Tuesday, July 19, 2005

Incompleteness & the Goedelian way

Edward Rothstein
New York Times/ The Hindu, Feb 15, 2005
Albert Einstein and Kurt Goedel objected to notions of relativism and incompleteness outside their work. They fled the politically absolute, but believed it its scientific possibility.
Relativity. Incompleteness. Uncertainty. Is there a more powerful modern Trinity? These reigning deities proclaim humanity's inability to thoroughly explain the world. They have been the touchstones of modernity, their presence an unwelcome burden at first, and later, in the name of post-modernism, welcome company. Their rule has also been affirmed by their once-sworn enemy: science. Three major discoveries in the 20th century even took on their names. Albert Einstein's famous Theory (Relativity), Kurt Goedel's famous Theorem (Incompleteness) and Werner Heisenberg's famous Principle (Uncertainty) declared that, henceforth, even science would be post-modern. Or so it has seemed. But as Rebecca Goldstein points out in her elegant new book, Incompleteness: The Proof and Paradox of Kurt Goedel (Atlas Books; Norton), of these three figures, only Heisenberg might have agreed with this characterisation. His uncertainty principle specified the inability to be too exact about small particles. "The idea of an objective real world whose smallest parts exist objectively," he wrote, "is impossible." Oddly, his allegiance to an absolute state, Nazi Germany, remained unquestioned even as his belief in absolute knowledge was quashed.

Einstein and Goedel had precisely the opposite perspective. Both fled the Nazis, both ended up in Princeton, New Jersey, at the Institute for Advanced Study, and both objected to notions of relativism and incompleteness outside their work. They fled the politically absolute, but believed in its scientific possibility. Einstein's convictions are fairly well known. He objected to quantum physics and its probabilistic clouds. God, he famously asserted, does not play dice. Also, he believed, not everything depends on the perspective of the observer. Relativity doesn't imply relativism. The conservative beliefs of an aging revolutionary? Perhaps, but Einstein really was a kind of Platonist: he paid tribute to science's liberating ability to understand what he called the "extra-personal world."

And Goedel? Most lay readers probably know of him from Douglas R. Hofstadter's playful best-seller, Goedel, Escher, Bach, a book that is more about the powers of self-referentiality than about the limits of knowledge. Goldstein's interpretation differs in some respects from that of another recent book about Goedel, A World Without Time: The Forgotten Legacy of Goedel and Einstein by Palle Yourgrau (Basic), which sees him as more of an iconoclastic visionary. But in both he is portrayed as someone widely misunderstood, with good reason perhaps, given his work's difficulty. Before Goedel's incompleteness theorem was published in 1931, it was believed that not only was everything proven by mathematics true, but also that within its conceptual universe everything true could be proven. Mathematics is thus complete: nothing true is beyond its reach. Goedel shattered that dream. He showed that there were true statements in certain mathematical systems that could not be proven. And he did this with astonishing sleight of hand, producing a mathematical assertion both true and unprovable.
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? Late in his life Goedel 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.

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