How Language Informs Mathematics

List of Figure and Tables

Acknowledgements

Brief Contents

Note on the Style of Referencing and the Use of Capitalisation and Emphasis in this Work

List of Symbols

Introduction

 1 On Marx’s and Hegel’s Dialectical Methods

 Introduction

 1 The Chronology of Hegel’s and Marx’s Historical and Systematic Dialectic

 2 Hegel’s Method

 3 Marx’s Comments on Hegel, Their Implications and Marx’s Twist on Hegel’s Dialectical Method

 4 Commentators on and Studies of Marx’s Dialectics

 Summary and Conclusions

 Preview

 2 The Dialectical Foundations of Mathematics

 Introduction

 1 Previous Literature on Hegel and Mathematics

 2 Hegel’s Determination of the Quantitative

 A Quality

 2.1 Being

 2.2 Nothing

 2.3 Becoming

 2.4 Presence

 2.5 Something and Other

 2.6 One and Many Ones

 2.7 Attraction and Repulsion

 B Quantity

 2.8 Quantity

 2.9 Continuous and Discrete Magnitude

 2.10 Quantum and Number

 2.11 Unit and Amount

 2.12 Limit

 2.13 Intensive and Extensive Magnitude

 C Measure

 2.14 Measure

 3 Hegel’s Determination of Mathematical Mechanics

 A Space and Time

 3.1 Space

 3.2 Spatial Dimensions

 3.3 The Point

 3.4 The Line

 3.5 The Plane

 3.6 Distinct Space

 3.7 Time

 3.8 Temporal Dimensions

 3.9 Now

 3.10 Place

 3.11 Motion

 3.12 Matter

 Summary and Conclusions: How This Dialectic Reflects on Mathematics

 Appendix: Comparison of the Determination of the Quantitative in the Wissenschaft and the Encyclopädie

 A1 Being, Nothing, Becoming, Presence, Something and Others

 A2 Qualitative Limit

 A3 Finitude and Infinity

 A4 True Infinite

 A5 Being-for-self

 A6 One, Many Ones, Repulsion, Attraction, Quantity, Continuous and Discrete Magnitude, Quantum, Number, Unit and Amount, Quantitative Limit and Intensive and Extensive Magnitude

 A7 Quantitative Infinity

 A8 Direct Ratio

 A9 Inverse Ratio

 A10 Ratio of Powers

 A11 Measure

 Concluding Remarks

 3 Marx’s Systematic Dialectics and Mathematics

 Introduction

 1 Marx’s Acquaintance with and Ideas on Mathematics

 2 Marx’s Exhibition of Capitalism as a System: The Systematic-Dialectical Position

 2.1 Sociation

 2.2 Dissociation

 2.3 Association: The Exchange Relation

 2.4 The Commodity, Exchangeability and the Bargain

 2.5 Value in Exchange

 2.6 The Simple, Expanded and General Commodity Form and the Money Form of Value

 2.7 Money as Measure of Value, Means of Circulation and End of Exchange

 2.8 Capital

 2.9 Constant and Variable Capital

 2.10 Accumulation

 2.11 The Money Capital, Production Capital and Commodity Capital Circuits

 2.12 Fixed and Circulating Capital

 2.13 Simple Reproduction, Means of Production, Consumption Goods, Total Social Capital and Expanded Reproduction

 2.14 General Rate of Profit, Many Capitals, Competition and Minimum Prices of Production

 3 The Role of Mathematics in Marx’s Investigation and Exhibition in Capital: the Case of Marx’s “Schemes of Reproduction”

 3.1 Simple Reproduction

  3.1.1 The Model

  3.1.2 Conclusions

 3.2 Expanded Reproduction

  3.2.1 The Model

  3.2.2 Conclusions

 Summary and Conclusions on the Role of Mathematics in Systematic-Dialectical Investigation and Exhibition

 4 A Formal Dynamic Reconstruction of Marx’s Schemes of Reproduction along Dialectical Lines

 Introduction

 1 The Model for Simple Reproduction

 2 Extensive Growth of Total Social Capital

 3 The Model for Expanded Reproduction

 Summary and Conclusions

 Appendix: Derivations

 A1 Accumulation and Growth Rate for Department c as a Function of Accumulation and Growth in Department p with Extensive Growth (expression 4.15 and 4.16)

 A2 Constant Capital’s Growth Rate for Department c for the Case of Expanded Reproduction (expression 4.19)

 A3 The Condition for Constant Rates of Accumulation in Case of Expanded Reproduction (expression 4.20)

 Summary and General Conclusions

 References

 Author Index

 Subject Index