Gottlob Frege: Foundations of Arithmetic

(Longman Library of Primary Sources in Philosophy)

Gottlob Frege

• Publisher: Routledge
• Year: 2020
• Print ISBN: 9780321241894
• eBook ISBN: 9781000154429
• Topic: Philosophy
• Subtopic: History & Theory of Philosophy

Part of theLongman Library of Primary Sources in Philosophy, this edition of Frege's Foundations of Arithmetic is framed by a pedagogical structure designed to make this important work of philosophy more accessible and meaningful for undergraduates.

The FOUNDATIONS of ARITHMETIC

A Logical-Mathematical Investigation into the Concept of Number

1884

GOTTLOB FREGE

Contents

§ 1. In mathematics in recent times an aspiration directed toward the rigor of proof and sharp delimitation of concepts is discernible.

§ 2. The inquiry must eventually extend even to the concept of number. The aim of proof.

§ 3. Philosophical motivations for such an investigation: the controversies, whether the laws of numbers are analytic or synthetic truths, a priori or a posteriori. The sense of these expressions.

§ 4. The task of this book.

I. OPINIONS OF SEVERAL WRITERS ON THE NATURE OF ARITHMETICAL PROPOSITIONS

Are numerical formulas provable?

§ 5. Kant denies that which Hankel rightly calls a paradox.

§ 6. Leibniz’s proof of 2 + 2 = 4 has a gap. Grassmann’s definition of a + b is defective.

§ 7. Mill’s opinion that the definitions of individual numbers assert observable facts from which calculations follow is ungrounded.

§ 8. Observation of facts is not required for the legitimacy of these definitions.

Are the laws of arithmetic inductive truths?

§ 9. Mill’s natural law. By calling arithmetical truths natural laws, Mill confuses them with their applications.

§ 10. Reasons against the proposition that the laws of addition are inductive truths: dissimilarity of numbers; by means of a definition we do not automatically possess a set of general properties of number; induction on the contrary is probably to be grounded on arithmetic.

§ 11. Leibniz’s ‘innate’.

Are the laws of arithmetic synthetic a priori or analytic?

§ 12. Kant. Baumann. Lipschitz. Hankel. Inner intuition as a ground of knowledge.

§ 13. Distinction of arithmetic and geometry.

§ 14. Comparison of truths in respect of the domains they govern.

§ 15. Views of Leibniz and W.S. Jevons.

§ 16. Against Mill’s disparagement of ‘artful manipulation of language’. Signs are not empty, therefore, because they do not signify something perceivable.

§ 17. Inadequacy of induction. Conjecture that the laws of number are analytic judgments; wherein then their use consists. Valuation of analytic judgments.

II. OPINIONS OF CERTAIN WRITERS ON THE CONCEPT OF NUMBER

§ 18. Necessity to investigate the general concept of number.

§ 19. The definition must not be geometrical.

§ 20. Is number definable? Hankel. Leibniz.

Is number a property of external things?

§ 21. Opinions of M. Cantor and E. Schröder.

§ 22. Against Baumann: external things represent no strict units. Number apparently depends on our conception.

§ 23. Mill’s opinion that number is a property of the aggregate of things is untenable.

§ 24. Extensive applicability of number. Mill. Locke. Leibniz’s immaterial metaphysical figure. If number were something sensible, it could not be attached to anything non-sensible.

§ 25. Mill’s physical distinction between 2 and 3. According to Berkeley number is not really in things, but rather created by mind.

Is number something subjective?

§ 26. Lipschitz’s description of the formation of numbers is not suitable, and cannot replace a determination of the concept.

§ 27. Number is not, as Schloemilch would have it, an idea of the place of an object in a series.

§ 28. Thomae’s naming.

III. OPINIONS ON UNITY AND ONE

Does the number word ‘one’ express a property of objects?

§ 29. Ambiguity of the expressions ‘μονασ’ and ‘unit’. E. Schröder’s definition of the unit as an object to be numbered is apparently futile. The adjective ‘one’ contains no qualification, cannot serve as a predicate.

§ 30. The proposed definitions of Leibniz and Baumann appear to entirely blur the concept of unity.

§ 31. Baumann’s criteria of indivisibility and delimitability. The idea of unity is not suggested to us by every object (Locke).

§ 32. Nevertheless language indicates a connection with indivisibility and delimitability, whereby, however, the sense is shifted.

§ 33. Indivisibility (G. Kopp) is not tenable as a criterion of unity.

Are units identical with one another?

§ 34. Identity as the ground of the word ‘unit’. E. Schröder. Hobbes. Hume. Thomae. One does not attain the concept of number by means of abstraction from the differences of things, and things are not thereby identical to one another.

§ 35. Diversity is if anything necessary, if there is to be talk of plurality. Descartes. E. Schröder. W.S. Jevons.

§ 36. The view concerning the differences of units also encounters difficulties. Distinct ones in W.S. Jevons.

§ 37. Locke’s, Leibniz’s, Hesse’s definitions of number in terms of unit or one.

§ 38. ‘One’ is a proper name, ‘unit’ a concept word. Number cannot be defined as units. Difference between ‘and’ and +.

§ 39. The difficulty of reconciling identity and distinguishability of units is disguised by the ambiguity of ‘unit’.

Attempts to overcome the difficulty

§ 40. Space and time as methods of distinction. Hobbes. Thomae. Against: Leibniz, Baumann, W.S. Jevons.

§ 41. The aim is not accomplished.

§ 42. Position in a series as a method of distinction. Hankel’s positioning.

