Before we can answer this question we need to agree what we mean by proof. (This is one of the reasons why its good to have mathematicians around. We like to begin by giving precise definitions of what we are going to talk about, a pedantic tendency that sometimes drives our physicist and engineering colleagues crazy.) For instance, following Descartes, I can prove to myself that I exist, but I can't prove it to anyone else. Even to those who know me well there is always the possibility, however remote, that I am merely a figment of their imagination. If it's rock solid certainty you want from a proof, there's almost nothing beyond our own existence (whatever that means and whatever we exist as) that we can prove to ourselves, and nothing at all we can prove to anyone else.
Mathematical proof is generally regarded as the most certain form of proof there is, and in the days when Euclid was writing his great geometry textElements that was surely true in an ideal sense. But many of the proofs of geometric theorems Euclid gave were subsequently found out to be incorrect—David Hilbert corrected many of them in the late nineteenth century, after centuries of mathematicians had believed them and passed them on to their students—so even in the case of a ten line proof in geometry it can be hard to tell right from wrong.
When you look at some of the proofs that have been developed in the last fifty years or so, using incredibly complicated reasoning that can stretch into hundreds of pages or more, certainty is even harder to maintain. Most mathematicians (including me) believe that Andrew Wiles proved Fermat's Last Theorem in 1994, but did he really? (I believe it because the experts in that branch of mathematics tell me they do.)
In late 2002, the Russian mathematician Grigori Perelman posted on the Internet what he claimed was an outline for a proof of the Poincare Conjecture, a famous, century old problem of the branch of mathematics known as topology. After examining the argument for two years now, mathematicians are still unsure whether it is right or not. (They think it "probably is.")
Or consider Thomas Hales, who has been waiting for six years to hear if the mathematical community accepts his 1998 proof of astronomer Johannes Keplers 360-year-old conjecture that the most efficient way to pack equal sized spheres (such as cannonballs on a ship, which is how the question arose) is to stack them in the familiar pyramid-like fashion that greengrocers use to stack oranges on a counter. After examining Hales' argument (part of which was carried out by computer) for five years, in spring of 2003 a panel of world experts declared that, whereas they had not found any irreparable error in the proof, they were still not sure it was correct.
With the idea of proof so shaky—in practice—even in mathematics,answering this year's Edge question becomes a tricky business. The best we can do is come up with something that we believe but cannot prove to our own satisfaction. Others will accept or reject what we say depending on how much credence they give us as a scientist, philosopher, or whatever, generally basing that decision on our scientific reputation and record of previous work. At times it can be hard to avoid the whole thing degenerating into a slanging match. For instance, I happen to believe, firmly, that staples of popular-science-books and breathless TV-specials such as ESP and morphic resonance are complete nonsense, but I can't prove they are false. (Nor, despite their repeated claims to the contrary, have the proponents of those crackpot theories proved they are true, or even worth serious study, and if they want the scientific community to take them seriously then the onus if very much on them to make a strong case, which they have so far failed to do.)
Once you recognize that proof is, in practical terms, an unachievable ideal, even the old mathematicians standby of GÏdel's Incompleteness Theorem (which on first blush would allow me to answer the Edge question with a statement of my belief that arithmetic is free of internal contradictions) is no longer available. GÏdel's theorem showed that you cannot prove an axiomatically based theory like arithmetic is free of contradiction within that theory itself. But that doesn't mean you can't prove it in some larger, richer theory. In fact, in the standard axiomatic set theory, you can prove arithmetic is free of contradictions. And personally, I buy that proof. For me, as a living, human mathematician, the consistency of arithmetic has beenproved—to my complete satisfaction.
So to answer the Edge question, you have to take a common sense approach to proof—in this case proof being, I suppose, an argument that would convince the intelligent, professionally skeptical, trained expert in the appropriate field. In that spirit, I could give any number of specific mathematical problems that I believe are true but cannot prove, starting with the famous Riemann Hypothesis. But I think I can be of more use by using my mathematician's perspective to point out the uncertainties in the idea of proof. Which I believe (but cannot prove) I have.