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Monday, December 19, 2005

Harris on mathematics

While in Paris for the 'Impact of Categories' conference in October I met up with Michael Harris, an American mathematician based at Jussieu. He's a number theorist who has worked with Richard Taylor, the mathematician who helped out his PhD supervisor Andrew Wiles fixing the holes in the proof of Fermat's Last Theorem. Harris has been asked to contribute to 'The Princeton Companion to Mathematics', which is being edited by Tim Gowers. It should be quite a book when it appears. (No website for it yet, but several contributions can be found with Google.) Harris has made available a draft of his essay “WHY MATHEMATICS?” YOU MIGHT ASK . I like it very much, not only because it discusses my work, but also because it lends support to the notion that philosophers should be thinking about mathematics as a body of ideas, rather than as a body of truths.

Gowers' Survey articles and general lectures are worth reading too.


John Baez said...

I was going to write a couple of essays for Gowers' Princeton Companion - one on quaternions and Clifford algebras, one on octonions, and a bigger one called "Struggles with the Continuum", explaining in detail the sad fact that very few of our most successful theories of physics have been given a fully rigorous foundation... not even Newtonian gravity for point particles!

(I hope someone asks me what I mean by that last remark.)

But, I'm just too darn busy - I start more projects than I can finish. So, after two years of procrastinating, I finally admitted I wouldn't be writing these essays. Too bad... it sounds like it'll be a cool book.

By the way, I wouldn't call Michael Harris a number theorist; I'd call him an algebraic geometer. But that's because my only encounter has been with the famous textbook Principles of Algebraic Geometry by Griffiths and Harris, which pays heavy attention to complex manifolds.

In a review of this book on Amazon, someone charmingly writes: "With their bare hands, Griffiths and Harris prove some of the greatest results in maths."

January 28, 2006 7:45 AM  
david said...

I'd like to know what you meant by "very few of our most successful theories of physics have been given a fully rigorous foundation... not even Newtonian gravity for point particles!"

There has been some philosophical work by Jeremy Butterfield (remember he did some work with Isham on topos theory and QM) arguing against what he calls pointillisme (here and (here.

January 31, 2006 12:28 PM  
John Baez said...

David wrote:

I'd like to know what you meant by "very few of our most successful theories of physics have been given a fully rigorous foundation... not even Newtonian gravity for point particles!"

Nobody has proved that for almost all initial conditions the Newtonian n-body problem has a global solution - i.e., a solution for all time.

There are clearly some initial conditions that run into singularities. The simplest case is a collision of two particles.

In fact, this sort of collision is not a big problem, since you can continue the solution through the singularity by letting the particles go through each other. (Or, alternatively, having them bounce off each other elastically.)

Collisions of three or more particles are trickier.

But, what's more shocking are the
non-collision singularities where several particles shoot off to infinity in a finite amount of time! For example, two pairs of heavy particles can toss a light one back and forth, faster and faster as the two pairs shoot off in opposite directions, with the speeds of all five particles becoming infinite at some moment.

Now, your natural reaction to these pathologies should be to say that they're very unlikely: they require carefully chosen initial conditions.

But, nobody knows if this is true!

In other words, nobody has proved that except for initial conditions forming a set of measure zero, the Newtonian n-body problem has a smooth solution that lasts for all time.

This is ironic because Newtonian gravity is supposed to be the paradigm of the deterministic world-view. We've all read that quote by Laplace where he says:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

But, for point particles obeying the laws of Newtonian gravity, nobody has shown that their present state determines a well-defined future state for all times! (Except perhaps for a set of measure zero: we certainly need at least this caveat.)

And, most of our more sophisticated later theories of physics suffer from even greater mathematical problems. The only one I feel quite satisfied with is the quantum version of the n-body problem: Schroedinger's equations for a bunch of charged point particles interacting by Coulomb's law. The Kato-Rellich theorem says this is okay. But this doesn't take relativity, spin, the finite size of nuclei, and a bunch of other stuff into account.

On a different note, I may have been thinking about a different "Harris" than yours....

February 01, 2006 10:52 AM  

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