Philosophy of Mathematics Handbook

700 Mark Colyvan [Darwin, 1958] C. Darwin, Autobiography and Selected Letters, Francis Darwin ed., Dover, New York (1958). [Davies, 1992] P. Davies, The Mind of God, Penguin, London (1992). [Devlin, 1977] K. Devlin, The Axiom of Constructibility, Lecture Notes in Mathematics, Vol. 617. Springer-Verlag, Berlin (1977). [Duhem, 1962] P. Duhem, The Aim and Structure of Physical Theory, Princeton University Press, Princeton (1954) (first published in 1906). [Dyson, 1964] F.J. Dyson, Mathematics in the physical sciences, Scientific American, 211 (3) (1964), 128-146. [Enderton, 1977] H.B. Enderton, Elements of Set Theory, Academic Press, New York (1977). [Feynman, 1965] R Feynman, The Character of Physical Law, BBC, London (1965). [Field, 1980] H. Field, Science Without Numbers: A Defence of Nominalism, Blackwell Pub- lishers, Oxford (1980). [Field, 1984/1989] H. Field, Can we dispense with space-time?, Realism, Mathematics and Modality, Blackwell Publishers, Oxford (1989) (first published in 1984), 171-226. [Field, 1985/1989] H. Field, On conservativeness and incompleteness, Realism, Mathematics and Modality, Blackwell Publishers, Oxford (1989) (first published in 1985), 125-146. [Field, 1988/1989] H. Field, Realism mathematics and modality, Realism, Mathematics and Modality, Blackwell Publishers, Oxford, 1989 (first published in 1988), 227-281. [Field, 1989] H. Field, Realism, Mathematics and Modality. Blackwell Publishers, Oxford (1989). [Field, 1990] H. Field, Mathematics without truth (a reply to Maddy), Pacific Philosophical Quarterly, 71 (3) (1990), 206-222. [Field, 1992] H. Field, A nominalistic proof of the conservativeness of set theory, Journal of Philosophical Logic, 21 (2) (1992), 111-123. [Field, 1998] H. Field, Mathematical objectivity and mathematical objects, Contemporary Read- ings in the Foundations of Metaphysics, S. Laurence and C. Macdonald, eds., Blackwell Pub- lishers, Oxford (1998), 387-403. [Fodor and Lepore, 1992] J. Fodor and E. Lepore, Holism: A Shopper's Guide, Blackwell Pub- lishers, Cambridge (1992). [Ginzburg and Colyvan, 2004] L. Ginzburg and M. Colyvan, Ecological Orbits: How Planets Move and Populations Grow, Oxford University Press (2004). [Hamming, 1980] RW. Hamming, The unreasonable effectiveness of mathematics, American Mathematics Monthly, 87 (1980), 81-90. [Hellman, 1989J G. Hellman, Mathematics without Numbers: Towards a Modal-Structural In- terpretation, Clarendon Press, Oxford (1989). [Hellman, 1992] G. Hellman, The boxer and his fists: the constructivist in the arena of quantum physics, Proceedings of the Aristotelian Society, Supplement, LXVI, (1992), 61-77. [Hellman, 1999] G. Hellman, Some ins and outs of indispensability: a modal-structural per- spective, Logic and Foundations of Mathematics, A. Cantini, E. Casari and P. Minari, eds., Kluwer, Dordrecht. 1999, 25-39. [Hersh, 1990J R Hersh, Inner vision outer truth, Mathematics and Science, in RE. Mickens, ed., World Scientific Press, Singapore (1990), 64-72. [Hilbert, 1899/1971] D. Hilbert, Foundations of Geometry, Open Court, La Salle ILL (1971) (first published in 1899). [Hunt, 1971] B.J. Hunt, The Maxwellians, Cornell University Press, Ithaca NY (1991). [Kitcher, 1981] P. Kitcher, Explanatory unification, Philosophy of Science, 48 (1981), 507-531. [Kitcher, 1984] P. Kitcher, The Nature of Mathematical Knowledge, Oxford University Press, New York (1984). [Lakatos, 1970] 1. Lakatos, Falsification and the methodology of scientific research programmes, Criticism and the Growth of Knowledge,!' Lakatos and A. Musgrave, eds., Cambridge Uni- versity Press, Cambridge (1970), 91-195. [Mac Lane, 1990] S. Mac Lane, The reasonable effectiveness of mathematics, Mathematics and Science, RE. Mickens, ed., World Scientific Press, Singapore (1990), 115-135. [Maddy, 1990a] P. Maddy. Realism in Mathematics, Clarendon Press, Oxford (1990). [Maddy, 1990bJ P. Maddy, Physicalistic Platonism, Physicalism in Mathematics, A.D. Irvine, ed., Kluwer, Dordrecht (1990), 259-289. [Maddy, 1990c] P. Maddy, Mathematics and Oliver Twist, Pacific Philosophical Quarterly,71 (3) (1990), 189-205.

