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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

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

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

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

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

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

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

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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