Sam Sanders (MCMP/LMU) gives a talk at the MCMP Colloquium (26 November, 2015) titled "The Unreasonable Effectiveness of Nonstandard Analysis". Abstract: There is a persistent belief, propagated by such luminaries as Errett Bishop and Alain Connes, that infinitesimals (in the sense of Robinson’s Nonstandard Analysis (NSA) ) somehow are fundamentally non-constructive and that NSA is devoid of numerical meaning, as Bishop was wont to say. In this talk, we disprove the Bishop-Connes claim regarding NSA.  To this end, we show that theorems of NSA are equivalent to their associated “highly constructive" theorems from numerical analysis (not involving NSA).  We shall focus on potential applications of these results in philosophy and computer science, especially concerning vagueness and ontology.

Sam Sanders (MCMP/LMU) gives a talk at the MCMP Colloquium (26 November, 2015) titled "The Unreasonable Effectiveness of Nonstandard Analysis". Abstract: There is a persistent belief, propagated by such luminaries as Errett Bishop and Alain Connes, that infinitesimals (in the sense of Robinson’s Nonstandard Analysis (NSA) ) somehow are fundamentally non-constructive and that NSA is devoid of numerical meaning, as Bishop was wont to say. In this talk, we disprove the Bishop-Connes claim regarding NSA. To this end, we show that theorems of NSA are equivalent to their associated “highly constructive" theorems from numerical analysis (not involving NSA). We shall focus on potential applications of these results in philosophy and computer science, especially concerning vagueness and ontology.

The Unreasonable Effectiveness of Nonstandard Analysis

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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The Unreasonable Effectiveness of Nonstandard Analysis

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