Mathematics and the World 701 [Maddy, 1992] P. Maddy, Indispensability and practice, Journal of Philosophy, 89 (1992), 275- 289. [Maddy, 1994] P. Maddy, Taking naturalism seriously, Logic, Methodology and Philosophy of Science IX, D. Prawitz, B. Skyrms and D. Westerstahl, eds., Elsevier, Amsterdam (1994), 383-407. [Maddy, 1995] P. Maddy, Naturalism and ontology, Philosophia Mathematica (3), 3 (3) (1995), 248-270. [Maddy, 1997] P. Maddy, Naturalism in Mathematics, Clarendon Press, Oxford (1997). [Maddy, 1998a] P. Maddy, Naturalizing mathematical methodology, Philosophy of Mathematics Today, M. Schirn, ed., Clarendon Press, Oxford (1998), 175-193. [Malament, 1982] D. Malament, Review of Field's Science Without Numbers, Journal of Phi- losophy, 79 (1982), 523-534. [May, 2004] RM. May, Uses and abuses of mathematics in biology, Science, 303 (6 February 2004), 790-793. [Melia, 2000] J. Melia, Weaseling away the indispensability argument, Mind, 109 (2000),455- 479. [Mickens, 1990] RE. Mickens, ed., Mathematics and Science, World Scientific Press, Singapore (1990). [Mortensen, 1995] C. Mortensen, Inconsistent Mathematics, Kluwer, Dordrecht (1995). [Papineau, 1993] D. Papineau, Philosophical Naturalism, Blackwell Publishers, Oxford (1993). [Penrose, 1989] R Penrose, The Emperor's New Mind: Concerning Computers, Minds and the Laws of Physics, Vintage Press, London (1990). [Putnam, 1971/1979] H. Putnam, Philosophy of logic, reprinted in Mathematics Matter and Method: Philosophical Papers Vol. I, second edition, Cambridge University Press, Cambridge (1979) (first published in 1971), 323-357. [Putnam, 1979] H. Putnam, What is mathematical truth?, Mathematics Matter and Method: Philosophical Papers Vol. 1, second edition, Cambridge University Press, Cambridge (1979), 60-78. [Quine, 1936/1983J W.V. Quine, Truth by convention, reprinted in Philosophy of Mathematics Selected Readings, second edition, P. Benacerraf and H. Putnam, eds., Cambridge University Press, Cambridge (1983) (first published in 1936), 329-354. [Quine, 1948/1980] W.V. Quine, On what there is, reprinted in From a Logical Point of View, second edition, Harvard University Press, Cambridge MA (1980) (first published 1948), 1-19. [Quine, 1951/1980] W.V. Quine, Two dogmas of empiricism, reprinted in From a Logical Point of View, second edition. Harvard University Press, Cambridge MA, 1980 (first published in 1951), 20-46. [Quine, 1953/1976] W.V. Quine, On mental entities, reprinted in The Ways of Paradox and Other Essays, revised edition, Harvard University Press, Cambridge, MA (1916) (first pub- lished in 1953), 221-227. [Quine, 1960J W.V. Quine, Word and Object, Massachusetts Institute of Technology Press and John Wiley and Sons, New York (1960). [Quine, 1963/1983] W.V. Quine, Carnap and logical truth, reprinted in Philosophy of Math- ematics Selected Readings, second edition, P. Benacerraf and H. Putnam, eds., Cambridge University Press, Cambridge (1983) (first published in 1963), 355-376. [Quine, 1981a] W.V. Quine, Five milestones of empiricism, Theories and Things, Harvard Uni- versity Press, Cambridge, MA (1981), 67-72. [Quine, 1981bJ W.V. Quine, Success and limits of mathematization, Theories and Things, Har- vard University Press, Cambridge, MA (1981), 148-155. [Quine, 1986] W.V. Quine, Reply to Charles Parsons, The Philosophy of W.V. Quine, L. Hahn and P. Schilpp, eds., Open Court, La Salle ILL (1986), 396-403. [Quine, 1992] W.V. Quine, Pursuit of Truth, revised edition, Harvard University Press, Cam- bridge MA (1992). [Quine, 1995J W.V. Quine, From Stimulus to Science, Harvard University Press, Cambridge MA (1995). [Resnik, 1983J M.D. Resnik, Review of Hartry Field's Science Without Numbers, Nous, 17 (1983), 514-519. [Resnik, 1985a] M.D. Resnik, How nominalist is Hartry Field's nominalism?, Philosophical Studies, 47 (1985), 163-181.

702 Mark Colyvan [Resnik, 1985b] M.D. Resnik, Ontology and logic: remarks on Bartry Field's anti-platonist philosophy of mathematics, History and Philosophy of Logic, 6 (1985), 191-209. [Resnik, 1995] M.D. Resnik, Scientific vs. mathematical realism: the indispensability argument, Philosophia Mathematica (3), 3 (2) (1995), 166-174. [Resnik, 1997J M.D. Resnik, Mathematics as a Science of Patterns, Clarendon Press, Oxford (1997). [Roseveare, 1983J N.T. Roseveare, Mercury's Perihelion from Le Verrier to Einstein, Clarendon Press, Oxford (1983). [Shapiro, 1983] S. Shapiro, Conservativeness and incompleteness, Journal of Philosophy, 80 (9) (1983), 521-531. [Shapiro, 1997J S. Shapiro, Philosophy of Mathematics: Structure and Ontology, Oxford Uni- versity Press, Oxford (1997). [Siegel, 1991J D.M. Siegel, Innovation in Maxwell's Electromagnetic Theory, Cambridge Uni- versity Press, Cambridge (1991). [Smart, 1963J J.J.C. Smart, Philosophy and Scientific Realism, Routledge and Kegan Paul, London (1963). [Sober, 1993] E. Sober, Mathematics and indispensability, Philosophical Review, 102 (1) (1993), 35-57. [Steiner, 1989J M. Steiner, The application of mathematics to natural science, Journal of Phi- losophy, 86 (9) (1989), 449-480. [Steiner, 1995] M. Steiner, The applicabilities of mathematics, Philosophia Mathematica (3), 3 (2) (1995), 129-156. [Steiner, 1998J M. Steiner, The Applicability of Mathematics as a Philosophical Problem, Har- vard University Press, Cambridge MA (1998). [Urquhart, 1990J A. Urquhart, The logic of physical theory Physicalism in Mathematics, A.D. Irvine, ed., Kluwer, Dordrecht (1990), 145-154. [van Fraassen, 1980] B.C. van Fraassen, The Scientific Image, Clarendon Press, Oxford (1980). [Weinberg, 1986J S. Weinberg, Lecture on the applicability of mathematics, Notices of the Amer- ican Mathematical Society, 33 (1986), 725-728. [Weinberg, 1993J S. Weinberg, Dreams of a Final Theory, Vintage Press, London (1993). [Wigner, 1960J E.P. Wigner, The unreasonable effectiveness of mathematics in the natural sci- ences, Communications on Pure and Applied Mathematics, 13 (1960), 1-14. [Wilson, 2000] M. Wilson, The unreasonable uncooperativeness of mathematics in the physical sciences, Monist, 83 (2000), 296-314.

