Thoughts following the Notices opinion article

by Orr Shalit

The December issue of the Notices of the AMS has quite a thought provoking opinion article by Doron Zeilberger. In fact I already read this piece earlier in Opinions of Dr. Z, but re-finding it in the notices re-kindled a feeling that I get so often: we (mathematicians) are lost. More precisely: we have lost contact with the ground.

I do not buy Zeilberger’s suggestion to abandon our “fanatical” insistence on rigorous proofs, nor do do I find his (fanatical?) vision of computerizing mathematics very alluring (this vision is not discussed in his opinion article. It is interesting in itself, more details can be found on his opinions page). Still I find his remarks refreshing and welcome; even though I don’t agree with every word he wrote, the math-world is so off balance that some extremely strong pulls in that direction are very needed.

Indeed and Amen, there is a lot more to mathematics than what pure mathematicians are trained to consider as important. I want to concentrate on one small example: computation. Most undergraduate and graduate students (of mathematics) I meet consider computations as a terribly boring chore, which is to be avoided at any cost, even at the cost of solving a different problem! By computation I do not mean-long multiplying numbers with dozens of digits by hand, I mean the science of obtaining quantitative values of interest, which requires understanding of the underlying mathematics, familiarity with the objects involved and their relations with other objects, it requires understanding precisely how transformations affect the problem, and it requires some good sense to see whether some transformation simplifies the problem or not; it requires sometimes thinking of a creative way of obtaining the quantitative values, and sometimes it requires some organisation and planning, and sometimes it requires the ability to think algorithmically, and sometimes – yes –  it requires also a basic set of skills (perhaps knowing how to long multiply). When I say computation I also mean the ability to make an approximate computation when an exact one cannot be made, and the ability to tell whether the approximate value obtained is any good, and how good. And by computation I do not mean only computing a number or representing analytically or asymptotically a solution to an ODE; computation also means computing a K-homology group, or computing the spectrum of an operator or a maximal ideal space, or computing a probability density.

It is not this particular skill or that particular trick which is the most important, but it is a state of mind, a point of view, a readiness. A mathematician should be ready and willing (lovingly) to compute.

People might shrug and say “a computer can do those computations”. Three answers to that. First: that’s not (entirely) true. Second: (to the extent that it is true) so what? A robot can play soccer or play music, that is not a reason not to learn how to do these things. And even before computers, many proofs were already written in books, so why learn them? And third: every day, people are programming computers to make computations, and tens of thousands (I don’t really know how much) are still employed and will continue doing this tomorrow.

It is very nice that students that come straight out of high-school are told that mathematics is more than just computations, but it looks like they are told that computations are not mathematics.

There are two problems in  undergraduate that programs put an anti-emphasis on computations. The first is that a graduate of such a program is completely useless from the point of view of any employer who is not academia (of course the individual might be clever and will be hired anyway). The second problem is that this graduate – who is trained to do one thing only and that is to continue studying pure mathematics – also has a lousy starting point for many (most?) kinds of research. Many graduate students learn secretly that computations are incredibly important in pure mathematics research, but they learn it “too late”, after they acquired a dislike to computation, and wasted the crucial early years in their training shaping their minds in a very unnatural way.

I bet that almost always, when scientists or engineers are stuck on a mathematical problem, and would like a mathematician to fall from the sky, they dream of a mathematician who knows how to compute things (how sad for them that even if two mathematicians would fall from the sky, neither would be able to help). I am not proposing at all to throw away rigorous proofs or the deep and beautiful conceptual constructions that we have in pure mathematics. But it is nice to recall once in a while where all this comes from, and what end does it serve.