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Shapiro S. — Thinking about Mathematics: The Philosophy of Mathematics
Shapiro S. — Thinking about Mathematics: The Philosophy of Mathematics

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Название: Thinking about Mathematics: The Philosophy of Mathematics

Автор: Shapiro S.

Аннотация:

This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that mathematics is logic (logicism), the view that the essence of mathematics is the rule-governed manipulation of characters (formalism), and a revisionist philosophy that focuses on the mental activity of mathematics (intuitionism). Finally, Part IV brings the reader up-to-date with a look at contemporary developments within the discipline.

This sweeping introductory guide to the philosophy of mathematics makes these fascinating concepts accessible to those with little background in either mathematics or philosophy.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 308

Добавлена в каталог: 18.10.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
A posteriori knowledge      22 91 101 2 108 9
A priori knowledge      22 4 27 9 57 75 76 90 91 2 99 108 9 124 5 130 135 6 211 215 218 19
Abstract counterpart      234 223 4 246 278 285
Abstract objects      26n 32 57 133 162 206 216 33 238 242 243 8 252 4 257-8 261 70 272
Abstract-concrete dichotomy      26n 162n 233
Abstraction      66 8 70 81 2 88 136 8 152 259 60 276 83
Algorithm      43 159 160 1 163 165 181n 183 196 7
Analytic proposition      77 81 84 5 92 107 9 124 30 133 6 190 212 15
Ante rem realism      262 263 70 275 6 279 80 282 285 6
Anti-realism      see realism
Anti-realism, semantic      33 194
Application of mathematics      16 18 19 33 9 40 57 8 69 71 76 7 91 113 146 8 149 50 216 20 234 7 252 255 6
Archimedean principle      101
Aristotle      4 54 63 71 76 95 262 3 269
Arithmetic      40 57 60 65 71 75 79n 81 86 9 95 101 109 15 119 125 134 7 147 165 8 210 258
Arithmetic, axioms of      43 95 100 111 120 1 125 127 134 250 284
Arithmetic, finitary      159 165 167 8 174 5 205 209 210 235 253
Arithmetic, primitive recursive      161
Astrology      19
Attributes      see concepts; properties
Ayer, A. J.      4 129 30
Azzouni, J.      250 2
Balaguer, M.      229n 252 6 286
Basic Law V.      1 14 15 1 16 134 137 8
Becoming, world of      51 9
Being, world of      51 9 62 3 64 70 205
Benacerraf, P.      31 3 257 265 267 271
Berkeley, G.      4 45 74
Bernays, P.      15 152 167
Bishop, E.      184n 187 189
Bivalence      24 30 33 53 142 173 4 185 193 194 202 209 law
Bolzano      3 4 124
Boolos, G.      137 8
Bounded      159 60 162
Bourbaki, N.      39
Brouwer, L E. J.      8 32 172 174 85 186 189
Burgess, J.      243-49
Caesar problem      111 12 137 157 264 6
Calculation      59 80 87 89 95 8 144 147 159 60 189
Cantor, G.      40 158 208 9
Cardinal structure      277 80
Carnap, R.      4 14 124 32 212 13
Category theory      4 272n
Cauchy sequence      181 3 281
Causal theory of knowledge      244 250 251 252
Certainty      22 61 2 74 76-7 99 102 129 158 161 2 165 211 218 19 252
ch      see continuum hypothesis
Character, typographical      140 142 8 161 5 175n 258 276 8 281
Chihara      33 227 237 43 246 8
Choice sequence      181 3 189
Choice, principle of      9n 122 3 182n
class      9 10 40 2 112 14 116 24 158 180 1 10 216 220-4 229 235 238 43 265
Classical logic      8 9 11 13 14 30 172 4 178 81 183 4 185-6 188 9 190 7 209n 242
Coffa, A.      24 91 124 133 184
Coherence principle      286 9
Completeness principle      101 122 181n 227 230 232 243
Completeness theorem      15 287
Computability      4 (see also calculation)
