Science faces the unthinkable

We have evolved minds capable of negotiating us through our
hunter-gatherer stages and up to our present level of civilization.
But are these abilities enough to ensure our future progress?
IMPOSSIBILITY: The Limits of Science and the Science of Limits

By John D. Barrow
Oxford, 279 pages, $37
The triumphs of 20th-century science have led many people to think that imperial science can get any result it wants. That is far from true. The most startling scientific achievements of the past 100 years have been demonstrations of fallability.

Only in the 19th century did mathematicians first prove the impossibility of certain geometric constructions, such as the trisection of an angle. Since then a succession of amazing impossibility results have dealt self-satisfied scientists, philosophers and politicians a barrage of dizzying smacks to the head.

Almost everyone knows that Einstein postulated that no signal can move faster than light. And that Werner Heisenberg's principle of indeterminacy has shown that in a single measurement you can't determine both the position and the momentum of any particle. But not so many have heard of Kurt Gödel's undecideability theorem, which seemed to upset the foundations of mathematics, and even fewer know of U.S. economist Kenneth Arrow's impossibility proof of democratic voting methods. Yet these inescapable features of reality keep us from knowing all the things we want to know, and going to all the places we'd like to go.

In his unsettling book, Impossibility: The Limits of Science and the Science of Limits, British cosmologist John D. Barrow sets out detailed maps of these and many other roadblocks to our mastery of the universe. Though a physical scientist himself, Barrow is too subtle to fall into the error of naive reductionists, who cherish the oafish superstition that everything reduces to physics. In refuting the reductionists he compares the scientific strategies of physicists with those of biologists. Since physicists deal only with the simplest organisms, such as electrons, they favour explanations found in the light of clear, brilliant mathematical symmetries. Biologists, on the other hand, struggle with complex organisms, such as mice, so they seek explanations in the messy processes of historical accident and natural selection. We cannot know for certain where the proper scope of each strategy lies.

Barrow says we have to consider "whether limits imposed by our humanity restrict our knowledge of the universe in significant ways." We have evolved brains and minds capable of negotiating us through our hunter-gatherer stages and up to our present level of civilization. But are these abilities enough to ensure our future progress?

There are so many uncertainties. How did the universe begin? Will it collapse, or expand forever? The speed limit of light makes it certain we will never be sure. Can there be a theory of everything? Barrow has dealt with this in an earlier book. Now he believes that our only hope of certainty dangles from the still untested string theory, which may show there is only one possible set of laws of physics. "But even if that were the case, we would not be able to prove it," Barrow says. A dark sea of creative mystery bathes us all around. So much lies beyond the reach of science.

Barrow sketches the plausible progress of civilizations up a ladder that begins with our present abilities. "We are nearly a low-level type II civilization," he says. That means we are almost capable of restructuring solar systems, among other things. Barrow's complex sequence of civilizations runs all the way up to the final type Omega civilization "which could manipulate the entire universe (and even other universes)."

However, even the Omegans would be unable to overcome the fundamental limits to knowledge imposed by Gödel's undecide-ability theorems. In 1931, Gödel shocked the philosophical world by proving that most logical systems cannot prove themselves to be both complete and consistent. He used a fiendishly clever method to prove a valid statement in arithmetic that says, "I am not provable." Because he studied a model system no more complex than ordinary arithmetic, his results would seem to inject pandemonium into any higher kind of thinking, such as philosophy.

Relax. There are many loopholes in Gödelian impossibility. Barrow cites the logician John Myhill, who was inspired by Gödel to declare, "No non-poetic account of reality can be complete." Barrow believes that Gödel's undecideability theorem tells us that sufficiently powerful forms of logical and technical manipulation must finally be confounded by their own strength. In fact, that's the ruling theme of his book. For instance, the all-powerful Omegan civilizations could never know for sure whether they were actually the ultimate Omegans.

This uncertainty could lead to strategic quarrels among the final beings. No matter how powerful the Omegans become, the paradoxes of democratic voting ensure that there is no satisfactory way, even for Omegans, to make social decisions. Democracy in any form can never satisfy everybody, so that "the will of the people" is doomed to flop around forever like a stranded fish, until some random fluctuation ends the agony.

Barrow explains this ludicrous endgame of politics with a lucid account of voting paradoxes that have been known since the 18th century. In 1950, Kenneth Arrow published his voting theorem, which put the kibosh on extravagant claims for the perfection of democratic justice.

The arguments behind Arrow's theorem are difficult, but not impossible to follow, and Barrow performs the public service of laying them out clearly in Impossibility, so that readers with the grit and wit to follow them with pencil and paper can see for themselves that uncritical idolatry of democracy can destroy it. "The only voting systems that are strategy-proof are dictatorial," says Barrow. In the end, we can best replace the dictator with random tie-breakers. So perhaps we would be wise to appoint the 6/49 lottery as our final constitutional arbiter.

"Far from being a harmless technicality," Barrow writes, "the problem of paradox enshrined in Arrow's Impossibility Theorem seems to be ubiquitous and common." The paradox infects computers just as it does human electorates. Arrow's theorem ensures that disputes among the omnipotent, god-like citizens of the Omegan civilizations will also produce indecision and quarrels. However, the gods will doubtless have the sagacity to consult some reliably random oracle, such as a lava lamp.

In Impossibility, John Barrow offers us an intriguing and detailed survey of many of the logical, philosophical and political limitations of the sciences. He shows that while power tends to corrupt, absolute power discombobulates absolutely.

Richard Lubbock writes on science, technology and philosophy.

Related Reading

Theories of Everything: The Quest for Ultimate Explanation, by John D. Barrow (Clarendon, 1991).