Written: March 16, 2010
In a recent BBC Horizon program (produced by Stephen Cooter and narrated by Steven Berkoff), I expressed, (21:30-24:52) my infinity-denying "controversial" mathematical world-view. Here is what I said:
"You can keep counting forever. The answer is infinity. But, quite frankly, I don't think I ever liked it. I always found something repulsive about it.
I prefer finite mathematics much more than infinite mathematics. I think that it is much more natural, much more appealing and the theory is much more beautiful. It is very concrete. It is something that you can touch and something you can feel and something to relate to.
Infinity mathematics, to me, is something that is meaningless, because it is abstract nonsense."
A few minutes later (30:45-31:25) came along the great infinitarian Hugh Woodin (of Woodin "cardinal" fame) who had the following reaction (my emphasis)
"To the person who does deny infinity and says that it doesn't exist, I feel sorry for them, I don't see how such view enriches the world. Infinity may be does not exist, but it is a beautiful subject. I can say that the stars do not exist and always look down, but then I don't see the beauty of the stars. Until one has a real reason to doubt the existence of mathematical infinity, I just don't see the point."
Of course, I didn't have a chance to rebut in that TV program, but luckily nowadays we have the internet, so I can reciprocate and express my extreme pity and hearfelt condolences to Professor Woodin for needing the fictional opiate (as Marx would put it) of the so-called infinity to keep him going.
Let me first pause and enlist an unlikely sympathizer with finitism, the greatest set-theorist of our time, Paul Cohen. In the second-to-last paragraph of "The Discovery of Forcing" (Rocky Mountain Journal of Mathematics, v.32 (2002), 1071-1100) he said (p. 1099)
"The only reality we truly comprehend is that of our own experience ... The laws of the infinite are extrapolations of our experiences with the finite"
So even the great "infinitarian" Paul Cohen was a devout finitist. But even though he strongly disagreed with Woodin (and Kurt Gödel and (practically!) infinitely many other people) about the ontology of the infinite, he totally agreed with him about the aesthetics, when he said (p. 1100):
"For me, it is the aesthetics which may well be the final arbiter. .... For me it is rather a paradise of beautiful results, in the end only dealing with the finite but living in the infinity of our own minds."
Here, I beg to differ even with my great hero Paul Cohen. I have studied some set theory and agree that it is pretty, but it is far from gorgeous. On a scale of 0 to 10, I would give it an 8. It is hideously ugly compared with some of the more beautiful combinatorial theorems. One does not need the "stars" to enjoy beauty. As Rutgers alumnus Joyce Kilmer has already said, he never saw a poem as lovely as a tree. [Hebrew translation], by Abraham Regelson (grandfather of Dror Bar Natan) I will go back to the subject of the beauty of trees later on, but first let me comment about the two "biggies" of "infinite" mathematics.
In my ultrafinitist weltanschauung, the great significance of both Gödel's famous undecidability meta-theorem, and Paul Cohen's independence proof is historical (or as Cohen would put it, "sociological"). Both are reductio proofs that anything to do with infinity is a priori utter nonsense, debunking the age-old erroneous belief of human-kind in the actual (and even potential) infinity. Granted, many statements: like "m+n=n+m for all (i.e. "infinitely" many) integers m and n" could be made a posteriori sensible, by replacing the phrase "for all" (when it ranges over "infinite" sets) by the phrase for "symbolic (commuting) variables (or rather letters) m and n". We have to kick the misleading word "undecidable" from the mathematical lingo, since it tacitly assumes that infinity is real. We should rather replace it by the phrase "not even wrong" (in other words utter nonsense), that cannot even be resurrected by talking about symbolic variables. Likewise, Cohen's celebrated meta-theorem that the continuum hypothesis is "independent" of ZFC is a great proof that none of Cantor's א-s make any (ontological) sense.
Going back to "beauty-bare", nothing in set theory rivals the beauty of André Joyal's lovely proof of Arthur Cayley's theorem that the number of labeled trees, Tn, on n vertices equals nn-2. It is so beautiful that I can say it in words.
Let's prove instead n2Tn=nn .
The left side counts doubly-rooted trees, which are labeled trees with a directed path (possibly of one vertex) of labeled vertices with some trees (possibly trivial one-vertex ones) hanging from them. On the other hand the right side counts the number of functions,f, from {1,...,n} into {1,...,n}. If you draw the directed graph of such a function joining vertex i to vertex f(i), you would get a collection of cycles with some trees (possibly trivial one-vertex ones) hanging from them. So the left-side counts "lines of numbers" with hanging trees, and the right side counts "collection of cycles" with hanging trees. But "lines of numbers" are permutations in one-line notation, and "collection of cycles" are permutations in cycle-notation. QED!
Now this is not just pretty, it is seminal. It lead to the gorgeous theory of Combinatorial species, that was the most significant use of Category theory, since it dealt with the only kind of categories worth studying, finite categories.
Of course, beauty is in the eyes of the beholder, and some parts of infinite mathematics are indeed a "9" (e.g. Greg Chaitin's utterly-fictional-yet-lovely Ω), but finite mathematics is both real (in the real sense of the word, not in the sense of so-called "real" numbers) and beautiful, while "infinite" mathematics is utterly fictional, and not-quite-as-pretty. I feel so sorry, and have infinite (pardon my French) pity and compassion for people who believe otherwise.