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date: 21 February 2020

(p. 811) Index

(p. 811) Index

A
absolute objects, 541
abstract algebra, 422
abstract concepts, 305–6
abstract entities, 483–84
abstraction principles, 167, 170, 182
and Bad Company objection, 180–84, 227
and impredicativity, 184–85
and Julius Caesar problem, 179–80
and naturalism, 465
and neo‐Fregean real analysis, 186
and neo‐Fregean set theory, 192–96
ontology and epistemology, 170–79
and Success by Default, 227
Abstraction Thesis, 225–26
abstractive domain, 190
abstractness, 332
abstract objects, 15, 173
abstract structures, 536
Academy (of Plato), 244n.12, 245
Ackermann, Wilhelm, 740–41
addition, 269–70, 629–30, 766
adequacy, 662, 664, 665, 666
aggregates, 64–65
algebra
abstract, 422
arithmetical, 275–77
Boolean, 378
Descartes on, 254–55, 263
early, 35
and formalism, 237
fundamental theorem of, 597
and group theory, 636
Heyting, 378
Leibniz on, 43n.40, 267n.45
Maseres on, 266
Peacock on, 272–77, 291n.85
Playfair on, 265
symbolical, 272–73, 275–77
and symbolic formalism, 263
usefulness of, 268–71
Wallis on, 258n.36, 281
algebraic language, 267
algebraic reasoning, 266
algorithmically generated sequence, 322
algorithms, 106, 108
alienated revolutionary nominalism, 520–23
alien epistemologists, 496–99
Amphinomus, 243
analogies, 644
analysis, 9, 10
analytic consequence, 657, 662
analyticity, 56–60, 65
analytic number theory, 9
analytic truth, 11
Anderson‐Belnap approach, 697, 698, 713, 717–22
antecedent justification, 224
ante rem structuralism, 22–23, 541–44, 576–77, 583, 584
antinomies, 490, 491
anti‐realism
and intuitionism, 379–82
in truth‐value, 6, 20
Apollonian theory of conic sections, 630
Apollonius, 245
application and applicability
and apriority, 29–49
of arithmetic, 137–38
canonical and noncanonical empirical, 627–32, 646–47
canonical nonempirical, 632–41
Frege's views, 137, 641, 642–43
logical, 82, 109, 641–45
mathematical, 109, 111, 212, 625–49
metaphysical problem of, 511
of numerical concepts, 141
Wittgenstein on, 109, 111
application function, 761
(p. 812) apriority
and application, 29–49
of cognition, 44–48
and implicit definition, 67
and intuition, 253, 332
Kant's conception of, 51–52, 55
of knowledge, 15, 53, 68
of math and logic, 4–5, 11, 14–15, 17, 29–49, 52–53, 78, 334, 420
of propositions, 55
and realism in ontology, 6
of space, 347
synthetic propositions, 51
and tautology, 66
Wittgenstein on, 55, 65–66, 84
arbitrary function, 104
Archimedes, 245, 257n.35
Argand, J.R., 270n.47
argument, 89, 90, 96, 657–60, 678, 687–94
Aristotle, 270, 644
division of mathematics, 238–39
on logical consequence, 654–55, 657, 663
and logically valid inferences, 672
as logician, 4, 289
on matter, 244n.13
notions of cause, 241, 246
statement of genetic ideal, 240
on syllogisms, 654–55, 659, 660, 661, 773
on viscosity, 346–47
arithmetic
applicability of, 137–38
application in physics, 631
basic principles of, 11, 13
cardinal, 629
and counting, 8, 642
first‐order, 768
foundational work in, 9
Frege on, 91, 95, 98, 166, 645
incompleteness theorem, 9, 766
induction principle, 777
Kant on, 5
language of, 137, 766
“logicist” analyses of, 81
nominalist analysis of, 64
order‐sensitivity in, 286
ordinal, 629–30, 633
Peacock on, 272, 275
pure, 645
Quine on, 647n.29
as science of multitude, 239, 240
standard inference, 508–9
standard laws of, 283–84
subject matter of, 21
truths of, 11, 130
Wittgenstein on, 99–100, 102 See also addition; multiplication
arithmetical analysis, 600
arithmetic progression, 257
assertability, 340–45, 349–50
assumption, 276, 687, 700–701, 714
atomic theory, 441, 469–70
attitude‐hermeneutic nominalism, 525–28, 532
Avigad, Jeremy, 590n.1
Awodey, S., 547, 548
axiomatic method, 293, 306
axiomatic theories, 784, 785–87
Axiom of Choice, 539, 597, 755–56, 765, 766, 793, 802
axiom of constructibility, 445, 803
Axiom of Countable Choice, 364
Axiom of Infinity, 594, 596
Axiom of Reducibility, 158, 595, 596, 601
Axiom of Solvability, 277–82, 284–89, 341, 345
axioms, 687, 738, 739
in ancient geometry, 245
and concepts, 296, 306
consequences of, 784
Gödel on, 306
and higher‐order logic, 772, 784, 787
Hilbert on, 294–95, 306
in mathematics, 462, 466, 687, 738, 739
proving theorems on basis of, 303, 772
in set theory, 801
in structuralism, 537–38 See also specific axioms
Azzouni, Jodi, 775
B
Bad Company objection, 180–84, 185, 227
Bar Rule, 609
Barwise, Jon, 774
Basic Laws of Arithmetic (Frege), 205
Bell, J.L., 199, 548
Benacerraf, Paul, 172, 484, 498, 542, 546, 581n.13
Berkeley, George, 237, 251, 263–68, 272, 626
Beth, Evert, 336, 338, 339
bipolarity, 88
Blancanus, Josephus, 247, 248
Bolzano, Bernard, 3, 11
Bonevac, Daniel, 484
(p. 813) Boole, G., 272
Boolean algebra, 378
Boolos, George
abstraction principles, 463
on Bad Company objection, 181
on Hume's principle, 169
and “limitation of size” idea, 193
on plural quantification, 806n.35
on second‐order languages, 763–64
on second‐order logic, 799
and second‐order quantification, 197, 763–64, 804
bounding principles, 404
Brouwer, L.E.J.
