Philosophy of Mathematics: Selected Readings

Cover......Page 00_0001.djvu
Title: Philosophy of Mathematics......Page 001_0001.djvu
Copyright......Page 002_0001.djvu
Contents ......Page 003_0001.djvu
Preface to the second edition ......Page 005_0001.djvu
1. General remarks ......Page 007_0001.djvu
1. The actual infinite and formalism ......Page 012_0001.djvu
3. The \"potential infinite\" and intuitionism ......Page 014_0001.djvu
4. Logicism ......Page 017_0001.djvu
5. Tautologies and sets ......Page 022_0001.djvu
6. Mathematical truth ......Page 027_0001.djvu
7. The iterative conception of set ......Page 033_0001.djvu
8. The problem of \"access\" ......Page 036_0001.djvu
9. Quine and Godel ......Page 039_0001.djvu
Part 1: The foundations of mathematics ......Page 045_0001.djvu
I. The derivation of mathematical concepts ......Page 047_0001.djvu
II. The derivation of the theorems of mathematics ......Page 050_0001.djvu
III. The problem of Impredlcitlve definition ......Page 052_0001.djvu
IV. Attempt at a solution ......Page 055_0001.djvu
[Die intuitionistische Grundlegung der Mathematik] ......Page 058_0001.djvu
3. The formalist foundations of mathematics ......Page 067_0001.djvu
Disputation ......Page 072_0001.djvu
Intuitionism and formalism ......Page 083_0001.djvu
Consciousness, Philosophy, and Mathematics......Page 096_0001.djvu
The Philosophical Basis of Intuitionistic Logic......Page 103_0001.djvu
Each individual number is an independent object ......Page 136_0001.djvu
In Order to Obtain Concept of Number, One Must Determine the Sense of a Numerical Equation ......Page 140_0001.djvu
Completion and confirmation of our definition ......Page 146_0001.djvu
Conclusion ......Page 159_0001.djvu
[Recapitulation] ......Page 163_0001.djvu
I. The series of natural numbers ......Page 166_0001.djvu
II. Definition of number ......Page 173_0001.djvu
III. Mathematics and logic ......Page 179_0001.djvu
On the infinite ......Page 189_0001.djvu
The problem of mathematical truth ......Page 208_0001.djvu
The formalist definition of mathematics ......Page 209_0001.djvu
Truth and acceptability ......Page 211_0001.djvu
Mathematics and logic ......Page 212_0001.djvu
Hilbert\'s Programme ......Page 213_0001.djvu
I. Syntactic analysis ......Page 220_0001.djvu
II. Significance of syntactic analysis ......Page 225_0001.djvu
Finitist Proof ......Page 230_0001.djvu
Completeness of the predicate calculus ......Page 235_0001.djvu
Postscript (Autumn 1978) to \"Hilbert\'s Programme\" ......Page 237_0001.djvu
Notes ......Page 240_0001.djvu
PART II : The Existence of Mathematical Objects ......Page 245_0001.djvu
1. The problem of abstract entities ......Page 247_0001.djvu
2. Linguistic frameworks ......Page 248_0001.djvu
3. What does acceptance of a kind of entities mean? ......Page 255_0001.djvu
4. Abstract entities in semantics ......Page 257_0001.djvu
5. Conclusion ......Page 262_0001.djvu
On platonism in mathematics ......Page 264_0001.djvu
I. The education ......Page 278_0001.djvu
II. The dilemma ......Page 284_0001.djvu
III. Way out ......Page 291_0001.djvu
Mathematics without foundations ......Page 301_0001.djvu
A sketch of my view ......Page 303_0001.djvu
The question of decidability ......Page 307_0001.djvu
The significance of the antinomies ......Page 314_0001.djvu
PART III: Mathematical Truth ......Page 319_0001.djvu
The A Priori ......Page 321_0001.djvu
Truth by Convention ......Page 335_0001.djvu
Carnap and Logical Truth ......Page 361_0001.djvu
2. Are the Propositions of Mathematics Self-Evident Truths? ......Page 383_0001.djvu
3. Is mathematics the most general empirical science? ......Page 384_0001.djvu
4. The analytic character of mathematical propositions ......Page 385_0001.djvu
5. Mathematics as an axiomatized deductive system ......Page 387_0001.djvu
7. Interpretations of Peano\'s primitives ......Page 390_0001.djvu
8. Definition of the customary meaning of the concepts of arithmetic in purely logical terms ......Page 391_0001.djvu
9. The truth of Peano\'s postulates in their customary interpretation ......Page 394_0001.djvu
10. Mathematics as a branch of logic ......Page 395_0001.djvu
11. On the applicability of mathematics to empirical subject matter ......Page 396_0001.djvu
On the Nature of Mathematical Reasoning ......Page 400_0001.djvu
Properties of addition ......Page 403_0001.djvu
Properties of multiplication ......Page 404_0001.djvu
Mathematical Truth ......Page 409_0001.djvu
I. Two kinds of account ......Page 411_0001.djvu
II. Two conditions ......Page 414_0001.djvu
III. The standard view ......Page 416_0001.djvu
IV. Knowledge ......Page 418_0001.djvu
V. Two examples ......Page 421_0001.djvu
Models and Reality ......Page 427_0001.djvu
PART IV: The Concept of Set ......Page 451_0001.djvu
Russell\'s Mathematical Logic ......Page 453_0001.djvu
1. The concept of cardinal number ......Page 476_0001.djvu
2. The continuum problem, the continuum hypothesis, and the partial results concerning its truth obtained so far ......Page 478_0001.djvu
3. Restatement of the problem on the basis of an analysis of the foundations of set theory and results obtained along these lines ......Page 479_0001.djvu
4. Some observations about the question: In what sense and in which direction may a solution of the continuum problem be expected? ......Page 484_0001.djvu
Definitions of some of the technical terms ......Page 486_0001.djvu
Supplement lo the second edition [1963] ......Page 488_0001.djvu
Postscript ......Page 491_0001.djvu
The Iterative Conception of Set ......Page 492_0001.djvu
I. Naive set theory ......Page 493_0001.djvu
Proof of the inconsistency of naive set theory (Russell\'s paradox) ......Page 494_0001.djvu
II. The iterative conception of set ......Page 495_0001.djvu
III. Zermelo set theory ......Page 501_0001.djvu
IV. ZermeJo-FraenkeJ set theory ......Page 505_0001.djvu
What is the Iterative Conception of Set? ......Page 509_0001.djvu
1. The (maximum) iterative concept ......Page 536_0001.djvu
2. Bankruptcy (contradiction) or misunderstanding (error)? ......Page 547_0001.djvu
3. Objectivism and formalism in set theory ......Page 550_0001.djvu
4. New axioms and crllerla of acceptability ......Page 557_0001.djvu
5. Comparisons with geometry and physics ......Page 562_0001.djvu
6. Digression on unbounded quantifications ......Page 565_0001.djvu
7. Extracting axioms of set theory from Cantor\'s writings ......Page 567_0001.djvu
8. The hierarchies of Cantor and Mirimanoff ......Page 570_0001.djvu
Bibliography ......Page 577_0001.djvu
Back Cover......Page 0_0001.djvu