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After Gödel: Platonism and Rationalism in Mathematics and - download pdf or read online

By Richard Tieszen

ISBN-10: 019960620X

ISBN-13: 9780199606207

Richard Tieszen provides an research, improvement, and protection of a couple of significant principles in Kurt Godel's writings at the philosophy and foundations of arithmetic and common sense. Tieszen constructions the argument round Godel's 3 philosophical heroes - Plato, Leibniz, and Husserl - and his engagement with Kant, and supplementations shut readings of Godel's texts on foundations with fabrics from Godel's Nachlass and from Hao Wang's discussions with Godel. in addition to delivering discussions of Godel's perspectives at the philosophical value of his technical effects on completeness, incompleteness, undecidability, consistency proofs, speed-up theorems, and independence proofs, Tieszen furnishes a close research of Godel's critique of Hilbert and Carnap, and of his next flip to Husserl's transcendental philosophy in 1959. in this foundation, a brand new form of platonic rationalism that calls for rational instinct, known as 'constituted platonism', is built and defended. Tieszen exhibits how constituted platonism addresses the matter of the objectivity of arithmetic and of the data of summary mathematical items. ultimately, he considers the consequences of this place for the declare that human minds ('monads') are machines, and discusses the problems of pragmatic holism and rationalism.

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The solution of at least some open problems in mathematics and logic is then held to depend on systematic and finite but non-mechanical methods of human reason. It is Husserl and Leibniz who emphasize the exactness of concepts of mathematics and logic, and the possibility of clarification of our grasp of these concepts, and Husserl (after 1900) and Plato who think of the concepts and objects of mathematics as ideal and abstract. Husserl, building on Kant’s transcendental philosophy, would interpret all of this in the context of transcendental phenomenology and, unlike Kant, he recognizes the possibility of a kind of rational intuition.

Results analogous to this, where the length of a proof is taken to be its Go¨del number (equivalently, the number of symbols) instead of the number of the lines in the proof, were obtained later. ) Speed-up results of this type are philosophically interesting because they indicate the benefits of ascending to more powerful formal systems. In terms of the interpretations of the formal systems involved and the related capacities of human reason, they indicate the benefits of ascending to higher-level conceptions in our thinking, and they suggest that it is not a good idea to insist that only the reasoning expressed in first-order logic is legitimate.

This fact opens up his views to a number of objections. I would like to cordon off some elements of Go¨delian platonic rationalism about mathematics and logic that I think can be defended. As for the rest, I am content to let Go¨del’s critics have their way. 1 Tarski’s theorem on the indefinability of arithmetical truth in arithmetical language is also mentioned. Most of the technical results for which Go¨del is famous were produced in the 1930s and early 1940s. The incompleteness theorems and other results I discuss are purely mathematical in character and are unassailable as such on scientific grounds.

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After Gödel: Platonism and Rationalism in Mathematics and Logic by Richard Tieszen

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