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From Mathematics to Philosophy


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Table of Contents

Preface; A note on the text; Introduction; 1 Substantial factualism as a method and an antidote 2 Against positivism 3 Against linguistic philosophy 4 Explanatory remarks on substantial factualism 5 Logic, mathematics, and the scope of this book; I Mathematical Logic and philosophy of mathematics; 1 Topics in the philosophy of mathematics 2 The axiomatic method and abstract structures 3 Questions of consistency 4 The deceptive appeal of mathematical logic to philosophers Notes; II Characterization of general mathematical concepts; 1 Natural numbers 2 The continuum 3 Mechanical procedures; III Russell’s logic and some general issues; Principles (1903) 2 Preludes to Principia (1903-10) 3 Principia 4 Wittgenstein and Ramsay 4 Logical truth and other philosophical matters 6 Predictive definitions and the vicious-circle principle Notes; IV Logical truth; 1 Presuppositions of Aristotle’s logic 2 Logical constraints and logical truths; V Metalogic; 1 Formal languages and formal systems 2 Origins and influences of metalogic 3 Exact results on formal mathematical systems 4 Exact results of logical calculi; VI The concept of the set; 1 The (maximum) iterative concept 2 Bankruptcy (contradiction) or misunderstanding (error)? 3 Objectivism and formalism set in theory 4 New axioms and criteria of acceptability 5 Comparisons with geometry and physics 6 Digression on unbounded quantifications 7 Extracting axioms of set theory from Cantor’s writings 8 The hierarchies of Cantor and Mirimanoff Notes; VII Theory and practice in mathematics; 1. Activity and feasibility 2 Reducing mathematics to logic 3 What is mathematics? 4 Practical aspects of mathematics Notes; VIII Necessity, analyticity, and apriority; 1 Homes and assimilations of the three concepts 2 Suggestions from Kant’s philosophy 3 From Frege to analytic philosophy 4 Notes on contemporary discussions Notes; IX Mathematics and computers; 1 New uses of computers 2 Influence of mathematics on the development of computers 3 Logical mathematics 4 Mathematical reasoning as mechanical 5 Finite computations and infinite mathematics 6 Logic and computers; X Minds and machines; 1 Aspects of mechanism 2 Computers and brains 3 Artificial or mechanical intelligence 4 Computer simulation of human thought 5 Protocols and theoretical psychology 6 Mathematical arguments 7 Gödel on minds and machines Notes; XI Notes on knowledge and life; 1 Intrinsic goals and large problems 2 Relevance and forms of life 3 Specialization and the unity of knowledge 4 Bertrand Russell as an example 5 Life and the pursuit of philosophy Notes; XII Themes and approaches; 1 Scientific studies of matter, mind, and machines 2 Science and philosophy 3 Remarks on contemporary philosophy 4 Respect for gross facts 5 Looking beyond; Appendix Exercises in Criticism; 1 Notes on the justification of induction 2 On scepticism about induction 3 The existence of material objects 4 A question of knowledge of knowledge 5 What is an individual?; Notes; Index

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Hao Wang

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