Foundational theories of classical and constructive mathematics:
Saved in:
| Format: | Book |
|---|---|
| Language: | German |
| Published: |
Dordrecht [u.a.]
Springer
2011
|
| Series: | The Western Ontario series in philosophy of science
76 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XI, 314 S. graph. Darst. |
| ISBN: | 9789400704305 9789400704312 9400704305 |
Staff View
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IMAGE 1
CONTENTS
INTRODUCTION .
GIOVANNI SOMMARUGA REFERENCES . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
PART I SENSES OF 'FOUNDATIONS OF MATHEMATICS'
FOUNDATIONAL FRAMEWORKS 53
GEOFFREY HELLMAN 1 1 INTRODUCTION: QUESTIONS OF JUSTIFICATION AND
RATIONAL
RECONSTRUCTION (BETWEEN HERMENEUTICS AND CULTURA1 REVOLUTION). 53 2
DESIDERATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 56
3 IMP1ICATIONS: SET THEORY AND CATEGORY THEORY . . . . . . . . . . . . .
. . . 57
4 MODAI-STRUCTURAI MATHEMATICS AND FOUNDATIONS . . . . . . . . . . . . .
. . 63
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 68
THE PROBLEM OF MATHEMATICAL OBJECTS . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 71
BOB HAIE 1 PARSONS ON MATHEMATICAL INTUITION. . . . . . . . . . . . . .
. . . . . . . . . . . . 72
1.1 INTUITION 0/ AND INTUITION THAT . . . . . . . . . . . . . . . . . .
. . . . 72
1.2 PURE ABSTRACT AND QUASI-CONCRETE OBJECTS . . . . . . . . . . . . 72
1.3 THE LANGUAGE OF STROKE STRINGS 73
2 FREGE'S PROOF . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 75
3 DUMMEU'S OBJECTIONS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 76
4 DUMMEU'S OBJECTION REFURBISHED . . . . . . . . . . . . . . . . . . . .
. . . . . . 80
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 84
SET THEORY AS A FOUNDATION. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 85
PENEIOPE MADDY REFERENCES . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
VII
IMAGE 2
VIII CONTENTS
FOUNDATIONS: STRUCTURES, SETS, AND CATEGORIES . . . . . . . . . . . . .
. . . . . . . . . . . 97
STEWART SHAPIRO 1 ONTOLOGY, MAYBE EVEN METAPHYSICS 97
2 EPISTEMOLOGY: WHAT WE KNOW AND HOW WE (CAN) KNOW 101
3 ORGANIZING THINGS 105
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 110
PART 11 FOUNDATIONS OF CLASSICAL MATHEMATICS
FROM SETS TO TYPES, TO CATEGORIES, TO SETS 113
STEVE AWODEY 1 SETS TO TYPES . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 113
1.1 IHOL 114
1.2 SEMANTIES 115
2 TYPES TO CATEGORIES 116
2.1 TOPOI 117
2.2 SYNTAETIE TOPOS 118
3 CATEGORIES TO SETS 119
3.1 CATEGORY OFIDEALS 120
3.2 BASIC INTUITIONISTIE SET THEORY 120
4 COMPOSITES 122
4.1 SETS TO CATEGORIES 122
4.2 TYPES TO SETS 122
4.3 CATEGORIES TO TYPES 123
5 CONELUSIONS 123
REFERENEES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 125
ENRICHED STRATIFIED SYSTEMS FOR THE FOUNDATIONS OF CATEGORY THEORY .
