Cambridge University Press 0521837820 - Phenomenology, Logic and the Philosophy of Mathematics Richard Tieszen Index More information Index absolutism, 44, 332 see also Frege, G.; natural numbers; abstraction, 34, 36–37, 78, 84, 325, 327, Peano arithmetic (PA); Poincar´e, H.; 328 primitive recursive arithmetic (PRA); formal and material, 28 Weyl, H. see also founding/founded distinction artificial intelligence, 288 absurdity associationism, 210 formal a priori, 27 authentic/inauthentic distinction, 3, 42 material a priori, 27 axiom systems algorithmic methods, 32 definite formal, 4, 11–12, 29 alienation, 42 analysis, paradox of, 320 Becker, O., 8, 9, 14, 62, 83, 126, 237, 247, analytic a priori 254, 260, 268 judgments, 28 Bell, D., 329 see also analyticity; analytic/synthetic Benacerraf, P., 58, 64, 172 distinction Bernays, P., 108, 153, 245 Analytic and Continental philosophy, 1–2, Beth models, 289, 293 44–45, 66 BHK interpretation, 229, 232, 238, 242, analyticity, 185–190 245 rational intuition and, 188–190 biologism, 23 see also Frege, G.; G¨odel, K.; Poincar´e, Bishop, E., 228, 288 H.; Quine,W.V.O. Bolzano, B., 24, 51, 154, 318 analytic/synthetic distinction, 318 Boolos, G., 187 in Husserl, 27–28 Brentano, F., 1, 22 antinomies, 131 Brouwer, L. E. J., 7, 8, 118, 227, 228, 231, see also paradoxes 234, 235, 248, 249, 253, 254, 266, 278, antireductionism 283, 296 in Husserl, 33 see also intuitionism see also reductionism apophantic analytics, 28–29 calculation Aristotelian realism, 127 in science, 36, 40, 41–42 Aristotle, 127 Cantor, G., 9, 86, 126, 285 arithmetic Carnap, R., 1, 9, 70, 178, 179, 182 in Husserl, 2–3, 22, 32 Carnap’s program, 136–138, 182 Husserlian transcendental view of, 319, Cartesian dualism, 224 325–336 Cayley, A., 77 349 © Cambridge University Press www.cambridge.org Cambridge University Press 0521837820 - Phenomenology, Logic and the Philosophy of Mathematics Richard Tieszen Index More information 350 Index Chihara, C., 211 substitutions of, 182–183, 193–194 choice, axiom of, 99, 196 see also G¨odel, K.; intentionality choice sequences, 229 Continental philosophy, 21, 36, 100 see also Weyl, H. continuum Church’s Thesis, 217, 230, 231, 288 atomistic, 265, 266 complexity intuitive, 229, 263–271 algorithmic, 157 Weyl and the, 263–271 concealment, 45 continuum hypothesis (CH), 98, 99–100, concepts, 127, 149–176 139, 144, 167, 187–188, 196 are not subjective ideas, 170–171 conventionalism, 50, 66, 81, 136, 166 as intensional entities, 157 countersense (Widersinn), 27, 28 and intentionality, 156 Couturat, L., 296 and particulars, 223 Curry, H., 312 properties of, 169 concrete mathematics, 150–152 Da Silva, J., 262 consequence logic (Konsequenzlogik), 28 Davidson, D., 224 consistency decidability, 158, 161, 274 and existence, 30 see also G¨odel, K.; incompleteness in geometry, 82 theorems see also G¨odel, K.; incompleteness Dedekind, R., 259, 284 proofs Dennett, D., 199 consistency proofs, 64, 286 Derrida, J., 21 forarithmetic, 133–135, 152, 221, Descartes, R., 25 222 Detlefsen, M. in geometry, 31, 87 on Poincar´e, 301–303 of ZF + CH, 146, 152 dual aspect theories, 224 see also G¨odel, K.; incompleteness Dummett, M., 14, 192, 235, 277, 290, 329 proofs critics of on intuitionism, 233 constitution and extensional view of meaning, 244 of