Philosophy of Arithmetic: Psychological and Logical Investigations - with Supplementary Texts from 1887-1901

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Springer Science & Business Media, 30 sep. 2003 - 513 páginas
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In his first book, Philosophy of Arithmetic, Edmund Husserl provides a carefully worked out account of number as a categorial or formal feature of the objective world, and of arithmetic as a symbolic technique for mastering the infinite field of numbers for knowledge. It is a realist account of numbers and number relations that interweaves them into the basic structure of the universe and into our knowledge of reality. It provides an answer to the question of how arithmetic applies to reality, and gives an account of how, in general, formalized systems of symbols work in providing access to the world. The "appendices" to this book provide some of Husserl's subsequent discussions of how formalisms work, involving David Hilbert's program of completeness for arithmetic. "Completeness" is integrated into Husserl's own problematic of the "imaginary", and allows him to move beyond the analysis of "representations" in his understanding of the logic of mathematics.
Husserl's work here provides an alternative model of what "conceptual analysis" should be - minus the "linguistic turn", but inclusive of language and linguistic meaning. In the process, he provides case after case of "Phenomenological Analysis" - fortunately unencumbered by that title - of the convincing type that made Husserl's life and thought a fountainhead of much of the most important philosophical work of the twentieth Century in Europe. Many Husserlian themes to be developed at length in later writings first emerge here: Abstraction, internal time consciousness, polythetic acts, acts of higher order ('founded' acts), Gestalt qualities and their role in knowledge, formalization (as opposed to generalization), essence analysis, and so forth.
This volume is a window on a period of rich and illuminating philosophical activity that has been rendered generally inaccessible by the supposed "revolution" attributed to "Analytic Philosophy" so-called. Careful exposition and critique is given to every serious alternative account of number and number relations available at the time. Husserl's extensive and trenchant criticisms of Gottlob Frege's theory of number and arithmetic reach far beyond those most commonly referred to in the literature on their views.
 

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Índice

THE AUTHENTIC CONCEPTS OF MULTIPLICITY UNITY AND WHOLE NUMBER
9
INTRODUCTION
11
THE ORIGINATION OF THE CONCEPT OF MULTIPLICITY THROUGH THAT OF THE COLLECTIVE COMBINATION The Analysis of the Co...
15
The Concrete Bases of the Abstraction Involved
16
Independence of the Abstraction from the Nature of the Contents Colligated
17
The Origination of the Concept of the Multiplicity through Reflexion on the Collective Mode of Combination
18
CRITICAL DEVELOPMENTS
23
The Collective Together and the Temporal Simultaneously
25
The Figural Moments
215
The Position Taken
223
The Psychological Function of the Focus upon Individual Members of the Group
225
What is It that Guarantees the Completeness of the Traversive Apprehension of the Individuals in a Group?
226
Apprehension of Authentically Representable Groups through Figural Moments
228
MISSING TOC element The Elemental Operations on and Relations between page 229 Infinite Groups
230
THE SYMBOLIC REPRESENTATIONS OF NUMBERS
235
The NonSystematic Symbolizations of Numbers
236