Computation      see calculation
concepts      10 22 26 7 40-4 65 68 77 82 84-6 88 90 108 110n 111 14 116 123 128 137 208 216 223 238 9 242 3 263
Concrete objects      10 12 26n 35 53 7 65 71 75 6 81 161 162 164 204 206 8 216 222 32 245 282n 286
Conservative extension      138 163 233-6
Consistency      13 134 138 153 7 158 163 4 166 8 169 235 253 4 274 286 8
Consistency. $\omega$-consistency      166
Consistency. proof of      134 153 7 164 8 169 70 185 209
Constructibility      240 3
Constructibility quantifier      240 3
Constructivism      184n 187 189
Contingency      see necessity
Continuum Hypothesis      42 3 208 10 232 243
Convention      126 30 237 251
Counterexample, weak      183
Creative subject      181 2
Cumulative hierarchy      118n 131 223 265 272 3 275 287 9
Curry, H.      168 70 188n
Dedekind cut      122 281
Dedekind, R.      11 12 122
Deductivism      148 57 158 164 209 285
Definition      76 92 99 108 9 111 119 124 135-6 154 7
Definition, functional      284n
Definition, implicit      129 152 3 154 5 157 283 9
Definition, impredicative      9 11 116 118 20 121n 122 8 135 180 203 4 241 2
Definition, ostensive      276 8
Descartes, R.      3 73 5 154 177
Diagrams      56 7 63 86 87 9 151
Dialectic      63
Divided line, Plato's      53 5
Double negation, principle of      173 178 9
Dummett, M.      13 32 33 125 190-7
Dynamic/static language      7 45 56 69 179-0
Effectiveness      43 4 131 159 60 165
Empiricism      4 18 19 63 69 74 6 91 2 100 2 124 6 130 1 212 20 Carnap Quine)
Epistemology      6 16 17 21 32 51 2 78 9 92 3 135-6 158 9 201 2 212 221 244 275-6
Equinumerosity      109 10 112 133-4 178 208
Equivalence relation      137n
Euclid's Elements      58n 69 84 7 94 156 231
Excluded middle, law of      8-9 11 13 172 4 178 84 186 9 192 7 201 209n intuitionism;
Existence condition      285 6
Explanation      34 7 64 135 6 217 264
Explanation, intrinsic      237
Extension of concepts      112 15 116 134 137 196
Extension, pure      75 81
External question      14 126 8 169n
Fermat's last theorem      132
Feynman, R.      39
Fiction      68 9 93 4 123-4 227
Fictionalism      227 37 255 6
Field, H.      32 226 7 227 37 246 255 260-1
Finitary arithmetic      see arithmetic finitary
Finitism      158-8 195 196
First-order logic/languages      15 40 1 236n 285 287
FORM      27 51 60 64 66 70 262
Formalism      140 70 175 184 188n 192 209 253
Formalism, game      144 8 148 50 158 163 169
Formalism, term      142 4 145 158 162 169
Framework principle      1 29 2 1 3
Free choice sequence      181 3 189
Frege's theorem      111 118 19 134 135 6
Frege, correspondence with Hilbert      154 7 164 253
Frege, G.      33 67 8 82 96 8 108 15 115 16 117 19 124 125n 133 7 143 4 145 8 149 150n 177n 191n 264 267
G$\ddot{o}$del      10 12 15 38 44 165 8 202 11 221 3 incompleteness)
Galileo, Galilei      6
Geometry      8 23 52 3 55 7 65 71 75 81-6 88 91 93 5 114 123 129 30 150 6 176 7 231
Geometry, analytic      44 73 150 153 4 177 231 234 261n
Geometry, non - Euclidean      89 90 130 150 153 176 177
Geometry, synthetic      230 2 261n
Goldbach conjecture      183
Good, The      52 53 54 63
Goodman, Nicolas      33 4
Hale      133 8
Hand, M.      269 70
Hebb, D.      222
Heine, E.      143
Hellman, G.      33 257 274 5
Hermeneutic nominalism      248 9
Heyting predicate calculus      185 187
Heyting semantics      185 6 193 194 5
Heyting, A.      8 9 32 185 9 191n 192
Higher-order logic/languages      see second-order logic/languages
Hilbert programme      158-68 210 235 6 253
Hilbert, correspondence with Frege      154 7 164 253
Hilbert, D.      4 5 8 148 65 170 174 5 209 210 253 285 6
