Wednesday, August 16, 2006

Smartest People You've Never Heard Of (Poincare Edition)

I'd been considering an occasional series about fascinating characters from intellectual history that are largely absent from the collective mind and with all of the fuss over the confirmation of the solution of Poincare's conjecture (for a nice discussion of the conjecture, see Good Math/Bad Math), it seemed to perfect not to start with Henri Poincare.

Jules Henri Poincare was born in 1854 in Nancy, into a prominent family -- his father was on the medical faculty at the University in Nancy and his cousin would become prime minister and President of the Republic. Where the word genius has become largely meaningless, Poincare fits the most rigorous of usages. He was a mathematical physicist, philosopher, and one of the greatest mathematicians in history. He made major and lasting contributions to many fields in mathematics -- indeed he is considered the last of the generalists, the last mathematician to be a force across the mathematical spectrum.

Indeed, it was this breadth of interest that led to one of his most famous contributions. He had been working on extending the result of his dissertation which dealt with differential equations, when he decided to go on a geological field trip for a diversion. The moment his foot hit the step to get on the bus, a thought shot through his mind, completely out of nowhere -- the equations he was playing with in order to solve a problem in algebra were actually identical in form to those which characterize the non-Euclidean geometry of Lobachevski. It was not a problem he had been working on. It was not a hunch that might be worth checking out. It was a sudden realization that arrived fully formed...and it was a biggie.

Euclidean geometry was the very epitome of what a mathematical system should be. It started with obvious undeniable truths (Euclid's five postulates) and through deductive logic derived with absolute certainty a wide range of stunningly intricate results. Philosophers for hundreds of years thought that it was the model that all other studies should follow. So when a new geometry was proposed by N. I. Lobachevski with a slightly different set of axioms and bizarre results that seemed like they couldn't be right, intellectuals across the academy treated it with open hostility. If it could be shown to be self-contradictory, then it would move from the annals of history to the anals of history. So a fullcourt press was on to find contradictory results derived from Lobachevski's postulates.

What Poincare's epiphany did was to create what is called a "relative consistency proof," that is he figured out a way of mapping the new axioms into the old system. The key is to understand that contradictions are a matter of the form of a sentence. "I have a bloog and I don't have a bloog" is a contradiction no matter what a bloog happens to be. Lobachevski's postulates are or are not self-contradictory because of how they relate their terms to each other, not because of what the terms mean. So Poincare figured out a way of giving the basic terms in Lobachevskian geometry new meanings derived from Euclidean geometry, but he did it in such a way that what Lobachevski's postulates said not only made sense in Euclid's system, but were true in Euclid's system. So since self-contradictoriness is a matter of form and sentences of the form of Lobachevski's postulates could be derived from Euclid's, the only way the new, hated non-Euclidean geometry could be self-contradictory is if Euclid was also self-contradictory...and no one would ever say that. So all of the hopes of saving the classical image of the construction of knowledge were vaporized the moment Poincare's foot hit the bus.

But Poincare was not only a mathematician, he also fancied himself a philosopher and thought about what his result meant. He realized that just as he could translate Lobachevskian into Euclidean geometry, so he could translate Euclidean into Lobachevskian. Any geometric fact could be moved back and forth between the systems, just as you could move distances between feet and meters, just as you could translate French sentences into German. The relative consistency proof gave Poincare a new picture for what mathematics was, the language for science. Choosing between mathematical systems is just choosing between languages, languages are not true or false they are what you use to express sentences which are true or false. As such, mathematical truths are mere grammatical conventions, free choices that have no deep meaning. Mathematical truths turn out not to be truths in any deep sense. So, not only had Poincare overturned the hopes of saving the classical approach to finding truth, but he had also radically undermined the classical notion of truth itself.

[This work was read by and highly influential on a certain Swiss patent clerk. Indeed, the claim has been made that if Einstein hadn't have and Poincare had lived longer, Poincare would have discovered the theory of relativity. We have no way of knowing whether such claims are true, indeed (contrary to what Hanno might say) such claims are likely not meaningful. But there it is.]

Poincare's work on the translation of geometry was based on his idea which he shared with/got from Felix Klein, that geometry is really a part of what mathematicians call group theory. A group is a set and an operation such that if you start with any member of the set and use operation, you get a member of the set. Take the even numbers and adding two -- add two to any even number and you still get an even number. Euclidean and non-Euclidean geometries just used different basic groups and it is a free choice which group you want to provide the language you use to describe reality. But Poincare realized that not all geometries could be put into the group mold. In an early piece, he derides these as not "real" geometries and mere oddities that are interesting to play with, but not meaningful. A few years later, suddenly they are real geometries...What happened?

What happened was that Poincare had virtually begun the study of topology, or analysis situs, as he called it. Topology is the study of what lies beneath geometry. Very roughly, think of geometry as the study of figures in space where topology studies the properties of the space itself. How many dimensions does space have? Is space infinite? Is it closed? These are all topological questions. Things like distances and angles that can be measured and have geometrical meaning are not worried about in topology. As long as you aren't ripping or connecting, you are dealing with topology. Squishing, stretching, twisting -- no problem.

It was in thinking about these topological relations that Poincare formulated his conjecture. Conjectures are guesses, but they are more like the stray dogs of the mathematical world. Conjectures may be true, they may be false, you just don't know. Some are mangy and ignored, but others are incredibly cute and you want to adopt them -- but you never know whether the dog belongs to someone and you never know whether someone is going to show up and take it away. Poincare's conjecture was like finding Lassie in a parking lot and what Russian mathematician Grigory Perelman just did was to let the mathematics community adopt her.