Harvey M. Friedman (OSU) gives a talk at the MCMP Colloquium (30 October, 2012) titled "Transcendental Proofs". Abstract: We discuss our Transcendental Proofs project, spanning approximately 45 years, which aims to uncover mathematically fundamental, rich, and diverse subareas of  athematics which can only be developed by going well beyond the usual ZFC axioms for mathematics. The discussion will have mathematical, computational, and philosophical components.The mathematical component focuses on long finite sequences, adjacent amsey theorems, finite trees and graphs, continuous maps between countable sets, Borel diagonalization and selection, Boolean Relation Theory, and Embedded Maximal Cliques. The computational component focuses on predictions from higher infinities of the result of actual computations. The philosophical component focuses on various formulations of the project, involving measures of simplicity and naturalness, as well as prospects for confirmation of the validity of transcendental methods.

Harvey M. Friedman (OSU) gives a talk at the MCMP Colloquium (30 October, 2012) titled "Transcendental Proofs". Abstract: We discuss our Transcendental Proofs project, spanning approximately 45 years, which aims to uncover mathematically fundamental, rich, and diverse subareas of athematics which can only be developed by going well beyond the usual ZFC axioms for mathematics. The discussion will have mathematical, computational, and philosophical components.The mathematical component focuses on long finite sequences, adjacent amsey theorems, finite trees and graphs, continuous maps between countable sets, Borel diagonalization and selection, Boolean Relation Theory, and Embedded Maximal Cliques. The computational component focuses on predictions from higher infinities of the result of actual computations. The philosophical component focuses on various formulations of the project, involving measures of simplicity and naturalness, as well as prospects for confirmation of the validity of transcendental methods.

Transcendental Proofs

Mathematical Philosophy - the application of logical and mathematical methods in philosophy - is about to experience a tremendous boom in various areas of philosophy. At the new Munich Center for Mathematical Philosophy, which is funded mostly by the German Alexander von Humboldt Foundation, philosophical research will be carried out mathematically, that is, by means of methods that are very close to those used by the scientists.
The purpose of doing philosophy in this way is not to reduce philosophy to mathematics or to natural science in any sense; rather mathematics is applied in order to derive philosophical conclusions from philosophical assumptions, just as in physics mathematical methods are used to derive physical predictions from physical laws.
Nor is the idea of mathematical philosophy to dismiss any of the ancient questions of philosophy as irrelevant or senseless: although modern mathematical philosophy owes a lot to the heritage of the Vienna and Berlin Circles of Logical Empiricism, unlike the Logical Empiricists most mathematical philosophers today are driven by the same traditional questions about truth, knowledge, rationality, the nature of objects, morality, and the like, which were driving the classical philosophers, and no area of traditional philosophy is taken to be intrinsically misguided or confused anymore. It is just that some of the traditional questions of philosophy can be made much clearer and much more precise in logical-mathematical terms, for some of these questions answers can be given by means of mathematical proofs or models, and on this basis new and more concrete philosophical questions emerge. This may then lead to philosophical progress, and ultimately that is the goal of the Center.

Mathematical Philosophy - the application of logical and mathematical methods in philosophy - is about to experience a tremendous boom in various areas of philosophy. At the new Munich Center for Mathematical Philosophy, which is funded mostly by the German Alexander von Humboldt Foundation, philosophical research will be carried out mathematically, that is, by means of methods that are very close to those used by the scientists. The purpose of doing philosophy in this way is not to reduce philosophy to mathematics or to natural science in any sense; rather mathematics is applied in order to derive philosophical conclusions from philosophical assumptions, just as in physics mathematical methods are used to derive physical predictions from physical laws. Nor is the idea of mathematical philosophy to dismiss any of the ancient questions of philosophy as irrelevant or senseless: although modern mathematical philosophy owes a lot to the heritage of the Vienna and Berlin Circles of Logical Empiricism, unlike the Logical Empiricists most mathematical philosophers today are driven by the same traditional questions about truth, knowledge, rationality, the nature of objects, morality, and the like, which were driving the classical philosophers, and no area of traditional philosophy is taken to be intrinsically misguided or confused anymore. It is just that some of the traditional questions of philosophy can be made much clearer and much more precise in logical-mathematical terms, for some of these questions answers can be given by means of mathematical proofs or models, and on this basis new and more concrete philosophical questions emerge. This may then lead to philosophical progress, and ultimately that is the goal of the Center.

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Transcendental Proofs

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