INDEX a-field, 495 mathematical, 35-98 apartness relation, 325 a posteriori knowledge, 3, 5, 6, 8, 157, Apostoli, P., x, 479, 481, 482, 484- 179-186, 199, 213-226 488 a priori knowledge, 3, 5-7, 9, 18, 22, applicability of mathematics, 133 33 to empirical science, 84-86 saying is believing, 18 approximation space, 479, 480 Abelard, A., 234 Archimedes, 164 aboutness Aristotelian realism (or Aristotelian- thick vs. thin, 46, 92 ism), viii, 103 'absolute' rest, motion, simultaneity, Aristotle, 1-2, 53, 105, 131, 135, 138, 223 160-168,172,176,178,192, absoluteness, 572-574, 576, 617 196, 203, 215-216, 225, 239, abstract algebra, 112 356 abstract objects, 94-98, 238 arithmetic, 239 abstraction axioms, 470 Armstrong's abstraction principle, 236 abstract objects, 664 abstraction scheme, 463 naturalism, 664 Ackermann function, 555, 559 Armstrong, D. M., 53, 132, 642 Ackermann, W., 302, 537, 543, 556 Aronszajn tree, 435 Aczel, P., 481 Australian school, 110 additivity, 493, 496 axiom, 241 agent-relative, 497 Axiom of Choice (AC), 314405, 406 aggregates, 256 Axiom of Constructibility, 431, 680 aleatory, 497 Axiom of Convergence, 497 Aleksandrov, P., 418 Axiom of Dependent Choice, 324 algebraic theories, 353-354 Axiom of Extensionality, 470, 476 algorithm, 586 Axiom of Foundation, 425 analysis, 214-216 Axiom of Independence, 499 analytic geometry, 241 Axiom of Infinity, 473 analytic sets, 419 Axiom of Randomness, 498 analytic truths, 169-170 Axiom of Reducibility, 336, 411 analyticity, 4, 5, 7, 8, 18, 20, 22, 24, Axiom of Replacement, 423, 431 25, 33 axiomatic method, 2 anti-foundation axioms, 466 axiomatic set theory, 279 anti-platonism axiomatization, 119, 241 mathematical, 76-86 axioms, 242 anti-realism, vii, 347 for arithmetic, 175, 178-179

704 Index for geometry, 171 boldness Frege-Hilbert dispute, 297, 298 of theories, 661 Ayer, A. J., 44, 213 Bolyai, J., 173 Ayer-Hempel-Carnap, 49 Bonevac, D., ix, 352-353, 378 Azzouni, J., 49, 85, 384, 676 BonJour, L., 182, 184 Borel, E., 412, 413 Baire Category Theorem, 413 Bostock, D., viii, 245 Baire property, 413 boundedness, 538, 598 Baire, R, 412, 413 condition, 585, 590, 593, 596, 599 Balaguer, M., vii, 349, 367, 373-374, Bourbaki, 114, 132 381,676 Bourbakism, 308 Banach, S., 381 Boyer, C. a, 198 Banach-Tarski Paradox, 415 Brady, R T., 472, 635 Bar Theorem, 326 bridges of Konigsberg, 111 Bar-Hillel, Y, 479 Brittan, G., 263 Barcan formula, 388 Brouwer, L. E. J., 39-40, 243, 320 Basic Law V, 462, 464, 465 Brown, r., 243 Bayes' theorem, 501 Burali-Forti Paradox, 410 Bayesian, 141, 500 Burgess, J., 348, 350, 357, 367, 371, Bayesian conditionalisation, 501 378,389 Bayesian net, 511 Bayesianism, x calculability, 535, 540, 555-577 Beall, JC, 49 Campbell, K., 667 Behmann, H., 543 Cantor's diagonal argument, 464 belief, 345-349 Cantor's paradox, 18, 410 belief function, 501 Cantor's theorem, vii, 402, 465 Bell, J. L., 480, 481 Cantor, Go, 14-16, 131, 198, 201, 246, Belnap, N., 641 248,300,317,359,379,381, Benacerraf, P., 42, 61-64, 66, 112, 396, 461, 464, 478, 679 113, 157, 199, 206, 352-353, cardinal characteristics, 423 355, 365, 373, 377, 384-385 cardinal comparability, 403 Bentham, J., 357-359 cardinal number, 244, 401 Berkeley models, 384, 388 Carnap, R, 44, 148, 263, 350, 375, Bernays, P., 302, 424, 431, 536, 543, 668 552, 555, 563, 568, 575, 576 Cartesian dualism, 52 Bernoulli, 504 Cartwright, N., 654 Bernstein, F., 421 Cassirer, E., 231, 249, 250, 266, 267 betting quotient, 501 Casullo, s; 184 BHK interpretation, 329 categorical concept, 233 Bigelow, J., 110,642 categorical theories, 210-212 Birkhoff, G., 480 Cauchy, A.-L., 114, 216, 364 Bishop, E. A., 311, 332 causal inertness of abstract objects, Blarney, S., 485 52, 85, 93 Boethius, 356 causal irrelevance principle, 513

