measuring pi with a ruler or induction. Neither does it replace rigorous proofs. I would say in fact it offers a path of discovery to a proof, and is complementary to traditional proof-only methods.
Wikipedia definition is good:
For example, math at stackoverflow has many examples people using high precision numerics to offer insight into whether certain integrals can be computed analytically.
After having read through some quantum information papers, I would suggest: Experimental mathematics already exists, it is a subsection of theoretical physics.
"Mathematics is the part of physics where experiments are cheap." V.I. Arnold
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