Written: June 6, 2008.
Arthur Benjamin is deservedly famous for his feats of mental calculations, that appeal both to mathematicians and to general audiences. But unlike many calculating prodigies, who often are idiot savants, Art is also a top-notch mathematician, and is as successful at his "day job", as a professor of mathematics at Harvey Mudd College, as he is in his "night job" of doing mental arithmetics. Unfotunately, as I will point out in this opinion, some of his talent is wasted. The same can be said about many other mathematicians, and I decided to "pick" him- as a case study- partly because of his celebrity, and partly because that I am sure that he can take some criticism.
Of course, Art is a real phenomenon, not only in his "mental arithmetics" but his general mathematical persona. His inimitable style of writing, teaching, and lecturing, conveys his contagious enthusiasm for doing mathematics, and shows, by example, that math should and could be fun.
When Art is doing one of his mathemagics shows, he does not explicitly state:
"All the calculations that I am doing in my head could be done even faster with electronic calculators, and the mathematical content of addition, multiplication, square-root extractions etc. is today, and has been for the last ten thousand years, purely routine. Doing sums or products in one's head is a challenging human sport, but it is not serious mathematics research, and hence should be labeled for entertainment only ."
Of course, it is hardly necessary, since it is so obvious, and one can't accuse him of intellectual dishonesty for not posting the above disclaimer. Everyone knows this, even the most "general" public. But, it is far less obvious, to many people, including Art himself, that many of the "theorems" given "elegant combinatorial proofs" in his attractive book, with Jenny Quinn, Proofs that Really Count, and many of his proofs in his articles with collaborators and students (see his website), are also, by today's knowledge, just as routine as, say, 11x14=154. While he is welcome to give a three-page "insightful" direct combinatorial proof of, for example, Σk binomial(2n,k) F2k=5nF2n , I believe that intellectual honesty requires that he should state, at the very beginning of the article, that this proof is "for entertainment only", since this identity-and tons of others like it- are today routinely provable by computers (and often even by humans). In fact, for this particular identity (and for many similar ones), as I show in pages 10 and 11 of my article, it suffices, in order to give a completely rigorous proof, to check the identity for only the first four special cases, n=0,n=1,n=2,n=3 .
Many years ago, I once said, in a talk,
"A Direct Combinatorial Proof is like Sex. If it is good, then it is great. If it is bad, then it is still better than nothing".I am no longer so sure. I still think that Direct Combinatorial Proofs are like Sex, except that both bad sex and bad combinatorial proofs are worse than nothing. A bad, long-winded combinatorial proof, for which there already exists a short and purely routine algebraic proof, is the biggest waste of time, both of the readers (if they exist), but especially of the authors. It is really pathetic to what lengths some people will go to "explain" and give "insight" to such routine trivialities, just because the proof belongs to the genre of "direct combinatorial proofs". It is also a great misunderstanding that algebraic and even "analytic" proofs are somehow philosophically inferior to direct combinatorial proofs, since the latter are more direct. Everything is combinatorics, since mathematics is just the manipulation of finite symbols. Also, as Greg Chaitin famously said, "if the explanation is longer than the things that is being explained, it is not much of an explanation".
Furthermore, there is a uniform method, described in the article, by Phillip Matchett Wood and myself, A Translation Method for Finding Combinatorial Bijections, that uniformly translates "ugly algebraic proofs" to "gorgeous combinatorial proofs".
One example of a great waste of time (in my opinion) is the
article, by Benjamin, Alex
Eustis and Sean Plott who go to great lengths in order to prove the above- mentioned identity.
( Σk binomial(2n,k) F2k=5nF2n , ).
But the "Ig Nobel" should go to the article
Combinatorially Composing Chebyshev Polynomials,
by Art Benjamin and Daniel Walton. They take eleven pages to find an "insightful"
combinatorial proof of the composition formula for Chebyshev polynomials
Tm (Tn(x))=Tmn(x), where Tn(x) are the Chebyshev polynomials, defined by T0(x)=1,T1(x)=x, and for n ≥2, Tn(x)=2xTn-1(x)-Tn-2(x) .
This is an utter triviality that can be proved in two lines. First observe that
Tn((z+1/z)/2)=(zn+z-n)/2 , since both sides satisfy the recurrence An=(z+1/z) An-1-An-2 , (and the initial values at n=0,1 match). Hence
Tm (Tn((z+1/z))/2)= Tm ((zn+ z-n)/2) =((zn)m+ (zn)-m)/2 = (zmn+z-mn)/2=Tmn((z+1/z)/2) QED.
Now don't get me wrong. There are lots of gorgeous proofs in the above-mentioned Benjamin-Quinn book, and the numerous papers by Benjamin and his coauthors (those that take less than one page). But those proofs that take more than one page are "too much of a good thing", and not worth the trouble. One should not fall in love with the hammer. In mitigation, one should admire the ingenuity of the authors (some of them are unergraduates), and in the sense of sharpening one's proof-muscles, and as etudes in that genre, there are not really a waste. Also, if one gets hooked on direct combinatorial proofs, and loves them for their own sake, then by all means, it is a harmless pastime to design them and to read them, but one should be honest and explicitly state, at the very first sentence and also in the abstract:
"We give an elegant, albeit long, direct combinatorial proof to the following result that can be quickly and/or routinely proved by plain algebra."