Ernst Schröder on Algebra and Logic

This volume offers English translations of three early works by Ernst Schröder (1841-1902), a mathematician and logician whose philosophical ruminations and pathbreaking contributions to algebraic logic attracted the admiration and ire of figures such as Dedekind, Frege, Husserl, and C. S. Peirce. Today he still engages the sympathetic interest of logicians and philosophers. The works translated record Schröder’s journey out of algebra into algebraic logic and document his transformation of George Boole’s opaque and unwieldy logical calculus into what we now recognize as Boolean algebra. Readers interested in algebraic logic and abstract algebra can look forward to a tour of the early history of those fields with a guide who was exceptionally thorough, unfailingly honest, and deeply reflective. Preface Translator's Introduction 0.1 The Lehrbuch Project 0.2 Doing Algebra with Classes 0.3 From Boole to Boolean Algebra References Contents 1 Numbers 1.1 The Mathematical Sciences 1.2 Numbers in General 1.3 What Makes Counting Possible? 1.4 When Do We Count? 1.5 The Emergence or Construction of the Natural Numbers 1.6 The Appellation of a Number 1.7 The Concept of Multiplicity 1.8 Numbers as Measures of Numerousness 1.9 Numbers as Rules 1.10 The General Purpose of Numbers 1.11 Cardinal and Ordinal Numbers 1.12 Independence of Numbers from the Ordering Imposed by the Counting Process 1.13 A Single Axiom 1.14 A Fundamental Proposition 1.15 Calculation: Expressions 1.16 Relations: Equations and Inequalities 1.17 Propositions About Equations 1.18 More on Equations 1.19 Ambiguous Expressions 1.20 Logical Subordination 1.21 Substitution 1.22 Parentheses 1.23 Literal and Numerical Numbers 1.24 Use of Letters Justified: Principles of Nomenclature 1.25 Use of Letters: Motivations and Benefits 1.26 Analytic and Synthetic Equations 1.27 Theorems, Rules 1.28 Constant and Variable Numbers, Functions 1.29 Conclusion References 2 The Three Direct Operations 2.1 Independent Treatment of Addition: Concepts and Terminology 2.2 First Law of Addition 2.3 Second Law of Addition 2.4 The Two Laws Combined 2.5 Addendum on Inequalities 2.6 Recursive Treatment of Addition: Number and Sum 2.7 The Associative Law for Trinomials 2.8 The Associative Law for Polynomials 2.9 The Commutative Law for Binomials 2.10 The Commutative Law for Polynomials 2.11 Independent Treatment of Multiplication: Basic Concepts 2.12 The Commutative Law of Multiplication 2.13 The Associative Law of Multiplication 2.14 Combination and Extension of the Two Laws 2.15 Expanded Definition of Product; Dirichlet's Proof of the Fundamental Theorem 2.16 The Distributive Laws 2.17 Extension of the Distributive Laws; Multiplying Out and Factoring 2.18 Fusion of the Distributive Laws in the Rule for Multiplying Polynomials 2.19 Toward an Alternative Treatment of the Foregoing 2.20 Additional Properties of Inequalities 2.21 Recursive Treatment of Multiplication: Concept and Laws 2.22 Another Proof of the Fundamental Theorem 2.23 Independent Treatment of Elevation: Potentiation and Exponentiation 2.24 Laws Governing These Operations; the Iteration Law 2.25 The Second Law 2.26 The Third Law 2.27 Addendum on Inequalities 2.28 Recursive Treatment of Elevation 2.29 Review; Iteration References 3 The Four Inverse Operations 3.1 Introduction; Inversion 3.2 Inversion of Univocal Operations: Subtraction 3.3 Division 3.4 Roots and Logarithms 3.5 Overview of the Operations, Expressions, and Operands 3.6 The Transposition Rules 3.7 Application to the Solution of Pure Equations 3.8 Necessity of the New Terminology 3.9 Concise Definition of the Outputs of Inverse Operations; Checking Inverse Operations 3.10 Realizability of the Inverse