Mathematics, Science and Epistemology

Cover
Title page
Editors' introduction
PART 1 PHILOSOPHY OF MATHEMATICS
1 Infinite regress and foundations of mathematics
Introduction
1 Stopping infinite regress in science
2 Stopping infinite regress by the logico-trivialization of mathematics
3 Stopping infinite regress by a trivial meta-theory
2 A renaissance of empiricism in the recent philosophy of mathematics?
Introduction
1 Empiricism and induction: the new vogue in mathematical philosophy?
2 Quasi-empirical versus Eudidean theories
3 Mathematics is quasi-empirical
4 'Potential falsifiers' in mathematics
5 Periods of stagnation in the growth of quasi-empirical theories
3 Cauchy and the continuum: the significance of non-standard analysis for the history and philosophy of mathematics (edited by 1. P. Cleave)
1 Non-standard analysis suggests a radical reassessment of the history of the infinitesimal calcul us
2 Cauchy and the problem of uniform convergence
3 A new solution
4 What caused the downfall of Leibniz's theory?
5 Was Cauchy a 'forerunner' of Robinson?
6 Metaphysical versus technical
7 Appraisal of mathematical theories
4 What does a mathematical proof prove?
5 The method of analysis-synthesis
1 Analysis-synthesis: a pattern of Eudidean heuristic and its criticism
(a) Prologue on analysis and synthesis
(b) Analysis-synthesis and heuristic
(c) The Cartesian Circuit and its breakdown
(c1) The Circuit is neither empiricist nor intellectualist. The source of knowledge is the Circuit as a whole
(c2) Induction and deduction in the Circuit
(c3) The continuity between Pappus and Descartes
(c4) The Cartesian Circuit in mathematics
(c5) The breakdown of the Cartesian Circuit
2 Analysis-synthesis: how failed attempts at refutations may be heuristic starting points of research programmes
(a) An analysis-synthesis in topology which does not prove what it sets out to prove
(b) An analysis-synthesis in physics which does not explain what it set out to explain
(c) Pappusian analyses-syntheses in Greek geometry
(d) [False awareness about analysis-synthesis]
PART 2 CRITICAL PAPERS
6 The problem of appraising scientific theories: three approaches
1 Three main schools of thought concerning the normative problem of appraising scientific theories
(a) Scepticism
(b) Demarcationism
(c) Elitism
2 Elitism and allied philosophical positions
(a) Elitists for psychologism and/or sociologism
(b) Elitists for authoritarianism and historicism
(c) Elitists for pragmatism
7 Necessity, Kneale and Popper
1 The ontological level
2 The epistemological-methodological level
3 The continuity of logical and natural necessity
8 Changes in the problem of inductive logic
Introduction
1 The two main problems of classical empiricism: inductive justification and inductive method
2 The one main problem of neoclassical empiricism: weak inductive justification (degree of confirmation)
3 The weak and strong atheoretical theses
(a) Carnap abandons the Jeffreys-Keynes postulate. Qualified instance confirmation versus confirmation
(b) The weak atheoretical thesis: confirmation theory without theories
(c) The conflation of the weak and the strong atheoretical theses
(d) The interconnection between the weak and strong atheoretical theses
(e) A Carnapian logic of discovery
4 Probability, evidential support, rational belief and betting quotients
(a) Are degrees of evidential support probabilities?
(b) Are 'degrees of rational belief' degrees of evidental support or are they rational betting quotients?
(c) Are rational betting quotients probabilities?
5 The collapse of the weak atheoretical thesis
(a) 'Correctness of language' and confirmation theory
(b) The abdication of the inductive judge
6 The one main problem of critical empiricism: method
(a) 'Acceptability₁'
(b) 'Acceptability₂'
(c) 'Acceptability₃'
7 Theoretical support for predictions versus (test)-evidential support for theories
Appendix. On Popper's three notes on degree of corroboration
9 On Popperian historiography
Appendix on 'ultra-falsificationism'
10 Anomalies versus 'crucial experiments' (a rejoinder to Professor Grünbaum)
Introduction
1 There have been no crucial experiments in science
2 The impossibility of Grünbaumian crucial experiments and the possibility of appraising scientific growth without them
3 On practical advice
4 The characteristic of science is not rational belief but rational replacement of propositions
11 Understanding Toulmin
Introduction
1 Three schools of thought on the normative problem of appraising scientific theories
2 Toulmin and the Wittgensteinian 'thought-police'
3 Toulmin's Darwinian synthesis of Hegel and Wittgenstein
4 Conclusion
PART 3 SCIENCE AND EDUCATION
12 A letter to the Director of the London School of Economics
13 The teaching of the history of science
14 The social responsibility of science
References
Lakatos bibliography
Indexes