and assumptions of actual infinite, 619
intuitionism, 19, 295n.95, 318–36, 341, 343–47, 356, 357, 366, 376, 387
and law of excluded middle, 389, 598
on logic, 334–35
methods of proof, 282, 389
reform of mathematics, 389
and Weyl, 601
Brouwerian sequence, 323–24
Brouwer's Continuity Theorem, 326–27, 345, 365–69, 381
Burali‐Forti paradox, 591
Burgess, John
and Anderson‐Belnap tradition, 697–98
on difference between mathematical and scientific terms, 457n.38
on epistemology, 495, 509–10
and indispensability argument, 454–56
on naturalism, 437, 438, 447–49, 456
on nominalism, 483–89, 497, 505–8, 512
on science, 446, 450–52
C
Caesar problem. See Julius Caesar problem
calculation, 106, 108
Cambridge algebraists, 271n.50, 272, 299
canonical arguments, 687–92, 694
canonical language, 425–27
canonical proofs, 684–87, 692, 694
canonical second‐order consequence, 781–808
Cantor‐Bendixson theorem, 612
Cantorian set theory, 321, 596–99, 619
Cantor's Continuum Hypothesis. See Continuum Hypothesis
Cantor's diagonal argument, 334
Cantor's diagonal construction, 591
Cantor's Paradox, 183
Cantor's Theorem, 358–60, 382, 428
Cantor's theory of higher cardinals, 323
Carbone, Ludovicus, 246, 247
Cardano's Rule, 296
cardinal arithmetic, 629
cardinality, 8, 100
cardinality operator, 134, 136, 152
cardinal number(s)
addition of, 629–30
as attribute of property, 504–5
Cantor's theory of, 323
Frege definitions, 135, 143, 167, 171
and Hume's principle, 13, 152, 171, 188
universal applicability of, 96
Wittgenstein's treatment of, 96, 98, 102, 103
cardinal‐ordinal equivalence, 629
Carnap, Rudolf, 55, 65, 68, 81, 478
Carnap conditional, 176–77
Cartesian coordinate system, 33n.10, 85
Cartesian dualism, 461
categorical component, 553
categoricity, 803–4
category theory, 546–51
Cauchy completeness, 598
Cauchy proof, 597
Cauchy sequence, 364, 365
causal attention, 329
causal laws, 648n.33
causal reasoning, 648n.33
causal theory, 484
cause, 237, 241, 244n.14, 246
Cayley, A., 261
certainty, 30, 78, 305
Chihara, Charles, 18–19, 419, 421, 422, 473, 570, 592
choice sequences, 366
Church, Alonzo, 3, 679, 775
Church's Thesis, 377, 662
circle, 648
classes, 131–34, 150–52, 496, 592
classical mathematics. See CR system; mathematics, classical
classical mechanics, 627n.15
classical reductio, 389, 391, 396
Clavius, Christopher, 247, 248
(p. 814) closed argument, 688–91
coefficients, 254
Coffa, Alberto, 11, 13, 68n.34, 656
cognition, 44–48, 52–53
cognitive verbs, 110
Cohen, Paul, 497n.35
coherence, 490
Coherence Axiom, 542, 546
collections, 65, 193, 549, 628, 642, 759
Collegio Romano, 247
combinatorial set, 797, 801
combinatorial subset, 802nn.29–30
common difference, 257n.33
common sense, 41, 42, 43, 46
commutativity, 285–86
compactness, 766, 767
compactness theorem, 8
completeness, 653, 665, 669, 766, 768, 807–8
completeness axiom, 776
completeness principle, 769, 776
completeness scheme, 768, 776
completeness theorem, 84n.10, 382, 653, 666, 668, 773
complex numbers, 269–71, 285, 286, 296, 297, 307
compositionality, 229–32, 549
compositional semantic theories, 20
compound sentence, 686
comprehension principle/axiom, 194, 783
comprehension scheme, 755, 765, 766, 769
computability, 376–78, 651
computers, 748
conceivability test, 63–64
concept(s), 9, 10, 81
abstract, 305–6
construction of, 292
denoting, 154–59
of direction, 172
extensions of, 12, 180, 496
finite, 152
formation, 296
Frege on, 89, 134, 136, 149–52
Hilbert on, 294–95
and language, 67
noncontentual role, 297
of number, 141, 153, 174–75, 179–81, 643
of set, 194
sortal, 179, 181, 229
and thinkable predication, 200
Wittgenstein on, 92–93, 101
concept possession, 76, 77, 109
conceptualism, 61, 62
conceptual set, 797
concrete definition, 59, 68
conic sections, 244n.12
connectedness, 346n.56
connotation, 60
consciousness, 329
Consequence‐Logicism, 203–5, 208–12, 215, 222
consequence relation, 781–83, 785–89, 805
conservation laws, 637
consistency, 304–5
constitutive completeness, 294
constructibility quantifier, 18, 499, 501
Constructibility Theory, 499–501, 506–12
constructible hierarchy, 603
constructional component, 241
constructive ideal, 240, 246
constructivism/constructivity, 117, 329–35, 671–94
content
in analytic sentences, 66
Kant on, 52
mathematical, 334
and paraphrase‐functions, 212
recarving, 216–23, 225, 231–32
of thought, 67
within framework, 53
within linguistic scheme, 55
content‐hermeneutic nominalism, 523–25
contentual reasoning, 272
context, 107, 784
Context Principle, 197, 199
continuity theorem. See Brouwer's Continuity Theorem
continuum, 8–9, 319, 321–25, 343, 345–47, 644
Continuum Hypothesis, 790–92, 795, 798
contradiction, 62, 491n.15, 660, 733, 749n.13
conventions, 53, 68, 70, 71
corollaries, 738
correctness, 664–65
correctness principle, 403
correspondence conception, 54–55
counterargument, 657, 658
counting, 141, 190, 239, 629, 642
couples, 502–3
Craig Interpolation Theorem, 734, 735
(p. 815) criterion of application, 179
criterion of identity, 135
Critique of Pure Reason (Kant), 348, 351, 626
CR system, 708, 711, 716–17, 723–24
crystallography, 636
cumulative deductive progress, 702–4
cumulative theory of types, 603
Curry, Haskell, 17
Cut Abstraction, 187, 188, 195
Cut Elimination Theorem, 707, 711, 739
Das Kontinuum (Weyl), 599, 601
D
Dedekind, R., 130, 140–41, 152–54, 161–62, 251
Dedekind abstraction, 545
Dedekind completeness (continuity), 598
Dedekind infinity, 160–61
Dedekind‐Peano axioms, 152–53, 169, 170, 210–11, 213, 228, 363, 465, 559, 804n.32
Dedekind sections, 597–98
Dedekind sets, 362
Dedekind Way, 186–89, 191
deduction
automated, 722
of conclusions from premises, 462
cumulative deductive progress, 702–4
of early mathematicians, 422
Frege's use of, 645
in logic, 57, 699, 701–2
in logical consequence, 24, 660
in mathematics, 6, 641
natural, 714–15, 716
as producer of knowledge, 60–61
in sentences or formulas, 687
unrestricted transitivity of, 706
Deduction Theorem, 708
deductive calculus, 772, 782, 792–93, 808
deductive consequence, 644n.24, 661, 664, 667, 669
deductive system, 652–54, 661–62, 666–68, 752, 754–57, 765, 772, 773
deductive validity, 653
default epistemology, 226, 228
definability theory, 603–6
definition, 7, 58–60, 237, 591
definition by property, 244n.15
Democritus, 257n.35, 648
demonstration, 247, 248, 255
De Morgan, A., 271n.50
denotation, 60
denotation function, 757, 758, 761
denoting concepts, 154–59
derivations, 371
de Rouilhan, Philippe, 592
Descartes, René, 3, 4, 30, 43, 438
and algebra, 255, 263
on ancient geometers, 281
on essence of material substance, 35–39
on extension, 35–39
on God, 37, 40, 626
and intuition, 252, 258
and invariantist ideal, 254, 255, 258
and representational methods, 33–34
on space, 38–39
“wax argument,” 37
description, 155–56, 158
