The Foundations of Arithmetic

, by Frege, Gottlob; Austin, J. L.

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ISBN: 9780810106055 | 0810106051
Cover: Paperback
Copyright: 6/1/1980

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Recent work in mathematics has shown a tendency towards rigour of proof and sharp definition of concepts

(1)

This critical examination must ultimately extend to the concept of Number itself. The aim of proof

(1)

Philosophical motives for such an enquiry: the controversies as to whether the laws of number are analytic or synthetic, a priori or a posteriori. Sense of these expressions

(1)

Task of the present work

(1)

I. Views of certain writers on the nature of arithmetical propositions

Are numerical formulae provable?

Kant denies this, which Hankel justly calls a paradox

(2)

Leibniz's proof that 2 + 2 = 4 contains a gap. Grassman's definition of a + b is faulty

(2)

Mill's view that the definitions of the individual numbers assert observed facts, from which the calculations follow, is without foundation

(2)

These definitions do not require, for their legitimacy, the observation of his facts

(1)

Are the laws of arithmetic inductive truths?

Mill's law of nature. In calling arithmetical truths laws of nature, Mill is confusing them with their applications

(2)

Grounds for denying that the laws of addition are inductive truths: numbers not all of the same sort; the definition of number does not of itself yield any set of common properties of numbers; probably the reverse is true and induction should be based on arithmetic

(3)

Leibniz's term ``innate''

(1)

Are the laws of arithmetic synthetic a priori or analytic?

Kant. Baumann. Lipschitz. Hankel. Inner intuition as the ground of knowledge

(2)

Distinction between arithmetic and geometry

(1)

Comparison between the various kinds of truths in respect of the domains that they govern

(1)

Views of Leibniz and W. S. Jevons

(1)

Against them, Mill's ridicule of the ``artful manipulation of language''. Symbols are not empty simply because not meaning anything with which we can be acquainted

(1)

Inadequacy of induction. Conjecture that the laws of number are analytic judgments; what in that case is the use of them. Estimate of the value of analytic judgments

(1)

II. Views of certain writers on the concept of Number

Necessity for an enquiry into the general concept of Number

(1)

Its definition not to be geometrical

(1)

Is number definable? Hankel. Leibniz

(1)

Is Number a property of external things?

Views of M. Cantor and E. Schroder

(1)

Opposite view of Baumann: external things present us with no strict units. Their number apparently dependent on our way of regarding them

(1)

Mill's view untenable, that the number is a property of the agglomeration of things

(1)

Wide range of applicability of number. Mill. Locke. Leibniz's immaterial metaphysical figure. If number were something sensible, it could not be ascribed to anything non-sensible

(2)

Mill's physical difference between 2 and 3. Number according to Berkeley not really existent in things but created by the mind

(1)

Is number something subjective?

Lipschitz's description of the construction of numbers will not do, and cannot take the place of a definition of the concept. Number not an object for psychology, but something objective

(3)

Number is not, as Schloemilch claims, the idea of the position of an item in a series

(2)

Numbers as sets

Thomae's name-giving

(1)

III. Views on unity and one

Does the number word ``one'' express a property of objects?

Ambiguity of the terms ``μoναs'' and ``unit''. E. Schroder's definition of the unit as an object to be numbered is apparently pointless. The adjective ``one'' does not add anything to a description, cannot serve as a predicate

(2)

Attempts to define unity by Leibniz and Baumann seem to blur the concept completely

(1)

Baumann's criteria, being undivided and being isolated. The notion of unity not suggested to us by every object (as Locke)

(1)

Still, language does indicate some connexion with being undivided and isolated, with a shift of meaning however

(1)

Indivisibility (G. Kopp) as a criterion of the unit is untenable

(1)

Are units identical with one another?

Identity as the reason for the name ``unit''. E. Schroder. Hobbes. Hume. Thomae. To abstract from the differences between things does not give us the concept of their Number, nor does it make the things identical with one another

(2)

Indeed diversity is actually necessary, if we are to speak of plurality. Descartes. E. Schroder. W. S. Jevons

(1)

The view that units are different also comes up against difficulties. Different distinct ones in W. S. Jevons

(2)

Definitions of number in terms of the unit or one by Locke, Leibniz and Hesse

(1)

``One'' is a proper name, ``unit'' a concept word. Number cannot be defined as units. Distinction between ``and'' and +

(2)

The difficulty of reconciling identity of units with distinguishability is concealed by the ambiguity of ``unit''

(1)

Attempts to overcome the difficulty

Space and time as means of distinguishing between units. Hobbes. Thomae. Against them: Leibniz, Baumann, W. S. Jevons

(2)

The purpose not achieved

(1)

Position in a series as a means of distinguishing between units. Hankel's putting

(1)

Schroder's copying of objects by the symbol I

(1)

