#### Dirk Damsma

*How Language Informs Mathematics*Dirk Damsma shows how Hegel’s and Marx’s systematic dialectical analysis of mathematical and economic language helps us understand the structure and nature of mathematical and capitalist systems. More importantly, Damsma shows how knowledge of the latter can inform model assumptions and help improve models.

His book provides a blueprint for an approach to economic model building that does away with arbitrarily chosen assumptions and is sensitive to the institutional structures of capitalism. In light of the failure of mainstream economics to understand systemic failures like the financial crisis and given the arbitrary character of most assumptions in mainstream models, such an approach is desperately needed.

#### Biographical Note

#### Readership

Anyone interested in the methodological relation between Hegel and Marx and/or the relation between language, systems and mathematics or mathematical models and anyone concerned with approaches to economic model building.

#### Table of Contents

**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**