At the turn of the 20th century, the presiding mind of the mathematical world, David Hilbert, drew up a list of the 23 major unsolved problems and unanswered questions to which his colleagues might best devote their efforts and their hopes. It was a list born of idealism; for dissimilar as they were on the surface, a thematic undercurrent ran through many of them: that mathematics could escape from its processes of intuition, insight, and guesses of genius to arrive at definite, methodical techniques to solve problems and prove theorems. Inspiration, always unreliable, would be replaced by mechanics, freeing mathematicians from the drudgery of separating truth from falsehood.
Mathematicians answered the call. Problem 3, for instance, ("Give two tetrahedra which cannot be decomposed into congruent tetrahedra directly or by adjoining congruent tetrahedra") was answered in the negative in 1902, with the ink hardly dry on Hilbert's list. But the resounding reply came when Gödel knocked problem #2 off its pedestal ("Can it be proven that the axioms of mathematical logic are consistent?") in 1931. The answer was a loud "No", which, together with the demise of #10 ("Does there exist a universal algorithm for solving Diophantine equations?"), put mathematics squarely back in the realm of men and not machines.
Perhaps in an effort to recapitulate the successes and surprises that were brought about by Hilbert's challenge, the Clay Mathematics Institute put forth a new list exactly a century later, in May of 2000: the seven Millennium Problems. The motivation was less idealistic than material this time around, since a prize of a million dollars has been assigned to each. Less thematic and more varied than Hilbert's, these seven nevertheless are at the peak of difficulty and mathematical interest, and seemed quite simply the hardest problems in the world at the time.
It now appears that the seven unsolved problems are to soon be six. Grigori Perelman's proposed proof of the Poincaré Conjecture (Clay problem #5) has withstood all scrutiny since it was put forth last year. Like the first of Hilbert's problems to fall, Poincaré's is also topological, concerning the basic nature of n-dimensional solids. He formulated his conjecture in 1904, just missing his chance to make the original list. If Perelman's proof endures another year without refutation (the Clay Institute is slow to part with its millions), the prize and the honor go to him.
If so, it will be a quiet victory indeed. Perelman hardly came out of his study long enough to explain his work, much less bask in the limelight. At the same time, the answer to the Conjecture ("Yes"), will not shake the world of mathematics as Gödel's did. These problems are indeed at the pinnacle of math, but this means they are not at its foundations the way some of Hilbert's questions were. They are hard to prove, but unlikely to harbor surprises.
There is at least one exception, though. It's the oldest problem on the short list of seven, and the only one to appear on Hilbert's list as well. It was unsolved then, and it's unsolved now. It's widely assumed to be true, but the discovery of just why holds forth the promise of new and interesting mathematics along the way.
It was #8 on Hilbert's list, and it reads simply "Prove the Riemann Hypothesis". Riemann proposed his fundamental connection between prime numbers and the complex roots of an infinite sum in 1859. Mathematicians have been at work on it ever since. Its attraction lies in the surprising bridge it builds between two very different types of mathematical objects, and the fact that on one side of that bridge are all the prime numbers, on which it may be fairly claimed that arithmetic itself is built. Its proof will be worth a million dollars, and require all the intuition and lucky guesswork that can be mustered. And it may bring some satisfaction to Dave Hilbert, whose hopes were so rudely dashed back in 1931, to see another one of his problems answered at last.
Poincaré Conjecture progress, and problems from Hilbert and CMI
Terrestrial life, in its exotic extremes, continues to outstrip in oddity anything that has been proposed for the extraterrestrial. Here's terrific story from Natural History Magazine on an enigmatic ocean floor life form, Paleodictyon nodosum, whose last known examples on Earth are fossils from fifty million years ago — fossils in the shape of a perfect hexagonal array. It survives in the company of undersea life prospering on nothing but geothermal action and the chemicals released by seawater encountering the heated crust beneath. It is so unlike anything else on the planet that it revives an old question: if we find life elsewhere in the universe, how will we recognize it?
Word Count is a web program I came across that visually displays the words of English in order of frequency. As such, it's a graph of language, and by implication a sort of map of the peaks in our mental landscape.
The list is a stable and familiar one. The stands alone at the top; of and and are close behind, words nearly indispensable for a sentence (for this one, at least).
A chart of the familiar seems, by definition, to contain no surprises, but I decided to browse down this list to find what it was that we talk about the most: the first word in the list that's an indisputable noun.
It's a long way down the list to things. We're not too eager to get specific in English, it seems. That is number 7, and it's just an indirect reference, followed by it and preceded by a gaggle (gaggle is at 41,738) of connectives such as in and to. Apparently we're builders; we use structuring words more than concepts, or even verbs; lots of nails and little lumber.
Pronouns start at 11, and start, of course, with I. But if we put ourselves first, you is not far behind. He is next (15), but it's a long way from there to she (30). And yet still there's nothing that could be called an idea, a topic of conversation. What is it that I, you, he and she are so busy being of, in, for, with, and about? What's on our minds?
It's surprisingly hard to predict. It's not money (227). It's not God (376). It is not good (116) or evil (3,274), nor day (141) or night (230). Yet it is everywhere and nowhere at once, invisible and ever-present. It is number 66 in the list.
It is time.
It seems obvious in retrospect: it is always time for something; we always want to know the time; there is never enough of it. Time comes even before me, and the next noun in the list, hard by at 74, is now. Time seems to be the atmosphere in which our thoughts move. It's not until number 81 that the second thing enters our list: people.
Time and people: there you have it, existence in a nutshell, a surprisingly abstract result to fall out of such cold, hard facts as a word list. Neat tool, with more no doubt waiting to be discovered in it.
See the Word Count program for yourself.