Hintikka, J.      81 83 4 85
Holism      17 19 37 130 1 192 3 213 16 218 20 224 239n 282
Homomorphism, representing      235 236n
Hume's Principle      110-12 117 118 134 8 284n
Hume, D.      4 74 75-6 92 99n 110n
Hypothetical-deductive method      17 102
Ideal mathematics      163-5 166 168 175 210 235
Idealism      25 6 28 9 29 30 32 36 38 180
Idealization      51 55-8 69 71 88 93 5 98 101-2 160 175 177 8 189 194 5 196
Identity      67 110 112 137 143 239 259 265 6 268n
Impredicativity      9 10 11 116 118 23 127 128 135 180 203 4 241 2 impredicative)
In re realism      262 3 270 1 275 279
Incompleteness      15 43-4 131 2 165 8 170 196 7 210 2 5 6
Indefinite extensibility      196 7
Independent propositions      42 4 153 208 10 224 232 243
Indispensability      37 8 216 18 220 1 227 34 244 6 254 6 282
Induction, empirical      37 92 4 96 100 1 210 212
Induction, mathematical principle of      100 102 111 120 258 264
infinity      10 25 28 9 79 86 7 98 101 2 111 117 19 135 158 159 181-2 197 232 260 272 279 81
Infinity, principle of      119 121 138 222 3 243
Instrumentalism      146 7 163 235
Internal question      126 8 266
Intrinsic explanation      237
Intuition      27 8 99 100 150 3 154 7 169 176 8 204 8 211 221 4 244 5
Intuition, Kantian      23 4 77 79 91 136 150 3 154 7 162 176 207
Intuitionism      8 9 32 87 172 97 207 209n
Intuitionistic logic      9 13 30 79n 172 4 180 1 83 185 187 190 193 7 law
Kant, I.      23 4 73 76 91 91-2 107 113n 161 162 174-8 207 Kantian)
Kitcher, P.      91 2 101 2 265 7
Klein, Jacob      59 60
Knowledge      see epistemology
Knowledge-of      205 6
Knowledge-that      205-6
Kreisel, G.      29
Kronecker, L.      12
Kuhn, T.      50
L$\ddot{o}$wenheim - Skolem theorem      40-2 285
Lakatos, I.      44 5
Language      11 13 14 16 25 6 29 31 2 41 124 33 5 153-4 184 5 186 7 190 5 212 15 239-42 248 256
Lear, J.      68
Leibniz, G.      3 4 73 74 230
Limit concept      93
Linguistic framework      126 33
Linguistic turn      125 184
Logic      4 6 24 29 40-1 45 79n 83 4 92 3 101 14 121 3 125n 127 129 30 134 6 6 149 53 155 7 159 61 184 5 187 190 202 214 15 219 2 8-9 246 274 5 285 288-9
Logical positivism      4 14 124 33 212 13 214 15
Logical terminology      108 110 111 122 3 136 149-50 190 193 4
Logicism      107 24 125 127 148n 150n 157 161 166 202
Logicism, neo      133 8 166n 264 5 283n
logistic      59 60
Maddy, P.      11n 18 20 32 97 8 220 4
Malament, D.      229n
Manifestation requirement      190-1 193 5
McLarty      273n
Meaning-is-use thesis      190-3 256
Mechanism      43 4 202n
Meta-mathematics      153-5 158 161n 164 5 169 70 187 253 4 proof model
Mill, J. S.      4 91 102 124 128 131 212 215 224
Modality      5 71 218 237 46 273-5
Model theory      5 40 2 123 150 153 197 234 238 253 4 264 272 275 287 9
Nagel, E.      130 151
National debt      279
Natural number structure      258 263 4 265 7 269 70 274 5 280 1 283n 287
naturalism      16 20 24 28 90 92 126 131 212 220 1 224 244 247 8
Necessity      21 23 4 27 28 9 75 6 81 89 99 113 124 5 129 135 211 218 19 229 30 246 252 258n 273 274
Neo-logicism      133 8 166n 264 5 283n
Neurath, ship of      17
Newton, I.      73 216 17 229
No class theory      123 4 238 239
Nominalism      26 27 29 33 216 17 218 226 56 260 1 263 273
Nominalistic language      216 17 228 37 240 2
Non - Euclidean geometry      see geometry non
Number, cardinal      41 1 5 6 277 80
Number, integer      121 2 144 267 8 281
Number, natural      25 40 1 57 60 65 8 71 75 80 82 86 7 95 100 109 14 115 117 21 131 2 134 5 142 159 62 164 176 7 222n 242 3 258 263 7 281
Number, rational      58 71 121 2 177 183
