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. Infinitymay bedoes not exist, but it is abeautifulsubject. 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.
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, T_{n}, on n vertices equals n^{n-2}. It is so beautiful that I can say it
in words.

Let's prove instead n^{2}T_{n}=n^{n} .

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.

Added July 25, 2010: People who Love or Hate or LoveAndHate or HateAndLove the infinity will probably enjoy Brian Rotman's masterpiece ad infinitum

Opinions of Doron Zeilberger