Index 705 Ceitin, G. S., 329 conservative extensions, 209--210 ceteris paribus principles, 361-363 conservativeness, 366-373, 672, 671- chance, 499 673 Chellas, B., 478 constraint graph, 525 Cheyne, C., 135, 676 constructed objects, 381 Chihara, C., 45-46, 80, 84, 159, 190, constructible objects, 381, 383-385 191, 194-195,201,212,346, constructible universe, 431 359, 669 construction, 232 choice sequences, 318 constructive empiricism, see empiri- Church, A., 28, 538, 564, 568, 570- cism, constructive 572, 575, 576, 586, 593, 611, constructive mathematics, 381-382, 622 685 Church's Thesis, 323, 537, 561, 564, constructive proof, 311 569, 572, 573, 576, 586 constructivism, viii, 167, 200, 311 Church-Turing Thesis, 314 constructivist, 243 classical logic, 469 contact with abstract objects, 51-54 closed set logic, 634 contingency, 371 closure principle, 473, 475 of mathematics, 56-58, 93-94 Cohen real, 442 continuity, 215-216, 242 Cohen, P., 441 continuum, 131, 253, 260 coherence, 501 hypothesis (CH), 17, 60, 68, 77- Coleman, E., 645 78, 91, 249, 367, 399, 404, collective, 497 679 Colyvan, M., xi, 75, 134, 371 problem, 399, 433 combinatory process, 538, 577, 580 contrastive empiricism, seeempiricism, Compactness Theorem, 428 contrastive Completeness Theorem, 428 contrivance, 359 compositionality, 376 control systems, 641 comprehension, 473 convenience, 529 axiom, 109, 471, 475 conventionalism, 44, 45, 81 scheme, 463 Copi, I. M., 200 computability, 535, 537, 539, 576, 577, countable additivity, 495 594,610,611,617 counterpart semantics, 484 theory, x Cowan, T., 639 computable function, 611 Craig, W., 660 computation, 538 creative activity, 347 computor, 584-587, 612, 618 cross-entropy, 516 conceivability, 171-173, 217-218 cumulative hierarchy, 425 concept, 462 Curry's paradox, 635 conceptualism, 158 Curry, H. B., 44 conditional probability function, 494 curvature of space, 173-175 confirmation holism, 55-56 cylinder sets, 496 conjectures, 142 conjunction, 495 Dalen, D. van, 318

706 Index dark matter, 654 Duhem, P., 667, 667, 668 Darwin, C., 689 Dummett, M., 40, 167, 222, 389 Davis, M., 547, 566, 568, 572, 573 Dunn, J. M., 635, 638 De Finetti, 502 Dutch book, 501 decidability, 535, 540-544 decision problem, 428, 536, 542, 543, Easton, W., 443 549,574 Eculid,211 Dedekind, R., 9, 12, 15, 26, 42, 178, Edidin, A., 184 198, 210, 215-216, 254, 364, effective calculability, 570, 571, 575, 396,408,412,535,537,545, 587 609 effectively calculable function, 540 deductivism, 38, 45, 46, 78-80, 346- Einstein, A., 115, 119, 172, 234 348, 351, 353, 388 Eklund, M., 350 definition, 175, 367 see analysisFrege- Eleatic principle, 134 Hilbert dispute, 297, 298 elegance, 661 implicit, 297, 298 elementary mathematics, 110 deflationary fictionalism, 349-350 eliminability, 367-368 Dehaene, S., 40 empirical, 504, 505 dependence of a variable, 277 scrutability, 353, 365 Descartes, R., 168, 172, 205, 240, 246, strategy, 359 643 -based subjective probability, 506 descriptions, 368 empiricism, 104 descriptive aid constructive, 682 mathematics, 86 contrastive, 682-683 descriptive set theory, 414, 418 empiricists, 246 Dever, J., 377 Entscheidungsproblem, 539, 582, 591, Devlin, K., 680 609 diagram, 111, 139, 240 epistemic, 529 dialethism, 631 objectivity, 517 Dirac Delta Function, 640 epistemological, 497 directed constraint graph, 525 argument against platonism, 50- discernibility of the disjoint, 486, 488 61 discovery vs. invention in mathemat- epistemology, 136 ics, 93 equational calculus, 564-566, 573, 577, discrete, 467 611, 612 disjunction, 496 Equivocator, 516 property, 316 Erdos, P., viii distributed computing, 599 Erdos, P., 435 doctrine of the limitation of size, 478 Erdmann, B., 39 double extension set theory, 476, 477 Escher, M. c., 171 double set theory, 468, 476, 478 Esser, 0., 473, 476 du Bois-Reymond, D. P. G., 316 Euclid, 1-2, 10-12, 19, 24, 112, 113, dualism, 52 138,164,171,173-175,234, duality, 644 239

Index 707 Euclidean geometry, x Field, see Field, fictionalism Euler, L., 111, 144 free-range, 363, 365, 373 event space, 495 Hermeneutic, 375-377 evolutionary theory, 682 hermeneutic, 348 ex contradictione quodlibet, 631 instrumentalist, 358-360, 366 exceptionalism, 346 mathematical, 35, 46-48, 76-81, exchangeable, 502 91-94, 98 existence relative reflexive, 375-377 mathematical - as consistency, representational, 360-363, 366 298, 299 revolutionary, 348 non-spatiotemporal, 95-98 fictitious objects, 358-359 existence property, 316 Field,495 existential theories, 353 conservativeness, 671, 671-673 experience, 159-160 consistency, 672 experiential equivalence, 350, 351 critics of, 675-676 experimental mathematics, 141 entailment, 672 explanation, 529 fictionalism, 670 as unification, 661 indispensability, 653, 669-670, 679 inference to the best, 654 motivation for nominalism, 670- intrinsic, 671 671 explanatory power, 661 nominalisation, 671, 673-675 extension of a concept, 465 Platonistic methods, 671 extension of an abstract set, 465 representation theorem, 675 extensionality, 471-473, 475 Field, H., 46-47, 56, 57, 77, 78, 84, principle, 472 92, 130, 158, 196, 202, 207- external, 521-523 213, 345, 348, 353, 359, 363, externalism (concerning knowledge), 366-373, 378, 695 182~185 figuralism, 374-377 Fine, K., 478 fabulous entities, 358-359 finitary arithmetic, 239 facts finitely additive, 496 of the matter, 94-98 finitism, 303, 311, 337, 552, 557-559, physical, 85 562 purely nominalistic, 85 finitist function, 557 purely platonisitc, 85 finitist mathematics, 540, 544-550, 562 Fan Theorem, 326 finitist proof, 558 Feferman, S., 201, 337, 443, 485 finitistically calculable functions, 553 Feigenbaum's bottleneck, 507 Finsler, P., 319 Fermat's last theorem, 142, 652 Fodor, J., 667 Fetzer, J., 125 forcing, 441 Feynman, R., 680 formal sciences, 123 fictionalism, ix, 211-213, 226, 345- formal system, 577, 610 389,652 formal theory, 538, 542, 548 deflationary, 349-350 formalism, ix, x, 36, 38, 44-45, 237

