Philosophy of Arithmetic Contents


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 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 “Authentic” Number Concepts.

XI: Symbolic Representations of Multiplicities. Authentic and Symbolic Representations. Sense Perceptible Groups. Attempts at an Explanation of How We Grasp Groups in an Instant. Symbolizations Mediated by the Full Process of Apprehending the Individual Elements. New Attempts at an Explanation of Instantaneous Apprehensions of Groups. Hypotheses. The Figural Moments. The Position Taken. The Psychological Function of the Focus upon Individual Members of the Group. What is it that Guarantees the Completeness of the Traversive Apprehension of the Individuals in a Group? Apprehension of Authentically Representable Groups through Figural Moments. The Elemental Operations on and Relations between Multiplicities Extended to Symbolically Represented Multiplicities. Infinite Groups.

XII: The Symbolic Representations of Numbers. The Symbolic Number Concepts and their Infinite Multiplicity. The Non-Systematic Symbolizations of Numbers. The Sequence of Natural Numbers. The System of Numbers. Relationship of the Number System to the Sequence of Natural Numbers. The Choice of the “Base Number” for the System. The Systematic of the Number Concepts and the Systematic of the Number Signs. The Process of Enumeration via Sense Perceptible Symbols. Expansion of the Domain of Symbolic Numbers through Sense Perceptible Symbolization. Differences between Sense Perceptible Means of Designation. The Natural Origination of the Number System. Appraisal of Number through Figural Moments.

XIII: The Logical Sources of Arithmetic. Calculation, Calculational Technique and Arithmetic. The Calculational Methods of Arithmetic and the Number Concepts. The Systematic Numbers as Surrogates for the Numbers in Themselves. The Symbolic Number Formations that Fall Outside the System, Viewed as Arithmetical Problems. The First Basic Task of Arithmetic. The Elemental Arithmetical Operations. Addition. Multiplication. Subtraction and Division. Methods of Calculation with the Abacus and in Columns. The Natural Origination of the Indic Numeral Calculation. Influence of the Means of Designation upon the Formation of the Methods of Calculation. The Higher Operations. Mixing of Operations. The Indirect Characterization of Numbers by Means of Equations. Result: The Logical Sources of General Arithmetic. Selbstanzeige – Philosophie Der Arithmetik.

Supplementary Texts (1887 – 1901).

A: Original Version of the Text through Chapter IV: On the Concept of Number: Psychological Analyses. Introduction. Chapter One. The Analysis of the Concept of Number as to its Origin and Content. 1. The Formation of the Concept of Multiplicity [Vielheit] Out of that of the Collective Combination. 2. Critical Exposition of Certain Theories. 3. Establishment of the “Psychological”. Nature of the Collective Combination. 4. The Analysis of the Concept of Number as to its Origin and Content. Appendix To “On the Concept of Number: Psychological Analyses” – Theses.

B: Essays. Essay I: (On the Theory of the Totality). I. The Definition of the Totality. II. Comparison of Numbers. III. Addenda. 1. Addendum to p. 367: Identity and Equality. 2. On the Definition of Number. IV. The Classification of the Cardinal Numbers. V. Remark. VI. Corrections. VII. Addenda. 1. Addendum to p. 369. 2. Addendum to p. 377. Essay II: On the Concept of the Operation. I. Arithmetical Determinations of Number. II. Combinations (or Operations). 1. Division. 2. On the Concept of Combination. III. Addendum. On the Concept of Basic Operation. Essay III: Double Lecture: On the Transition through the Impossible (“Imaginary”) and the Completeness of an Axiom System. I. For a Lecture before the Mathematical Society of Göttingen 1901. 1. Introduction. 2. Theories Concerning the Imaginary. 3. The Transition through the Imaginary. Appendix I: Appendix II: Appendix III: Notes on a Lecture by Hilbert. Husserl’s Excerpts from an Exchange of Letters between Hilbert and Frege. Essay IV: (The Domain of an Axiom System /Axiom System – Operation System. System of Numbers. Arithmetizability of a Manifold. On the Concept of an Operation System. Essay V: The Question about the Clarification of the Concept of the “Natural” Numbers as “Given,” as “Individually Determinant”. Essay VI: On the Formal Determination of Manifold.



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