Skip to comments.Truth, Incompleteness and the Goedelian Way
Posted on 02/15/2005 2:39:04 PM PST by snarks_when_bored
John Patrick NaughtonRebecca Goldstein's new book is about the mathematician Kurt Gödel.
elativity. 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 postmodernism, 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 Gödel's famous Theorem (Incompleteness) and Werner Heisenberg's famous Principle (Uncertainty) declared that, henceforth, even science would be postmodern.
Or so it has seemed. But as Rebecca Goldstein points out in her elegant new book, "Incompleteness: The Proof and Paradox of Kurt Gödel" (Atlas Books; Norton), of these three figures, only Heisenberg might have agreed with this characterization.
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 Gödel had precisely the opposite perspective. Both fled the Nazis, both ended up in Princeton, N.J., 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.
And therein lies Ms. Goldstein's tale. From the late 1930's until Einstein's death in 1955, Einstein and Gödel, the physicist and the mathematician, would take long walks, finding companionship in each other's ideas. Late in his life, in fact, Einstein said he would go to his office just to have the "privilege" of walking with Gödel. What was their common ground? In Ms. Goldstein's interpretation, they both felt marginalized, "disaffected and dismissed in profoundly similar ways." Both thought that their work was being invoked to support unacceptable positions.
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 Gödel? Most lay readers probably know of him from Douglas R. Hofstadter's playful best-seller "Gödel, Escher, Bach," a book that is more about the powers of self-referentiality than about the limits of knowledge. But the latter is the more standard association. "If you have heard of him," Ms. Goldstein writes, perhaps too cautiously, "then there is a good chance that, through no fault of your own, you associate him with the sorts of ideas - subversively hostile to the enterprises of rationality, objectivity, truth - that he not only vehemently rejected but thought he had conclusively, mathematically, discredited."
Ms. Goldstein's interpretation differs in some respects from that of another recent book about Gödel, "A World Without Time: The Forgotten Legacy of Gödel 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 Gödel'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. Gödel 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 that was both true and unprovable.
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.
Perhaps you'll be more specific about what it is you're calling 'nonsense'?
Completeness is a vital, crucial, essential theoretical attribute that is necessary in the design of algorithms. Without proof of its existence, algorithms are rendered useless.
Goedel's ideas surrounding his proof of the incompleteness of formal reasoning systems are as important if not more important than any philosopher of any age including Einstein.
His philosophy is not anti-science but is rather a view on its limitations inside a particular theory. Theories can be expanded to be more general but they may never capture everything.
Completeness then defines what a theory can ascertain and ensures that it ascertains all that is specified in its scope. To say a theory is complete is to say at once that it is reliable and valid within its ***scope***.
I would think that Goedel's theorem would keep scientists humble, as mentioned in post #6.
Rebecca Goldstein is the author of four novels, including THE MIND-BODY PROBLEM, and a collection of short stories, STRANGE ATTRACTORS. Her work has won numerous prizes, including two Whiting Awards. In 1996, she was named a MacArthur Foundation Fellow. She holds a Ph.D. in philosophy from Princeton University, where her work was concentrated in the philosophy of science and was supported by a National Science Foundation fellowship. She resides in Cambridge, Massachusetts.
I'm reasonably certain that you are unfamiliar with Godel's Proof.
There is nothing left-wing or nihilistic in Godel's lucid, brief proof for the incompleteness of any non-contradictory formal system.
It was part of Symbolic Logic II when I was an undergrad.
Nothing communist about Godel.
I can't believe that I have to post this.
js1138's Law: The first to post a negative comment on a science thread hasn't read the article.
Depressing, innit? ;)
Math threads never get very many pings. They're worth posting from time to time, but you can't expect a whole lot of action.
One function it is impossible to calculate is the interaction (or interference, if you will) upon the observed by the observer. Since you can't predict the latter wouldn't you end up with a model that is unstable over time? Or is it merely a function with infinite solutions (or do they only seem infinite?).Either way, it seem incomplete to me.
...Godel's lucid, brief proof for the incompleteness of any non-contradictory formal system...
Two (friendly) comments. First, Gödel's proof was stunning, but not brief (nor was it lucid in the ordinary meaning of that word). Second, the formal system has to be strong enough to express natural number arithmetic with multiplication; systems weaker than thatsay, systems that include only the operation of additioncan be proved to be both complete and consistent.
Here's a useful link to an article by R.B. Braithwaite which provides some background and details on Gödel's Incompleteness Theorem:
Russel countered with his Theory of Types, but it was thoroughly unsatisfactory and you could tell his heart wasn't in it. That idea essentially ruled out self-reference; that is, metastatements of the order of "this statement is false" were illegal. But in fact there turned out to be no reliable way of deciding where the boundary between statement and metastatement really lay; in fact, there isn't one, as Russell concluded himself.
But that does not invalidate logical systems per se, as the postmodernists fervently believe, it simply dictates that they have boundaries. It does not state that there is no truth or logic, or that all points of view are therefore equally valid, quite the opposite, in fact. It simply states that formal logical systems have limitations. Most adults should be able to live with that.
What Godel showed was twofold: first, that within any formal logical system of sufficient power, a statement could be made that was true but undemonstrable (incompleteness) and second, that a statement could be formulated following that system's rules that could be shown both true and false (incoherence).
Two (friendly) comments. First, it's important to say "undemonstrable within the system" rather than simply "undemonstrable". Second, it's not quite accurate to say that "a statement could be formulated following that system's rules that could be shown both true and false (incoherence)"if that were true, the system would be inconsistent, which nobody desires. The aim of the constructor of a formal logical system is to rule out the possibility of contradiction, but at the same time to insure the possibility of proving the greatest number of true propositions. What Gödel showed was that if a system has the expressive resources needed to formulate the axioms of natural number arithmetic with multiplication, then it is impossible to prove within the system every true proposition without incurring the penalty of inconsistency, i.e., the ability to 'prove' a contradiction (such as, 1=2). Since no formal system constructor wants an inconsistent system, (s)he's forced to give up the hope of being able to prove every true proposition within that system.
In short, in a 'strong enough' system, the penalty for completeness is inconsistency, and the penalty for consistency is incompleteness.
Of course, by expanding the system to include new axioms, propositions which were previously known to be true but unprovable, become provable. But, again, in the expanded system, new propositions can be formulated which can be known to be true, but turn out to be unprovable within the expanded system. And so on ad infinitum.
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