The Origin of the Logic of Symbolic Mathematics

Burt C. Hopkins

Description

The logic of modern symbolic mathematics

Burt C. Hopkins presents the first in-depth study of the work of Edmund Husserl and Jacob Klein on the philosophical foundations of the logic of modern symbolic mathematics. Accounts of the philosophical origins of formalized concepts—especially mathematical concepts and the process of mathematical abstraction that generates them—have been paramount to the development of phenomenology. Both Husserl and Klein independently concluded that it is impossible to separate the historical origin of the thought that generates the basic concepts of mathematics from their philosophical meanings. Hopkins explores how Husserl and Klein arrived at their conclusion and its philosophical implications for the modern project of formalizing all knowledge.

Preface by Eva Brann

Introduction: The Subject Matter, Thesis, and Structure of the Study
Part One. Klein on Husserl's Phenomenology and the History of Science
1. Klein's and Husserl's Investigations of the Origination of Mathematical Physics
2. Klein's Account of the Essential Connection between Intentional and Actual History
3. The Liberation of the Problem of Origin from its Naturalistic Distortion: The Phenomenological Problem of Constitution
4. The Essential Connection between Intentional History and Actual History
5. The Historicity of the Intelligibility of Ideal Significations and the Possibility of Actual History
6. Sedimentation and the Link between Intentional History and the Constitution of a Historical Tradition
7. Klein's Departure from the Content but Not the Method of Husserl's Intentional-Historical Analysis of Modern Science
Part Two. Husserl and Klein on the Method and Task of Desedimenting the Mathematization of Nature
8. Klein's Historical-Mathematical Investigations in the Context of Husserl's Phenomenology of Science
9. The Basic Problem and Method of Klein's Mathematical Investigations
10.Husserl's Formulation of the Nature and Roots of the Crisis of European Sciences
11. The "Zigzag" Movement Implicit in Klein's Mathematical Investigations
12. Husserl and Klein on the Logic of Symbolic Mathematics
Part Three. Non-Symbolic and Symbolic Numbers in Husserl and Klein
13. Authentic and Symbolic Numbers in Husserl's Philosophy of Arithmetic
14. Klein's Desedimentation of the Origin Algebra and Husserl's Failure to Ground Symbolic Calculation
15. Logistic and Arithmetic in Neoplatonic Mathematics and in Plato
16. Theoretical Logistic and the Problem of Fractions
17. The Concept of
18. Plato's Ontological Conception of
19. Klein's Reactivation of Plato's Theory of
20. Aristotle's Critique of the Platonic Chorismos Thesis and the Possibility of a Theoretical Logistic
21. Klein's Interpretation of Diophantus's Arithmetic
22. Klein's Account of Vieta's Reinterpretation of the Diophantine Procedure and the Consequent Establishment of Algebra as the General Analytical Art
23. Klein's Account of the Concept of Number and the Number Concepts in Stevin, Descartes, and Wallis
Part Four. Husserl and Klein on the Origination of the Logic of Symbolic Mathematics
24. Husserl and Klein on the Fundamental Difference between Symbolic and Non-Symbolic Numbers
25. Husserl and Klein on the Origin and Structure of Non-Symbolic Numbers
26. Structural Differences in Husserl's and Klein's Accounts of the Mode of Being of Non-Symbolic Numbers
27. Digression: The Development of Husserl's Thought, after Philosophy of Arithmetic, on the "Logical" Status of the Symbolic Calculus, the Constitution of Collective Unity, and the Phenomenological Foundation of the Mathesis Universalis
28. Husserl's Accounts of the Symbolic Calculus, the Critique of Psychologism, and the
29. Husserl's Critique of Symbolic Calculation in his Schröder Review
30. The Separation of Logic from Symbolic Calculation in Husserl's Later Works
31. Husserl on the Shortcomings of the Appeal to the "Reflexion" on Acts to Account for the Origin of Logical Relations in the Works Leading Up to the Logical Investigations
32. Husserl's Attempt in the Logical Investigations to Establish a Relationship between "Mere" Thought and the "In Itself " of Pure Logical Validity by Appealing to Concrete, Universal, and Formalizing Modes of Abstraction and Categorial Intuition
33. Husserl's Account of the Constitution of the Collection, Number, and the 'Universal Whatever' in
Experience and Judgment
34. Husserl's Investigation of the Unitary Domain of Formal Logic and Formal Ontology in Formal and Transcendental Logic
35. Klein and Husserl on the Origination of the Logic of Symbolic Numbers
Coda: Husserl's "Platonism" within the Context of Plato's Own Platonism
Glossary
Bibliography
Index of Names
Index of Subjects

Sujets

• Philosophie
• Épistémologie
• Philosophie de l'esprit
• Movements
• PHILOSOPHY
• Philosophie et théologie
• Logic
• Phenomenology
• Continental

