## Mathematics, Science and Epistemology: Volume 2, Philosophical PapersImre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics (much of it unpublished), together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues. |

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### Contents

Infinite regress and foundations of mathematics | 3 |

1 Stopping infinite regress in science | 4 |

by the logicotrivialization of mathematics | 10 |

3 by a trivial metatheory | 20 |

A renaissance of empiricism in the recent philosophy of mathematics? | 24 |

the new vogue in mathematical philosophy? | 25 |

2 Quasiempirical versus Euclidean theories | 28 |

3 Mathematics is quasiempirical | 30 |

Changes in the problem of inductive logic | 128 |

inductive justification and inductive method | 129 |

weak inductive justification degree of confirmation | 131 |

3 The weak and strong atheoretical theses | 138 |

confirmation theory without theories | 142 |

c The conflation of the weak and the strong atheoretical theses | 145 |

d The interconnection between the weak and strong atheoretical theses | 147 |

e A Carnapian logic of discovery | 149 |

4 Potential falsifiers in mathematics | 35 |

5 Periods of stagnation in the growth of quasiempirical theories | 41 |

Cauchy and the continuum the significance of nonstandard analysis for the history and philosophy of mathematics | 43 |

2 Cauchy and the problem of uniform convergence | 45 |

3 A new solution | 47 |

4 What caused the downfall of Leibnizs theory? | 53 |

5 Was Cauchy a forerunner of Robinson? | 55 |

6 Metaphysical versus technical | 58 |

7 Appraisal of mathematical theories | 59 |

What does a mathematical proof prove? | 61 |

The method of analysissynthesis | 70 |

b Analysissynthesis and heuristic | 72 |

c The Cartesian Circuit and its breakdown | 75 |

c1 The Circuit is neither empiricist nor intellectualist The source of knowledge is the Circuit as a whole | 77 |

c2 Induction and deduction in the Circuit | 79 |

c3 The continuity between Pappus and Descartes | 83 |

c4The Cartesian Circuit in mathematics | 88 |

how failed attempts at refutations may be heuristic starting points of research programmes | 93 |

b An analysissynthesis in physics which does not explain what it set out to explain | 97 |

c Pappusian analysessyntheses in Greek geometry | 99 |

d False awareness about analysissynthesis | 101 |

The problem of appraising scientific theories three approaches | 107 |

b Demarcationism | 108 |

c Elitism | 111 |

2 Elitism and allied philosophical positions | 112 |

b Elitists for authoritarianism and historicism | 116 |

c Elitists for pragmatism | 117 |

Necessity Kneale and Popper | 121 |

2 The epistemologicalmethodological level | 124 |

3 The continuity of logical and natural necessity | 126 |

4 Probability evidential support rational belief and betting quotients | 151 |

a Are degrees of evidential support probabilities? | 152 |

b Are degrees of rational belief degrees of evidential support or are they rational betting quotients? | 157 |

c Are rational betting quotients probabilities? | 159 |

5 The collapse of the weak atheoretical thesis | 160 |

b The abdication of the inductive judge | 165 |

method | 170 |

b Acceptability | 173 |

c Acceptability | 181 |

7 Theoretical support for predictions versus testevidential support for theories | 192 |

Appendix On Poppers three notes on degree of corroboration | 193 |

On Popperian historiography | 201 |

Appendix on ultrafalsificationism | 208 |

Anomalies versus crucial experiments a rejoinder to Professor Grunbaum | 211 |

2 The impossibility of Grunbaumian crucial experiments and the possibility of appraising scientific growth without them | 216 |

3 On practical advice | 218 |

4 The characteristic of science is not rational belief but rational replacement of propositions | 220 |

Understanding Toulmin | 224 |

1 Three schools of thought on the normative problem of appraising scientific theories | 225 |

2 Toulmin and the Wittgensteinian thoughtpolice | 228 |

3 Toulmins Darwinian synthesis of Hegel and Wittgenstein | 235 |

4 Conclusion | 241 |

A letter to the Director of the London School of Economics | 247 |

The teaching of the history of science | 254 |

The social responsibility of science | 256 |

259 | |

Lakatos bibliography | 274 |

277 | |

### Common terms and phrases

accept according Agassi analysis analysis-synthesis appraisal argument arithmetic axiomatic axioms basic statements body of science calculus Carnap Carnapian Cartesian Circuit Cauchy Cauchy's chapter claim classical concept conjecture consistent convergence course criticism deductive degree of confirmation degree of corroboration demarcation demarcationists Descartes Einstein's elite elitist empiricism empiricist epistemological evidence evidential support fact fallible false formal Godel Griinbaum growth heuristic hypotheses Ibid inductive logic inductivist infallible inference infinite infinitesimal intuition Kneale knowledge Lakatos language game Leibniz lemmas logic of discovery mathematics meta-mathematics method methodology neoclassical Newton non-standard analysis paper Pappusian philosophy of mathematics philosophy of science Popper Popperian principles problem problem of inductive problemshift proof propositions prove qualified instance confirmation quasi-empirical rational belief rational betting quotients rational reconstruction refuted reliability research programme Russell scepticism scientific theories scientist sense set theory theorem thesis touchstone theory Toulmin trivial true truth truth-value uniform convergence universal volume Wittgenstein