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Tables of Contents for The Conceptual Roots of Mathematics
Chapter/Section Title
Page #
Page Count
Plato's Philosophies of Mathematics
Meno
1
2
A Priori
3
2
Relevance
5
4
What Are We Talking About?
9
4
How Do We Know?
13
7
Modality
20
2
Cogency
22
2
Deduction
24
3
Whence the Premisses?
27
6
Geometry
Euclid
33
1
The Fifth Postulate
34
5
Non-Euclidean Geometries
39
5
Formal and Physical Geometry
44
5
Conceptual Constraints
49
5
Which Geometry?
54
3
The Theory of Groups
57
3
Pythagorean Geometry has a Better Metric
60
3
Desargues
63
1
Conclusions
64
4
Formalism
More Geometrico
68
2
Formalising
70
1
Maximum Cogency
70
5
The Theory of Formal Systems
75
6
Meaning and Interpretation
81
3
Epistemological Formalism
84
3
The Logicist Programme
87
3
Numbers: The Cardinal Approach
Etymology
90
2
The Uses of Numbers
92
3
How Many?
95
6
Nought
101
1
Quotifiers and Quotities
102
4
Frege's Extensions and Sets
106
3
Paradigm Sets
109
5
Numbers: The Ordinal Approach
The Superlative Approach
114
1
Dedekind's Successor
115
3
And So On
118
2
Grounding the Ordinals
120
2
How to Count
122
1
Ordinals and Cardinals
123
1
Conclusion
124
2
Numbers: The Abstract Approach
The Third Approach
126
1
Peano
127
5
Monomorphism and Non-standard Models
132
6
Immigration Control
138
5
The Fifth Postulate
143
2
Sorites Arithmetic
145
1
Dialogues
146
6
Recursive Reasoning
152
2
The Natural Numbers
154
3
The Infinite
In Defence of Doubt
157
2
Cardinality
159
6
The Mostest
165
4
Characterization of Intuitionism
169
3
Proofs and Dialogues
172
9
Verificationist Arguments for Intuitionism
181
6
Selective Scepticism
187
2
Ultrafinitism
189
3
Lax Finitism
192
1
Actualising Potentiality
193
3
All
196
3
The Implications of Godel's Theorem
Pons Asinorum
199
1
The Flavour of the Godelian Argument
200
1
Godel Numbering
201
2
Translation
203
3
Diagonalization
206
4
Conditions
210
2
Corollaries and Consequences
212
2
Church's Theorem and Turing's Theorem
214
2
Godel's Second Theorem
216
2
Mechanism
218
1
Godel's Theorem and Provability
219
5
Transitive Relations
Logic
224
2
Equivalence Relations
226
6
Functions
232
2
Identity in Difference
234
2
Ordering Relations
236
3
Macrostructure and Microstruture
239
6
The Continuum
245
4
The Marriage of Equivalence with Order
249
4
Converse Transitivity
253
2
Paradigm Partial Orderings
255
2
Lattices and Set Theory
257
4
Trees and Mereology
261
10
Prototopology
Togetherness
271
1
Axiomatic Approaches
272
2
Whitehead's Programme
274
3
Failure
277
2
Pointed and Linear Hopes
279
3
Alternatives
282
1
Mooreology
283
2
More Mereology
285
6
Magnitude and Measure
Quantum? and Quot?
291
1
The Point of Measuring
292
2
Equivalence Structures
294
4
Addition Rules
298
3
Limits and Zero
301
1
Zeno
302
4
Measures and Numbers
306
5
Down With Set Theory
Multitudes and Magnitudes
311
3
Russell's Paradox
314
1
Responses to Paradox
315
4
Formal Approaches
319
3
The Skolem Paradox
322
1
The Axiom of Choice
323
5
The Continuum Hypotheses
328
2
Axioms and Existence
330
3
The Axiom of Extensionality
333
4
Conclusion
337
3
Chastened Logicism?
Logicism
340
2
What is Logic?
342
1
Boolean Plus
343
5
Iterated Modalities
348
5
Completeness
353
3
Paradox
356
2
Second-order Logic
358
3
Analytic and A Priori Truth
361
2
Mathematical Knowledge
Synthetic A Priori?
363
3
Not Seeing But Doing
366
2
Pattern Recognition
368
4
Lakatos
372
5
Cogency and Dialogue
377
2
On Behalf of the Fool
379
4
Hilbert
383
5
The Bed Theory of Truth
388
6
Mathematical Knowledge
394
5
Realism Revisited
Existence and Reality
399
4
Self-Subsistent Objects
403
4
Meaning and Impredicativity
407
6
Bivalence and Determinacy
413
2
Competing Truths
415
3
Contingency and Structure
418
4
Coherence and Depth
422
2
Laws of the Laws of Nature
424
3
Chastened Isms
427
6
Envoi
433
2
Summaries
435
7
Index
442