127 SOLOMON FEFERMAN 1 INTRODUCTION 127
2 WHAT THE VARIOUS PROPOSALS 00 AND DON'T 00 128
3 THE SYSTEM NFU WITH STRATIFIED PAIRING 130
4 FIRST-ORDER STRUETURES IN NFUP 132
5 MEETING REQUIREMENTS (RL) AND (R2) IN NFUP 134
6 THE REQUIREMENT (R3); TYPE-SHIFTING PROBLEMS IN NFUP 135
7 THE REQUIREMENT (R3), CONTINUED; BUILDING IN ZFC 137
8 CANTORIAN CLASSES AND EXTENSION OF NFU IN ZFC 139
REFERENEES 142
RECENT DEBATE OVER CATEGORICAL FOUNDATIONS 145
COLIN MELARTY 1 THE FOUNDING IDEAS 146
2 FEFERMAN AND RAO 150
3 THE DIFFERENEES . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 151
REFERENEES 153
IMAGE 3
CONTENTS
PART 111 BETWEEN FOUNDATIONS OF CLASSICAL AND FOUNDATIONS OF
CONSTRUCTIVE
MATHEMATICS
IX
THE AXIOM OF CHOICE IN THE FOUNDATIONS OF MATHEMATICS 157
JOHN L. BELL REFERENCES . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
REECTIONS ON THE CATEGORICAL FOUNDATIONS OF MATHEMATICS 171 JOACHIM
LAMBEK AND PHI1IP J. SCOTT 1 INTRODUCTION 171
2 TYPE THEORY 172
3 ELEMENTARY TOPOSES 173
4 COMPARING TYPE THEORIES AND TOPOSES 174
5 MODELS AND COMPLETENESS 175
6 GDEL'S INCOMPLETENESS THEOREM 177
7 RECONCILING FOUNDATIONS 179
7.1 CONSTRUCTIVE NORNINALISM 179
7.2 WHAT IS THE CATEGORY OF SETS? 180
8 WHAT IS TRUTH? 181
9 CONTINUOUSLY VARIABLE SETS 182
10 SOME INTUITIONISTIC PRINCIPLES 183
11 CONCLUDING REMARKS 184
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 185
PART IV FOUNDATIONS OF CONSTRUCTIVE MATHEMATICS
LOCAL CONSTRUCTIVE SET THEORY AND INDUCTIVE DEFINITIONS 189
PETER ACZEL 1 INTRODUCTION 189
2 INDUCTIVE DEFINITIONS IN CST 192
2.1 INDUCTIVE DEFINITIONS IN CZF 192
2.2 INDUCTIVE DEFINITIONS IN CZF+ 195
3 THE FREE VERSION OF CST 196
3.1 A FREE LOGIC 196
3.2 THE AXIOM SYSTEM CZF F 196
3.3 THE AXIOM SYSTEMS CZF F-, CZFFI AND CZFF' 199
4 LOCAL INTUITIONISTIC ZERMELO SET THEORY 200
5 SOME AXIOM SYSTEMS FOR LOCAL CST 202
5.1 MANY-SORTED FREE LOGIC 202
5.2 THE AXIOM SYSTEM LCZFF- 203
5.3 THE AXIOM SYSTEM LCZFFI 204
5.4 THE AXIOM SYSTEM LCZF F' 204
6 WELL-FOUNDED TREES IN LOCAL CST 205
REFERENCES 207
IMAGE 4
X
CONTENTS
PROOFS AND CONSTRUCTIONS 209
CHARLES MCCARTY I PREAMBLE 209
2 BROUWER, HILBERT AND MATHEMATICAL PRACTICE 209
3 INTERNAL AND EXTERNAL NEGATIONS 212
4 THERE IS ONLY ONE NEGATION 213
5 INTUITIONISM AND MEANING 216
6 FATALLY WEAK COUNTEREXAMPLES 216
7 PROOFS AND CONSTRUCTIONS 219
8 A REALIZABILITY THEORY OF CONSTRUCTIONS 221
REFERENCES 225
EUCLIDEAN ARITHMETIC: THE FINITARY THEORY OF FINITE SETS 227
I.P. MAYBERRY 1 THE SORITES FALLACY 227
2 THE ANCIENT CONCEPT OF NUMBER 228
3 EUCLIDEAN ARITHMETIC 229
4 INDUCTION AND RECURSION 232
5 ARITHMETICAL FUNCTIONS AND RELATIONS 234
6 NATURAL NUMBER SYSTEMS 235
7 BINARY EXPANSIONS 240
8 CONCLUSIONS 241
REFERENCES 242
INTENTIONAIITY, INTUITION, AND PROOF IN MATHEMATICS 245
RICHARD TIESZEN 1 INTENTIONALITY 246
2 INTUITION AS FULFILLMENT OF MEANING-INTENTION 247
3 A GENERAL CONCEPTION OF PROOFS AS FULFILLMENTS OF MATHEMATICAL
MEANING-INTENTIONS 248
4 PROOFS AND PURELY FORMAL PROOFS 250
5 PROOFS, PRACTICE, AND AXIOMS 253
