Written: Sept. 17, 2006.
I recently got back from a two-day conference in honor of Pierre Leroux, that took place in Montreal last weekend. All the talks were great, but the one I particularly liked was by Adriano Garsia, who talked about his current work with collaborators.
All of Adriano's sentences I liked, but I particularly liked the sentence (that he felt so strongly about that it was uttered in English, the rest of the talk being in French)
``Cauchy ruined mathematics! Let's throw out all that epsilon-delta nonsense.''
He was referring to the brilliant young mathematician Guoce Xin's work on
A Fast Algorithm for Macmahon's Partition Analysis , that uses iterated formal (and hence elementary!) Laurent series as the basis for MacMahon's Partition Analysis. MacMahon's Partition Analysis was revived and systematized by George Andrews who was later joined by Peter Paule and Axel Riese who developed a beautiful Mathematica package OMEGA that solved lots of interesting enumeration problems.
I liked this work very much in general, and naturally and justifiably, so did George Andrews. But there was one aspect of it that I didn't like, and sure enough, it was that aspect that George Andrews seemed to like the best. George claimed that one can not get away with formal power series, but needs the full heavy gun of analytical convergence, a la Cauchy and Weirerstrass. Of course, deep in my heart, I knew that he was wrong, and that while you may need the latter (for now!) in order to do asymptotic enumeration, it does not make philosophical (or any) sense that you would need it for exact enumeration. But gut feelings are no proofs, so I had to give him the benefit of the doubt.
Then came the brilliant Guoce Xin (that I am willing to bet you would hear about in years to come, if you haven't yet), and showed that indeed one does not need all that heavy (and boring!) artillery. Even more amazingly, using his new, elementary, approach, is much more computationally efficient than the original approach of Andrews-Paule-Riese that used "analytical" power-series.
When I told this to my young colleague and collaborator at Rutgers, Drew Sills, he told me that I am not the only one who had the gut feeling that analysis could be dispensed with. Drew was co-advised, at the University of Kentucky, by both George Andrews (from Penn State, his "real" advisor) and Avinash Sathaye (his local, "co-" advisor). When George came, back in 1999, to participate in Drew's oral exam, he was asked to use this opportunity to give a colloquium talk, where he talked about MacMahon's Partition Analysis and that it is a great example that analytical convergence is sometimes indispensable. After the talk, Drew's co-advisor, Avinash, said that while he can't prove it formally, he is sure that there is a way to stay within good-old high-school-algebra. In that, Avinash was being a true disciple of his illustrious academic father, Shreeram Abhyankar, the modern prophet of high-school-algebra, who coined that term.
Speaking of Abhyankar (one of my great heroes), I vaguely recall that he once said that in Indian mythology there is a character who always thought that he was a slave, and only very late in life realized that he was free all along (I am paraphrasing obviously, but I hope that it is roughly right), and this is analogous to the long misconception that for power-series proofs in combinatorics, number theory, and algebra, one needs analytical convergence, as opposed to purely high-school-algebra formal-power-series convergence.
BUT this is just one example, of our servile belief that we are slaves of analytical and other high-powered high-brow paradigms and tools. I am willing to bet that in the future, especially with the help of our computer friends, there would be more and more examples that we are free after all, and we don't have to bow down to those arrogant masters.
Now, what if you like to be a slave (what the bible calls eved nirtza)? Then good for you. You are welcome to wear a tie, if it makes you happy, and makes you more comfortable. But please don't impose your dress-code on me, and claim that a tie is an indispensable article of clothing, since myself, I am much more comfortable with an open collar.