By Richard Tieszen
Richard Tieszen offers an research, improvement, and safeguard of a couple of valuable principles in Kurt Godel's writings at the philosophy and foundations of arithmetic and good judgment. Tieszen buildings the argument round Godel's 3 philosophical heroes - Plato, Leibniz, and Husserl - and his engagement with Kant, and vitamins 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 supplying discussions of Godel's perspectives at the philosophical importance of his technical effects on completeness, incompleteness, undecidability, consistency proofs, speed-up theorems, and independence proofs, Tieszen furnishes an in depth 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 kind of platonic rationalism that calls for rational instinct, known as 'constituted platonism', is built and defended. Tieszen indicates how constituted platonism addresses the matter of the objectivity of arithmetic and of the information of summary mathematical items. ultimately, he considers the results 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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Additional info for After Gödel: Platonism and Rationalism in Mathematics and Logic
If we consider formulas that can be proved in both systems, however, then the result says that if the length of a proof is deﬁned to be the number of lines in the proof, then the proof of a given formula in Si + 1 will be much shorter than the shortest proof in Si. To be more precise, it says that for each function ç that is computable in Si there exist inﬁnitely many formulas f such that if k is the length of the shortest proof of f in Si and l is the length of the shortest proof of f in Si + 1, then k > f( l ).
Husserl, as we saw, refers to the transcendental ego in its full concreteness as a monad. Kant, like Leibniz and Plato, does not put the intentionality of human consciousness at the center of his philosophy. Kant is also not a platonist about mathematical objects or facts, and he mounts a critique of classical rationalism (including Leibniz). For Kant, knowledge is restricted to sensory intuition and the two forms of sensory intuition, space and time. Kant, unlike Husserl, distinguishes phenomena from noumena (which is what Wang calls Kant’s dualism in the ﬁrst passage quoted above), and is able to develop the transcendental method far enough to show how empirical realism is compatible with transcendental idealism (see chapter 4), but in his work there is no question of showing how a kind of platonism or mathematical objectivism is compatible with transcendental idealism.
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. George Boolos (Boolos 1987) gives an example of the premises and conclusion of an argument written in the language of ﬁrst-order predicate logic with identity and function symbols that is logically valid and short, in the sense that the premises and conclusion contain only about 60 symbols.