Introduction .................................................... 1
Perception
1 Where We Begin .............................................. 19
1.1 The Limits of Cartesianism ............................. 20
1.2 Biological Evolution and the Concept of Life ........... 27
1.3 The Emergence of Human Culture and Social
Significances .......................................... 35
1.4 The World in View ...................................... 41
1.5 The Nature of Natural Language ......................... 50
1.6 Conclusion ............................................. 56
2 Ancient Greek Diagrammatic Practice ......................... 58
2.1 Some Preliminary Distinctions .......................... 60
2.2 Euclid's Constructions ................................. 68
2.3 Propaedeutic to the Practice ........................... 72
2.4 Generality in Euclid's Demonstrations .................. 78
2.5 Diagrammatic Reasoning in the Elements ................. 87
2.6 Ancient Greek Philosophy of Mathematics ................ 99
2.7 Conclusion ............................................ 104
3 A New World Order .......................................... 107
3.1 The Clockwork Universe ................................ 110
3.2 Viete's Analytical Art ................................ 118
3.3 Mathesis Universalis .................................. 127
3.4 The Order of Things ................................... 135
3.5 Descartes' Metaphysical Turn .......................... 141
3.6 Conclusion ............................................ 148
Understanding
4 Kant's Critical Turn ....................................... 153
4.1 The Nature of Mathematical Practice ................... 157
4.2 An Advance in Logic ................................... 166
4.3 Kant's Transcendental Logic ........................... 176
4.4 The Forms of Judgment ................................. 182
4.5 Kant's Metaphysics of Judgment ........................ 188
4.6 The Limits of Reflection .............................. 194
4.7 Conclusion ............................................ 199
5 Mathematics Transformed, Again ............................. 202
5.1 Intuition Banished .................................... 204
5.2 Constructions Banished ................................ 212
5.3 The Peculiar Purity of Modern Mathematics ............. 220
5.4 Prospects for the Philosophy of Mathematics ........... 231
5.5 Conclusion ............................................ 243
6 Mathematics and Language ................................... 247
6.1 Quantifiers in Mathematical Logic ..................... 250
6.2 The Model-Theoretic Conception of Language ............ 262
6.3 Meaning and Truth ..................................... 272
6.4 The Role of Writing in Mathematical Reasoning ......... 276
6.5 The Leibnizian Ideal of a Universal Language .......... 285
6.6 Conclusion ............................................ 292
7 Reasoning in Frege's Begriffsschrift ....................... 297
7.1 The Idea of a Begriffsschrift ......................... 300
7.2 The Basics ............................................ 308
7.3 A Second Pass Through ................................. 326
7.4 Seeing How It Really Goes ............................. 348
7.5 Conclusion ............................................ 361
8 Truth and Knowledge in Mathematics ......................... 364
8.1 What We Have Seen ..................................... 367
8.2 The Science of Mathematics ............................ 372
8.3 The Nature of Ampliative Deductive Proof .............. 383
8.4 Frege's Logical Advance ............................... 400
8.5 The Achievement of Reason ............................. 411
8.6 Conclusion ............................................ 419
9 The View from Here ......................................... 422
9.1 Einstein's Revolutionary Physics ...................... 425
9.2 The Quantum Revolution ................................ 433
9.3 Completing the Project of Modernity ................... 445
9.4 Conclusion ............................................ 451
Afterword ..................................................... 452
Bibliography .................................................. 454
Name Index .................................................... 473
Subject Index ................................................. 477
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