Science faces the unthinkable
We have evolved minds capable of negotiating us through our
Saturday, August 8, 1998
hunter-gatherer stages and up to our present level of civilization.
these abilities enough to ensure our future progress?
Limits of Science and the Science of Limits
By Richard Lubbock
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
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
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
Richard Lubbock writes on science, technology and philosophy.
Theories of Everything: The Quest for Ultimate Explanation, by John
D. Barrow (Clarendon, 1991).
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