Mathematics and the Image of ReasonA thorough account of the philosophy of mathematics. In a cogent account the author argues against the view that mathematics is solely logic. |
Contents
INTRODUCTION | 1 |
AXIOMATIZATION RIGOUR AND REASON | 7 |
DESERTING EUCLIDEAN STANDARDS | 10 |
THE RETURN TO EUCLIDEAN STANDARDS | 25 |
FREGE ARITHMETIC AS LOGIC | 33 |
NUMBERS AND THE NATURE OF ARITHMETICAL TRUTHS | 38 |
NUMBERS AS OBJECTS | 42 |
THE NATURAL NUMBERS | 49 |
FORGING THE FORMAL CHAINS OF REASON | 103 |
SUCCESSES AND FAILURES | 110 |
LOGIC AND ITS LIMITATIONS | 123 |
APPENDIX RECURSIVE FUNCTIONS | 128 |
IDEAL ELEMENTS AND RATIONAL IDEALS | 129 |
FORMULAE SYMBOLS AND FORMS | 130 |
IDEAL ELEMENTS AND IDEALS | 138 |
DIAGRAMS AND RIGOUR | 142 |
WORD GAMES? | 54 |
RUSSELL MATHEMATICS AS LOGIC | 59 |
GEOMETRY AND RELATIONAL STRUCTURES | 60 |
PARADOXES AND LOGICAL TYPES | 70 |
EMPIRICISM LOGICAL POSITIVISM AND THE STERILITY OF REASON | 78 |
HILBERT MATHEMATICS AS A FORMULAGAME? | 89 |
FORMALISM AND HILBERTS PROGRAMME | 90 |
GEOMETRICAL RIGOUR | 92 |
PRAGMATISM AXIOMATIZATION AND IDEALS | 147 |
LOGIC AND THE OBJECTS OF MATHEMATICAL KNOWLEDGE | 154 |
LACK OF CLOSURE AND THE POWER OF REASON | 166 |
GLOSSARY OF SYMBOLS | 175 |
FURTHER READING | 176 |
| 178 | |
| 184 | |
Other editions - View all
Common terms and phrases
absolute consistency proof abstract application arithmetic Arnauld assumption axiomatization calculation claim complete computational conservative extension construction deduction defined definition Descartes determine diagrams domain empirical world empiricist entities Euclid Euclidean geometry Euclidean space Euclidean standards example existence experience expressed finitary formal consistency formal system formalist formula framework Frege function given Gödel's hence Hilbert Hilbert's Programme ideal elements ideas image of reason imaginarius independent infinite interpretation introduced intuition judgement justified Kant Kant's language logicist mathematical reasoning mathematicians means method natural numbers non-Euclidean geometries notion number of Fs number series one-one correspondence particular Peano axioms philosophy of mathematics position possible predicate problem procedure proof theory propositions provable proved pure quantification rational real numbers reality reduced relations rules Russell Russell's sense sentence sequence space statement structure symbols theorem theory things tion triangle true truth values universal universal quantifier Vicious Circle Principle well-formed formula
References to this book
Scientific Knowledge: A Sociological Analysis Barry Barnes,David Bloor,John Henry Limited preview - 1996 |
Constructing Mathematical Knowledge: Epistemology and Mathematics Education Paul Ernest No preview available - 1994 |