Collection and Temporal Succession
26
The Collective Synthesis and the Spatial Synthesis
35
B Baumanns Theory
45
Colligating Enumerating and Distinguishing
49
Critical Supplement
61
THE PSYCHOLOGICAL NATURE OF THE COLLECTIVE COMBINATION
67
The Collection as a Special Type of Combination
68
On The Theory of Relations
69
Psychological Characterization of the Collective Combination
74
ANALYSIS OF THE CONCEPT OF NUMBER IN TERMS OF ITS ORIGIN AND CONTENT
81
The Concept Something
84
The Cardinal Numbers and the Generic Concept of Number
85
Relationship Between the Concepts Cardinal Number and Multiplicity
87
One and Something
88
Critical Supplement
89
THE RELATIONS MORE AND LESS
95
Comparison of Arbitrary Multiplicities as well as of Numbers in Terms of More and Less
98
The Segregation of the Number Species Conditioned upon the Knowledge of More and Less
99
THE DEFINITION OF NUMBEREQUALITY THROUGH THE CONCEPT OF RECIPROCAL ONETOONE CORRELATION
101
The Definition of NumberEquality
103
Concerning Definitions of Equality for Special Cases
105
Application to the Equality of Arbitrary Multiplicities
106
Comparison of Multiplicities of One Genus
108
The True Sense of the Equality Definition under Discussion
110
Reciprocal Correlation and Collective Combination
111
The Independence of NumberEquality from the Type of Linkage
114
DEFINITIONS OF NUMBER IN TERMS OF EQUIVALENCE
117
Illustrations
120
Critique
121
Freges Attempt
123
MISSING TOC element KERRYS ATTEMPT page 129 Concluding Remark
131
DISCUSSIONS CONCERNING UNITY AND MULTIPLICITY
133
One and Zero as Numbers
136
The Concept of the Unit and the Concept of the Number One
141
Further Distinctions Concerning One and Unit
143
Sameness and Distinctness of the Units
146
Further Misunderstandings
157
Equivocations of the Name Unit
159
The Arbitrary Character of the Distinction between Unit and Multiplicity The Multiplicity Regarded as One Multiplicity as One Enumerated Unit as O...
162
Herbartian Arguments
164
THE SENSE OF THE STATEMENT OF NUMBER
169
Refutation and the Position Taken
170
The Nominalist Attempts of Helmholtz and Kronecker
179
THE SYMBOLIC NUMBER CONCEPTS AND THE LOGICAL SOURCES OF CARDINAL ARITHMETIC
189
OPERATIONS ON NUMBERS AND THE AUTHENTIC NUMBER CONCEPTS
191
MISSING TOC element The Fundamental Activities on Numbers page 192 Addition
193
Partition
198
SYMBOLIC REPRESENTATIONS OF MULTIPLICITIES
205
Sense Perceptible Groups
207
Attempts at an Explanation of How We Grasp Groups in an Instant
208
Symbolizations Mediated by the Full Process of Apprehending the Individual Elements
210
New Attempts at an Explanation of Instantaneous Apprehensions of Groups
211
Hypotheses
213
The Sequence of Natural Numbers
238
The System of Numbers
241
Relationship of the Number System to the Sequence of Natural Numbers
247
MISSING TOC element The Choice of the Base Number page 249 The Systematic of the Number Concepts and the Systematic of the Number Signs
251
The Process of Enumeration via Sense Perceptible Symbols
253
through Sense Perceptible Symbolization
254
Differences between Sense Perceptible Means of Designation
257
The Natural Origination of the Number System
258
Appraisal of Number through Figural Moments
267
Chapter XIII THE LOGICAL SOURCES OF ARITHMETIC
271
The Calculational Methods of Arithmetic and the Number Concepts
274
The Systematic Numbers as Surrogates for the Numbers in Themselves
275
The First Basic Task of Arithmetic
277
Addition
279
Multiplication
283
Subtraction and Division
284
Methods of Calculation with the Abacus and in Columns The Natural Origination of the Indic Numeral Calculation
288
Influence of the Means of Designation upon the Formation of the Methods of Calculation
290
The Higher Operations
292
MISSING TOC element The Indirect Characterization of Numbers by Means of Equations page 296 Mixing of Operations
294
The Logical Sources of General Arithmetic
298
ON THE CONCEPT OF NUMBER PSYCHOLOGICAL ANALYSES
305
Chapter One
312
Critical Exposition of Certain Theories
318
The Analysis of the Concept of Number as to its Origin and Content
352
PSYCHOLOGICAL ANALYSES THESES
357
B ESSAYS
359
II Comparison of Numbers
364
III Addenda
368
2 On the Definition of Number
369
V Remark
374
VI Corrections
375
VII Addenda
379
2 Addendum to p 377
380
ON THE CONCEPT OF THE OPERATION
385
II Combinations or Operations
397
2 On the Concept of Combination Verknüpfung
400
III Addendum
405
DOUBLE LECTURE ON THE TRANSITION THROUGH THE IMPOSSIBLE IMAGINARY AND THE COMPLETENESS OF AN AXIOM SYSTEM
409
2 Theories Concerning the Imaginary
413
3 The Transition through the Imaginary
427
APPENDIX I
453
APPENDIX II
459
APPENDIX III
464
Husserls Excerpts from an Exchange of Letters between Hilbert and Frege
468
THE DOMAIN OF AN AXIOM SYSTEM AXIOM SYSTEM OPERATION SYSTEM
475
System of Numbers
477
Arithmetizability of a Manifold
479
On the Concept of an Operation System
482
ON THE FORMAL DETERMINATION OF MANIFOLD
497
INDEX
505
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Página xxv - Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
Página xxxiii - If the number belonging to the concept F is defined as the extension of the concept equinumerous with the concept F, this means to Husserl that what we intend or mean (essentially have before the mind) in thinking of the number belonging to F is: the totality or set of concepts which have extensions that can be correlated one-to-one with the extension ofF. Given this, he says that "Further commentary is surely pointless.

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Sobre el autor (2003)

Philip Steele is an experienced writer and editor who has been writing books for children and adults on a wide range of topics including history, geography,cultural, and social themes for over 20 years.

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