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It is difficult to overstate the impact of his theorem and the
possibilities that opened up from Gödel's extraordinary methods, in
which he discovered a way for mathematics to talk about itself. (Ms.
Goldstein compares it to a painting that could also explain the
principles of aesthetics.)
The theorem has generally been understood negatively because it
asserts that there are limits to mathematics' powers. It shows that
certain formal systems cannot accomplish what their creators
hoped.
But what if the theorem is interpreted to reveal something
positive: not proving a limitation but disclosing a possibility?
Instead of "You can't prove everything," it would say: "This is what
can be done: you can discover other kinds of truths. They may be
beyond your mathematical formalisms, but they are nevertheless
indubitable."
In this, Gödel was elevating the nature of the world, rather than
celebrating powers of the mind. There were indeed timeless truths.
The mind would discover them not by following the futile
methodologies of formal systems, but by taking astonishing leaps,
making unusual connections, revealing hidden meanings.
Like Einstein, Gödel was, Ms. Goldstein suggests, a
Platonist.
Of course, those leaps and connections could go awry. Gödel was
an intermittent paranoiac, whose twisted visions often left his
colleagues in dismay. He spent his later years working on a proof of
the existence of God. He even died in the grip of a perverse
esotericism. He feared eating, imagined elaborate plots, and
literally wasted away. At his death in 1978, he weighed 65 pounds.
But he was no postmodernist. Late in his life Gödel said of
mathematics: "It is given to us in its entirety and does not change,
unlike the Milky Way. That part of it of which we have a perfect
view seems beautiful, suggesting harmony." That beauty, he proposed,
would be mirrored by the world itself. These are not exactly the
views of an acolyte devoted to Relativity, Incompleteness and
Uncertainty. And Einstein was his fellow dissenter.
The Connections column will appear every other
Monday.