708 Index game, 44 Codel's first incompleteness theorem, metamathematical, 44 249,306 Forrest, P., 642 Codel, K., viii, 26-27, 40, 41, 51, 52, Forti, M., 472 69, 15~ 159, 191-193, 200, Fosen, G., 348 243, 319, 564, 370-569, 572, foundation, 426 574,576,586,608-610,613- foundationalism, 632 618, 620, 622, 680 Fraenkel, A., 421, 424, 479 Gabbay, D., xi Fraenkel-Mostowski models, 421, 443 Gaifman, 502 Francis, G., 639 Galileo, 361 Gambling system, 498 Franklin, J., viii game formalism, 45 free-range fictionalism, 363, 365, 373 Gandy Machine, 538, 597, 599-601, freedom, 379 606 Frege structure, 465, 481 Gandy's Thesis, 596 Frege, G., 8-10,14-17,23-25,39-41, Gandy, R., 537, 538, 572, 579, 584, 44,45,62,76,82,83,89,90, 586, 593-596, 608, 622 109, 127, 129, 157, 161, 163, general recursive function, 561, 611 176-178, 188-189, 194-196, general recursiveness, 575 203-204, 207, 235, 254, 294, generic set, 442 410,416,462,464,481,536, Gentzen, G., 307, 308 548,684 geometry, 104, 239, 367, 387, 639 Fregean problem, 465, 467 Gilmore, P., 469-472, 478, 485 frequency, 497 Goldbach's conjecture, 148 fruitfulness (of axioms), 192-193 Goldman, A., 182 full conception of the natural num- Goodstein, R. L., 339 bers (FCNN), 63-68, 71-75 Gosse, E., 682, 684 full objectivity, 517 Gray, J., 265. full-blooded platonism (FBP), 35, 40- Greek mathematics, 164-166,211-212 41, 49, 59-61, 68-75, 91-94, group, 132, 642 98, 373-374 group theory, ix, 112 function /-l recursive, 567 Hajnal, A., 438 primitive, 537 Hale, B., 52, 55, 57, 184, 236 calculable, 561 Hallett, M., 248 effectively calculable, 588 Halpern, 523 finitist, 561 Hamel, G., 421 finitistically calculable, 537, 561 Hanf, W., 439 general recursive, 564, 565, 569, Hankel, H., 293 572 Hardy, G. H., viii primitive recursive, 555, 561 Hart, W. D., vii recursive, 537, 569 Hausdorff's paradox, 415 Turing computable, 579 Hausdorff, F., 412, 414, 416, 417 functionality, 636 Hawthorne, N., 346, 358

Index 709 Heath, T. L., 171 Husserl, E., 39 Heine, H. E., 293 Hyper Freqe, 475 Hellman, E., 689 hyper-continuous function, 485 Hellman, G., 42, 45, 46, 79, 346, 378, hyperuniverse, 468 684 Hempel, C., 44, 346 idea, 168, 233 Henkin, L., 437 ideal, 233 Herbrand, J., 537, 538, 540, 543-544, ideal elements, 239 549,551,556,559,564,566, idealisation, 118, 162-163, 189-191, 611 220, 360-363 hereditarily finite sets, 192-196, 201 idealism, 365 hermeneutic fictionalism, 348, 375- idealist, 243 377 identity, 235, 237 Hersh, R, 40, 693 of indiscernibles, 235, 478 Heyting, A., 39, 40, 316, 327 if-thenism, see deductivism, 45 Hilbert program, 637, 646 Ignorabimusstreit, 317 Hilbert space, 481 implication, 495 Hilbert, D., 16, 25, 41, 44, 45, 177, inaccessible numbers, 678, 678 211,237,239,337,367,369, incommensurability of the diagonal, 379,381,387,535,540,544, 113 546-548, 551, 552, 555, 575, incompleteness, 556 576, 608, 613, 615, 673-674 Incompleteness Theorem, 428, 559, Grundlagen der Geometrie, 296 610, 614 Hinnion, R, x, 468, 470-472, 474, Gi::idel, K., 306 475 inconsistency, 363~365, 374 Hintikka, J., viii inconsistent mathematics, 631, 698 Hoare, C. A. R, 125 indescribable cardinals, 439 Hodes, H., 375 indeterminacy, 355-356, 373 Hofstadter, D., 247 indiscernibility, 466 Hofweber, T., 347, 350 of locations, 481 holism, 657, 667 indispensability, 134, 197-202 confirmational, 667 of mathematics to empirical sci- moderate, see holism, semantic ence, 84-86 semantic, 667 indispensability argument Holland, R A., 263 general, 653 Holmes, M. R, 476, 477 pragmatic, 658-659 homeomorphism, 466 Quine-Putnam, 656 homoiomerous, 109, 129 scientific, 654 Honsell, F., 472 infant cognition, 137 Horgan, T., 367 infinite, 232, 241, 245 Howson, 510, 522 infinitesimals, 29, 31, 32 Hume, D., 7, 25, 114, 168~169, 233, infinity, 164~168, 190-191, 196-198, 235 203-204, 219, 363, 374, 378, Hunter, 512 387

