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Logos and Máthēma

Studies in the Philosophy of Mathematics and History of Logic

Series:

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) Mathematical Knowledge . . . . . . . . . . . . . . . . . . . . . . . 15 On the Power and Weaknesses of the Axiomatic Method . . . . . . . . 53 Remarks on the Mathematical Universe . . . . . . . . . . . . . . . . 63 Structuralism and Category Theory in the Contemporary Philosophy of Mathematics (with Izabela Bondecka-Krzykowska) . . . . . . . . . . 73 Part II: Hilbert’s Program vs. Incompleteness Phenomenon Hilbert’s Program: Incompleteness Theorems vs. Partial Realizations . 83 On the Distinction Proof-Truth in Mathematics . . . . . . . . . . . . 101 Reactions to the Discovery of the Incompleteness Phenomenon . . . . 115 Go¨del’s Incompleteness Theorems and Computer Science . . . . . . . 127 The Present State of Mechanized Deduction, and the Present Knowledge of Its Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 On Proofs of the Consistency of Arithmetic . . . . . . . . . . . . . . 159 Decidability vs. Undecidability. Logico-Philosophico-Historical Remarks 165 Undefinability of Truth. The Problem of the Priority: Tarski vs. Go¨del . 177 Troubles With (the Concept of) Truth in Mathematics . . . . . . . . . 187 Part III: Philosophy of Mathematics in Poland The Philosophy of Hoene-Wroński . . . . . . . . . . . . . . . . . . . 205 Philosophical Reflection on Mathematics in Poland in the Interwar Period 215 Philosophy of Mathematics in the Warsaw Mathematical School . . . . 227 Andrzej Mostowski on the Foundations and Philosophy of Mathematics (with Jan Woleński) . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Part IV: Mathematical Logic in Poland Stanisław Piątkiewicz and the Beginnings of Mathematical Logic in Poland (with Tadeusz Batóg) . . . . . . . . . . . . . . . . . . . . . . . . . 261 Contribution of Polish Logicians to Recursion Theory . . . . . . . . . 267 Logical Investigations at the University of Poznań in 1945–1955 (with Jerzy Pogonowski) . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Editorial Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Index of Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

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