Philosophy of Arithmetic

This volume is a window on a period of rich and illuminating philosophical activity that has been rendered generally inaccessible by the supposed "revolution" attributed to "Analytic Philosophy" so-called. Careful exposition and critique is given to every serious alternative account of number and number relations available at the time.

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Foreword. First Part: The Authentic Concepts of Multiplicity, Unity and Whole Number. Introduction. I: The Origination of the Concept of Multiplicity through that of the Collective Combination. The Analysis of the Concept of the Whole Number Presupposes that of the Concept of Multiplicity. The Concrete Bases of the Abstraction Involved. Independence of the Abstraction from the Nature of the Contents Colligated. The Origination of the Concept of the Multiplicity through Reflexion on the Collective Mode of Combination. II: Critical Developments. The Collective Unification and the Unification of Partial Phenomena in the Total Field of Consciousness at a Given Moment. The Collective 'Together' and the Temporal 'Simultaneously'. Collection and Temporal Succession. The Collective Synthesis and the Spatial Synthesis. A: F.A. Lange's Theory. B: Baumann's Theory. Colligating, Enumerating and Distinguishing. Critical Supplement. III: The Psychological Nature of the Collective Combination. Review. The Collection as a Special Type of Combination. On the Theory of Relations. Psychological Characterization of the Collective Combination. IV: Analysis of the Concept of Number in Terms of its Origin and Content. Completion of the Analysis of the Concept of Multiplicity. The Concept Something'. The Cardinal Numbers and the Generic Concept of Number. Relationship between the ConceptsCardinal Number' and `Multiplicity'. One and Something. Critical Supplement. V: The Relations 'More' and 'Less'. The Psychological Origin of these Relations. Comparison of Arbitrary Multiplicities, as well as of Numbers, in Terms of More and Less. The Segregation of the Number Species Conditioned upon the Knowledge of More and Less. VI: The Definition of Number-Equality through the Concept of Reciprocal One-to-One Correlation. Leibniz's Definition of the General Concept of Equality. The Definition of Number-Equality. Concerning Definitions of Equality for Special Cases. Application to the Equality of Arbitrary Multiplicities. Comparison of Multiplicities of One Genus. Comparison of Multiplicities with Respect to their Number. The True Sense of the Equality Definition under Discussion. Reciprocal Correlation and Collective Combination. The Independence of Number-Equality from the Type of Linkage. VII: Definitions of Number in Terms of Equivalence. Structure of the Equivalence Theory. Illustrations. Critique. Frege's Attempt. Kerry's Attempt. Concluding Remark. VIII: Discussions Concerning Unity and Multiplicity. The Definition of Number as a Multiplicity of Units. One as an Abstract, Positive Partial Content. One as Mere Sign. One and Zero as Numbers. The Concept of the Unit and the Concept of the Number One. Further Distinctions Concerning One and Unit. Sameness and Distinctness of the Units. Further Misunderstandings. Equivocations of the Name 'Unit'. The Arbitrary Character of the Distinction between Unit and Multiplicity. The Multiplicity Regarded as One Multiplicity, as One Enumerated Unit, as One Whole. Herbartian Arguments. IX: The Sense of the Statement of Number. Contradictory Views. Refutation, and the Position Taken. Appendix to the First Part: The Nominalist Attempts of Helmholtz and Kronecker. Second Part: The Symbolic Number Concepts and the Logical Sources of Cardinal Arithmetic. X: Operations on Numbers and the Authentic Number Concepts. The Numbers in Arithmetic are Not Abstracta. The Fundamental Activities on Numbers. Addition. Partition. Arithmetic Does Not Operate with