Mathematics

Number Theory and Discrete Mathematics
 Philosophy of Arithmetic  Psychological and Logical Investigations with Supplementary Texts from
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In his first book, Philosophy of Arithmetic, Edmund Husserl provides a carefully worked out account of number as a categorial or formal feature of the objective world, and of arithmetic as a symbolic technique for mastering the infinite field of numbers for knowledge. It is a realist account of numbers and number relations that interweaves them into the basic structure of the universe and into our knowledge of reality. It provides an answer to the question of how arithmetic applies to reality, and gives an account of how, in general, formalized systems of symbols work in providing access to the world. The "appendices" to this book provide some of Husserl's subsequent discussions of how formalisms work, involving David Hilbert's program of completeness for arithmetic. "Completeness" is integrated into Husserl's own problematic of the "imaginary", and allows him to move beyond the analysis of "representations" in his understanding of the logic of mathematics. Husserl's work here provides an alternative model of what "conceptual analysis" should be  minus the "linguistic turn", but inclusive of language and linguistic meaning. In the process, he provides case after case of "Phenomenological Analysis"  fortunately unencumbered by that title  of the convincing type that made Husserl's life and thought a fountainhead of much of the most important philosophical work of the twentieth Century in Europe. Many Husserlian themes to be developed at length in later writings first emerge here: Abstraction, internal time consciousness, polythetic acts, acts of higher order ('founded' acts), Gestalt qualities and their role in knowledge, formalization (as opposed to generalization), essence analysis, and so forth. 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" socalled. Careful exposition and critique is given to every serious alternative account of number and number relations available at the time. Husserl's extensive and trenchant criticisms of Gottlob Frege's theory of number and arithmetic reach far beyond those most commonly referred to in the literature on their views.
Content Level »Research
Keywords »Analysis  Edmund Husserl  Gottlob Frege  Hermann von Helmholtz  body  concept  finite field  language  logic
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 Concepts `Cardinal 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 NumberEquality through the Concept of Reciprocal OnetoOne Correlation. Leibniz's Definition of the General Concept of Equality. The Definition of NumberEquality. 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 NumberEquality 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