Gödel, Escher, Bach: An Eternal Golden Braid
“Well now, would you like to hear of a race-course, that most people fancy they can get to the end of in two or three steps, while it really consists of an infinite number of distances, each one longer than the previous one?”
What the Tortoise Said to Achilles
During the nineteenth century, mathematics, the acknowledged foundation of all the sciences, began to turn its attention to its own foundations. With the work of Boole, De Morgan, Frege, Peano, Russell, and others, a great project began to take shape: mathematics would examine its own structure with all the rigor it had brought to other explorations. The goal was a system that would be unquestionable and secure; a mathematical language would be developed whose simplicity and clarity would dispel all doubts, reveal all mathematical truths.
The goal of the project was similar to the dream of many seventeenth-century thinkers who wished to create or discover a “universal language” in which a set of symbols would be used to describe all of man’s knowledge. Its structure would be so tied to the structure of the universe that the language itself could be used to discover truths. Its syntax would prevent falsity. There would be no ambiguous meanings. Leibniz, with his ideas of a precise symbolic language and a calculus of reasoning, was, in fact, a major influence on the nineteenth-century systemization of logic which was to provide the syntax of the new mathematical language.
In 1884, for example, Gottlob Frege claimed that arithmetic is “simply a development of logic,” and in the Principia Mathematica Russell and Whitehead attempted to show that all of “pure mathematics” might be derived from the propositions of logic. The full extent of the project was articulated in 1904 by David Hilbert: mathematics would be based upon a system of axioms similar to the one later elaborated in Principia Mathematica. New mathematical statements would be produced according to a set of rules; no two statements produced would contradict each other, making mathematics consistent. Moreover, the system would “fill” its chosen universe; it would be complete, producing every truth. Mathematical knowledge would then be secure. Just by manipulating signs of mathematics according to the rules of logic—its “grammar”—one could eventually discover all necessarily true statements, and one would never produce a falsehood.
The dream of producing such a language was shattered in 1931 by a twenty-five-year-old Austrian mathematician, Kurt Gödel. In a paper entitled (in English translation) “On Formally Undecidable Propositions of Principia Mathematica and Related Systems I,” he showed that in sufficiently powerful mathematical systems, consistency could not be proven in the way Hilbert wished, and more importantly, that such systems were incomplete: there are true statements in mathematics that the system will not produce, true statements, that is, that cannot be proven. This result has since taken its place among the other…
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