§ 43. Schröder’s picturing of objects by means of the sign 1.

§ 44. Jevons’s abstraction from the character of the differences while adhering to their existence. 0 and 1 are numbers in the same way as the others. The difficulty persists.

Solution of the difficulty

§ 45. Retrospect.

§ 46. The number statement contains an assertion about a concept. Objection that the number varies while the concept is unchanged.

§ 47. The actuality of a number statement is explained by the objectivity of concepts.

§ 48. Resolution of some difficulties.

§ 49. Confirmation in Spinoza.

§ 50. E. Schröder’s achievement.

§ 51. Correction of the same.

§ 52. Confirmation in German linguistic usage.

§ 53. Distinction between characteristics and properties of a concept. Existence and number.

§ 54. One can refer to unit as the subject of a number statement. Indivisibility and delimitability of the unit. Identity and indistinguishability.

IV. THE CONCEPT OF NUMBER

Every individual number is a self-subsistent object

§ 55. Attempt to complete the Leibnizian definitions of the individual numbers.

§ 56. The attempted definitions are unusable, because they define an assertion of which number is only a part.

§ 57. A number statement is to be regarded as an identity between numbers.

§ 58. Objection to the inconceivability of number as a self-subsistent object. Number is generally inconceivable.

§ 59. An object is not therefore excluded from the investigation because it is inconceivable.

§ 60. Besides, concrete things are not always conceivable. One must consider the words in a proposition when one inquires after their meaning.

§ 61. Objection to the aspatiality of numbers. Not every objective object is spatial.

To acquire the concept of number, one must establish the sense of a numerical identity

§ 62. We need a distinguishing mark for numerical identity.

§ 63. The possibility of a univocal correlation as such. Logical misgivings that the identity will be specially defined for this case.

§ 64. Examples for a similar procedure: direction, position of a plane, the shape of a triangle.

§ 65. Attempt at a definition. A second misgiving: whether the laws of identity will suffice.

§ 66. Third misgiving: the distinguishing mark of identity is insufficient.

§ 67. The completion cannot be achieved by one’s taking it as a criterion of a concept how an object is introduced.

§ 68. Number as the extension of a concept.

§ 69. Comment.

Completion and proof of our definition

§ 70. The relation-concept.

§ 71. Correlation by means of a relation.

§ 72. Mutually univocal [one-one] corelation. Concept of number.

§ 73. The number that belongs to the concept F is identical to the number that belongs to the concept G, if there exists a relation that mutually univocally [one-one] correlates the objects that fall under F with those that fall under G.

§ 74. Zero is the number that belongs to the concept ‘not identical to itself’.

§ 75. Zero is the number that belongs to a concept under which nothing falls. No object falls under a concept if zero is the number belonging to it.

§ 76. Definition of the expression ‘n follows in the natural number series immediately after m’.

§ 77. 1 is the number that belongs to the concept ‘identical with 0’.

§ 78. Propositions that are to be proved by means of our definitions.

§ 79. Definition of following in a series.

§ 80. Remarks on this. Objectivity of following.

§ 81. Definition of the expression ‘x belongs to the ϕ-series ending with y’.

§ 82. Indication of a proof that there exists no last member of the natural number series.

§ 83. Definition of finite number. No finite number follows after itself in the natural number series.

Infinite numbers

§ 84. The number that belongs to the concept ‘finite number’ is an infinite one.

§ 85. The Cantorian infinite numbers; ‘Cardinal number [Power]’. Deviation in the nomenclature.

§ 86. Cantor’s following in a succession and my following in a series.

V. CONCLUSION

§ 87. The nature of arithmetical laws.

§ 88. Kant’s underestimation of analytic judgments.

§ 89. Kant’s assertion: ‘Without sensibility no object would be given to us’. Kant’s importance for mathematics.

§ 90. Toward a complete verification of the analytic nature of arithmetical laws there is lacking a gap-less chain of deductions.

§ 91. Remedy for this deficiency is possible by means of my concept-script.

Other numbers

§ 92. The sense of the question about the possibility of numbers, according to Hankel.

§ 93. Numbers are neither spatially outside us nor subjective.

§ 94. The non-contradictoriness of a concept is no guarantee that something falls under it, and itself requires a proof.

§ 95. One must not regard (c - b) without further ado as a sign that solves the problem of subtraction.

§ 96. Even the mathematician cannot create something arbitrarily.

§ 97. Concepts are to be distinguished from objects.

§ 98. Hankel’s definition of addition.

§ 99. Defectiveness of the formal theory.

§ 100. Attempt to verify thereby that the meaning of multiplication is extended for complex numbers in a special way.

§ 101. The possibility of such a verification is not indifferent to the power of a proof.

§ 102. The mere postulation that an operation should be executable is not its fulfillment.

§ 103. Kossak’s definition of complex numbers is only a guideline for a definition and does not avoid the interference of foreign elements. Geometrical representation.

§ 104. It is essential to establish the sense of a recognition-judgment for the new numbers.

§ 105. The charm of arithmetic lies in its rational character.

§ 106–109. Retrospect.

INTRODUCTION

To the question, what the number one is, or what the sign 1 refers to, one mostly receives the answer, ‘Well, now, a thing’. And if one then thereupon calls attention to the fact that the proposition

‘The number one is a thing’

is not a definition, because on the one side it has the definite article an...

Table of contents

• Cover
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Translator’s Introduction and Critical Commentary
• The Foundations of Arithmetic: A Logical-Mathematical Investigation into the Concept of Number
• Recommendations for Further Reading
• Index