Logic and Philosophy of Mathematics in the Early Husserl (eBook)

Stefania Centrone obtained her master degree in Philosophy at the University of Florence. She was a PhD student at the Scuola Normale Superiore in Pisa, where she defended her doctoral dissertation in 2004. She was then awarded post-doctoral research grants at the Scuola Normale Superiore in Pisa until 2008. Since January 2009, Stefania Centrone has a two-year Alexander-von-Humboldt fellowship for research on Logical Objectivism, Inference and Foundational Proof in Bernard Bolzano’s ‘Wissenschaftslehre’ at the Philosophy Department of the University of Hamburg. She has given contributed talks at various workshops and conferences all over Europe, as well as invited talks at the Universities of Dublin, Koeln and Genève.

185616_1_En_BookFrontmatter_OnlinePDF 1
185616_1_En_1_Chapter_OnlinePDF 21
Chapter 1: Philosophy of Arithmetic 21
Introduction 21
`Many As One´: The Concept of Multiplicity (or Set) 26
The Collective Connection (kollektive Verbindung) 29
The Concept of Cardinal Number (Anzahl) 30
Chapters VI and VII of the Philosophy of Arithmetic 33
Husserl and Frege´s Theory 41
Three Further Issues: Unity, Zero and One, Numbers and Numerical Signs 45
Arithmetic Does Not Operate with Proper Numerical Concepts 49
Symbolic Presentations 51
`Sensuous Sets´ and Infinite Sets 53
Unsystematic Number Symbolizations and the Natural Number Series 55
The Numerical System 57
The Symbolic Aspect of the System 60
The Concept of Computation 63
The Fundamental Task of Arithmetic 65
The Taxonomy of Arithmetical Operations 67
Appendix1: Husserl´s Computable Functions200200This appendix is excerpted from Centrone 2006. 74
Appendix2: On Operations, Algorithmic Systems, and Computation 81
On the Concept of the Operation 82
Combinations and Operations 82
Equivalences and Algorithmic-Deductive Systems 87
A Historical Postlude on Leibniz 92
On the Notion of Computation and on Boole 95
Appendix3: Sets and Finite Numbers in ``Zur Lehre vom Inbegriff´´ 101
Introduction 101
Sets and Operations on Sets 103
Definition of the General Concept of Cardinal Number 107
Comparison of Two Sets Relative to Their Cardinal Number 108
Infinite and Finite Numbers, Natural Numbers and Their Classification 111
Concluding Remarks 117
185616_1_En_2_Chapter_OnlinePDF 119
Chapter 2: The Idea of Pure Logic 119
Introduction 119
The Concept of a Theory 122
The Concept of Begründung 124
The Interconnection of Things and the Interconnection of Truths 129
The Idea of Pure Logic 130
Logical Morphology and Logic of Non-Contradiction in the Fourth Investigation 134
Appendix4: On Bolzano 138
The Relation of Derivability (Ableitbarkeit) 138
The Relation of Exact Derivability (genaue Ableitbarkeit) 141
The Relation of Consecutivity (Abfolge) 143
Some Remarks on the Structure of Etiological Proofs 146
Appendix5. The Theory of Propositional and Conceptual Inferences in the Logikvorlesung of 1896 148
The Concept of a Calculus 148
On Propositional Inferences 150
The General Principles 151
The Notation 153
The Calculus 154
On Predication and Conceptual Inferences 161
Concluding Remarks 166
185616_1_En_3_Chapter_OnlinePDF 168
Chapter 3: The Imaginary in Mathematics 168
Introduction 168
The Einleitung 171
Universal Arithmetic 178
Theories of the Imaginary 180
Passage Through the Imaginary 186
On Different Interpretations of Husserl´s Notion of Definiteness 195
Husserl´s Two Notions of Definiteness 195
Husserl´s Definitheit and Hilbert´s Vollständigkeit 196
Did the Doppelvortrag Ever Confront the Problem of Semantic Completeness? 198
More on the Conservativity of Expansions 200
Definite Manifolds 202
The Concept of `Mathematical Manifolds´ 204
On the Concept of an Operation System 207
Arithmetizability of a Manifold 210
Husserl´s Reappraisal of His Early Theory of Definite Manifolds 211
Formal Aspects of the Theory of Manifolds 214
Ways of Generalization 220
Generalization by Weakening Axioms 220
Generalization by Removals 221
Generalization ``Tout Court´´ 221
Appendix 6: Husserl´s Existential Axiomatics 222
Concluding Remarks 230
General Conclusion 230
[s_chaptitle]Bibliography 168
Works by Other Authors 1
Studies 1
185616_1_En_BookBasckmatter_OnlinePDF 233
: Author Index 241
: Subject Index 244