Metamathematics and the Philosophical Tradition

Metamathematics and the Philosophical Tradition

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Overview

Metamathematics and the Philosophical Tradition is the first work to explore in such historical depth the relationship between fundamental philosophical quandaries regarding self-reference and meta-mathematical notions of consistency and incompleteness.

Using the insights of twentieth-century logicians from Gödel through Hilbert and their successors, this volume revisits the writings of Aristotle, the ancient skeptics, Anselm, and enlightenment and seventeenth and eighteenth century philosophers Leibniz, Berkeley, Hume, Pascal, Descartes, and Kant to identify ways in which these both encode and evade problems of a priori definition and self-reference. The final chapters critique and extend more recent insights of late 20th-century logicians and quantum physicists, and offer new applications of the completeness theorem as a means of exploring "metatheoretical ascent" and the limitations of scientific certainty.

Broadly syncretic in range, Metamathematics and the Philosophical Tradition addresses central and recurring problems within epistemology. The volume’s elegant, condensed writing style renders accessible its wealth of citations and allusions from varied traditions and in several languages. Its arguments will be of special interest to historians and philosophers of science and mathematics, particularly scholars of classical skepticism, the Enlightenment, Kant, ethics, and mathematical logic.

Product Details

ISBN-13: 9783110572216
Publisher: De Gruyter
Publication date: 12/17/2018
Pages: 492
Sales rank: 798,034
Product dimensions: 6.10(w) x 9.06(h) x (d)
Age Range: 18 Years

About the Author

William Boos (†), University of Iowa, USA.

Table of Contents

Preface ix

Editorial Remarks xi

1 Introductions Boundaries of Experience 1

1 "Wisdom", the "Desire to Know" and the Primacy of the Practical 1

2 "Hieratic" Inquiry and Its "Limits" 5

3 "Zetetic" Inquiry and its "Limits" 7

4 "Ta Meta Ta Logika" 9

5 "Semantic" Paradoxes 10

6 Dialogue, "Diakritik" and "Diagonalisatiom" 12

7 Interpretative Plurality and Metatheoretic Ascent 18

8 Alternative Views of "Secular Design" 23

9 "Transcendental" and "Proto-transcendental" Arguments 26

10 "Proto-transcendental" Arguments and "Hieratic" "Design" 28

11 "Coherentist Idealism" in the Hieratic Ideals of Leibniz and Berkeley 32

12 The "Second-order Idealism" of Hume's "Empiricism" 39

13 "Transcendental" Design 44

14 The Zetetic Incomplétude of "Merely" Regulative Ideals 52

15 Borgesian Maps 58

16 "Philosophia" and "Scientia" 64

2 "Was Blind, But Now I See": Ramifications of Plato's "Line" 68

1 Introduction 68

2 Metalogicians' "Intended Interpretations" 70

3 Metaphysical "Forms" 78

4 "The Very Idea" ("To Eidos Auton") of Unitary Bivalent "Truth" 88

5 Reflective Theories! 96

3 The Stoics, the Skeptics and Aporetic Autonomy: Is "What Is In Our Power" In Our Power? 105

1 Introduction 105

2 An Early Dialectical "Refutation" 106

3 "Semantic" Paradox(es) 108

4 "Diagonalisation" 109

5 "The World", "The Whole" and "Hypothetical Necessity" 115

6 "Our" Power over (What Is In) "Our" Power 121

7 Freedom, Autonomy and Aporetic "Self-Legislation" 125

8 Envoi 129

4 Anselm, Fides Quaerens Interpretationem, and Grenzideen as Generators of Metatheoretic Ascent 131

1 Outline and Introduction 131

2 "Gaunilonian" Relativisation(s) of "Anselmian" Argument(s) 134

3 Anselm of Canterbury and Georg von Halle 141

4 Metalogically "Intelligible" Concept-Formation 143

5 "Existence" and "Predication" in the First Critique 146

6 "Necessary Existence", and the "Existence" of Many "Necessities" 149

7 Conclusion: Anselmian "Numino" and Kantian "Noumena" 151

5 "Parfaits Miroirs de I'Univers": A "Virtual" Interpretation of Leibnizian Metaphysics 160

1 Introduction 160

2 "Miroirs", "Notions" et "Predicats" 163

3 "Necessit(é)" et "Raison(s) Suffisante(s)" 170

4 "Perfection" and "Plenitude" 173

5 Conclusion 180

6 "One Who Works for the People" 184

6 Berkeleyan Metalogical "Signs" and "Master Arguments" 186

1 Introduction 186

2 Iterated Abstraction and Metatheoretic Ascent 192

3 Semantic Paradox(es) and "Master Arguments" 197

4 Semeiotic "Coherentism"? 207

5 Conclusion(s) 223

7 The Second-order Idealism of David Hume 233

1 Introduction 233

2 Some Dogmas of Empiricism 235

3 "An Establish'd Maxim in Metaphysics" 240

4 Immaterialism and "Irrelationism" 252

5 "Whatever Is Clearly Conceived … 261

6 "[Metatheoretic] Matters of [Object-theoretic] Fact" 271

7 Hypotheses Non Fingimus? 275

8 "A Kind of Pre-established Harmony?" 283

9 Conclusion 299

8 Kantian Ethics and "the Fate of Reason" 306

1 Introduction 307

2 [Reflective Inquiry into "Ultimate" Limits of [Reflective inquiry into "Ultimate" Limits of [Reflective inquiry into "Ultimate" Limits of […]]]] 312

3 Reflective Inquiry and "Transcendent" "[Self]-Validation" 320

4 Reflective Inquiry and "das Schicksal der Vernunft" 328

5 Reflective Inquiry and "Das" Reich "der" Zwecke 338

6 Reflective Inquiry and "Systeme von Zwecken" 348

7 Reflective Inquiry and Kant's Two "Unermeßlichkeiten" 356

8 Reflective Inquiry as a Form of Skeptical "Theories" 365

9 Metamathematical Interpretations of Free Will and Determinism 382

1 Three Venerable Quotations 382

2 Preliminary Observations 384

3 Three Distinctions 386

4 "Schematic Induction" and Metalogical "Nonstandardness" 390

5 Gödelian "Incompleteness" and "Indeterminism" 392

6 Mathematically Exiguous Interpretations of Physical "Continuity" 396

7 The "Mystery" of Essentially Incomplete Inquiry 400

10 Time-Evolution in Random "Universes" 406

1 Introduction 406

2 Four Heuristic Premises 409

3 "Linguistic" and "Theoretical" Processes 409

4 "Canonical" Measures and Topologies 412

5 "Observational Measures" and "Observational Frames" 414

6 A Continuous Representation of Wave-Function "Collapse" 418

7 "Virtual" Set-Theoretic Universes 428

8 The "Direction" of "Time" 433

9 The Elusive Ideals of a "Closed" and "Isolated System" 437

10 A Promissory Note and Postscript 440

Bibliography 443

Index of Names 455

Main Index 459

Foreign Words Index 477

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