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The Unknown and The Unknowable
A Talk With Joseph Traub

JB: JB: To what extent is your interest in the limits of scientific knowledge influenced by the work of Gdel?

TRAUB: In 1931 a logician named Kurt Gdel announced a result that astonished the scientific world. Gdel said that there are statements about arithmetic that can never be proved or disproved. This impossibility result is about elementary arithmetic, not some arcane corner of mathematics. To the educated lay person, Gdel's undecidability theorem may be the single most widely-known mathematical result of the 20th century.

Gdel's theorem is just one of numerous impossibility results established in the last 60 years stating what cannot be done. Another famous negative result, due to the British genius, Alan Turing, states that you cannot tell in advance if a certain abstraction of a digital computer called a Turing machine will ever halt with the correct answer. Now, what all these impossibility results have in common is that they are about the manipulation of symbols, that is, they are about mathematics.

I've spent some of my time for most of a decade asking myself what does this tell us about the unknowable in science. Indeed, the first time that I spoke publicly about this subject was on February 1, 1989 at a panel discussion in memory of Heinz Pagels organized by John Brockman.

Science is about understanding the universe and everything in it. Examples of scientific questions are: Will the universe expand forever, or will it collapse?; Will there be major global changes due to human activities, and what will be the effects on earth's ocean levels, and on agriculture and biodiversity? Note that there are, a priori, no mathematical models that accompany these questions. Science uses mathematics, but it is also very different from mathematics. Can we up the ante from mathematics and prove impossibility results in science?

Ralph Gomory, the President of the Alfred P. Sloan Foundation, proposes a tripartite division of science: the known, the unknown, and the unknowable. The known is taught in the schools and universities and is exhibited in the science museums. But scientists are excited by the unknown. Parenthetically, artists go to art museums to learn; scientists do not go to science museums because those museums act as if it's all known and preordained. That may be changing; exemplars are the Exploratorium in San Francisco and the American Museum of Natural History.

Gomory's tripartite division proposes three distinct areas: the known, the unknown which may someday become known, and the unknowable, which will never be known. The unknown and the unknowable form the boundary of science. Here are examples of questions for which the answers are today unknown.

How do physical processes in the brain give rise to subjective experience? That is, explain consciousness. Can the healthy, active lives of humans be significantly prolonged by, say, a factor of two or three? How did life originate on earth? Will the universe expand forever, or will it collapse? Can we develop a grand unified theory of the fundamental physical laws? Why do fundamental constants, such as the speed of light, have their particular values? Is there life elsewhere in the universe? Is it intelligent? How do children acquire language?

For which of these are the answers unknowable? We cannot prove scientific unknowability. That can only be done in mathematics. This is sometimes not understood, even by professionals. I expressed my interest in the unknowable to a very senior European scientist. He immediately responded that this had been, of course, settled by Gdel's theorem. Not so; Gdel's theorem limits the power of mathematics and does not establish that certain scientific questions are unanswerable.

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