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The logician Kurt Gödel (1906-1978) published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Gödel's work on the foundations and philosophy of mathematics. Their essays explore almost every aspect of Godel's intellectual legacy including his concepts of intuition and analyticity, the Completeness Theorem, the set-theoretic multiverse, and the state of mathematical logic today. This groundbreaking volume will be invaluable to students, historians, logicians and philosophers of mathematics who wish to understand the current thinking on these issues.

Logic, Symbolic and mathematical. --- Mathematics --- Logic of mathematics --- Mathematics, Logic of --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Philosophy. --- Gödel, Kurt. --- Gkentel, Kourt --- גדל --- Logic, symbolic and mathematical --- Mathematics - Philosophy --- Gödel, Kurt

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This volume tackles Gödel's two-stage project of first using Husserl's transcendental phenomenology to reconstruct and develop Leibniz' monadology, and then founding classical mathematics on the metaphysics thus obtained. The author analyses the historical and systematic aspects of that project, and then evaluates it, with an emphasis on the second stage. The book is organised around Gödel's use of Leibniz, Husserl and Brouwer. Far from considering past philosophers irrelevant to actual systematic concerns, Gödel embraced the use of historical authors to frame his own philosophical perspective. The philosophies of Leibniz and Husserl define his project, while Brouwer's intuitionism is its principal foil: the close affinities between phenomenology and intuitionism set the bar for Gödel's attempt to go far beyond intuitionism. The four central essays are `Monads and sets', `On the philosophical development of Kurt Gödel', `Gödel and intuitionism', and `Construction and constitution in mathematics'. The first analyses and criticises Gödel's attempt to justify, by an argument from analogy with the monadology, the reflection principle in set theory. It also provides further support for Gödel's idea that the monadology needs to be reconstructed phenomenologically, by showing that the unsupplemented monadology is not able to found mathematics directly. The second studies Gödel's reading of Husserl, its relation to Leibniz' monadology, and its influence on his publishe d writings. The third discusses how on various occasions Brouwer's intuitionism actually inspired Gödel's work, in particular the Dialectica Interpretation. The fourth addresses the question whether classical mathematics admits of the phenomenological foundation that Gödel envisaged, and concludes that it does not. The remaining essays provide further context.  The essays collected here were written and published over the last decade. Notes have been added to record further thoughts, changes of mind, connections between the essays, and updates of references.

Philosophy. --- Phenomenology. --- Mathematical Logic and Foundations. --- Philosophy of Science. --- Philosophy (General). --- Science --- Logic, Symbolic and mathematical. --- Phénoménologie --- Sciences --- Logique symbolique et mathématique --- Philosophie --- Science_xPhilosophy. --- Philosophy & Religion --- Philosophy --- Mathematics --- Gödel, Kurt --- Logic of mathematics --- Mathematics, Logic of --- Gkentel, Kourt --- גדל --- Philosophy and science. --- Mathematical logic. --- Phenomenology . --- Normal science --- Philosophy of science --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Philosophy, Modern --- Science and philosophy

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The first systematic examination of Hilary Putnam's arguments against computational functionalism challenges each of Putnam's main arguments.

Cognitive psychology --- Putnam, Hilary --- Gödel, Kurt --- Computers. --- Functionalism (Psychology) --- Mind-brain identity theory. --- Realism. --- Gödel, Kurt. --- Putnam, Hilary. --- Functionalism (Psychology). --- Gödel, Kurt. --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic brains --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Brain-mind identity theory --- Functional psychology --- Gkentel, Kourt --- גדל --- Computer systems --- Cybernetics --- Machine theory --- Calculators --- Cyberspace --- Brain --- Mind and body --- Psychology --- Empiricism --- Philosophy --- Universals (Philosophy) --- Conceptualism --- Dualism --- Idealism --- Materialism --- Nominalism --- Positivism --- Rationalism --- Computers --- Mind-brain identity theory --- Realism --- PHILOSOPHY/General --- COGNITIVE SCIENCES/General --- Godel, Kurt.