idealities, 4, 13, 26 and intuitionism, 227–247 of natural numbers, 327–329 meaning-theoretic argument for of space, 31 intuitionism, 231–233 constructibility, axiom of, 109, 196, 197 constructions, 228, 230, 331 eidetic reduction, 69, 95 as full manifestations of knowledge of see also free variation meaning, 233–237 empiricism, 197, 333 in intuitionism as fulfilled intentions, Husserl’s critique of, 23 237–247, 254 essences, 127–130 and mechanical (Turing) computation, formal and material, 128 230, 237, 240 in geometry, 69–88 see also intuitionism; proof hierarchies of geometric, 76–80 constructive recursive mathematics as invariants, 71, 74 (CRM), 228, 231 and meaning, 12–13 constructivism, 14, 61–62, 107, 227–247, morphological, 31, 69, 84 276–293 reductionism about, 130 and classical mathematics, 273–275, and sets, 76 284–286 Euclid, 316 and weak counterexamples, 284 Euclidean geometry in Weyl, 248–275 see geometry see also intuitionism evidence, 30, 61, 96, 103, 245, 252, 332, 333 content and proof, 277–293 intentional, 52–53, 96, 199, 210, 279, excluded middle, principle of, 229, 238, 280, 327 269, 284 mathematical, 190–199, 318 as an idealization, 284, 301, 332 © Cambridge University Press www.cambridge.org Cambridge University Press 0521837820 - Phenomenology, Logic and the Philosophy of Mathematics Richard Tieszen Index More information Index 351 existentialists, 100 notion of sense (Sinn), 139, 154, 298, extensions 321, 330 determined by intentions, 243–245, on psychologism, 316, 318 280–281 and realism (platonism), 316 see also Dummett, M.; Frege, G.; Quine, and role of intuition in arithmetic, 316, W. V. O.; Weyl, H. 324 extensionality, axiom of, 99 telescope analogy in, 171 on thoughts and judgments, 329–330 Feferman, S., 111, 217, 249, 266 Freudenthal, H., 239 Fichte, G., 248, 250–251, 253, 261, 272, fulfillment, 3, 60, 252 273 and intuitionistic constructions, see fictionalism, 48, 64–65, 285 constructions Field, H., 48, 212 and mathematical intentions, see proof finitism, 63 see also evidence; intentionality and intuitionism, 233, 234 functions Folina, J., 306, 307 as rules or graphs, 230 Føllesdal, D., 97, 183 formalism, 66, 230, 295, 310–313 Gadamer, H-G., 44 in Husserl, 3, 298 Galileo, 39, 42 see also Hilbertian formalism genetic phenomenology, 42–44 formalization, 298, 312, 333, 336 see also origins role of in science, 40 Gentzen, G., 232, 290 formalization/generalization distinction, genus/species relations, 28 28, 181, 191, 192 geometry, 31–32, 43, 69–88 formal systems, 4–6, 230, 234 non-Euclidean, 79 see also Hilbertian formalism projective, 77, 79 founding/founded distinction, 14, transcendental, 88 35, 54–55, 159, 258, 259, 304–305, see also space 330 George, A., 234, 235 in arithmetic, 327–329, 330 G¨odel, K., 51, 214, 258, 273, 289, 308 in geometry, 83–88 on absolutely unsolvable problems, 119, see also origins 124, 145, 151, 161, 167, 186 free variation (in imagination), 43, on analyticity, 117, 120, 123, 139, 185, 66, 69–89, 98–100, 165–167, 274, 186–190 334 on Aristotelian realism, 120, 176 Frege, G., 1, 9, 14, 25, 51, 60, 126, 147, on Carnap, 120–121, 179 165, 208, 209, 214, 238, 242, 243, on the concept of proof, 114, 116 283, 310 and concepts, 97–100, 154–155, analyticity in, 296, 318 180–181, 185 and the definition of number, 