descriptive criticism, 729, 731
designative occurrence, 146
determinacy, 797–800, 807–8
deviance, 345
Dewey, John, 340n.47
dialectical consequence, 747
dialectical valuation, 746–47
dialetheism, 696, 749n.13
difference equations, 644
differential equations, 644–45
Dilution, 704–5, 711, 714, 724
Dilution Elimination Theorem, 710
dimensionality, 33, 285–86, 648
Direction Equivalence, 167, 172
discharge of assumptions, 714
discrete mathematics. See separable mathematics
Disjunctive Syllogism, 697–98, 707, 732–33, 740–44, 748
disjunctive weakening, 732–33
distinctness, 179
Distributivity, 719
Diversity of the Dissimilar, 571
Division Problem, 306–9
domain, 652, 757, 759, 796
domain of discourse, 751
double negation elimination, 399
double valuation, 745–46
downward Löwenheim‐Skolem theorem, 209n.10, 765, 768, 769, 784
du Bois‐Reymond, Emil, 280, 283
du Bois‐Reymond, Paul, 280, 283, 366
Duhem, Pierre, 414–15, 419, 421, 440, 475
Duhem‐Quine problem, 700
(p. 816) Dummett, Michael
coining of linguistic turn, 11
on Field, 646
on Hume's principle, 184–85
on indefinitely extensible domains, 157n.22, 195
on induction, 777
intuitionism, 20, 21, 318–20, 342–44, 379, 387–91, 395–98, 400
on meaning, 682n.6
on radical conventionalism, 70
on sortal concepts, 179
on truth conditions, 680
E
Einstein, Albert, 630, 648n.33
electrons, 638
elementary formulas, 730
Elements of Geometry (Euclid), 4, 250
eliminations, 391, 684–85, 754, 765
eliminative induction, 63
eliminative structuralism, 22–23
empiricism
canonical and noncanonical empirical applications, 627–32
conflict with rationalism, 5
criticisms of, 69–73
historical and philosophical context, 51–56
with holism, 413
of Mill, 14, 51, 53, 55–65, 71–72, 626, 642, 647
orientation to knowledge, 4
empiricist formalism, 299–302
enumerative induction, 63
epistemically reductive philosophy, 109
epistemic gain, 706, 711, 723–24
epistemic truth, 747
epistemology
abstraction principles, 170–79
of Brouwer, 319
default, 226, 228
and logical consequence, 659–60
and mathematical intuition, 331–32
of Mill, 72
naturalized, 15, 54, 494–95
normative and descriptive, 417–18
of Quine, 72, 417–18, 439, 446, 772
and truth, 681 See also knowledge
equations, 97, 104, 106–8, 110, 141, 254, 297, 636, 644–45
equinumerosity, 12, 97, 108
equivalence, 274
Erlanger Program, 262
Etchemendy, John, 675n.3, 676, 677
Euclid, 245n.17, 247
Euclidean geometry, 4, 31, 58, 242, 423, 640n.20, 785
Eudoxus, 240n.6, 257n.35
Examination of Sir William Hamilton's Philosophy (Mill), 54, 61, 62
excluded middle, 7, 19–21, 334, 335, 383, 387, 389, 397, 401, 406, 408–9, 598
existence, 14, 68, 71, 333–34, 427–28
existence theorems, 516–17
experience, 42, 45–46, 47, 71
extendability, 540
extension, 12, 13, 35–40, 61, 82, 137, 138, 180
external relations, 84
extramathematical linguistic context, 107
extramental reality, 39–40
F
facts, 55, 644n.24
faithful models, 765, 769, 771, 776
fallacy of relevance, 721
Feferman, Soloman, 497n.35, 549–50, 602
Fermat, Pierre de, 30
fermions, 640
Feynman, Richard, 645
fictionalism, 17, 473, 643
Field, Hartry, 480
fictionalism, 17, 473, 643–44
and Hume's principle, 176–77
nominalism, 17–18, 175, 182, 484, 643–45
on Platonism, 172, 474, 645, 646
figures, 244, 255–56, 262–63
Fine, Kit, 177n.21, 501n.42
finite cardinal structures, 545n.10
finite concepts, 152
finite description, 322
finite lines, 40n.30
finite mathematics, 323
finite model property, 375
finite numbers, 171, 629
finitism, 118, 618–19
finitude, 768
(p. 817) First Lewis Paradox. See Lewis's First Paradox
first‐order abstraction, 190
first‐order languages. See language(s), first‐order
first‐order logic. See logic, first‐order
first‐order model theory. See model theory
first‐order variables, 751, 763, 796
fixed collection, 759
fleeing properties, 324
formalism, 16–17, 236–309
challenges to, 299–309
complications concerning, 282–87
creativist component, 237
emergence of, 249–63
empiricist, 299–302
framework of, 236–38
of Hilbert, 287–99, 300n.102, 301, 303–6, 319, 335, 491
of Leibniz, 42–43
and logicism, 592
of Peacock, 271–77
and retreat from intuition, 252–62
of simple theory of types, 594
symbolic, 263–99
traditional viewpoint, 238–46
formality, 787
formulas, 302, 303, 687, 730, 734
foundational work, 9
Foundations of Arithmetic (Frege), 205, 216
fourth‐order variables, 754
Frege, Gottlob, 3, 64, 78
account of number, 153
account of reasoning by mathematical induction, 138–40
analysis of natural numbers, 12, 134–37
ancestral constructions, 99, 138–39, 142, 143
and application of mathematics, 137, 641, 642–43
on arithmetic, 91, 95, 98, 166, 645
challenges to formalism, 299–305
on combinatorial sets, 797
on concepts, 89, 134, 136, 149–52
context principle, 90
contributions and errors of, 101
definition of numbers, 643n.23
dependence relations among definitions, 142–43
distinction between function and argument, 89, 90
hierarchy of functions and senses, 149
Hilbert's criticism of, 251
and Hume's principle, 12, 13, 97n.23, 135–37, 185
interest in irreducible case, 297n.98
and mathematical objects, 511
on necessity, 84
and numbers as objects, 170–71
philosophy of math, 166–70
Platonism, 166–67, 170, 179
on quantity, 189–90
as realist in ontology, 11
on symbolical reasoning, 298
theory of classes, 131–34, 150–52
theory of thoughts, 149–52
on truth, 82, 83, 85, 112
and Wittgenstein, 80, 81, 91, 95, 97, 101, 102 See also Kant/Frege conception; neo‐Fregeanism
Frege‐Russell definition of cardinal number, 135
Frege's constraint, 191
Frege's Theorem, 169n.9, 170, 182, 187, 191, 199
French, Steve, 639
Frend, William, 269
Friedman, Harvey, 703n.1
full comprehension axiom, 609
full mathematics, 325–28
full models, 760, 766
function/argument distinction, 89, 90, 96
function(s)
denotation, 757, 758, 761
descriptive, 158
Frege's hierarchy of, 149
in full models, 766
material, 93
mathematical notion to structure of sentences, 82
paraphrase, 206–8, 212, 213, 214n.20, 217, 219
predicative, 158
propositional, 149, 157
successor, 11
in theories, 14
value ranges of, 168
Wittgenstein on, 92, 93, 97
(p. 818) Functions of a Complex Variable (Pierpont), 307
function variables, 752–53, 760
functors, 549
Fundamental Theorem of Algebra, 279, 284, 289, 296, 297
G
Galileo, 248, 648
Galois, Evariste, 636
Galois group of an equation, 636
game formalism, 16
Gauss, C., 269–71, 279n.64, 286, 296, 307, 309
Gell‐Mann, Murray, 639
general cubic, 297
generality
of construction of infinite system, 153
and denoting concept, 154
of form, nature, and value, 274
Wittgenstein's views on, 84, 90–92, 95, 99–100, 102, 104, 105, 109
general proof, 686
General Theory of Knowledge (Schlick), 55, 58
General Theory of Relativity, 630, 648n.33
Gentzen, Gerhard
consistency proof for arithmetic, 601
Cut Elimination Theorem, 707, 711, 739
