The main project of the thesis is to provide a comprehensive examination of Gödel’s philosophy of mathematics and logic, in defence of his Platonism and the idea of mathematical intuition. The whole work is considered to be partly historical, explaining Gödel's ideas and arguments in comparison with his contemporaries such as Carnap, Russell, Turing and Hilbert; partly mathematical, relying especially on his incompleteness theorem to refute certain philosophical positions; and partly philosophical, shedding light on debates in current discussions of philosophy of mathematics and logic such as the nature of paradoxes, the indispensability argument, the feasibility of formalism, etc.
After the introduction, chapter one deals with Gödel’s critique and refutation of Carnap’s syntactical view of mathematics, which interprets mathematics purely in terms of linguistic rules of symbols, and thus void of content. Chapter two discusses Gödel’s criticism of Russell’s constructivistic view towards logic, especially his no-class theory and the vicious circle principle. Chapter three turns attention to Gödel’s role in the development of computability theory, using it as a case study for conceptual analysis and tries to reconcile the apparent conflict concerning Gödel’s remarks on Turing, who gave a “precise and adequate definition” of mechanical computability and yet committed “a philosophical error” in his argument. The last chapter focuses on Gödel’s relation to Hilbert, especially the relation of his incompleteness theorems for Hilbert’s finitary consistency proof program and the significance of Gödel’s
serious engagement with finitism. The thread uniting all the discussions in the chapters is the idea of the incompletability or inexhaustibility of mathematics by pure formalism and the indispensability of abstract concepts and a kind of mathematical intuition in providing a satisfactory foundation for mathematics and in enabling the human mind to surpass any particular mechanism, which Gödel himself considered to be the major philosophical conclusion to be drawn from his logical results.
Original language | English |
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Award date | 2018 |
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