Number, real      40 44 45 71 101 113 122-3 138 144 148n 158 177 181 2 208 217 228 231 232 3 237 243 260 1 281
Object      see singular term
One-over-many      see Form; universals
Ontological commitment      228 240 248 9 250
Ontological option      272 273
Ontology      see realism in ontology
Ontology, relativity of      41 2 267 8 269
Open sentence      239 43
Ostension      276 8
Parsons      86 7 268n 271 282n
Pasch, M.      151
Pattern      257 258 60 261 3 ante
Pattern recognition      276 83
Peano axioms      see arithmetic axioms
Perception      18 23 27 64 5 75 80 2 83 4 87 9 93 4 97 8 99 100 162 176 8 204 7 221 3
Philosophy-first principle      9 16 17-18 20 90 91 2 212 244
Philosophy-last principle      13 15 17 18 20
Physical objects      see concrete objects
Physics      34 38 67 89 90 99n 183 204 5 208 211 215 20 222 228 229 32 235-6 246 9
Places-are-objects perspective      268-70 284
Places-are-offices perspective      268 70 271
Plato      3 5 6 7 8 50 63 66 69 74 88 179 205 262
Platonism      8 15 27 8 70 1 75 221 233 234 257 8 264 5
Platonism, full-blooded      252 4 256 286
Poincare, H.      9-10 176
Positivism, logical      see logical positivism
Possible worlds      5 238
Postulate system      250 2
Postulation      52 68 119 121 282
Posy      79n 195
Primitive notion      108 129 151 209 230 254 275 287 8
Proclus      56
proof      4 22 3 36 43 45 56-7 62 3 79n 86 9 151 164 5 167 8 185 7 192 193
Proof theory      161n 187 253 4
Properties      10 26 7 44 65 68 108 109 110n 111 14 116 123 4 128 137 157 208 216 222n 238 9 242 3 262 268
Psychology      66 76 219 222 276
Putnam, H.      41 214 216-18 221 238
Quantifier      127 194 5 197 216 240 3
Question, external      14 126 8 169n
Question, internal      126 8 266
Quine, W. V.      4 16-20 37 92 130 1 136 192 3 212 20 237 8 239 240 251 267
Ramsey, FP      121n 128
Rationalism      3 4 50 63 74 6 88
Real analysis      see number real
Realism      9 12 15
Realism in ontology      25 9 31-3 36 53 66 8 113 125 133 143 174 201 56 257 8 264 5
Realism in truth-value      29 33 37 8 42 3 53 66 8 174 185 202 216 220 226 7
Realism, ante rem      see ante rem realism
Realism, in re      see in re realism
Reducibility, principle of      120 1 122 3 127
Reference      5 25 29 41 203-4 250 1
Reference, thick/thin/ultrathin      251
Relationalism      229 30
Relativity of ontology      41 2 267 8 269
Resnik, M.      32 230 257 60 264 267 8 269 282 285-6 287
Revisionism      13-14 16 30 172 97 227 241
Revolution      21 2 50
Revolutionary nominalism      247 8 249
Rigour      49 50 63 74 149 150 168
Rosen, G.      243 9
Rosser, J.      166n
Rule-following      145n
Rule-governed procedure      140 181 3
Russell      3 114 15 115 24 125 127 202 3 204 8 238 9 268n
Russell's paradox      114 15 116 158 9 206
Satisfaction      40 134 5 138 153 238 239 43 246 253 4 272 275 285 9
Schroder - Bernstein theorem      178
Second-order logic/languages      41 2 110 120n 134 135 6 178 239n 285 287
Semantic tradition      124 5 133 184
Semantics      16 29 30 31 36 45 173 185 6 190 5 240
Semantics, compositional      193
Semantics, molecular      193 4
Set      9 10 40 2 52 101 116 18 158 178 181 10 222 3 229 235 238 43 265
Set theory      18 27 40 102 118 136 158 10 219 20 222 4 253 4 265 272 275 287 9
Set theory, Zermelo - Fraenkel      42 3 118 209 224
Shapes      142 3 262 277
Shapiro, S.      32 42 232 235 236n 267 269 282 283n 284 5
Singular term      31 45 83 4 110 134 137 239 251 270 284
Skolem paradox      40 2
Skolem, T.      15 40 41
Socrates      60 2
Socratic method      51 60 1 62 63
1 2
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