on inferences, 684, 687
on introductions and eliminations, 391, 684–85, 689, 692
natural deduction presentation, 370
negative translation theorem, 371
sequent calculus, 738
geometric cognition, 44, 46, 47
geometric figures, 255–57, 262–63
geometry
axioms of, 42, 58, 66, 68, 304
classical, 237, 255, 262, 265, 269
construction in, 244–46
constructive ideal in, 241
Descartes’ methods, 30, 34, 35
Euclidean, 4, 31, 58, 242, 423, 640n.20, 785
Kant on, 44, 48
Maseres on, 266
nominalist analysis of, 64
non‐Euclidean, 347n.66, 422, 630
Pasch on, 250–51
Plato on, 243
Playfair on, 265
Poncelet on, 265
problems in, 35
projective, 258–62
as science of magnitude, 239, 240
of space‐time, 640
symplectic, 637n.14
for two‐dimensional vector space, 640n.20
Global Reflection Principle, 211, 215
God, 37, 40, 483, 626
Gödel, Kurt, 3, 112, 512
axiom of constructibility, 445, 803
challenge to formalism, 305–6
on combinatorial subsets, 802nn.29–30
completeness theorem, 84n.10, 653, 666
and intuitionism, 335, 338, 341, 373, 375, 399
negative translation theorem, 371
on sets, 497n.32, 498n.36, 615
theorem of unprovability of consistency, 601
and vicious circle principle, 592 See also incompleteness theorems
Goldbach's conjecture, 328
Goldfarb, Warren, 596–97
Goodman, Nelson, 647
Goodwin, William, 506n.43
Gottlieb, Dale, 484
grammar, 68, 69, 87, 90, 101, 107, 108, 110
gravitation, 630, 632, 648
gravitational field theory, 646
Gregory, D.F., 271n.50
group, 536, 647
group theory, 625, 636–41
H
hadrons, 639
Hahn, Hans, 249–50
Hale, Bob, 12, 463
Hallett, Michael, 597
Hamilton, W.R., 285–86
Hankel, H., 286
Hankel's Theorem, 286, 287
Hardy, G.H., 631n.7
Hebb, Donald, 496n.29
Heine, Heinrich Eduard, 300–301, 303
Heine‐Borel covering theorem, 300n.103
Heisenberg, Werner, 638
Hellman, Geoffrey, 473, 590n.1
Henkin consequence, 760, 783, 784, 795, 808
Henkin construction, 766n.5
Henkin interpretation, 783
Henkin model, 759, 760, 762, 765–66
Henkin satisfaction, 760, 762
(p. 819) Henkin semantics, 654, 759–60, 763, 769–71, 773, 774, 776, 777, 783, 803
Henkin validity, 760, 762
hermeneutic nominalism, 486, 487, 505, 509, 517, 523–28
Heyting, Arend
intuitionism, 318, 319–20, 336, 339–44, 379, 387
on law of excluded middle, 389, 406
on mathematical objects, 19
and mathematical sentences, 380–81
Neighborhood Theorem, 366
Heyting algebra, 378
Heyting Arithmetic, 337, 338, 370, 376, 377, 614
higher‐dimensional quantities, 286, 288
higher‐order abstractions, 167
higher‐order identity, 760
higher‐order logic, 170, 751–77
axiomatic theories, 784, 785–87
and canonical second‐order consequence, 781–808
categoricity, 803–4
completeness and determinacy, 807–8
deductive system, 754–57
and existence of structures, 794–95
and formal languages, 752–54
for Frege's Theorem, 199
logical versus iterative sets, 796–97
and logicism, 206–7, 209–14
metatheory, 764–69
model theory, 757–64
ontology of, 197, 198
plural interpretation, 804–7
plural quantification, 762–64
Quine on, 197, 770–72
reconsidered, 781–808
standard semantics, 757–59, 762–63, 766–69
and subsets, 796, 800–804
higher‐order nonlogical constants, 754
higher‐order variables, 763, 770
Hilbert, David, 3, 376, 433, 601, 619
axioms for Euclidean geometry, 58
Axiom of Solvability, 278–82, 341, 345
border between finitary and nonfinitary methods, 308
criticism of Dedekind and Frege, 251
on intuitionism, 382–83
metamathematics, 106
on proof, 282–83
Hilbert space, 637nn.15–16, 640n.20
Hilbert's Program, 299
Hippasus of Metapontum, 239n.5
Hippocrates, 257n.35
Hodes, Harold, 212, 213
holism, 444n.15
basic idea of, 414–5
with empiricism, 413
and logic, 418–19
objections to, 419–23
of Quine, 414–23, 434, 440, 443, 494n.26
and web of belief, 414–23
Holism‐Naturalism Indispensability Argument, 430–32
homogeneity, 593
homomorphism, 549, 627
Hume's principle, 12–13, 97, 98n.24, 135–37, 152, 168–91, 196, 199, 463, 465, 478–79
Husserl, Edmund, 305, 329n.25
hyperarithmetic, 604, 612
Hyperarithmetic Comprehension Rule, 608
hypotheses, 419–22, 462, 466, 511n.48, 687, 700
hypothetical component, 553
hypothetical proof, 686, 693
I
ideal propositions, 289n.80
ideal reasoning, 298
identity, 96n.19, 97, 100, 135, 179, 333
identity‐isomorphism, 231–32
identity of structural indiscernibles, 544
identity‐predicate, 231–32
identity relation, 770
image, 78
imaginary numbers, 102, 269–70, 296, 297, 307–8
imagination, 41, 45, 61, 244–45
implicit definitions, 56, 58–60, 67, 68, 71
impredicative definitions, 7
impredicative propositional function, 157
impredicative separation, 802
impredicativity, 184–85, 591, 592
incommensurables, 239, 240, 346
incompleteness
of mathematical objects, 565–76
Resnik on, 565–76, 578
Shapiro on, 576–86
(p. 820) incompleteness theorems
and Consequence‐Logicism, 208, 210
and formalism, 309
Gödel's first, 169n.7
and Hilbert's Program, 299
on intuitionism, 372–73
and logicism, 463
and proof theory, 683
and role of truth in mathematics, 112
and second‐order logic, 559, 653
on set of arithmetic truths as noneffective, 9
and standard semantics, 766, 767
on true nonprovable propositions, 70
inconceivability, 62, 63–64
indecomposability, 358–60, 368
indefinitely extensible domains, 157n.22, 195
Indeterminacy Challenge, 228, 229
Indispensability Argument for Mathematical Objects, 430, 493, 494
Indispensability Argument for Mathematical Realism, 14, 17, 429–32, 444, 454–56
indispensability arguments, 613–14
induction, 61, 63, 104, 138–40, 142, 159, 633
induction axiom/principle, 98, 609, 767, 769, 776–77
induction scheme, 609
inequations, 141
inertia, 415
inference
Frege on, 82, 83, 302
gap‐free chains of, 772
logical, 462, 672
Mill/Kant, 67
rules, 463, 685
in sentences, 687
standard arithmetical, 508–9
validity of, 250–51, 675, 693
Wittgenstein on, 66, 109
inferentialism, 390
infinite sequences, 321–22, 325
infinite sets, 321–22, 323, 801, 802
infinite systems, 152–54
infinity, 619, 803
discrete, 331
of numbers, 136, 152–54, 160–61
of objects, 199
Poincaré on, 598
Russell's axiom of, 96n.19, 103
Wittgenstein on, 105
instrumentalism, 477n.18, 486, 626
intellect, 38
intelligible matter, 244n.13, 248
internal mathematics, 357, 369–79
internal relations, 84–86, 90, 101
interpretational semantics, 676
interpretation function, 757
interpretations, 652–54, 662, 663, 673, 773, 782–83
In the Light of Logic (Feferman), 599
introductions, 391, 684–85, 692, 754, 765
intuition
in conceptualism, 61
decline of, 249–52
and deductive systems, 661–62
Descartes on, 252
Kant on, 44, 48, 141, 252–53, 262, 292
logical positivists on, 53
Maseres on, 266
mathematical and epistemology, 331–32
Mill on, 63–64
non‐empirical, 292
perceptual, 321
and proofs, 729
retreat from and formalism, 252–62
Wittgenstein on, 99
intuitionism, 19–21
and anti‐realism, 379–82
