Kurt Godel Quotes on Knowledge
Kurt Gödel (1906–1978), whose 1931 incompleteness theorems established that any consistent formal system rich enough to express elementary arithmetic contains true statements it cannot prove, gave twentieth-century philosophy of mathematics its sharpest result on the limits of formalization as a route to mathematical knowledge. Gödel's own philosophical interpretation of the result, developed in the late essays "Russell's Mathematical Logic" (1944) and "What Is Cantor's Continuum Problem?" (1947, expanded 1964), defends an explicit mathematical Platonism: the truths the formal system fails to capture are nevertheless accessible to mathematical intuition, which Gödel treated as a cognitive faculty analogous to perception in the natural domain.
Quotes
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“Either mathematics is too big for the human mind, or the human mind is more than a machine.”
As quoted in Topoi : The Categorial Analysis of Logic (1979) by Robert Goldblatt , p. 13 -
Attributed to Kurt Godel:
“Any consistent formal system that contains elementary arithmetic is incomplete.”
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Attributed to Kurt Godel:
“I have come to the conclusion that the world is rational.”
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“The more I think about language, the more it amazes me that people ever understand each other at all.”
Reflections on Kurt Gödel , MIT Press, Hao Wang 1987, page 95, according to Karl Menger -
“Proposition VI, On Formally Undecidable Propositions in Principia Mathematica and Related Systems I (1931); Informally, recursive systems of axioms cannot be complete.”
To every ω-consistent recursive class κ of formulae there correspond recursive class signs r , such that neither v Gen r nor Neg ( v Gen r ) belongs to Flg (κ) (where v is the free variable of r ). -
“Attributed as a remark of 29th November 1972, in Incompleteness (2005) by Rebecca Goldstein”
But every error is due to extraneous factors (such as emotion and education ); reason itself does not err.