317, 318, concern for constructivity, 113 319, 321 on constructivity and impredicativity, and extensionality and intensionality in 306, 307 arithmetic, 317, 320–321, 322 and content, 120, 138, 139, 178, and gapless formal proofs, 316 179–181, 186, 190–199 on geometric foundations of arithmetic, on conventionalism, 119, 174 322–325 on empiricism, 120, 175–176 on Hilbertian formalism, 312 on Hilbert’s program, 114–115, 121, and Husserl on logic and meanings, 131–132, 138 297–299, 300 and Husserl, 10–13, 93–107, 121–122, and Husserl on number, 14–16, 67, 299, 123, 125–148, 149–176, 334 314–336 on the incompleteness theorems, laterview of number, 322–325 113–115, 118–122, 186–187 and logicism, 295, 296, 298, 314–322, and inexhaustibility of mathematics, 333, 334 139, 144, 173, 197 © Cambridge University Press www.cambridge.org Cambridge University Press 0521837820 - Phenomenology, Logic and the Philosophy of Mathematics Richard Tieszen Index More information 352 Index G¨odel, K. (cont.) Hilbert’s program, 146, 150, 182, 220, and the intuition of concepts, 149–176, 273 178, 189 see also G¨odel, K. on mathematical intuition, 101–102, Hintikka, J., 105 138, 140 historical a priori, 42–44 and meaning clarification, 141–142, 153, historicism, 23, 43–44, 328 154, 155, 177, 178–180, 181–183, 184, holism, 233 223 in Quine, see Quine, W. V. O. on minds and machines, 106, 110, 114, scope of, 183, 184, 195 145, 153, 173–174 horizons, 160 on nominalism, 120, 174 of acts, 63, 99, 158, 256, 268, 298, 334 philosophical writings on logic and Hume, D., 185 mathematics, 112–124 Husserl, E. and platonism, 116–118, 119 stages of his work, 2–4, 6–7 on positivism, 175 hyletic data, 38 on psychologism, 120, 175 and Quine, 177–200 idealism, 249–255, 270 and rational intuition, 122, 138–140, subjective, 62 144, 179, 188–190, 192 transcendental, 4, 248, 249, 253 realism and idealism in, 101, 105–107 idealization(s), 31, 39, 63, 160 on Russell, 116–117 classical real numbers as, 265 on set theory, 118, 139–140, 144–145, in geometry, 84–85 187, 286 in intuitionism, 245 and Turing, 115 principle of excluded middle as, 284, Goldfarb, W., 296 301, 332 Grassmann, H., 31, 70 ideal objects, 2, 3, 25–26, 56, 57–58, 168, Griss, G. F. C., 239 169, 325–327, 330 group theory, 76 bound, 57 free, 57 Habermas, J., 21 individual, 168 Hahn, H., 136 as omnitemporal, 25 Hallet, M., 307 universal, 168 Hebb, D., 204, 210 ideation, 69 Heidegger, M., 1, 21, 23, 44, 83, 130 see also free variation Heinzmann, G., 303 illusion, 60, 104, 139, 326 Helmholtz, H., 70 imagination, 72–73 hermeneutics, 44 see also free variation Heyting, A., 8, 14, 62, 126, 227, 228, 231, impredicativity, 258, 305–310 234, 237 and limits of constructive intuition, on degrees of evidence in intuitionism, 306–308 245 see also vicious circle principle and Freudenthal correspondence, inauthentic presentations 238–239 see authentic/inauthentic distinction on intuitionistic constructions, 237–239, incompleteness theorems, 6, 32, 131–138, 242, 247, 254, 279, 287 151, 153, 173, 179, 192–194, 216–224, Hilbert, D., 106, 126, 135, 153, 178, 179, 230, 273, 276, 314, 333 249, 253, 271–273, 284, 312 and Husserl’s definite axiom systems, optimism in the work of, 135–136 11–12 Hilbertian formalism, 48–49, 173, 295, 301, see also G¨odel, K.; Penrose, R.