6 FRUSTRATED MEANING-INTENTIONS 255
7 MULTIPLE PROOFS FOR THE SAME MEANING-INTENTION 256
8 PROOFS THAT EXCEED MEANING-INTENTION, AND MISMATCHES BETWEEN PROOFS
AND MEANING-INTENTIONS 257
9 INTERNAL AND EXTERNAL PROOFS FOR MEANING-INTENTIONS 257
10 MISTAKEN PROOFS (INTUITIONS) 259
11 CONSTRUCTIVE PROOF 260
12 CONCLUSION 262
REFERENCES 262
IMAGE 5
CONTENTS
XI
FOUNDATIONSFOR COMPUTABLE TOPOLOGY 265
PAUL TAYLOR I FOUNDATIONS FOT MATHEMATIES 266
2 CATEGORY THEORY AND TYPE THEORY 269
3 METHOD AND CRITIQUE 276
4 STONE DUALITY 281
5 ALWAYS TOPOLOGIZE 284
6 THE MONADIE FRAMEWORK 289
7 THE SIERPIILSKI SPAEE 295
8 TOPOLOGY USING THE PHOA PRINEIPLE 298
9 CONCLUSION 305
REFERENEES 307
CONCLUSION: A PERSPECTIVE ON FUTURE RESEARCH IN FOM 311
GIOVANNI SOMMARUGA AND LOHN BELL REFERENEES 314 |
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| series2 | The Western Ontario series in philosophy of science |
| spelling | Foundational theories of classical and constructive mathematics Giovanni Sommaruga (ed.) Dordrecht [u.a.] Springer 2011 XI, 314 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier The Western Ontario series in philosophy of science 76 Konstruktive Mathematik (DE-588)4165105-4 gnd rswk-swf Theorie (DE-588)4059787-8 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Konstruktive Mathematik (DE-588)4165105-4 s Mathematik (DE-588)4037944-9 s Theorie (DE-588)4059787-8 s DE-604 Sommaruga, Giovanni Sonstige oth The Western Ontario series in philosophy of science 76 (DE-604)BV024452061 76 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3546324&prov=M&dok_var=1&dok_ext=htm Inhaltstext V:DE-604 application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020674621&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Foundational theories of classical and constructive mathematics The Western Ontario series in philosophy of science Konstruktive Mathematik (DE-588)4165105-4 gnd Theorie (DE-588)4059787-8 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4165105-4 (DE-588)4059787-8 (DE-588)4037944-9 (DE-588)1071861417 |
| title | Foundational theories of classical and constructive mathematics |
| title_auth | Foundational theories of classical and constructive mathematics |
| title_exact_search | Foundational theories of classical and constructive mathematics |
| title_full | Foundational theories of classical and constructive mathematics Giovanni Sommaruga (ed.) |
| title_fullStr | Foundational theories of classical and constructive mathematics Giovanni Sommaruga (ed.) |
| title_full_unstemmed | Foundational theories of classical and constructive mathematics Giovanni Sommaruga (ed.) |
| title_short | Foundational theories of classical and constructive mathematics |
| title_sort | foundational theories of classical and constructive mathematics |
| topic | Konstruktive Mathematik (DE-588)4165105-4 gnd Theorie (DE-588)4059787-8 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Konstruktive Mathematik Theorie Mathematik Konferenzschrift |
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