of Brouwer, 295n.95, 318–36, 343–45, 356, 366, 376, 387
contemporary, 343–44, 357
of Dummett, 318, 344, 387
first act of, 330, 331
formal logic and internal mathematics, 369–79
of Heyting, 318, 340, 343, 344, 366, 379–82, 387, 406–7
as house divided, 344–45
logical domains, 357–62
in mathematics, 322–28, 330–32, 356–83
models and modality, 373–75
and naturalism, 473n.12
negative doctrines, 334–35
ontology, 333–34
phases of, 318–20
phenomenology of, 329–35, 344
and philosophy, 318–51
reconsidered, 387–409
second act of, 330–31
technical side of, 344, 345–49
and unknowable, 350–51 See also IR system
(p. 821) intuitionistic logic
connectives, 394–95
disagreements on meaning, 394–98
disagreements on preservation, 398–400
disagreements on truth, 400–402
of Dummett, 342–43, 388–89, 395–98, 400
epistemic argument for, 388–90
formal system, 336–39
of Heyting, 320, 336, 339–42, 389, 394
and Kant, 347–49
and metalogic, 336–39
philosophy of, 336–43
proof‐theoretic argument for, 390–92
semantic properties, 338
syntactic properties, 337–38
invariantist ideals, 238, 253–58, 262, 263
inventions, 108, 112
inverse square law, 632
investigation, 255
irrational numbers, 34
irrational quantities, 240
irreducible case, 296n.97, 297n.98
irreducible representation, 638–39, 647
IR system, 708, 711, 712–15, 717, 723–24
isomorphism property, 663
isotopic spin, 638–39
iterative sets, 796–97
J
Jesuits, 246–47
judgment, 61, 72, 86n.12, 87, 303
Julius Caesar problem, 12, 130, 137–38, 179–80, 185
K
Kant, Immanuel, 5, 32, 38, 40, 138, 263
conception of apriority, 51–52, 55
critical epistemology, 280
doctrine of judgment as synthesis, 86n.12
on experience, 61
and intuition(ism), 44, 48, 130, 141, 252–53, 262, 292, 346, 347–49
on inverse square law, 632
on logic, 85
philosophy of math, 44–49
on schematization, 141
transcendental idealism, 31, 44, 348, 349, 626
transcendental realism, 348, 350, 351
and truth, 112
and Wittgenstein, 78, 79, 85
Kant/Frege conception, 56–58
Kant/Mill conception, 56–58, 60, 65–67, 71
Kepler, Johannes, 630
Kitcher, Philip, 473, 510
Kleene, S., 376–77, 381, 382, 604, 620
Klein, Felix, 262
knowledge, 3–4, 51–53, 55, 236
apriority of, 15, 53, 68
of cause, 237
geometrical, 243
of infinity of numbers, 152–54
Kant on, 52
limits on, 280
logical, 61
and logical consequence, 659–60
mathematical, 41–43, 47, 61, 241, 247, 248, 292, 331
metalinguistic, 66
metaphysical, 41, 43, 46, 47
“mirror” idea of, 52, 62
naturalized epistemology, 15, 54
Plato on, 243, 642
produced by deduction, 61
Wittgenstein on, 79–80 See also epistemology
König's paradox, 591
Kreisel, Georg, 6, 614–15, 619, 620, 665, 703n.1, 775–76
Kripke, Saul, 336, 338, 339, 341, 348, 633
Kripke models, 373–75, 376, 378
Kronecker, Leopold, 282, 291, 300, 308–9, 322n.8, 598, 619
L
Lambert, J.H., 250, 252
language(s)
algebraic, 267
of arithmetic, 137, 766
Berkeleyan conception of, 263–68
Boolos on, 806n.35
Brouwer on, 335
canonical, 425–27, 789
in communication, 20
consequence relation, 788
Dummett on, 342
equation as rule of, 110
first‐order, 209n.10, 665–66, 752, 761, 765, 769–71, 773, 774, 776, 782
Hilbert on, 291
linguistic rules, 67–68
mathematical, 17, 342, 640, 774
natural, 661–66, 669
(p. 822)
necessity and a priori knowledge in, 11
nonrepresentational role in mathematical reasoning, 237
philosophy of, 490
pictures, 87, 93, 107
second‐order, 653–54, 751, 752, 754, 757, 763–73, 775–77, 782–84, 789, 796, 799, 803
Wittgenstein on, 66, 71, 76, 78, 80, 86, 99, 103 See also semantics; sentence(s)
language‐games, 78, 101, 110
Language‐Logicism, 203–5, 208–10, 212–13, 216, 222, 226
Laudan, Larry, 511
Lavine, Shaughan, 777
Law of Continuity, 259n.37
law of excluded middle. See excluded middle
law of multiplicative commutativity, 285–86
law of noncontradiction, 42, 443
law of the excluded third. See tertium non datur
Law of the Permanence of Algebraic Forms, 274n.56
learning, 626
least upper bound, 7
Lebesgue measure, 611, 612
Leibniz, Gottfried, 3, 30, 32, 40–43, 45, 259n.37, 267n.45, 268, 677
Leibniz's Law, 571
lemma, 702–3, 738, 739
levels, 753–54
Lewis, David, 495, 741, 806n.35
Lewis's First Paradox, 696, 704–5, 706–10, 714
limitation of size, 193
limitative theorems, 8
Lindström, Per, 769, 789
lines, 245
linguistic anti‐realism, 320
linguistic context, 107
linguistic deficit, 656
linguistic rules, 67–68
linguistic turn, 11, 67
lists, 65
Locke, John, 252, 649
logic
as analytic, 57
application of, 82, 109, 641–45
apriority in, 4–5, 11, 52–53
background of, 748
basic principles of, 4
Brouwer on, 334–35
classical laws of, 288, 290
deductive, 701–2
definitions of, 651
descriptive criticism of classical, 729, 731
failure of classical, 749
first‐order, 206–8, 426, 539, 653, 665, 752, 769, 774, 783
formal and internal mathematics, 369–79
Frege on, 799n.26
and holism, 418–19
as instrument of reasoning, 699
internal, 377
intuitionist attack on classical, 729
intuitionistic, 320, 336–43
of mathematics, 42, 772
as modeling, 402–5
modern, 641
and naturalism, 450
neo‐Fregean, 196–200
“paradoxes” of classical, 727–30
and paraphrase, 206–8
and philosophy of math, 3–24
philosophy of, 8, 23–24
Quine on, 442, 443, 450, 465
requirements of, 699–702
second‐order, 12, 131, 144, 178n.22, 186, 228, 539, 559, 654, 763, 769–75, 800, 803, 806
as synthetic, 60–61, 130
logical consequence
canonical arguments, 687–92
canonical proofs, 684–87
canonical second‐order consequence, 781–808
constructivist viewpoint, 671–94
epistemic matters, 659–60
Field on, 644n.24
forms, 657–59
intuitive notion of, 88, 398, 403
and logics, 392–94
and meaning of logical constants, 679–81
modality, 654–56
and necessity of thought, 677–79
as normative notion, 660
and proofs, 682–87
(p. 823)
proof theory and model theory, 651–54, 661–69, 773
rigorization of notion of, 83
semantics, 656–57
Tarskian analysis of, 675–77
and theory, 464
and ultraholism, 465
variations of specific contents, 672–74
logical constants, 679–81, 682
logical equivalence, 88
logical falsehoods, 700–701
logical grammar, 68
logically unrestricted quantifiers, 213–15
logical monism, 393, 409
logical objects, 152
logical pluralism, 388n.1, 393
logical positivism, 65–69, 412
analyticity in, 56–60
criticisms of, 69–73
historical and philosophical context, 51–56
Tractatus as bridge to, 81
logical sets, 796–97
logical syntax, 68, 80
logical terms, 652, 658–59, 785
logical truth, 674, 676, 701
logical validity, 83
logicism, 11–13, 129–62, 412
assessment of, 206–16
Consequence‐Logicism, 203–5, 208–12, 215, 222
content recarving, 216–23, 225, 231–32
of Dedekind, 130, 140–41, 152–54, 161–62
definition of, 203–5
first‐order view, 208, 215
and formalism, 592
higher‐order view, 209–14, 215, 216
Language‐Logicism, 203–5, 208–10, 212–13, 216, 222, 226
and naturalism, 462–63
