On Mon, 27 Jul 2009, mart wrote:
>> Here is the problem with the deep platonism of this idea. Crudely put
>> it was destroyed by Goedel's dual theorems on consistancy and
>> completeness.
>
> despite his theorems, godel actually was a kind of Platonist
Yes indeed. Jim Holt had a wonderful article on Goedel's deep Platonism in the New Yorker a couple of years ago:
http://www.newyorker.com/archive/2005/02/28/050228crat_atlarge
If you're into this sort of thing, the whole article is very worth while, for the anecdotes as well as the idea.
But as for his abiding platonism:
<begin excerpt>
Gödel entered the University of Vienna in 1924. He had intended to
study physics, but he was soon seduced by the beauties of mathematics,
and especially by the notion that abstractions like numbers and circles
had a perfect, timeless existence independent of the human mind. This
doctrine, which is called Platonism, because it descends from Plato's
theory of ideas, has always been popular among mathematicians. In the
philosophical world of nineteen-twenties Vienna, however, it was
considered distinctly old-fashioned.
Among the many intellectual
movements that flourished in the city's rich café culture, one of the
most prominent was the Vienna Circle, a group of thinkers united in
their belief that philosophy must be cleansed of metaphysics and made
over in the image of science. Under the influence of Ludwig
Wittgenstein, their reluctant guru, the members of the Vienna Circle
regarded mathematics as a game played with symbols, a more intricate
version of chess. What made a proposition like "2 + 2 = 4" true, they
held, was not that it correctly described some abstract world of
numbers but that it could be derived in a logical system according to
certain rules.
Gödel was introduced into the Vienna Circle by one of his professors,
but he kept quiet about his Platonist views. Being both rigorous and
averse to controversy, he did not like to argue his convictions unless
he had an airtight way of demonstrating that they were valid. But how
could one demonstrate that mathematics could not be reduced to the
artifices of logic? Gödel's strategy--one of "heart-stopping beauty,"
as Goldstein justly observes--was to use logic against itself.
Beginning with a logical system for mathematics, one presumed to be
free of contradictions, he invented an ingenious scheme that allowed
the formulas in it to engage in a sort of double speak. A formula that
said something about numbers could also, in this scheme, be interpreted
as saying something about other formulas and how they were logically
related to one another. In fact, as Gödel showed, a numerical formula
could even be made to say something about itself. (Goldstein compares
this to a play in which the characters are also actors in a play within
the play; if the playwright is sufficiently clever, the lines the
actors speak in the play within the play can be interpreted as having a
"real life" meaning in the play proper.) Having painstakingly built
this apparatus of mathematical self-reference, Gödel came up with an
astonishing twist: he produced a formula that, while ostensibly saying
something about numbers, also says, "I am not provable." At first, this
looks like a paradox, recalling as it does the proverbial Cretan who
announces, "All Cretans are liars." But Gödel's self-referential
formula comments on its provability, not on its truthfulness. Could it
be lying? No, because if it were, that would mean it could be proved,
which would make it true. So, in asserting that it cannot be proved, it
has to be telling the truth. But the truth of this proposition can be
seen only from outside the logical system. Inside the system, it is
neither provable nor disprovable. The system, then, is incomplete. The
conclusion--that no logical system can capture all the truths of
mathematics--is known as the first incompleteness theorem. Gödel also
proved that no logical system for mathematics could, by its own
devices, be shown to be free from inconsistency, a result known as the
second incompleteness theorem.
Wittgenstein once averred that "there can never be surprises in logic."
But Gödel's incompleteness theorems did come as a surprise. In fact,
when the fledgling logician presented them at a conference in the
German city of Königsberg in 1930, almost no one was able to make any
sense of them. What could it mean to say that a mathematical
proposition was true if there was no possibility of proving it? The
very idea seemed absurd. Even the once great logician Bertrand Russell
was baffled; he seems to have been under the misapprehension that Gödel
had detected an inconsistency in mathematics. "Are we to think that 2 +
2 is not 4, but 4.001?" Russell asked decades later in dismay, adding
that he was "glad [he] was no longer working at mathematical logic." As
the significance of Gödel's theorems began to sink in, words like
"debacle," "catastrophe," and "nightmare" were bandied about. It had
been an article of faith that, armed with logic, mathematicians could
in principle resolve any conundrum at all--that in mathematics, as it
had been famously declared, there was no ignorabimus. Gödel's theorems
seemed to have shattered this ideal of complete knowledge.
That was not the way Gödel saw it. He believed he had shown that
mathematics has a robust reality that transcends any system of logic.
But logic, he was convinced, is not the only route to knowledge of this
reality; we also have something like an extrasensory perception of it,
which he called "mathematical intuition." It is this faculty of
intuition that allows us to see, for example, that the formula saying
"I am not provable" must be true, even though it defies proof within
the system where it lives.
<end excerpt>
And as an example of the anecdotes:
<begin excerpt>
So naïve and otherworldly was the great logician that Einstein felt
obliged to help look after the practical aspects of his life. One much
retailed story concerns Gödel's decision after the war to become an
American citizen. The character witnesses at his hearing were to be
Einstein and Oskar Morgenstern, one of the founders of game theory.
Gödel took the matter of citizenship with great solemnity, preparing
for the exam by making a close study of the United States Constitution.
On the eve of the hearing, he called Morgenstern in an agitated state,
saying he had found an "inconsistency" in the Constitution, one that
could allow a dictatorship to arise. Morgenstern was amused, but he
realized that Gödel was serious and urged him not to mention it to the
judge, fearing that it would jeopardize Gödel's citizenship bid. On the
short drive to Trenton the next day, with Morgenstern serving as
chauffeur, Einstein tried to distract Gödel with jokes. When they
arrived at the courthouse, the judge was impressed by Gödel's eminent
witnesses, and he invited the trio into his chambers. After some small
talk, he said to Gödel, "Up to now you have held German citizenship."
No, Gödel corrected, Austrian.
"In any case, it was under an evil dictatorship," the judge continued.
"Fortunately that's not possible in America."
"On the contrary, I can prove it is possible!" Gödel exclaimed, and he
began describing the constitutional loophole he had descried. But the
judge told the examinee that "he needn't go into that," and Einstein
and Morgenstern succeeded in quieting him down. A few months later,
Gödel took his oath of citizenship.
<end excerpt>
Michael