710 Index axiom, 473 Kleene's normal form theorem, 566, infintesimal nearness, 481 569,572 inner penumbra, 484 Kleene, So Co, 324, 538, 566, 568, 576, insight, 138 582,596 instantiation, 274 Kline, M., 198 rules, 275 knowledge of mathematical objects, instrumentalism, 207-209 50-61 instrumentalist fictionalism, 358-360, Kock, s.. 481 366 Kolmogorov, A., 329, 592 instrumentalist strategy, 359 Konig, D., 434 intended objects or structures, 67- Kripke, So, 170, 378, 380, 381, 478 69,71-75,77 Kronecker,L., 314, 319, 535, 544-548 intensionality, 471 Kuratowski, x., 418, 421 internal, 521-523 Kurepa tree, 435 internal properties of mathematical Kurepa, R., 436 objects, 42-43 internalism (concerning knowledge), Loweinheirn-Skolem Theorem, 425 182-185 Lowenheim, r.. 425, 542 intuition, 232, 243 Lakatos, I., 640, 668 mathematical, 52-55 language (Godel's),192-193 definition of, 97 intuitionism, x, 39-40, 167,201,311, relativity, 508 633 Laplace, 508 intuitionistic logic, 381-382, 389 lattice, 642 invariance, 516 Law of extensions, 462, 463 invention vs, discovery in mathemat- Lebesgue measure, 413 ics, 93 Lebesgue, H., 412, 413, 679, 680 Inwagen, P. van, 48 Leibniz, G. W., 172, 216, 234, 541- 542,610 Jaynes, J., 148, 506 Lepore, Eo, 667 Juhl, Co, 367 Levy collapse, 443 Kalderon, Mo, 347 Levy, A., 438 Kamp, n., 379 Lewis, D., 38, 52, 56-59, 132, 484, Kanamori, A., x 519, 520 Kanda, «, x, 479, 481, 482, 484-488 Libert, To, x, 472, 474, 475 Kant, I., viii, 3-7, 135, 168-170, 176, Liebniz, G., 114 213, 363-364, 374 likelihood principle, 682 Kantianism, viii limitation of size doctrine, 465, 466, Katz, J., 52, 56-59 468 Keynes, J. Mo, 508, 513 Lindenbaum, A., 421 Kisielewicz, A., 476 Link, G., 346 Kitcher, P., 37-38, 83-84, 158, 179- Linsky, B., 41, 70, 684 191, 203, 225, 661 Liouville, J., 398 Kladeron, M., 345 Lobatchevsky, N. I., 173

Index 711 local observations, 467 anti-platonism, 76-86 locality condition, 538, 585, 590, 593, anti-realism, 35-98 596, 599 fictionalism, 695 Locke, J., 39 intuition, 52-55 logic, ix knowledge, vii, viii, x, xi, 50-61 alternative, 218-222 physicalism, 36-38 nature of, 213, 216-225 physics, 233 second order, 209 platonism, 40-44, 50-75 logical, 504, 505 realism, 35-98 logicism, viii, x, 40, 205, 271, 346, triviality, 635 347, 632 truth, 68 logicists, 235 mathematics Lowenheim, L., 544 as descriptive aid, 86 lower approximation, 480 as invention or discovery, 93 Luzin set, 418 in biology, 697 Luzin, N., 418 Maximum Entropy Principle, 505-510, 512-515, 523 Machover, M., 481 Maxwell, J. C., 691 Maddy's Mayberry, J., 244, 257 indispensability, 679 McCarty, C., viii mathematical fictions, 677-678 measurable cardinal, 427 mathematical practice, 678-680 measurement, 114, 133, 369 problems with indispensability, 669, mechanical 676 computability, 610 scientific fictions, 677 procedure, 538, 574, 608, 611, set theoretic realism, 653 615, 617, 619 V = L, 680 process, 618 Maddy, P., 37, 51, 53-54, 91, 158, Meinong, A., 37, 48-49, 81 179,191-196,203,225,243, Meinongianism, 37, 48-50, 81 352 Melia, J., 676 Mahlo cardinals, 415, 439 Mental, 497 Mahlo, P., 415 Mental/Physical, 497 make-believe, 345, 375-377 mereology, 115 Malament, D., 84, 367 metamathematics, 282, 304 Malitz, R. J., 472 Meyer, R. K., 633, 635, 637 Mancosu, P., 198 Mill, J. S., viii, 37, 83, 87, 91, 158, Manfredi, P. A., 184 168-359 manifold, 254, 272 Minervan constructions, 383-384 Marginal probability function, 494 minimum perceptibility, 481 Markov's Principle, 330 Mirimanoff, D., 423 Markov, A. A., 329 mixed,521 Markovian constructivism, 311 modal fictionalism, 377 marsupial constructions, 384-385 modal strategy, 359 mathematical modal structuralism, 346

712 Index model theory, 242 nominalization of empirical science, models, 496 46,84 of PFS, 484 non-deductive logic, 142 Montague, R., 438 non-Euclidean geometry, ix, 234 Moore's paradox, 658 non-spatiotemporal existence, 95-98 Moore, A., 245 non-uniqueness objection to platon- Mortensen, C., x ism, 61-69 Mostowski, A. M., 421, 437 non-uniqueness platonism (NUP) , 67- multiple reductions objection to pla- 69, 73-75 tonism, 61-69 noncognitivism, 345 notions, 296 Nozik, R., 182 naive notion of set, 461 null set, 194 naive set theory, 463, 465, 468 number, 113, 238 natural sciences, ix number theory, 462 naturalism, 657, 664 numerals, 238 Quinean, see Quine, naturalism numerical ordinals, 214-215 naturalized epistemology, 352 numerical quantifiers, 203-205, 209- naturalized platonism, 53-54 210,214 NBC set theory, 205 numerically definite comparisons, 206, necessary truth, 6 215-216 necessity, 204, 362, 372-373 of mathematics, 56-58, 93-94 object-platonism, 41-44 negation, 495 objective, 497 neighbourhood, 467 Objective Bayesian net, 510, 512 Nerlich, G., 251 objective Bayesian semantics, 522 Neugebauer, 0., 208 objective Bayesianism, 501 Neumann, J. von, 423 objective credal nets, 526 new colours, 171, 184-185 objectivity, 529 new constructivism, 311 obprogic, 524 new foundations, 472 Ockham's Razor, 87-90 Newton, 1., 114, 120, 172, 201, 211- Ockham, William of, 350, 357-358 212, 216, 223-685 ontological commitment, 346, 349-353, Newton-Smith, W. H., 201 357, 377-379, 387 Newtonian mechanics, 234 ontological parsimony, 87-90 niminalization of empirical science, 77 open models, 386-387 no-class theory, 349, 350 operations (physical vs. mathemati- nominalism, 106, 130, 158, 202-207, cal), 176, 187-191 335, 348, 350, 356, 366-373 operations research, 124 easy road to, 669 ordinal number, 244, 404 hard road to, 669 ortholattice, 483, 484 nominalistic content of empirical sci- of exact sets, 483 ence, 85-86 ostensible commitment, 351, 358, 377 nominalistic scientific realism, 85-86 Ostwald, J., 677, 681

