Kurt Gödel Philosophy
Czech-American Logician and Philosopher Famous for Gödel’s Theorem
Nov 14, 2009
Kurt Gödel (1906-1978) was a famous Czech-American mathematician and philosopher, considered one of the most significant logicians of all time. He is famous for “Gödel’s Theorem” in the 1930s, and contributed to various developments in mathematics.
He is known for his quote: “The human mind is capable of working out truths that no formal or mechanical procedure can decide.”
Gödel’s Theorem
Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, a year after finishing his doctorate at the University of Vienna. For what is often referred to as “Gödel’s Theorem,” it is really two related theorems of incompleteness.
- The first theorem states that in any formal, mathematical or logical, system that is internally consistent (that is, there are no contradictions), there will be some well-formed proposition that cannot be proven either true or false. In other words, these will be formally undecidable. Gödel shows that such a proposition is equivalent to an instance of the “liar paradox,” a statement such as “This sentence is not provable,” which if true, is false, and if false is true.
- The second incompleteness theorem shows that one cannot prove within the system that the system is actually internally consistent.
Gödel's two proofs had remarkable effects. In mathematics, it effectively put an end to the “formalist” programme which is derived from Immanuel Kant’s metaphysics taken up by mathematician David Hilbert.
Hilbert attempted to show that classical mathematics does not consist in the description of an independently real by abstract realm of entities, “numbers,” but rather, in a system of signs which are constructed out of perceptual experience. The key to the formalist programme was the ability to give an account of infinite quantities, which are never part of experience but are indispensable to mathematics.
Hilbert worked a theory in which infinite quantities could be taken as assumptions for their instrumental value. Since Hilbert needed a means of justifying and distinguishing valid assumptions against those which are invalid, he made consistency a condition. Gödel’s work showed that the demand for a proof of consistency could never be met. Gödel's theorem of incompleteness disproved David Hilbert’s programme.
Gödel’s Philosophy
Gödel was a believer in Platonism as his work reaffirmed it, as well as to the proof of the impossibility of artificial intelligence. Derived from the great ancient Greek philosopher Plato, the idea of Platonism is that abstract objects exist independently in a “third realm” – that they are neither mental nor physical, but occupy a distinct and eternal world by the effort of the intellect, described by mathematics, logic and geometry.
Alan Turing Disagreement to Gödel's Theorem
One significant disagreement to Gödel’s theorem came from Alan Turing, regarded the father of modern computer science. Turing pointed out a weakness in Gödel’s theorem that although it is correct to say there is a limitation to the power of any machine that uses a formal language, it assumes without any kind of proof that the human intellect does not suffer from the same kind of limitation.
It is not known, but perhaps it is from Turing’s laid out argument that Gödel did not pursue in great lengths further research in artificial intelligence, nor did he revive Platonism outside of mathematics.
Gödel Best Works
His two best known works are Incompleteness Theorems (1931) and the article “An Example of New Type of Cosmological Solution to Einstein’s Field Equations of Gravitation” (1949.)
The article provides Gödel’s findings of a new solution to the general relativity equation of his friend, Albert Einstein, that the universe is not expanding, but rather, rotating. It is this rotation that keeps things stable as it causes the centrifugal force a kind of gravity balance.
Insight to Kurt Gödel Philosophy
Gödel proved that in all logical or mathematical systems, there would always be some propositions that could not be proven true or false. Later famous physicist and mathematician, Roger Penrose, echoed Gödel.
According to Penrose and other Gödel advocates, this gives hope to pursue artificial intelligence programs, since all such machines are formal finite systems, however complex they may be.
Sources:
Moore, Pete. E=MC²: The Great Ideas That Shaped Our World. London: Quintet Publishing, 2002.
Stokes, Philip. Philosophy, the Great Thinkers. London: Capella, 2007.
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