Jevons' abstraction from the character of the differences while retaining the fact of their existence. 0 and 1 are numbers like the rest. The difficulty still remains

(3)

Solution of the difficulty

Recapitulation

(1)

A statement of number contains an assertion about a concept. Objection that the number varies while the concept does not

(1)

That statements of number are statements of fact explained by the objectivity of concepts

(1)

Removal of certain difficulties

(1)

Corroboration found in Spinoza

(1)

E. Schroder's account quoted

(1)

Correction of the same

(1)

Corroboration found in a German idiom

(1)

Distinction between component characteristics of a concept and its properties. Existence and number

(1)

Unit the name given to the subject of a statement of number. How indivisible and isolated. How identical and distinguishable

(2)

IV. The concept of Number

Every individual number is a self-subsistent object

Attempt to complete the definitions of the individual numbers as given by Leibniz

(1)

The attempted definitions are unusable, because what they define is a predicate in which the number is only an element

(1)

A statement of number should be regarded as an identity between numbers

(1)

Objection that we can form no idea of number as a self-subsistent object. In principle number cannot be imagined

(1)

Because we cannot imagine an object, we are not to be debarred from investigating it

(1)

Even concrete things are not always imaginable. In seeking the meaning of a word, we must consider it in the context of a proposition

(1)

Objection that numbers are not spatial. Not every objective object is spatial

(1)

To obtain the concept of Number, we must fix the sense of a numerical identity

We need a criterion for numerical identity

(1)

Possible criterion in one-one correlation. Doubt as to the logic of defining identity specially for the case of numbers

(1)

Examples of similar procedures: direction of a line, orientation of a plane, shape of a triangle

(2)

Attempt at a definition. A second doubt: are the laws of identity satisfied?

(1)

Third doubt: the criterion of identity fails to cover all cases

(1)

We cannot supplement it by taking as a defining characteristic of a concept the way in which an object is introduced

(1)

Number as the extension of a concept

(1)

Elucidation

(1)

Our definition completed and its worth proved

The relation-concept

(2)

Correlation by means of a relation

(1)

One-one relations. The concept of Number

(1)

The Number which belongs to the concept F is identical with the Number which belongs to the concept G, if there exists a relation which correlates one to one the objects falling under F with those falling under G

(1)

Nought is the Number which belongs to the concept ``not identical with itself''

(2)

Nought is the Number which belongs to a concept under which nothing falls. No object falls under a concept if nought is the Number belonging to that concept

(1)

Definition of the expression ``n follows in the series of natural numbers directly after m''

(1)

I is the Number which belongs to the concept ``identical with o''

(1)

Propositions to be proved by means of our definitions

(1)

Definition of following in a series

(1)

Comments on the same. Following is objective

(2)

Definition of the expression ``x is a member of the &phis;-series ending with y''

(1)

Outline of the proof that there is no last member of the series of natural numbers

(1)

Definition of finite Number. No finite Number follows in the series of natural numbers after itself

(1)

Infinite Numbers

The Number which belongs to the concept ``finite Number'' is an infinite Number

(1)

Cantor's infinite Numbers; ``power.'' Divergence in terminology

(1)

Cantor's following in the succession and my following in the series

(1)

V. Conclusion

Nature of the laws of arithmetic

(1)

Kant's underestimate of the value of analytic judgments

(2)

Kant's dictum: ``Without sensibility no object would be given to us.'' Kant's services to mathematics

101

(1)

For the complete proof of the analytic nature of the laws of arithmetic we still need a chain of deductions with no link missing

102

(1)

My concept writing makes it possible to supply this lack

103

(1)

Other numbers

The sense, according to Hankel, of asking whether some number is possible

104

(1)

Numbers are neither outside us in space nor subjective

105

(1)

That a concept is free from contradiction is no guarantee that anything falls under it, and itself requires to be proved

105

(1)

We cannot regard (c-b) without more ado as a symbol which solves the problem of subtraction

106

(1)

Not even the mathematician can create things at will

107

(1)

Concepts are to be distinguished from objects

108

(1)

Hankel's definition of addition

108

(1)

The formalist theory defective

109

(1)

Attempt to produce an interpretation of complex numbers by extending the meaning of multiplication in some special way

110

(1)

The cogency of proofs is affected, unless it is possible to produce such an interpretation

111

(1)

The mere postulate that it shall be possible to carry out some operation is not the same as its own fulfilment

111

(1)

Kossack's definition of complex numbers is only a guide towards a definition, and fails to avoid the importation of foreign elements. Geometrical representation of complex numbers

112

(2)

What is needed is to fix the sense of a recognition-judgment for the case of the new numbers

114

(1)

The charm of arithmetic lies in its rationality

115

(1)

Recapitulation

115

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The Foundations of Arithmetic

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