杏吧原创

G枚del: A life of logic, the mind, and mathematics by John Casti and
Werner Depauli, Perseus, 拢18.50, ISBN 0738202746

A MAN is walking by the banks of the Nile with his child. A crocodile jumps
out of the river, seizes the child, then says, 鈥淚 will return the child if you
guess correctly whether or not I will return the child.鈥� The father replies:
鈥淵ou will not return the child.鈥� No one recorded the crocodile鈥檚 reply.

No wonder. I suspect that what distinguishes us from other animals鈥攅ven
all other hominids鈥攊s our ability to grasp these puzzles and understand
that they matter. We may well have been losing sleep over them ever since we
became Homo sapiens sapiens. At any rate written versions of them
appear at the earliest possible stage in the record, roughly at the point where
people started using writing for thought and not merely the archaic Linear A and
B accounts that kept track of ox hides and grain for Minoan tax collectors.

But what exactly is the connection between crocodiles, puzzles and Kurt
G枚del? Among mathematicians G枚del is famous. He proved three of the
major fundamental theorems of 20th-century maths. A lot of work stands on the
foundations he built. Most of this is so esoteric that only mathematicians know
or care about its significance. But, one of his ideas broke out into the wider
world. It caught the imagination of the public. And it is on this, the
incompleteness theorem, that this rockumentary by two Vienna-based professors
dwells.

Both the strands to this theorem stretch far back into the past. You鈥檒l find
the first in the crocodile puzzle from ancient Greece. The second dates back
just as far. Around the 3rd century BC, the Greek geometer Euclid proposed
deducing all of geometry from a few self-evident axioms. About a hundred years
ago mathematicians began warming to the idea that Euclid鈥檚 notion could be
extended to the whole of mathematics and set to work. This project became known
as the Hilbert programme, after the mathematician David Hilbert. The three
volumes of Bertrand Russell and Alfred Whitehead鈥檚 colossal Principia
Mathematica were among the first bricks to be laid for this Tower of
Babel.

All in vain, as it turned out. G枚del鈥檚 incompleteness theorem showed
that in any consistent formal system that encompasses arithmetic there鈥檒l be
propositions that it cannot decide no matter what set of axioms you pick.

G枚del showed that any set of axioms for such a system would be
pathological in the sense that not only would there be infinitely many axioms in
it, but that there would not even be a general rule for recognising the axioms
when you saw them. The moral is that the Hilbert programme is doomed. The proof
is too long to provide here, but at its heart is the same idea that lurks in the
riddle of the crocodile.

He was able to use this idea because he found a way of formalising certain
hitherto unformalised concepts, such as 鈥減roof鈥� and 鈥渃omputable鈥�, in such a way
that we could reason about them and give the crocodile riddle mathematical
teeth. It was immediately clear to mathematicians and philosophers at the
time鈥�1931鈥攖hat this was a development in our intellectual
understanding on a scale beyond anything in their collective memories. It took
technological developments in electronic machinery, however, to compel a wider
public to take notice. As computers became more powerful and more numerous, more
and more people became interested in the question of what they could and
couldn鈥檛 do.

Once people start reading about G枚del鈥檚 proof, they find the danger
posed by the crocodile at its heart exciting. The possible ramifications of the
general constraint announced by the theorem are obviously important. It leads
many鈥攕ome of whom ought to know better, Roger Penrose among them鈥攖o
jump to extreme conclusions about how it may or may not constrain the reach of
human thought or computer capability.

G枚del鈥檚 incompleteness theorem surely has something to tell us. But
G枚del himself was well-known to be extremely cautious about drawing hasty
conclusions from his theorem. The only way to guard against the haste is to
master the material. Uniquely for such ground-breaking work, G枚del鈥檚
original article is an excellent introduction for the mathematically confident,
but there is still a need for books like G枚del that explain the
ideas in straightforward terms.

There are other such books, some of them very good. When Douglas Hofstader
wrote G枚del, Escher, Bach, many mathematicians thought it was so
easy they wondered why they hadn鈥檛 done it themselves. John Casti and Werner de
Pauli have written a similar book, but have chopped out Escher and Bach and
added a thumbnail biography of the remaining hero.

G枚del himself emerges from it as an enigmatic and shadowy figure. A
happy childhood in provincial Austria-Hungary, where he was born in 1906, was
followed by membership of the Vienna Circle in the 1930s. Exiled to the US, he
lived there for the rest of his life. He worked at the Princeton Institute for
Advanced Study, and it was there that he proved the third of the theorems for
which mathematicians remember him.

Stories did not accrete round him the way they did around the batty and
gregarious Flying Hungarian Paul Erd贸s (see the review of Paul Hoffman鈥檚
The Man Who Loved Numbers, 1 August, 1998, p 40). But one story about
G枚del is well-known. He nearly landed himself in trouble: at his American
citizenship interview, he insisted that the US constitution allows for
dictatorship in certain circumstances. With a two-thirds majority in the Senate,
there isn鈥檛 much you can鈥檛 do. G枚del was always a shy man, but it seems he
became more and more suspicious and reclusive in old age, eventually dying of
hunger because he wouldn鈥檛 feed himself properly. A sad end for an enigmatic
genius.