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The present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles, is undertaken in Parts II.—VII. of this Volume, and will be established by strict symbolic reasoning in Volume II. The demonstration of this thesis has, if I am not mistaken, all the certainty and precision of which mathematical demonstrations are capable. As the thesis is very recent among mathematicians, and is almost universally denied by philosophers, I have undertaken, in this volume, to defend its various parts, as occasion arose, against such adverse theories as appeared most widely held or most difficult to disprove. I have also endeavoured to present, in language as untechnical as possible, the more important stages in the deductions by which the thesis is established.
Content:-
introduction to the 1992 edition
introduction to the second edition
preface
PART I: THE INDEFINABLES OF MATHEMATICS
introduction to the second edition
preface
PART I: THE INDEFINABLES OF MATHEMATICS
1. Definition of Pure Mathematics
2. Symbolic Logic
3. Implication and Formal Implication
4. Proper Names, Adjectives and Verbs
5. Denoting
6. Classes
7. Propositional Functions
8. The Variable
9. Relations
10. The Contradiction
PART II: NUMBER
11. Definition of Cardinal Numbers
11. Definition of Cardinal Numbers
12. Addition and Multiplication
13. Finite and Infinite
14. Theory of Finite Numbers
15. Addition of Terms and Addition of Classes
16 Whole and Part
17. Infinite Wholes
18. Ratios and Fractions
PART III: QUANTITY
19. The Meaning of Magnitude
19. The Meaning of Magnitude
20. The Range of Quantity
21. Numbers as Expressing Magnitudes: Measurement
22. Zero
23. Infinity, the Infinitesimal and Continuity
PART IV: ORDER
24. The Genesis of Series
24. The Genesis of Series
25. The Meaning of Order
26. Asymmetrical Relations
27. Difference of Sense and Difference of Sign
28. On the Difference Between Open and Closed Series
29. Progressions and Ordinal Numbers
30. Dedekind’s Theory of Number
31. Distance
PART V: INFINITY AND CONTINUITY
32. The Correlation of Series
32. The Correlation of Series
33. Real Numbers
34. Limits and Irrational Numbers
35. Cantor’s First Definition of Continuity
36. Ordinal Continuity
37. Transfinite Cardinals
38.. Transfinite Ordinals
39. The Infinitesimal Calculus
40. The Infinitesimal and the Improper Infinite
41. Philosophical Arguments Concerning the Infinitesimal
42. The Philosophy of the Continuum
43. The Philosophy of the Infinite
PART VI: SPACE
44. Dimensions and Complex Numbers
44. Dimensions and Complex Numbers
45. Projective Geometry
46. Descriptive Geometry
47. Metrical Geometry
48. Relation of Metrical to Projective and Descriptive Geometry
49. Definitions of Various Spaces
50. The Continuity of Space
51. Logical Arguments Against Points
52. Kant’s Theory of Space
PART VII: MATTER AND MOTION
53. Matter
53. Matter
54. Motion
55. Causality
56. Definition of a Dynamical World
57. Newton’s Laws of Motion
58. Absolute and Relative Motion
59. Hertz’s Dynamics
APPENDICES
APPENDIX A. The Logical and Arithmetical Doctrines of Frege
APPENDIX B. The Doctrine of Types
index
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