Index 713 outcome space, 495 full-blooded (FBP), 35, 40-41, 49, outer penumbra, 484 68-75, 91-94, 98 mathematical, 40-44, 50-75 Papineau, D., 664 naturalized, 53-54 paraconsistent, 632 non-uniqueness (NUP) , 67-69, 73- paradox, 364, 631 75 paradoxical case axioms, 475 object, 41-44 paradoxical set theory, 468, 472, 474, physicalistic, 37, 53-54 478 plenitudinous, 41, 49, 68-75 parallel computation, 578, 599, 607 plenitude, 488 Paris, 523 plenitudinous platonism, 41, 49, 68- parsimony, 661 75 ontological, 87-90 plenum, 488 Quine, see Quine, parsimony pleonastic propositions, 346 Parsons, C., 43, 52, 55, 184, 191, 192, Poincare, n., 41, 200, 313, 334, 359, 198-199, 263, 478 379-381, 677, 681 Parsons, T., 378 Polya, Go, 142, 151 part-whole relation, 235 Popper, 498-500, 503 partial set, 468, 469, 471, 478 Porphyry, 356 partition property, 434 positive comprehension, 472 pattern recognition, 136 positive set, 468, 472, 478 patterns, 363 Post worker, 580 mathematical, 42 Post, Eo, 28, 536, 538, 576, 578-582, Peano arithmetic, 487 589-591 Peano, G., 178, 364, 388, 410, 416 Posterior, 507 Peirce, C. S., 360, 366, 542, 646 potential infinite, 246 Penrose, R., 639 powers of relations, 205-206 penumbral modality, 484 pragmatism, 372 perfect set property, 401, 418 predicament, 354-355 permutable models, 385 predicate calculus, 462 personalist, 497 predicative theories, 200-201, 212 PFS, 484, 486, 488 predicativism, 311 physical, 497 pretense, 367, 375 physicalism Priest, Go, 49, 631 mathematical, 36-38 Prime Number Theorem, vii physicalistic platonism, 37, 53-54 Principal Principle, 519, 520 Pigden, Co, 135 Principia Mathematica, 312, 410, 416, place selection, 498 427, 645 Plato, 1, 24, 40, 61, 157, 160-161, Principle of extensionality, 463 192, 213, 239, 356, 378 Principle of Indifference, 508-510 Platonic realism (or Platonism), vii Principle of naive comprehension, 463 Platonism, viii, 106, 107, 127, 135, Prior, 507 352-355, 369, 373-374, see probability function, 493, 495, 496 realism, mathematical probability logic, 521

714 Index probability space, 495 new foundation, 464, 477 probability theory, x ontic commitments, 665 Progic, 521 -Putnam indispensability argu- progression, 112 ment, 84-86 projective sets, 419 quantification, 659 proof, 139, 632 realism, 693 proof theory, 307 semantic holism, 667 proofs of correctness of computer pro- unapplied mathematics, 678-679 grams, 125 V = L, 680 propensity, 499 Quine, W. V. 0., 35, 37, 40, 48, 52, propositional language, 495 55-56, 76-78, 82, 84-86, 88, propositional variable, 495 90, 134, 158, 169, 191, 195- proximal Frege structure, 479, 481 199, 201, 202, 218, 226, 353, proximity space, 480 355, 359, 378, 381-383, 472 proximity structure, 479 psychologism, 36-50, 81-86 Rado, R., 436 Ptolemy, 208 Ramsey, F., 434 Putnam ranging-over idea, 277 goals of science, 656 ratio, 112 indispensability, 655-656, 658 rationalists, 246 intellectual dishonesty, 656 real number system Putnam, H., 35, 45, 76-78, 82, 84-86, axioms, 299 88, 90, 134, 158, 191, 196- real numbers, 197-202, 211-213, 396 199, 201, 220-222, 226, 346, realism, 158, 191-192 348 mathematical, 35-98, 652 Pythagorean Theorem, vii metaphysical, 652 selective, 667 quantification, 379 set theoretic, 653 quantifiers, 277 realist, 243 quantity, 104, 110 reckonable function, 538, 575-577 quantum recollection (Plato's theory), 159, 161 field theory, 367 recursive, 498 logic, 219-222, 481 function, 559 mechanics, 641 reduction, 346, 347, 351, 357, 377, nominalization of, 84 388 theory, 172 reference class problem, 499, 502 Quine(an) Reflection Principle for ZF, 438 confirmational holism, 667 regularity property, 414 continuity thesis, 666 Reichenbach, H., 218, 219 -Duhem Thesis, see holism, con- Reichenbach, R., 250, 263 firmational relation, 108, 296 first philosophy, 665 relational structures, 243 indispensability, 655 relative reflexive fictionalism, 375-377 naturalism, 657-658, 664, 664 repeatable, 496