neo‐Fregean program, 223–29
partial, 559
Platonist version of, 166–67
Poincaré versus logicists, 596–99
of Russell, 79, 81–82, 85, 86, 130, 154–59, 161–62
Truth‐Logicism, 204–6, 208–10, 213–15
in twenty‐first century, 166–200
logics, 392–94
Löwenheim‐Skolem theorems, 8, 765–69, 774, 784
Lyndon Interpolation Theorem, 735
M
Mac Lane, S., 549, 551
MacLaurin, C., 268–69
Maddy, Penelope, 431, 432, 467–72, 496n.29, 498n.36, 518–20, 613
magnitude (quantity), 42, 189–91
definition of, 238–39
Descartes on, 36–37, 39, 254
of geometrical figure, 256
irrational quantities, 240
mathematical notion of, 29–30
multidimensional, 285–86
representational methods, 32–34, 35n.13
science of, 239, 240
manifold, 637n.15
mapping, 78
Markov's Principle, 377
Martin‐Löf, Per, 685n.8
Maseres, Baron Francis, 266, 269
mass, 641, 648
material objects, 35–37, 40
mathematical cognition, 44–48
mathematical deviance, 345
mathematical induction, 138–40
mathematical knowledge, 41–43, 47, 61, 241, 247, 248, 292, 331
mathematical language, 17, 342, 640
mathematical notation, 645
mathematical objects. See object(s), mathematical
mathematical realism, 429–32, 444
mathematical reasoning. See reason and reasoning, mathematical
mathematical truth. See truth, mathematical
mathematics
analytic principles of, 11
application and applicability, 29–49, 111, 212, 625–49
apriority and necessity in, 4–6, 11, 14–15, 17, 29–49, 52–53, 78, 334, 420, 656
Aristotelian division of, 238–40
classical, 319, 328, 334, 507
as “confirmed,” 14
definition of, 238
demonstration in, 247, 248
discovery in, 108
as following of meaningless rules, 16
foundations of, 641
(p. 824)
full, 325–28
Hilbert on, 289–99
incoherent ideas in, 626
internal, 357, 369–79
intuitionistic, 322–28, 330, 331–32, 356–83
logic of, 772
meaning in, 9
mixed, 31n.3
nominalistic reconstructions of, 492–505
phenomenology of, 330–31
pure, 30, 31n.3, 35, 39–41, 641, 642, 645
relative notions of, 8–9
representational methods, 32–35
separable, 322–23, 346
standard, 626
as synthetic, 60–61, 130
matter, 244n.13
maxim of minimum mutilation, 441, 443
maxim of narrow analysis, 722–23
McCarty, David, 387
McKinsey, J., 378, 399
meaning, 77, 656, 679, 681n.6, 682, 684
means, 239
measurement, 107–8, 189–90, 239
mechanics, 637
mechanism, 9
Meditations on First Philosophy (Descartes), 35–36, 438, 626
Menaechmus, 243, 244
metabasis, 258
metalanguage, 103, 107
metalinguistic knowledge, 66
metalogic, 70
metamathematics, 106, 298, 651–54, 661–69
metaphysical knowledge, 41, 43, 46, 47
metaphysical problem of applicability, 511
metatheory, 764–69, 771
Method of Exhaustion, 257n.35
methodological naturalism, 460–61, 462, 473
metric geometries, 261
metric space, 368
Meyer, R.K., 698
Middle Ages, 246–49
Mill, John Stuart, 71–72
analyticity in, 56–60
applicability of math to nature, 626
empiricism, 14, 51, 53, 55–65, 71–72, 626, 642, 647
on math and logic, 52
naturalized epistemology, 54 See also Kant/Mill conception
mind‐body problem, 626
minimal anti‐nominalism, 485
mixed line, 245
mixed motion, 245
modal operators, 426
modal statements, 18–19
modal structuralism, 23, 551–60
models, 652, 757, 770, 773, 783
model‐theoretic consequence, 652–53, 661–63, 665–69, 772–73
model‐theoretic semantics, 663, 664, 666, 757–64
model theory, 8, 83, 651–54, 661–69, 757–64, 770, 773
moderate realism, 485, 486
monads, 40–41
monism, 85
Morgenbesser, Sidney, 627
morphism, 549
motion, 245, 346, 415
Mourey, C.V., 270n.47
multidimensional quantities, 285–86
multiplication, 270, 632–35, 766
multiplicative commutativity, 285–86
multitude, 42, 239, 240
N
Naive Comprehension Axiom, 132–33
natural deduction, 714–15, 716
naturalism
distinguished from realism, 54
forms of, 437–58
and logic, 450
of Maddy, 468–72
and mathematics, 453–54
methodological, 460–61, 462, 473
Millian, 54, 63
ontological, 427–28, 460, 461, 472–73
reconsidered, 460–80
and science, 446–50
naturalized epistemology, 15, 54, 494–95
naturalized revolutionary nominalism, 518–20
(p. 825) natural light of reason, 37–38, 40, 42, 43
natural number(s), 11
categorical axiomizations of, 766
definition of, 139, 140, 159, 594, 596
embedding in complex plane, 9
Frege on, 12, 134–37, 143, 144, 166
and induction principle, 98, 777
infinite sequences of, 325
infinite sets of, 323
and intuitionism, 363–64
as mental constructions, 19
ordering of, 545
progression of, 577
reasoning on, 138
sets of, 604
and simple theory of types, 594
singular terms for, 172
specific attributes, 505
structure of, 21–22
Uniformity Principle, 358
Wittgenstein on, 98–100, 102
natural sciences, 5, 700
natural world, 32, 40, 43, 45, 47, 48–49
nature, 274, 625–26, 627
necessity, 4–5, 14, 30
of mathematics, 656
Mill on, 61–62
in sentences, 84
of thought, 677–79, 692
Wittgenstein's views, 84, 86, 88, 102, 109
Ne'eman, Yuval, 639, 641
negation, 685
negation‐completeness, 294
negative numbers, 307
negative translation theorem, 371
negative uniform continuity theorem, 345n.55
Neighborhood Theorem, 366–67, 376, 377
neo‐Fregeanism, 166–70, 174–75, 181–84, 220, 222–29, 463
neo‐Fregean logic, 196–200
neo‐Fregean real analysis, 186–91
neo‐Fregean set theory, 192–96
neologicism, 12–13, 167n.2, 223, 463, 465
Neoplatonism, 246
neutrons, 638–39
Newton, Sir Isaac, 30, 32, 38–40, 43, 46, 627, 628n.4, 632
Noether, Emmy, 637
nominalism
alienated revolutionary, 520–23
attitude‐hermeneutic, 525–28, 532
Burgess‐Rosen account, 16, 483–89, 505–7
Constructibility Theory, 499–501, 506–12
content‐hermeneutic, 523–25
and fictionalism, 17, 643–44
of Field, 17–18, 182, 484, 643–44, 645
hermeneutic, 486, 487, 505, 509, 517, 523–28
and higher‐order logic, 770–71
and logicism, 174, 182
Millian, 60, 61, 63, 64
naturalized revolutionary, 518–20
philosophical view, 489–92
Quine's challenge, 493–94, 506, 519
reconsidered, 515–34
reconstructions of mathematics, 492–505
revolutionary, 486, 487, 505, 506, 517–23
and varieties, 515–18
Vineberg defense, 510–11
Yablo's figuralism, 528–34
noncontentual role, 297
non‐Euclidean geometries, 347n.66, 422, 630
noninterference, 789
nonlogical terms, 785
non‐standard models, 769
normal‐form deducibility, 719
notational definition, 503–4
nucleons, 638, 639
Nuisance Principle, 181–82
number(s), 14
applicability of, 141
concept of, 13, 34, 141, 153, 174–75, 179–81, 643
definition of, 168
finite, 171, 629
Frege's definition of, 643n.23
infinity of, 136, 152–54, 160–61
Leibniz on, 41
as objects, 151, 170–71
original laws of, 288
number‐concept, 283–84, 286
number theory, 64–65, 103
number words, 97, 103, 170
numerical identity, 98
numerical property, 134
O
(p. 826) object(s), 89, 98
absolute, 541
abstract, 15, 173
and construction, 237, 333
Fregean views, 151–52, 304–5
geometrical, 141
infinity of, 199
introducting new, 428–29
of intuition, 141
logical, 152