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During his lifetime, Kurt Gödel was not well known outside the professional world of mathematicians, philosophers and theoretical physicists. Early in his career, for his doctoral thesis and then for his Habilitation (Dr.Sci.), he wrote earthshaking articles on the completeness and provability of mathematical-logical systems, upsetting the hypotheses of the most famous mathematicians/philosophers of the time. He later delved into theoretical physics, finding a unique solution to Einstein’s equations for gravity, the ‘Gödel Universe’, and made contributions to philosophy, the guiding theme of his life. This book includes more details about the context of Gödel’s life than are found in earlier biographies, while avoiding an elaborate treatment of his mathematical/scientific/philosophical works, which have been described in great detail in other books. In this way, it makes him and his times more accessible to general readers, and will allow them to appreciate the lasting effects of Gödel’s contributions (the latter in a more up-to-date context than in previous biographies, many of which were written 15–25 years ago). His work spans or is relevant to a wide spectrum of intellectual endeavor, and this is emphasized in the book, with recent examples. This biography also examines possible sources of his unusual personality, which combined mathematical genius with an almost childlike naiveté concerning everyday life, and striking scientific innovations with timidity and hesitancy in practical matters. How he nevertheless had a long and successful career, inspiring many younger scholars along the way, with the help of his loyal wife Adele and some of his friends, is a fascinating story in human nature.

Matemàtics --- Gödel, Kurt. --- Àustria --- Científics --- Dones matemàtiques --- Matemàtica --- Gödel, Kurt, --- Autriche --- Österreich --- República d'Àustria --- Republik Österreich --- Europa central --- Països de la Unió Europea --- Països de parla alemanya --- Alta Àustria (Àustria) --- Baixa Àustria (Àustria) --- Estíria (Àustria) --- Salzkammergut (Àustria) --- Semmering (Àustria : Port de muntanya) --- Tirol (Àustria) --- Viena (Àustria) --- Vorarlberg (Àustria) --- Àustria-Hongria --- Sacre Imperi Romanogermànic --- Mathematicians --- Mathematics. --- Gödel, Kurt. --- Math --- Science --- Gkentel, Kourt --- גדל --- Gödel, Kurt --- History. --- Mathematical logic. --- Physicists --- Astronomers --- Philosophy --- History of Mathematical Sciences. --- Mathematical Logic and Foundations. --- Biographies of Physicists and Astronomers. --- History of Science. --- History of Philosophy. --- Biography. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism

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Mathematical logic --- Philosophy of science --- Escher, Maurits Cornelis --- Gödel, Kurt --- Bach, Johann Sebastian --- Escher, Maurits Cornelis, --- Gödel, Kurt, --- Escher, M. C. --- Godel, Kurt --- Gkentel, Kourt --- גדל --- Metamathematics. --- Artificial intelligence --- Metamathematics --- Symmetry --- #WSCH:ETOS --- Aesthetics --- Proportion --- Logic, Symbolic and mathematical --- Mathematics --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Philosophy --- Bach, Johann Sebastian, --- Gödel, Kurt. --- Escher, Mauricio, --- Bach, Jean-Sébastien --- Artificial intelligence. --- Symmetry. --- Bakh, Iogann Sebastian, --- Bakh, Y. S., --- Bach, Jean Sébastien, --- Bach, G. S., --- Bach, Jan Sebastian, --- Bachas, J. S., --- Bach, J. S. --- Bahs, Johans Sebatjans, --- Pa-ha, Te, --- Bakh, Ĭ. S. --- Bakh, Ĭokhan Sebastian, --- Bach, Joh. Seb. --- Bakh, Yohan Sebasṭyan, --- Bach, Iohann Sebastian, --- Bahha, J. S., --- Bahha, Yohan Sebasutian, --- Bach, I. S., --- Bach, Juan S., --- Bach, John Sebastian, --- Bach, Giovanni Sebastiano, --- באך, יוהן סבסטיאן --- Bach, Johann Sebastian, - 1685-1750 --- Escher, Maurits Cornelis, - 1898-1972 --- Gödel, Kurt, - 1906-1978 --- Escher, Maurits Cornelis, - 1898-1971