In 1931 at the age of just 25 years, the young Austrian mathematician Kurt Goedel proved an astonishing mathematical theorem that made him instantly famous and a celebrity in mathematical circles around the world (see picture; in the Anglo-Saxon literature Goedel is usually referred to as "Godel" skipping the Umlaut in his German name). Despite its very abstract nature and the lack of any every day practical use, Goedel's theorem - the so called Incompleteness Theorem - has had a dramatic and deep impact on mathematics itself and its foundations. It also had a substantial impact on the philosophy of the 20th century and our understanding of the general limitations of computers and algorithms. I will explain here how Goedel's theorem actually caused the emergence of the new science of Artificial Intelligence (AI) and theoretical computer science in the 1940s and 1950s and how it motivated such key AI pioneers like Alan Turing to get involved. The birth of AI and the course AI has taken ...
One of the most amazing features of human languages is their capacity for self-reference. The consequences of this feature were explored by Eubulides, a 4th-century BCE Greek philosopher, who formulated the Liar's Paradox, "What I am saying now is a lie." Is this a lie or not? For over 2,000 years, the Liar's Paradox was a philosophical oddity. In 1902, in a letter to the mathematician Friedrich Frege, the philosopher Bertrand Russell reformulated the Liar's Paradox as a paradox in set theory, arguing that "the collection of all sets that do not include themselves as members" both cannot be a set and cannot fail to be a set.
We give some lectures on the work on formal logic of Jacques Herbrand, and sketch his life and his influence on automated theorem proving. The intended audience ranges from students interested in logic over historians to logicians. Besides the well-known correction of Herbrand's False Lemma by Goedel and Dreben, we also present the hardly known unpublished correction of Heijenoort and its consequences on Herbrand's Modus Ponens Elimination. Besides Herbrand's Fundamental Theorem and its relation to the Loewenheim-Skolem-Theorem, we carefully investigate Herbrand's notion of intuitionism in connection with his notion of falsehood in an infinite domain. We sketch Herbrand's two proofs of the consistency of arithmetic and his notion of a recursive function, and last but not least, present the correct original text of his unification algorithm with a new translation.
Leibniz's logical essays have been published only slowly and very incompletely. Even today they are scattered piecemeal among various partial editions, most notably the two in which Gerhardt has so unfortunately separated the mathematical and philosophical writings,2 as if one could dissect the work of an encyclopedic savant whose philosophy was nourished by the study of all the sciences and in turn inspired all of his scientific discoveries. If there is one thinker whose thought cannot be divided with impunity in this way, it is certainly the one who said, "My metaphysics is entirely mathematical,"3 or again, "Mathematicians have as much need to be philosophers as philosophers have to be mathematicians."4 ... Louis CoutruratTim Monroe and I set out to translate La Logique de Leibniz because we believed that, in the English-speaking world, Couturat's contributions had not been sufficiently appreciated. Although Russell's Critical Exposition offers a more rigorous analysis of Leibniz's metaphysics, Couturat's book remains unsurpassed as a survey of the full range of Leibniz's work in logic and the philosophy of logic, the philosophy of language, and the philosophy of mathematics. Couturat's success in this regard was the result of archival research that brought to light a wealth of unpublished writings on these topics that had been omitted from previous editions of Leibniz's works. These texts were published by Couturat in the collection Opuscules et fragments inédits de Leibniz (Paris, 1903), which serves as a companion to La Logique de Leibniz.... Donald Rutherford, University of California, San Diego (2002).
On the other hand, there are physicists and philosophers who say there's something more about human behavior that cannot be computed by a machine. Creativity, for example, and the sense of freedom people possess don't appear to come from logic or calculations. Yet these are not the only views of what consciousness is, or whether machines could ever achieve it. Another viewpoint on consciousness comes from quantum theory, which is the deepest theory of physics. According to the orthodox Copenhagen Interpretation, consciousness and the physical world are complementary aspects of the same reality.