I have had the pleasure of reading over the years a great many of Doron Zeilberger’s notorious Opinions, and I have a great appreciation for his devotion to experimental mathematics! I agree in particular with his opinions that much of mathematics is empty symbol pushing, leading to nothing but expanding the egos of its authors. However, Dr. Z identifies the reason behind this to be the embrace of infinity in mainstream mathematics. This I believe is preposterous. It is entirely possible to do absolutely useless mathematics while staying firmly in the realm of the finite!
For instance, in What is Mathematics and What Should it Be, Zeilberger holds up Ramanujan’s identity
where (a)_k=a(a+1)\ldots(a+k-1) as “in the eyes of God”, much prettier than Euler’s formula
In defense of this, he says that Ramanujan’s formula enables a very fast computation of \pi to billions of decimals. However, NASA uses only 15 digits of pi even in computations which need to be astronomically precise! Computing billions of digits of \pi is an utter waste of time, useful only to provide entertainment to GPU clusters or give children practice on how to memorize numbers!
On the other hand, while Euler’s formula is an embarrassing triviality, the theory behind it is a beautiful connection between complex arithmetic and planar geometry, as Tristam Needham’s excellent book Visual Complex Analysis explains. I use this connection in Semagrams to avoid having to remember any names for rotating, translating, scaling in 2d – I just use complex numbers! Although it is not a traditional criterion of beauty, I see saving the working programmer keystrokes as a noble end.
Dr. Z’s criterion in “What is Mathematics and What Should it Be” for what is good mathematics is the following.
Every statement that has a proof from the book, is ipso facto, trivial, at least, a posteriori, since all deep statements have long proofs. In my eyes the most beautiful theorems are those with succinct statements for which the shortest known (and hopefully any) proof is very long.
However, it is precisely this criterion that leads to all of the ridiculous ego-driven symbol pushing in mathematics. People want to find these “chestnuts”, these simple statements with tricky solutions, to show off how smart they are. This criterion implies that a proof in mathematics is only worthwhile if I can come up with it, but you can’t. Any mathematics that naturally follows from easily understandable principles is “trivial” and hence worthless!
With this in mind, where does Zeilberger see the future of mathematics?
Mathematics should become a science, and its main raison-d’etre should be the discovery of mathematical truth (broadly defined!).
I disagree! I think that the goal of mathematics is not to become a science, but rather to serve the sciences, to provide scaffolding and rigor so that the scientist may rest her feet upon solid ground as she reaches towards the stars, and have “abstract molds” at hand in which she can pour concrete structures into. The most useful mathematics for the scientist may have “trivial” proofs, which do not serve the purpose of aggrandizing the ego of the mathematician who discovers them. But that is not the point of mathematics; mathematicians should be happy to be forgotten so long as their theories drive the creations of wonders for the eons ahead.
And it is by this criterion that I also warn against a dogmatic rejection of infinity. I suspect that I agree with Dr. Z that the ultimate test of a mathematical theory is whether it can be implemented on a computer. In this case, I would claim that there is plenty of “infinity-based mathematics” that is perfectly applicable to computation! A large part of mathematical physics, which depends on infinities for the continuum and for randomness, falls into this category. If it is convenient to work with infinity, and the results end up being useful to computation, who is to say something wrong has happened? It is often easier to work with things “in the limit”, with the finite-approximation case analyzed afterwards, as finite approximation brings its own complications that can clutter the intuition.
Finally, I would like to defend my own area of study, category theory, from an imagined dismissal on grounds that all of the theorems are trivial. I got into category theory, and through category theory, the rest of mathematics, via getting into the programming language Haskell in highschool. Category theory is an incredibly powerful tool for organizing computation, and I continue to use it daily in my work in AlgebraicJulia to serve the scientist. I would encourage Dr. Z to reevaluate what he finds important in mathematics, and to encourage his students to use their skills not just to prove impressive theorems in order to get tenure, but to look outwards and see how their clever minds can be put to work on the problems of humanity.