Tagged: Kurt Gödel

Gödel: The modern development of the foundations of mathematics in the light of philosophy.

Source: Kurt Gödel, Collected Works, Volume III (1961) publ. Oxford University Press, 1981. The Complete lecture reproduced here. I would like to attempt here to describe, in terms of philosophical concepts, the development of foundational research in mathematics since around the turn of the century, and to fit it into a general schema of possible philosophical world-views [Weltanschauungen]. For this, it is necessary first of all to become clear about the schema itself. I believe that the most fruitful principle for gaining an overall view of the possible world-views will be to divide them up according to the degree and...

Kurt Gödel

Here is the biographical sketch of Kurt Gödel’s life from the Stanford Encyclopedia of Philosophy: Biographical Sketch Kurt Gödel was born on April 28, 1906 in what was then the Austro-Hungarian city of Brünn, and what is now Brno in the Czech Republic. Gödel’s father Rudolf August was a businessman, and his mother Marianne was a well-educated and cultured woman to whom Gödel remained close throughout his life, as witnessed by the long and wide-ranging correspondence between them. The family was well off, and Gödel’s childhood was an uneventful one, with one important exception; namely, from about the age of four...

Incompleteness Theorem

The Incompleteness Theorem is Gödel’s main contribution to 20th century thought. Gödel showed that within a logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions. It appears to destroy the hope of mathematical certitude through the use of the obvious methods. It also deconstructs an ideal of science, that we...

Kurt Gödel’s philosophical viewpoint, and his proof of the existence of God.

Explanation of the terms  in the image above. This is Godel’s formalized proof of the existence of God. P(psi) P is “positive” G(x) x have the property God ess. essential E existing • (bullet) Necessary (Kurt Gödel (1995). “Ontological Proof”. Collected Works: Unpublished Essays & Lectures, Volume III. pp. 403–404. Oxford University Press. Gödel left in his papers a  fourteen-point outline of his philosophical beliefs, that are dated around 1960. They show his deep belief in the rational structure of the world. Here are his 14 points: The world is rational. Human reason can, in principle, be developed more highly (through certain...