material, 35–37, 40
mathematical, 6, 19, 41–42, 241, 246, 444, 492, 507, 510–11, 515, 520, 565–76, 642, 645, 775
numbers as, 151, 170–71
Plato on, 243
Quine on, 442, 455
recognizing, 423–29
objectivity, 111
obviousness, 80
Ockham's razor, 645
octonions, 283
omega minus particle, 639
“On the Gravity and Equilibrium of Fluids” (Newton), 38
ontological commitment, 423–30, 444, 454, 470, 493, 519
ontological naturalism, 427–28, 460, 461, 472–73
ontological relativity, 434
ontology, 5–6, 30
abstraction principles, 170–79
and assertability, 349–50
of higher‐order logic, 197, 198
and intuitionism, 319, 333–34
and logicism, 170–79, 197–99
mathematical, 453, 454
and naturalism, 452, 462
and ontological relativity, 432–35
positing, 429
Quine on, 432–33, 439
realism in, 6–7, 11, 14–17
Resnik on, 565–76
Shapiro on, 576–86
Ontology and the Vicious‐Circle Principle (Chihara), 493, 498
open argument, 688, 690–91
open‐endedness, 616
open sentences, 18, 499–500, 501, 502
open‐sentence tokens, 500
open‐sentence types, 500
open‐sentence variables, 18
operations, 90, 92, 94–95, 102–4, 549, 616–17
optimism, 280–81, 341, 345
order, 628n.4
ordered field, 768
ordered pair, 503
ordinal arithmetic, 629–30, 633
P
Pairs abstraction, 186
paraconsistent system, 696
parallelogram law, 270
parallels postulate, 63
paraphrase‐functions, 206–8, 212, 213, 214n.20, 217, 219
Parity Principle, 181n.29
Parsons, Charles, 419, 421, 422, 500
partial logicism, 559
participation, 626
particles, 639–40, 647
Pascal, Blaise, 3
Pasch, M., 250–51
past/future distinction, 332n.30
patterns, 21, 541
Peacock, George, 271–77, 282, 291
Peano, G., 287, 288, 289, 295n.96, 592
Peano Arithmetic, 337, 338, 398n.19, 610, 614, 618, 682–83, 698
Peano postulates, 11, 13, 279 See also Dedekind‐Peano axioms
Peirce, Charles, 632
Pererius, Benedictus, 246–47
perfectionism, 734–40
perfect validity, 701
pessimism, 280
phase space, 637n.15
phenomenology, 329–35, 344
philosophy of math
Fregean, 166–70
interpretive nature of, 10
of Kant, 44–49
and logic, 3–24
in modern period, 29–49
post‐Quinean, 456–58
of Quine, 429, 474
rationalist, 37–38
realism in, 492
of Wittgenstein, 75–118
physical world, 5
physics, 625, 630–32, 639–41, 644, 647
Piccolomini, Alessandro, 247
(p. 827) pictures, 87, 93, 107, 108, 109
Pierpont, James, 307, 309
pions, 639
planetary orbits, 630
Platonic Forms, 6, 626
Platonism/platonism
and concept of direction, 172–73
and constructive ideal, 241, 245
of Field, 172, 474, 645, 646
and Frege, 166, 170, 172–73, 179, 643, 645–46
and mathematical objects, 170, 492, 507
position on empirical science, 497
and power sets, 802
and realism in ontology, 6
and truth‐conditions of mathematical statements, 172
version of logicism, 166–67
view of mathematics, 497–98
and Wittgenstein, 78
Playfair, John, 265
plurality and pluralism, 141, 142, 239, 388n.1, 393, 405
plural quantification, 762–64, 804–7
Poincaré, Henri
and coordinations, 59
and geometry, 58
and predicativity, 591–92, 593, 601, 619
on symbol creation, 295n.96
versus logicists, 99, 596–99
vicious circle principle, 156, 491, 597
Poincaré phenomenon, 731–32
points, 245
Pólya, George, 602
polyhedron, 10
Poncelet, Jean‐Victor, 259, 265
positing, 428–29
positivism, 466
possible, meaning of, 500–501
postulates, 242, 245, 276
postulation, 293
Posy, Carl, 345n.55, 387
Potter, Michael, 138
power sets, 796, 800, 802–5
practical pursuits, 626
Pragmatic Indispensability Argument, 431–32
Prawitz, Dag, 675n.3, 676n.4
predicative functions, 158
predicativity, 590–621
as chapter of definability theory, 603–6
definition of, 590, 619–21
emerging of, 591–92
and indispensability arguments, 613–14
mathematical reach of, 610–12
outer mathematical bounds of, 612
of Poincaré, 591–92, 601, 619
Poincaré versus logicists and Cantorians, 596–99
provability in 1960s, 606–7
reducible systems, 607–10
rethinking of (1970–96), 614–15
of Russell, 83, 591–96, 601
sidelining of (1920–50), 601–3
summarized, 619–21
in transition, 603–6
as unfolding, 615–19
Weyl's development of analysis, 599–600
predictions, 641
pre‐intuitionists, 322
premises, 302, 303, 462, 772
prescriptive criticism, 729–30
presentist conception, 265, 266
prime numbers, 631
Primitive Recursive Arithmetic, 610
primitive terms, 785
primordial consciousness, 329
Principle of Continuity, 258–62
principle of exclusion, 63
principle of invalidity recognition, 408
Principles of Mathematics, The (Russell), 145–47, 154, 587
Principles of Mechanics (Hertz), 79
Principle of the Permanence of Equivalent Forms, 274–75, 277–79, 282–89, 291n.87
Prior, Arthur, 390, 667, 683
problems, 242, 244n.12
Proclus, 241–45
projectible predicates, 647
projective geometry, 258–62
proof(s), 10, 422
canonical, 684–87, 694
by cases, 714
cumulative deductive progress, 702–4
formalized, 109
Frege on, 302
Gauss on, 270–71
general, 686
on a genetic model, 236–37, 269
Hilbert on, 282, 283
(p. 828)
hypothetical, 686, 693
imaginary and complex numbers as means of, 270
inferences in, 251
and intuitionism, 340–41, 369–70, 390–92
intuitionistically acceptable, 729
Lambert on, 250
and logical consequence, 682–84
Maseres on, 266
by mathematical induction, 633
perfectionist, 739
premises of, 303
traditional ideals of, 240–46
Wallis on, 256–57
Wittgenstein on, 110
proof theory. See metamathematics
proportion, 33, 239
proposition(s), 230–31
analogy to pictures, 87–88, 93
and apriority, 51–52, 55, 61, 67
Aristotle on, 654
content of, 90
correspondence conception, 54
elementary, 103
Hilbert on, 289
ideal, 289n.80
as internally structured entities, 198
logical, 11, 95
Millian view, 56, 60, 65
“necessary,” 84
as pseudo concept, 85, 94
Russell on, 156
systems of, 102–3
tautologies, 89
Wittgenstein on, 81, 85, 90, 94–95, 100, 102–5
propositional function, 552, 594
propositional paradox. See Russell's paradox
proto‐elementary entailment, 740, 744–45
protons, 638–39, 648
proxy function, 434
pseudo concept, 85, 94
Putnam, Hilary, 8, 14, 17, 430, 431, 519, 613
Pythagoras, 648
Pythagoreans, 239, 346, 631, 641, 649
Q
quadratic equations, 254
quadrature, 256
quantification theory, 91, 95
quantifiers, 212, 502, 554, 763–65, 776, 789, 804–7
quantity. See magnitude
quantum mechanics, 637, 644, 647
quarks, 639
quaternions, 283
Quine, W.V.O., 13
on arithmetic, 647n.29
challenge to nominalism, 493–94, 498, 506, 519
criterion of ontological commitment, 423–25, 444
epistemology, 72, 417–18, 439, 446, 772
on higher‐order logic, 197, 770–72
holism, 414–23, 434, 440, 443, 494n.26
indispensability argument, 14, 17, 429–32
on language, 14, 70–72, 423–27, 434–35, 464
on ontology of mathematics, 432–33
philosophy of mathematics, 429, 474
on science, 413, 439, 442, 446–51, 613
on set theory, 445, 613, 770, 771
on synonymy, 70–71
web of belief, 15, 412–35, 440, 464
Quine phenomenon, 731–32
