ViennaFounds042811

ViennaFounds042811 - PAST, PRESENT, AND FUTURE DIRECTIONS...

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PAST, PRESENT, AND FUTURE DIRECTIONS IN THE FOUNDATIONS OF MATHEMATICS by Harvey M. Friedman Ohio State University Distinguished University Professor of Mathematics, Philosophy, and Computer Science New Trends in Logic Kurt Godel Society April 29, 2011
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AGENDA 1. The Foundational Life. 2. Profound Uneasiness. 3. Grand Unification. 4. Assertions and Proofs. 5. Fundamental Foundational Moves: consistency, completeness, incompleteness. 6. Consistency, and Incorporation of New Notions. 7. Completeness, and Delicate Choice of Fragments. 8. Incompleteness, and Concreteness, Simplicity, Naturalness.
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THE FOUNDATIONAL LIFE Kurt Gödel is the great practitioner of the Foundational Life who has had such a profound influence on me and so many others. In the Foundational Life, there are two principal goals. i. The founding of new systematic disciplines that have the same general features as the great systematic disciplines that have emerged over the centuries - such as mathematics, physics, statistics, computing science, electrical engineering, etc. ii. The redevelopment, reorganization, and imaginative exposition of existing systematic disciplines in the direction of creating deeper relevance to the original purposes for which they were founded. In the Foundational Life, philosophy is commonly used as a method for choosing and analyzing fundamental concepts, and mathematics is commonly used for rigorous development. The mathematics informs the philosophy and the philosophy informs the mathematics. In the Foundational Life, there is a delicate balance between the philosophical and the mathematical, both subservient to the principal focus on i,ii above.
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FOUNDATIONAL LIFE In the Foundational Life, no premium is paid, per se, to careful philosophical statements and arguments, and no premium is paid, per se, to complicated or deep mathematical developments. However, careful philosophy and deep mathematics are used if they facilitate the main focus on i,ii - and they often do. In the Foundational Life, there is constant assessment of the prospects for developing new systematic disciplines, or redevelopment of existing systematic disciplines. Generally, this requires that there be a wide range of deep phenomena available for analysis. The Foundational Life as I practice it, is both highly mathematical and highly philosophical. But it differs profoundly from the Mathematical Life and the Philosophical Life in various ways.
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PHILOSOPHICAL LIFE In the Philosophical Life, there is a focus on careful analysis of apparently fundamental concepts, and making careful arguments that can be defended against attacks, as well as attacking other’s careful arguments. In the Philosophical Life, the choice of these apparently fundamental concepts is not generally measured in terms of their appropriateness for the creation of new systematic disciplines that have the same general features as the great systematic disciplines that have emerged over the centuries - mathematics, physics, statistics, computing science, etc. The clearest exception to the normal modus operandi in the
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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ViennaFounds042811 - PAST, PRESENT, AND FUTURE DIRECTIONS...

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