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Miscellaneous Comments About Mathematics
ULAM: A French philosopher whose name I forget said that nowhere has the human mind shown itself so inventive as in devising new games.
ROTA: Inventors of games are always anonymous. Why? What is your philosophy of the anonymity of games?
ULAM: Probably other people quickly perfected the original invention, and it is difficult to find out who thought of it first.
Are games part of combinatorics or the other way around? I claim that much of mathematics can be “paisaised," a Greek word which means to play.
Here is an example of a problem inspired by a game. Suppose n is a given integer and we are to build, you and I, two permutations of n letters. We construct them in turn as follows: For the first permutation I take n1, you take nɔ, I take nз, and so on. Finally we get a permutation. Then we play for the second permutation. If the two permutations generate the group of all permutations, I win; if not you win. Who has a winning strategy in this game? I don't know.
If we do it at random, what is the chance [that there is a winner]? This then becomes a combination of measure, probability, and combinatorics. I talk about this racket in my book of problems. It is amusing, isn't it? It can be done in any branch of mathematics.
ULAM: Combinatorics is devoid of general methods. It is full of nice individual curiosities, it is Erdösian. I have nothing against it, it is amusing. But it throws no light on anything else.
ROTA: You are not being fair.
ULAM: Complex functions, the idea of entropy are broader. Ramsey's theorem, interesting as it is, is like progress in zoology when a new species of insects with one red eye and one green eye has been discovered!
ROTA: Ramsey's theorem tells more about the nature of sets than all the axioms of set theory!
ULAM: It is one of numerous properties of infinity. Why take two sets of pairs and divide them into two classes? My master's thesis already contains that sort of thing.
Some problems, big or small, are solved with a bang; they open new vistas. Others are solved with a whimper, in a way which is very specific and leaves nothing to be said or asked, regardless of whether the problems are important or interesting.
Ramsey's Theorem One consequence of Ramsey's theorem is the following: Among a gathering of 6 people, there 'will be at least 3 all of whom know one another or else there will be 3 none of whom know one another. This is not true if only 5 are gathered together. In general, for each positive integer k there is a positive integer n = n(k) such that if n people are gathered together, then there will be k all of whom know one another or else none of whom know each other. To this date we know only that n(k) exists but not its value for arbitrary k. It is known, however, that n(2) = 2, 1 n(3) = 6, and n(4) = 18.
ULAM: Set theory revolutionized mathematics. It is largely the work of Cantor. What made set theory is the fact that Cantor proved that the continuum is not countable. It is hard to imagine that a field that arose from trigonometric series quickly transformed the shape and flavor of math.
Paris April 1972
ULAM: A second landmark on the scale of centuries was Gödel's undecidability theorem.
Mathematics is not a finished object based on some axioms. It evolves genetically.
Paris May 1972
ROTA: Can you list ten unsolved problems in mathematics which you consider important?
ULAM: First, the continuum hypothesis. If you take the existing axioms for set theory, then it is independent.
ULAM: But the existing axioms are probably not enough to give expression to our intuitions about sets. In that sense the continuum hypothesis is not a closed story.
Two. In number theory, any problem is as good as any other. I don't know which to choose, the infiniteness of twin primes or the Goldbach conjecture. The fact that they are very difficult and so simple makes them in my opinion very important. I have to list the Riemann hypothesis because it has so many consequences, although it is not one of my favorite problems, for a reason which I cannot express.
ROTA: Would you list the Riemann hypothesis as third?
ULAM: I don't like to order them. Snobbism plays a role in the ranking of mathematical problems. By chance some so-called great mathematician mentions something. For example, out of Hilbert's marvelous twenty-three problems, several would not be considered important if it were not for the fact that it was Hilbert who proposed them! Now what would you say besides these?
It is like asking someone to please mention ten best dishes or paintings! I don't know whether any single problem is really important, except in foundations of set theory. They are mainly important for what they suggest or allude to. Think of Fermat's conjecture. It is important because it is difficult but probably also because whoever will solve it will have found some new trick or method. The important thing
is that the break is simple and difficult. I came to this conclusion sort of gradually. I am being honest, which most people are not.
A great problem is: Why are some problems sometimes difficult to solve? That is metamathematical, but it may some day be mathematized. The notion of complexity is beginning to be made precise, and what I just said will become a super problem. ROTA: Why should Goldbach's conjecture be more interesting than a Chinese puzzle? ULAM: Because it is simple. Any child can understand it. Isn't it curious that a child can ask questions about numbers that no mathematician can answer?
ULAM: Why is it that calculus, which deals with limits, is so effective? Or why are asymptotic theorems so much simpler than finite approximations? Infinity does not correspond to the popular image. It is a guiding light, a star that draws us to finite ways of thinking, God knows why.
ROTA: What is the value of mathematics?
ULAM: Value? In what sense? In what market?
It has value because it trains the brain. Just like in any other game, practice sharpens the organ. I don't know if today mathematicians' brains are any sharper than in the time of the Greeks. Yet I think mathematics plays a genetic role. It is one of the few ways to perfect the brain, to perhaps develop new connections in the brain. It has a peculiar sharpening value. Nothing could be more important. I don't know if any other science plays the same role. Another value is the aesthetic one, which is for the practitioners.
ROTA: What is its ugliness? Could you state an ugly theorem?
ULAM: Ugliness lies in the fact that one has to be punctilious, make sure of every step.
ROTA: Compare mathematics to the classics as an educational technique.
ULAM: I would say they are complementary. Latin grammar is good training in logic, not Boolean logic, but relational logic.