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

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

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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

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Task of the present work

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

Are numerical formulae provable?

Kant denies this, which Hankel justly calls a paradox

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

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

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

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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

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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

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Leibniz's term ``innate''

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Are the laws of arithmetic synthetic a priori or analytic?

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

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Distinction between arithmetic and geometry

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

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Views of Leibniz and W. S. Jevons

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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

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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

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II. Views of certain writers on the concept of Number

Necessity for an enquiry into the general concept of Number

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Its definition not to be geometrical

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Is number definable? Hankel. Leibniz

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Is Number a property of external things?

Views of M. Cantor and E. Schroder

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

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

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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

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

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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

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

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Numbers as sets

Thomae's name-giving

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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

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

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

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

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

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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

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

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

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

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

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

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Attempts to overcome the difficulty

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

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The purpose not achieved

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

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Schroder's copying of objects by the symbol I

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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

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Solution of the difficulty

Recapitulation

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

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

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Removal of certain difficulties

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Corroboration found in Spinoza

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E. Schroder's account quoted

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Correction of the same

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Corroboration found in a German idiom

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

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

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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

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

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

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

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

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

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

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

We need a criterion for numerical identity

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

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

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

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

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

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Number as the extension of a concept

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Elucidation

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Our definition completed and its worth proved

The relation-concept

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Correlation by means of a relation

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One-one relations. The concept of Number

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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

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

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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

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

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

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Propositions to be proved by means of our definitions

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Definition of following in a series

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Comments on the same. Following is objective

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

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

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

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Infinite Numbers

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

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

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

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V. Conclusion

Nature of the laws of arithmetic

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Kant's underestimate of the value of analytic judgments

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

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

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My concept writing makes it possible to supply this lack

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Other numbers

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

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Numbers are neither outside us in space nor subjective

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

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

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Not even the mathematician can create things at will

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Concepts are to be distinguished from objects

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Hankel's definition of addition

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The formalist theory defective

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

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

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

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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

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

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The charm of arithmetic lies in its rationality

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Recapitulation

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