Operations 3.11 Univocity of the Inverse Operations 3.12 Special Cases (Singularities) 3.13 Additional Fundamental Properties; Using an Inverse to Check a Direct Operation 3.14 Another View of Transposition 3.15 The Final Fundamental Property; Using an Inverse Operation to Check an Inverse Operation 3.16 Simplifying Equations 3.17 Inversion of Ambiguous Operations: Non-commutative Multiplication 3.18 Operations on Ambiguous Expressions 3.19 Relations Between Expressions Involving a Community of Values 3.20 Linkage of Propositions of All Types by Univocal and Ambiguous Operations 3.21 Accommodating Ambiguous Operations 3.22 Using Univocal/Ambiguous Operations to Transpose Univocal/Ambiguous Expressions 3.23 What Inferences Are Permitted and Which of Them Are Reversible? 3.24 Substitutions Involving Ambiguous Expressions; Diagonals and Functions References 4 Rules for Linking All Our Operations 4.1 The Road Ahead 4.2 Rules for Transforming Algebraic Expressions: Resolution and Reduction 4.3 Additional Rules: Transitions from One Logarithmic System to Another 4.4 Rules for Connecting First-Order Operations with One Another: A Précis of Those Rules 4.5 Proof and Detailed Formulation of the Propositions 4.6 Another Proof and More Propositions 4.7 Alternative Proofs 4.8 Overview of the Preceding Propositions 4.9 Rules for Connecting Second-Order Operations with One Another 4.10 Overview of the Propositions 4.11 Rules for Connecting First- and Second-Order Operations with One Another 4.12 The First Group of Third-Order Rules 4.13 The Second Group of Third-Order Rules 4.14 The Third Group of Third-Order Rules 4.15 Review and Final Remarks 4.16 Addendum on Inequalities 4.17 More on Inequalities 4.18 Conventions Involving Parentheses 4.19 First Convention 4.20 Second Convention 4.21 Identification of Expressions and Operands 4.22 Linking up More than Three Numbers: Four Numbers 4.23 Arbitrarily Many Numbers 4.24 Conclusion 4.25 The Formal Approach to Algebra 4.26 The Elementary Assumptions 4.27 Separating the Elementary Assumptions 4.28 A Comprehensive Algorithm for Pure Associative Operations 4.29 More Properties of Pure Associative Operations 4.30 Alternatives to the Associative Law 4.31 Type-Two Algorithms: Conjugation 4.32 Type-Three Algorithms 4.33 Type-Four Algorithms 4.34 Type-Five Algorithms: Cyclic Multiplication 4.35 Type-Six Algorithms: The Pure Third-Order Laws 4.36 Completeness of the Cycle of Formulas 4.37 Division of the 100 Equations of the Second Class into Elementary Cycles 4.38 Division of the Equations of the First and Third Classes into Elementary Cycles 4.39 Specific Third-Order Laws 4.40 Consequences of the Distributive Laws for All Three Orders 4.41 Review and Final Thoughts References 5 The Operations of the Logical Calculus 5.1 Introduction 5.2 Five Operations 5.3 Axioms 5.4 Solution of a Boolean Problem 5.5 The Inverse Operations 5.6 Note on the Operations of the Logical Calculus References Index This volume offers English translations of three early works by Ernst Schroder (1841-1902), a mathematician and logician whose philosophical ruminations and pathbreaking contributions to algebraic logic attracted the admiration and ire of figures such as Dedekind, Frege, Husserl, and C.S. Peirce. Today he still engages the sympathetic interest of logicians and philosophers. The works translated record Schroders journey out of algebra into algebraic logic and document his transformation of George Booles opaque and unwieldy logical calculus into what we now recognize as Boolean algebra. Readers interested in algebraic logic and abstract algebra can look forward to a tour of the early history of those fields with a guide who was exceptionally thorough, unfailingly honest, and deeply reflective