Mechanism, Mentalism and Metamathematics: An Essay on FinitismSpringer Science & Business Media, 9 de març 2013 - 285 pàgines This book grew out of a graduate student paper [261] in which I set down some criticisms of J. R. Lucas' attempt to refute mechanism by means of G6del's theorem. I had made several such abortive attempts myself and had become familiar with their pitfalls, and especially with the double edged nature of incompleteness arguments. My original idea was to model the refutation of mechanism on the almost universally accepted G6delian refutation of Hilbert's formalism, but I kept getting stuck on questions of mathematical philosophy which I found myself having to beg. A thorough study of the foundational works of Hilbert and Bernays finally convinced me that I had all too naively and uncritically bought this refutation of formalism. I did indeed discover points of surprisingly close contact between formalism and mechanism, but also that it was possible to under mine certain strong arguments against these positions precisely by invok ing G6del's and related work. I also began to realize that the Church Turing thesis itself is the principal bastion protecting mechanism, and that G6del's work was perhaps the best thing that ever happened to both mechanism and formalism. I pushed these lines of argument in my dis sertation with the patient help of my readers, Raymond Nelson and Howard Stein. I would especially like to thank the latter for many valuable criticisms of my dissertation as well as some helpful suggestions for reor ganizing it in the direction of the present book. |
Continguts
1 | |
CHAPTER IIMIND NUMBER AND THE INFINITE | 33 |
Mind and Philosophy of Number | 40 |
Dedekinds Theory of Arithmetic | 46 |
Propositioning the Infinite | 57 |
THE MENTAL THE FINITE AND THE FORMAL | 72 |
72 | 119 |
Frege Later Hilbert and the Problem of Formalization | 132 |
Hilberts Rule and the Phenomenology of Infinity | 151 |
EFFECTIVENESS MECHANIZED | 175 |
CONCLUDING SUMMARY | 244 |
Altres edicions - Mostra-ho tot
Mechanism, Mentalism and Metamathematics: An Essay on Finitism J. Webb Previsualització no disponible - 2010 |
Mechanism, Mentalism and Metamathematics: An Essay on Finitism J. Webb Previsualització no disponible - 1980 |
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algebraic algorithm analysis applied argued arithmetic axiomatic axiomatic set theory axioms basic behavior Bernays binary function calculation Cantor's Church's thesis claimed classical complete concept consistency proof constructive defined definition demon Desargues Descartes elementary Entscheidungsproblem enumeration equations essential Euclid's euclidean euclidean geometry fact finite number Finsler's first-order logic formal system Frege geometry given Gödel's Gödel's incompleteness theorem halting problem hence Hilbert Hilbert's axioms ibid idea induction infinite sets infinity intuition intuitionistic isomorphism Kleene Kleene's Kreisel Laplace's demon laws logic mathematical Mathematik means mechanism metamathematical method mind natural number notion number theory objects ordinal logic paradox partial recursive Peano model Peirce physical plane Poincaré possible postulate predicate principle problem processes propositions provable purely question recursion theorem recursive functions refute relation result Richard's Paradox rules satisfy second-order logic sense sentences sequence set theory simulate space symbols thought tion transfinite triangles Turing machine Turing's undecidable variables