Informations

• Publié par: Indiana University Press
• Date de parution: 07 septembre 2011
• Nombre de lectures: 0
• EAN13: 9780253005274
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

The Origin of the Logic of Symbolic Mathematics
STUDIES IN CONTINENTAL THOUGHT John Sallis, editor
CONSULTING EDITORS
Robert Bernasconi
William L. McBride
Rudolph Bernet
J. N. Mohanty
John D. Caputo
Mary Rawlinson
David Carr
Tom Rockmore
Edward S. Casey
Calvin O. Schrag
Hubert Dreyfus
Reiner Sch rmann
Don Ihde
Charles E. Scott
David Farrell Krell
Thomas Sheehan
Lenore Langsdorf
Robert Sokolowski
Alphonso Lingis
Bruce W. Wilshire
David Wood
THE ORIGIN OF THE LOGIC OF SYMBOLIC MATHEMATICS
EDMUND HUSSERL AND JACOB KLEIN
BURT C. HOPKINS
This book is a publication of Indiana University Press 601 North Morton Street Bloomington, Indiana 47404-3797 USA iupress.indiana.edu Telephone orders 800-842-6796 Fax orders 812-855-7931 Orders by e-mail [e-mail]
2011 by Burt Hopkins Published with assistance from the Theiline Piggot McCone Chair in Humanities, the College of Arts Sciences, Seattle University. All rights reserved
No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, without permission in writing from the publisher. The Association of American University Presses Resolution on Permissions constitutes the only exception to this prohibition.
The paper used in this publication meets the minimum requirements of the American National Standard for Information Sciences-Permanence of Paper for Printed Library Materials, ANSI Z39.48-1992. Manufactured in the United States of America Library of Congress Cataloging-in-Publication Data
Hopkins, Burt C. The origin of the logic of symbolic mathematics : Edmund Husserl and Jacob Klein / Burt C. Hopkins. p. cm. - (Studies in continental thought) Includes bibliographical references and index. ISBN 978-0-253-35671-0 (cloth : alk. paper) - ISBN 978-0-253-00527-4 (electronic book : alk. paper) 1. Logic, Symbolic and mathematical. 2. Mathematics-Philosophy. I. Title. QA9.H66 2011 511.3-dc23 2011022942
1 2 3 4 5 16 15 14 13 12 11
Die Zeiten, in welche die Entstehung der Zahl- und Zahlzeichensysteme f llt, kannten keine historische berlieferung, und so ist den an eine Reproduktion der historischen Entwicklung nicht zu denken. Gleichwohl besitzen wir Anhaltspunkte genug . . . , um die psychologische Entwicklung derartiger Systembildungen a posteriori und doch in allen wesentlichen Punkten zutreffend zu rekonstruieren.
The periods within which the origination of number systems and number sign systems falls are unknown to any historical tradition. Therefore there can be no thought of a reproduction of the historical development. We nevertheless possess sufficient clues . . . in order to reconstruct the psychological development of such systematic formations in an a posteriori fashion that is still correct in all essential points.
- Edmund Husserl, Philosophy of Arithmetic (1891)
Es kommt also darauf an, die Rezeption der griechischen Mathematik im 16. Jahrh. nicht von ihren Ergebnissen aus zu beurteilen, sondern sie sich in ihrem faktischen Vollzuge zu vergegenw rtigen.
Hence our object is not to evaluate the revival of Greek mathematics in the sixteenth century in terms of its results retrospectively, but to rehearse the actual course of its genesis prospectively.
- Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1934)
Gewi ist die historische R ckbeziehung niemandem eingefallen; und gewi ist die Erkenntnistheorie nie als eine eigent mlich historische Aufgabe angesehen worden.
Certainly the historical backward reference [from present-day geometrical knowledge to its genesis] has not occurred to anyone; certainly theory of knowledge has never been seen as a peculiarly historical task.
- Edmund Husserl, The Origin of Geometry (1936)
Contents
Preface by Eva Brann
Acknowledgments
List of Abbreviations
Introduction. The Subject Matter, Thesis, and Structure of This Study
Part One. Klein on Husserl s Phenomenology and the History of Science
Chapter One. Klein s and Husserl s Investigations of the Origination of Mathematical Physics
1. The Problem of History in Husserl s Last Writings
2. The Priority of Klein s Research on the Historical Origination of the Meaning of Mathematical Physics over Husserl s
3. The Importance of Husserl s Last Writings for Understanding Klein s Nontraditional Investigations of the History and Philosophy of Science
4. Klein s Commentary on Husserl s Investigation of the History of the Origin of Modern Science
5. The Curious Relation between Klein s Historical Investigation of Greek and Modern Mathematics and Husserl s Phenomenology
Chapter Two. Klein s Account of the Essential Connection between Intentional and Actual History
6. The Problem of Origin and History in Husserl s Phenomenology
7. The Internal Motivation for Husserl s Seemingly Late Turn to History
