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Logos and Máthēma

Studies in the Philosophy of Mathematics and History of Logic

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Roman Murawski

The volume contains twenty essays devoted to the philosophy of mathematics and the history of logic. They have been divided into four parts: general philosophical problems of mathematics, Hilbert’s program vs. the incompleteness phenomenon, philosophy of mathematics in Poland, mathematical logic in Poland. Among considered problems are: epistemology of mathematics, the meaning of the axiomatic method, existence of mathematical objects, distinction between proof and truth, undefinability of truth, Gödel’s theorems and computer science, philosophy of mathematics in Polish mathematical and logical schools, beginnings of mathematical logic in Poland, contribution of Polish logicians to recursion theory.

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Part I: Philosophy of Mathematics (in General)

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Part I Philosophy of Mathematics (in General) Mathematical Knowledge Since its very beginnings mathematics played a special and distinguished roˆle in the human knowledge. It was close to the ideal of a scientific theory, even more, it established such an ideal and served as a pattern of a theory. It has played an important roˆle also in the development of the epistemology. In fact mathematics has been through ages a pattern of any rational knowledge and the paradigm of a priori knowledge. Hence the importance and meaning of philosophical and methodological reflections on mathematics as a science. Such reflections have accompanied mathematics since ancient Greece. In philosophical reflections on mathematics one can distinguish two principal groups of problems: ontological and epistemological. Among main questions of the first group are the following ones: what is the subject of mathematics, in particular what is the nature of mathematical objects, where and how do they exist, what are the criteria of their existence, what is the source and origin of mathema- tical objects, what is the nature and properties of the mathematical infinity. Epistemological problems concerning mathematics (which are the main subject of the present article) can be divided into four groups: – the problem of cognitive methods used and accepted in mathematics. In particular one considers here the problem of sources and origin of mathematical knowledge, the problem of the process of arriving at new results, the problem of methods of justifying mathematical statements and theorems, the problem of the validity...

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