Realizing Reason: A Narrative of Truth and Knowing

Description

Realizing Reason pursues three interrelated themes. First, it traces the essential moments in the historical unfolding―from the ancient Greeks, through Descartes, Kant, and developments in the nineteenth century, to the present―that culminates in the realization of pure reason as a power of knowing. Second, it provides a cogent account of mathematical practice as a mode of inquiry into objective truth. And finally, it develops and defends a new conception of our being in the world, one that builds on and transforms the now standard conception according to which our experience of reality arises out of brain activity due, in part, to merely causal impacts on our sense organs. Danielle Macbeth shows that to achieve an adequate understanding of the striving for truth in the exact sciences we must overcome this standard conception and that the way to do that is through a more adequate understanding of the nature of mathematical practice and the profound transformations it has undergone over the course of its history, the history through which reason is first realized as a power of knowing. Because we can understand mathematical practice only if we attend to the systems of written signs within which to do mathematics, Macbeth provides an account of the nature and role of written notations, specifically, of the principal systems that have been developed within which to reason in mathematics: Euclidean diagrams, the symbolic language of arithmetic and algebra, and Frege's concept-script, Begriffsschrift. Cover Realizing Reason: A Narrative of Truth and Knowing Copyright Dedication Preface Acknowledgements Contents Introduction This has not happened Perception 1: Where We Begin 1.1 The Limits of Cartesianism 1.2 Biological Evolution and the Concept of Life 1.3 The Emergence of Human Culture and Social Significances 1.4 The World in View 1.5 The Nature of Natural Language 1.6 Conclusion 2: Ancient Greek Diagrammatic Practice 2.1 Some Preliminary Distinctions 2.2 Euclid’s Constructions 2.3 Propaedeutic to the Practice 2.4 Generality in Euclid’s Demonstrations 2.5 Diagrammatic Reasoning in the Elements 2.6 Ancient Greek Philosophy of Mathematics 2.7 Conclusion 3: A New World Order 3.1 The Clockwork Universe 3.2 Vie`te’s Analytical Art 3.3 Mathesis Universalis 3.4 The Order of Things 3.5 Descartes’ Metaphysical Turn 3.6 Conclusion Understanding 4: Kant’s Critical Turn 4.1 The Nature of Mathematical Practice 4.2 An Advance in Logic 4.3 Kant’s Transcendental Logic 4.4 The Forms of Judgment 4.5 Kant’s Metaphysics of Judgment 4.6 The Limits of Reflection 4.7 Conclusion 5: Mathematics Transformed, Again 5.1 Intuition Banished 5.2 Constructions Banished 5.3 The Peculiar Purity of Modern Mathematics 5.4 Prospects for the Philosophy of Mathematics 5.5 Conclusion 6: Mathematics and Language 6.1 Quantifiers in Mathematical Logic 6.2 The Model-Theoretic Conception of Language 6.3 Meaning and Truth 6.4 The Role of Writing in Mathematical Reasoning 6.5 The Leibnizian Ideal of a Universal Language 6.6 Conclusion Reason 7: Reasoning in Frege’s Begriffsschrift 7.1 The Idea of a Begriffsschrift 7.2 The Basics 7.3 A Second Pass Through 7.4 Seeing How It Really Goes 7.5 Conclusion 8: Truth and Knowledge in Mathematics 8.1 What We Have Seen 8.2 The Science of Mathematics 8.3 The Nature of Ampliative Deductive Proof 8.4 Frege’s Logical Advance 8.5 The Achievement of Reason 8.6 Conclusion 9: The View from Here 9.1 Einstein’s Revolutionary Physics 9.2 The Quantum Revolution 9.3 Completing the Project of Modernity 9.4 Conclusion Afterword Bibliography Name Index Subject Index