Chapter Three. The Liberation of the Problem of Origin from Its Naturalistic Distortion: The Phenomenological Problem of Constitution
8. Psychologism and the Problem of History
9. Internal Temporality and the Problem of the Sedimented History of Significance
Chapter Four. The Essential Connection between Intentional and Actual History
10. The Two Limits of the Investigation of the Temporal Genesis Proper to the Intrinsic Possibility of the Intentional Object
11. The Transcendental Constitution of an Identical Object Exceeds the Sedimented Genesis of Its Temporal Form
12. The Distinction between the Sedimented History of the Immediate Presence of an Intentional Object and the Sedimented History of Its Original Presentation
Chapter Five. The Historicity of the Intelligibility of Ideal Significations and the Possibility of Actual History
13. The Problem of I underlying Husserl s Concept of Intentional History
14. Two Senses of Historicity and the Meaning of the Historical Apriori
15. Historicity as Distinct from Both Historicism and the History of the Ego
Chapter Six. Sedimentation and the Link between Intentional History and the Constitution of a Historical Tradition
16. Maintaining the Integrity of Knowledge Requires Inquiry into Its Original Historical Discovery
17. Two Presuppositions Are Necessary to Account for the Historicity of the Discovery of the Ideal Objects of a Science Such as Geometry
18. Sedimentation and the Constitution of a Geometrical Tradition
19. The Historical Apriori of Ideal Objects and Historical Facts
20. The Historical Apriori Is Not a Concession to Historicism
Chapter Seven. Klein s Departure from the Content but Not the Method of Husserl s Intentional-Historical Analysis of Modern Science
21. The Contrast between Klein s Account of the Actual Development of Modern Science and Husserl s Intentional Account
22. Sedimentation and the Method of Symbolic Abstraction
23. The Establishment of Modern Physics on the Foundation of a Radical Reinterpretation of Ancient Mathematics
24. Vieta s and Descartes s Inauguration of the Development of the Symbolic Science of Nature: Mathematical Physics
25. Open Questions in Klein s Account of the Actual Development of Modern Science
Part Two. Husserl and Klein on the Method and Task of Desedimenting the Mathematization of Nature
Chapter Eight. Klein s Historical-Mathematical Investigations in the Context of Husserl s Phenomenology of Science
26. Summary of Part One
27. Klein s Failure to Refer to Husserl in Greek Mathematical Thought and the Origin of Algebra
28. Critical Implications of Klein s Historical Research for Husserl s Phenomenology
Chapter Nine. The Basic Problem and Method of Klein s Mathematical Investigations
29. Klein s Account of the Limited Task of Recovering the Hidden Sources of Modern Symbolic Mathematics
30. Klein s Motivation for the Radical Investigation of the Origins of Mathematical Physics
31. The Conceptual Battleground on Which the Scholastic and the New Science Fought
Chapter Ten. Husserl s Formulation of the Nature and Roots of the Crisis of European Sciences
32. Klein s Uncanny Anticipation of Husserl s Treatment of the Historical Origins of Scientific Concepts in the Crisis
33. Historical Reference Back to Origins and the Crisis of Modern Science
34. Husserl s Reactivation of the Sedimented Origins of the Modern Spirit
35. Husserl s Fragmentary Analyses of the Sedimentation Responsible for the Formalized Meaning Formations of Modern Mathematics and Klein s Inquiry into Their Origin and Conceptual Structure
Chapter Eleven. The Zigzag Movement Implicit in Klein s Mathematical Investigations
36. The Structure of Klein s Method of Historical Reflection in Greek Mathematical Thought and the Origin of Algebra
Chapter Twelve. Husserl and Klein on the Logic of Symbolic Mathematics
37. Husserl s Systematic Attempt to Ground the Symbolic Concept of Number in the Concept of Anzahl
38. Klein on the Transformation of the Ancient Concept of A ( Anzahl ) into the Modern Concept of Symbolic Number
39. Transition to Part Three of This Study
Part Three. Non-symbolic and Symbolic Numbers in Husserl and Klein
Chapter Thirteen. Authentic and Symbolic Numbers in Husserl s Philosophy of Arithmetic
40. The Shortcomings of Philosophy of Arithmetic and Our Basic Concern
41. Husserl on the Authentic Concepts of Multiplicity and Cardinal Number Concepts, and Inauthentic (Symbolic) Number Concepts
42. The Basic Logical Problem in Philosophy of Arithmetic
43. The Fundamental Shift in Husserl s Account of Calculational Technique
44. Husserl s Account of the Logical Requirements behind Both Calculational Technique and Symbolic Numbers
45. Husserl s Psychological Account of the Logical Whole Proper to the Concept of Multiplicity and Authentic Cardinal Number Concepts
46. Husserl on the Psychological Basis for Symbolic Numbers and Logical Technique
47. Husserl on the Symbolic Presentation of Multitudes
48. Husserl o