The Uncertainty Principle

In his fascinating piece “Mathematicians Face Uncertainty” [The Year in Science, January], mathematician Keith Devlin points out that a number of ironclad proofs have proved to be not so ironclad—for example, the proofs of Euclid that contained unrecognized assumptions were only confirmed by mathematician David Hilbert centuries later. Devlin’s point is that even now we can’t be certain that there aren’t unrecognized flaws, fallacies, or assumptions in Hilbert’s or any other proofs. No doubt Devlin is correct, but his point would have been made more forcefully by examples of proofs that were long accepted and then shown to contain fallacies or unjustified assumptions. Problems like Euclid’s—problems that can be corrected while the proof is maintained—don’t strike me as lethal to our dependence on proofs in the way that demonstrating the falseness of a long-accepted proof would. The only example I know of the acceptance of a false proof is the “four color” problem [the theory that any map requires only four colors in order to avoid using the same color for adjacent areas], which was accepted for more than a decade before being shown to be faulty. Are there other examples? If not, then while Devlin is correct that we can never be certain that a proof is flawless, it seems that our traditional acceptance of proofs as ironclad has served us pretty well.

STEVEN GOLDBERG

Chairman, City College

City University of New York

New York, New York

Keith Devlin responds: I agree that the accepted notion of proof has served us well and will continue to do so; indeed, in mathematics it’s the only game in town when it comes to establishing truth. But there have been many instances where it took a while for a proof to be found false or declared true: Some years ago a proof of the Poincaré conjecture was announced that was subsequently found to be false, and a claimed proof of the Bieberbach conjecture was at first dismissed as false by experts but later shown to be correct. With famous problems like these, there was enough interest to ensure that the matter was eventually resolved to the satisfaction of the profession, but for less stellar results we may never know for sure.

Political Science

I was thrilled to hear examples from The Year in Science on our local public radio station. I had just read “Bush vs. Science” that morning and have to admit that the extent of the administration’s attempts to underhandedly support business and mislead Congress and the public in pursuit of its aims was quite disturbing. I was also disturbed to learn that some of your readers wish that Discover would stay out of politics. Without Discover’s willingness to tell the facts, I would not have seen the president’s plan so clearly.

ROBERT BRANDOM

Harvard, Massachusetts

The number five story of the top 100 stories of the year [“Bush vs. Science”] was listed under politics. I’m no politician, so I cannot say how accurate the story was, but I do know I enjoy reading articles about science and technology, and this was basically neither. This is not the first time I have seen this magazine step into the realms of politics and religion, but it will be the last. Consider my subscription canceled.

PJ DARCY

Poughkeepsie, New York

Turkey Guts Edged Out by Guinea Pig

With dismay I realized that the thermal depolymerization process (“Anything Into Oil,” May 2003) was not the number one story of the top 100 science stories of 2003, nor, to my horror, was it even in the top five. How could the most revolutionary concept of the century have been ignored—a process so empowering to the civilized world, so absolutely timely? Then it came to me. A full-color spread of tons of turkey offal must have so put off the selection committee that they had an Oh, my! moment and excluded a process that would create a balance between runaway consumption and the corresponding waste that threatens to bankrupt civilization, a process that would be affordable and available worldwide, turn old TVs and computers into oil and fertilizer, and stave off global warming! Please, folks, this is surely as important as number 83, “Prehistoric Guinea Pig Weighed Nearly a Ton.”

ROBERT MAGILL

Sarasota, Florida

Earth, Mooned

I was amazed at the apparent lack of scientific knowledge and common sense reflected in Bob Berman’s December Sky Lights column regarding the differences between the near and far sides of the moon. He points out that “the face [of the moon] we see has fewer large craters and far greater areas of smooth, dark, frozen lava. Nobody really knows why.” Duh! It doesn’t take a professional astronomer to know that the same side of the moon always faces Earth. Since Earth does not throw off comets and asteroids, it stands to reason that the side of the moon facing Earth would be protected by Earth’s mass from asteroid and comet impacts more than the side facing away from Earth.

PAUL TAYLOR

Asheville, North Carolina

The editors respond: One of the most crucial aspects of the scientific method is testing a theory against the evidence. With that in mind, let’s look at your theory that the lunar near side is protected from bombardment by Earth’s bulk. How much of a solid angle of the sky does Earth block as seen from the moon? The apparent diameter of Earth is two degrees. That is about 1/7,000 of the Earth-facing half of the sky—hardly enough to explain a significant difference in cratering. In fact, Earth’s gravity might act as a slight gravitational magnifier, causing more meteorites to strike the Earth-facing side of the moon. The bulk of lunar cratering took place more than 3.8 billion years ago, when Earth and the moon were closer together. That means the solid angle of Earth was larger (although still fairly small), but the gravitational effect was also greater. Most important, your theory does not explain the lunar seas, vast plains of relatively young volcanic rock that appear primarily on the lunar near side. Internal geologic processes, possibly influenced by external events, erased many craters and gave the moon its split personality; the real question is why those processes were so lopsided.