Index 715 repeatably instantiatable, 496 semantics, 352-353, 375-377 replacement, 426 sequences of finite projections, see SFP representational fictionalism, 360-363, series, 214-215 366 set theory, x, 243, 272, 366-367 Resnik set-theoretic indiscernibility, 483 indispensability, 659 sets, 109, 132 Resnik, M., 37, 39, 41-43, 52, 55-56, settled models, 383, 386 58, 59, 63, 64, 84, 105, 110, Shanin, N. A., 329 346,367 Shannon, C., 507 Restall, G., 41, 70-74 Shapiro, S., 41, 42, 52, 58, 59, 64, 84, revisionism, 644 105, 110, 115, 210, 370-378 revolutionary fictionalsim, 348 Shepherdson, J., 437, 438 Riemann hypothesis, 142, 145 Sieg, W., x, 597-610, 621-623 Riemann, B., 173 Sierpiriski, W., 418, 421 Robertson, H. P., 265 Simons, P., ix Robinson, A., 437, 638 simple random variable, 495 Roscelin, 356 simplicity, 661 Rosen, G., 46, 85, 350, 356, 367, 376, simply infinite system, 258 378,389 simulation, 140 Rosser, B., 538, 568 single-case / repeatable, 496 Rothberger, F., 421 Singular Cardinals Hypothesis, 444 Rotman, B., 255 Singular Cardinals Problem, 444 rough set theory, 479 Skolem's Paradox, 425 Routley, R, 49, 472, 631, 635 Skolem, T., 337, 338, 424, 425, 551, rules, 238 553 Russell set, 468, 471, 472, 476, 477 Smart, J. C. C., 655, 694 Russell's paradox, xi, 17,24,295,300, Snir, M., 502 301, 410, 464 Sober, E., 367, 669, 682-689 Russell, B., 19-21, 109, 119, 193, 198, social challenges, 180-181, 183-187 203-204, 225, 250, 300, 312, Solovay, R M., 443, 444; 446 334, 346, 349, 350, 357-358, space and time, 232, 235, 369 364,368,410,416,462,464, Specker, E., 444, 477 536, 632, 643 Spurr, J., xi Stalnaker, R, 350 Sainsbury, R M., 200 standard probabilistic semantics, 522 Salmon, N., 48 Stanley, J., 367, 375-377 schema, 233 Steiner, M., 42, 52, 55-56, 690-696 Schiffer, S., 346-347 Steinitz, E., 421 scholastics, 138 Steps 1, 2, 3, and 4, 525 Schroder, E., 412, 542 strong nets, 384 sciences of complexity, 124 structural property, 109 Scott, D. S., 439, 440, 446, 484, 485 structuralism, 41-44, 58-59, 64-66, second-order logic, 462, 463 110, 114, 365, 653 second-order set theory, 366-367, 387 structuralist models, 385-386, 388

716 Index structure, 110, 114 truth by convention, 668 subitization, 137 Turing computor, 538, 597, 599, 601, subjective, 497 606 subjective / Objective, 497 Turing machine, 313, 538, 579, 586, subjective Bayesianism, 501 606, 609, 612, 622 substitutional quantification, 378-379 Turing's Thesis, 537, 564, 578, 587- success, 345 592 Summerfield, D. M., 184 Turing, A., 28, 538, 580, 584-586, sundials, x 588,590,596,608,609,611, supervenience, 346, 347, 368 617-620, 622 Suppes, Po, 367 type theoretic, 464 Suslin tree, 435 type-level, 496 Suslin, M., 419 type-neutrality, 205-206 symbolic manipulation, 133, 145 Ulam, S., 427 symmetry, 112, 115, 117 ultimate belief, 503, 518 synthesis, 232 ultrafinitist, 133 synthetic a priori, 232, 263, 264, 266 ultraproduct construction, 440 Szabo, Z., 348, 352, 378 undecidability, 610 undecidable mathematical sentences, Tarski, A., 283, 352-354, 369, 379, see continuum hypothesis, 91- 421, 574 92 Tautology, 496 undefinability theorem, 283 tertium non datur, 312 underdetermination, 522 Thagard, P., xi undermining, 520 The Maximum Entropy Principle, 504 understanding, 138 theoretic virtues, 661-662 understudy properties, 385-386 theories, 633 undogmatic, 502 theories of inconsistent mathematics, unificatory power, 661 x Uniform Continuity Theorem, 326 Third Man Argument, 135 uninstantiated universals, 106 Thomas, C. r., 293 unit-making properties, 109 Thomasson, A., 48 universals, 105 Tiles, M., viii unreasonable effectiveness of mathe- Token-level, 496 matics, 689-698 topological set theory, 466 upper approximation, 480 topology, 112, 466, 468 Urquhart, s., 366 transfinite arithmetic, 300 transfinite numbers, 399 vagueness, 360-361 transparency, 635 Vaihinger, H., 363-365, 373 tree property, 435 van Fraassen, B., 654, 665, 682, 683 Troelstra, A., 318 Velleman, D., 355 truth, 346-348, 352-353, 366, 377- vicious circle principle, 200, 313, 335 378 Vitali, Go, 421 mathematical, 68 Von Mises, R., 498, 499

Index 717 von Neumann, J., 306, 536, 550, 556, 557,653 Walton, K., 345 Wang, H., 550, 615, 616 warrants (for knowledge), 179-180,182- 185 weak counterexamples, 318 weak nets, 385 Weierstrass, K., 114, 216 Weinberg, S., 689 well-founded set, 425, 473 well-ordering, 399 Well-ordering Theorem, 405 Weydert, E., 472 Weyl, H., 336 Whedon, J., 380-382 Whitehead, A. N., 263, 312, 536 Wiener, N., 417 Wigner, E., 689 William of Ockham, 350, 357-358 Williamson, J., x Wittgenstein, L., 45, 205, 237 Woods, C., xi Woods, J., xi Wright, C., 52, 55, 57, 236, 371 Yablo, S., 46, 85, 348, 354-356, 371, 374-378, 384 Yessenin-Volpin, A. S., 337, 339 Zalta, E., 41, 48, 70, 75 Zermelo's set theory, 407 Zermelo, E., 21-22, 405, 407, 487, 653 Zermelo-Fraenkel axiom, 479 Zermelo-Fraenkel set theory, see ZF ZF, 193, 198, 199, 465, 466, 468, 471 ZFC, 478 ZFU, 366-373 Zorn's Lemma, 415