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Gödel believed that CH is false, and that his proof that CH is consistent with ZFC only shows that the Zermelo–Fraenkel axioms do not adequately characterize the universe of sets. Gödel was a Platonist and therefore had no problems with asserting the truth and falsehood of statements independent of their provability.
Chapter 9: The Drunken Vandal; William Rowan Hamilton extended the field of complex numbers to the quaternions. Chapter 10: The Would-Be Soldier and the Weakly Bookworm; Marius Sophus Lie formalized Lie groups and Lie algebras. Wilhelm Killing classified all simple Lie algebras (in what Ian Stewart calls the "greatest mathematical paper of all ...
An "enduring and powerful product" of Cohen's work on the continuum hypothesis, and one that has been used by "countless mathematicians" [16] is known as "forcing", and it is used to construct mathematical models to test a given hypothesis for truth or falsehood.
What "truth" or objectivity can be ascribed to this theoretic construction of the world, which presses far beyond the given, is a profound philosophical problem. It is closely connected with the further question: what impels us to take as a basis precisely the particular axiom system developed by Hilbert?
The human mind has no special claim on reality or approaches to it built out of math. If such constructs as Euler's identity are true then they are true as a map of the human mind and cognition . Embodied mind theorists thus explain the effectiveness of mathematics—mathematics was constructed by the brain in order to be effective in this ...
Logicomix: An Epic Search for Truth is a graphic novel about the foundational quest in mathematics, written by Apostolos Doxiadis, author of Uncle Petros and Goldbach's Conjecture, and theoretical computer scientist Christos Papadimitriou.
Logical Intuition, or mathematical intuition or rational intuition, is a series of instinctive foresight, know-how, and savviness often associated with the ability to perceive logical or mathematical truth—and the ability to solve mathematical challenges efficiently. [1]
The next year, the translation was completed and still greater interest in Bergson and his work ensued. By coincidence, in that same year (1911), Bergson wrote a 16-page preface, Truth and Reality, to the French translation of James's book Pragmatism. In it, he expressed sympathetic appreciation of James's work, together with certain important ...