R
radical conventionalism, 70, 71
ramified analytic hierarchy, 604
ramified theory of types, 148, 156, 594, 754
Ramsey, F.P., 104, 105, 156
Ramsey‐conditionals, 212
Ramsey sentence, 176–77, 209n.10, 479
ratio, 33, 239, 240
rationalism, 3–4, 5, 78
rationality. See reason and reasoning
rational numbers, 34, 321, 324–26, 363–64
real abstraction, 190
real analysis, 8, 166, 186–91, 768, 769, 772, 776
realism, 14, 512
mathematical, 429–32, 444
Mill's position on, 61, 63
moderate, 485, 486
naturalism distinguished from, 54
in ontology, 6–7, 11, 14–17
in philosophy of math, 492
transcendental, 348, 350, 351
in truth‐value, 6–7
realizability, 376–78, 381, 382
real numbers
and Brouwer's Theorem, 365–69
categorical axiomizations of, 766
and completeness scheme, 776
Cut Abstraction, 187
and Frege, 166, 189, 190
and intuitionism, 19, 321, 324–26, 333, 363–69
as mental constructions, 19
neo‐Fregeanism, 167
as relations of quantities, 189, 190
and Wittgenstein, 102, 103
reason and reasoning
algebraic, 266
Berkeleyan conception of, 263–68
causal, 648n.33
contentual, 272
“correct,” 651, 659
Frege's account of, 138–40, 142
ideal, 298
Leibniz on, 41
logic as instrument of, 699
mathematical, 31, 37–38, 43, 45, 237, 250, 272, 295, 298, 302, 508–12
natural light of, 37–38, 40, 42, 43
on natural numbers, 138
relevance in, 696–725
Wittgenstein on, 107 See also deduction; induction; logic
rectangle, 256–57
reducibility, 158 See also Axiom of Reducibility
reducible systems, 607–10
refinement, 326
Reflection Principle, 542
regimentation, 425–27
relations, 42, 502, 503, 766
relation variables, 752–53, 758–60, 763
relevance
Anderson‐Belnap approach, 697, 698, 713, 717–22
banning Dilution, 704–5
CR system, 708, 711, 716–17, 723–24
Disjunctive Syllogism, 697–98, 707, 732–33, 740–44
epistemic gain, 706, 711, 723–24
fallacy of, 721
IR system, 708, 711, 712–15, 717, 723–24
Lewis's First Paradox, 696, 704–5, 706–10
maxim of narrow analysis, 722–23
option of rejecting transitivity, 734–40
“paradoxes” of classical logic, 727–30
potential utility of relevantism, 748
in reasoning, 696–725
relevantist's options, 730–34
stringency, 740–46, 748
relevance logic, 747
reliability thesis, 484
Renaissance, 246–49
Replacement Axiom, 540–41
representational methods, 32–35
representational semantics, 676
representations, 637–39, 647
Republic (Plato), 626
Resnik, Michael, 507, 559n.24, 564–76, 578–80, 586–87
restricted functional calculus, 752
restriction to faithful models. See faithful models
reverse mathematics, 161
Reverse Mathematics program, 610, 611
revolutionary nominalism, 486, 487, 505, 506, 517–23
Richard paradox, 591
Riemann's Hypothesis, 380
roots, 254, 297
Rosen, Gideon, 446–49, 452, 454, 456, 483–89, 495, 497, 505–7, 512
Routley‐Routley valuation, 745
rule of Cut. See Cut
rules, 106, 109
Russell, Bertrand, 11, 78
account of reasoning by induction, 159
analysis of arithmetic, 98
ancestral construction of, 99
axiom of infinity, 96n.19, 103
on causality, 648
contributions and errors of, 101
on Dedekind numbers, 160–61, 545
distinction between function and argument, 89, 90
and Hume's principle, 136
on intensional entities, 753
logicism, 79, 81–82, 85, 86, 130, 154–59, 161–62
and mathematical reality, 112
on necessity, 84
notion of incomplete symbol, 90
predicativity, 83, 591–96, 601
(p. 830)
rejection of denoting concepts, 154–59
and structuralism, 551–53, 587
theory of types, 83
on truth, 82, 83, 85
and Wittgenstein, 80, 81, 91, 95, 97, 101, 102
Russell‐Myhill paradox, 151
Russell's paradox, 12, 81, 93, 145–52, 168, 491, 492n.22, 643n.23
Russell‐Zermelo paradox, 133, 161
S
satisfaction relation, 499, 662, 663, 757
schematization, 141
scheme/content distinction, 54–55
Schlick, Moritz, 55, 58–60, 68, 69
Schubert, Hermann, 287
science
Burgess on, 446–47
classical use of, 239n.3
as “confirmed,” 14
as empirical, 5
Hilbert on, 293
and holism, 414
in modern period, 29
naturalistic study of, 446–50
as prior to philosophy, 15, 16
Quine on, 413, 439, 442, 446–51, 613
revolutions in, 490
scientific study of, 451–54, 456
speculative and practical, 246–47
as ultimate arbiter of existence, 427–28
scientific method, 460–61, 462
scientific testing, 419–21, 431
Scott, D., 378
second‐degree equations, 254
second‐order consequence, 781–808
second‐order languages. See language(s), second‐order
second‐order logic. See logic, second‐order
second‐order quantification, 197, 763–64, 796, 804–5
second‐order validity, 772
second‐order variables, 751, 770, 771, 804
self‐consciousness, 54
self‐evidence, 80
self‐identity, 183
semantic consequence, 757
semantics
canonical, 783
compositional theories, 20
and deductive system, 653
first‐order, 760–63, 765–66, 776, 777
Heyting, 20–21, 380–82, 394
of informal mathematical languages, 774
and logical consequence, 656–57, 663–66, 676
Mill on, 54
model‐theoretic, 663, 664, 666, 757–64
multisorted first‐order, 760–62
Tarskian, 20
semantic tradition, 11
senses, 149–50
sensible matter, 244n.13, 248
sentence(s)
analytic, 66
apriority of, 67
arithmetic, 302, 682–83
compound, 686
content recarving, 216–23, 225, 231–32
context of, 107
deduction composed of, 687
first‐order, 207, 209n.10, 787
following of one another, 83–84, 782
in formal languages, 752
and function, 82
and inference, 672n.2
internal relations between, 90
in linguistic framework, 70
logicality of, 91
mathematical, 110, 380–81, 641
meaning of, 20, 67, 68, 679–80, 684, 692
open, 18, 499–500, 501, 502
as pictures or models, 87, 93
with plural quantifiers, 763
quantificational forms of, 92
Quine on, 70–72, 423–24, 475
Ramsey, 176–77, 209n.10, 479
second‐order, 283, 798, 804
truth and existence in, 71, 93
truth conditions of, 662, 679–81, 684, 692
truth‐value of, 14, 566, 676
understanding of, 111
as verification, 682
and Wittgenstein's philosophy, 87, 90–93, 107, 109–11, 679
separable mathematics, 322–23, 346
sequences, 321–25, 364–66
sequent calculus, 704, 715–17, 738
(p. 831) set theory, 8–9
axiomatic, 601
and canonical second‐order consequence, 797–800
Cantorian, 321, 596–99, 619
categoricity, 803–4
connection to “correct reasoning,” 651
and full mathematics, 325–28
Gödel's views, 497n.32, 498n.36, 615
logical versus iterative sets, 796–97
neo‐Fregean, 192–96, 463
and nominalism, 496, 499
paradoxes of, 491
and Principle of Permanence, 288
Quine on, 445, 613, 770, 771
second‐order, 211, 763
in second‐order language, 771
structuralism in, 538–41
subsets, 800–804
Zermelo‐Fränkel, 184, 211, 214n.20, 491, 539–40, 601–3, 788n.8, 790
shape, 41
Shapiro, Stewart, 406n.29, 489, 541–42, 545n.10, 564, 576–87, 676
show/say distinction, 103, 104
signs, 296, 297, 298
simple hierarchy, 158
simple theory of types, 594
simply infinite system, 152–53
singular terms, 171, 172
Skolemite relativism, 774
Sober, Elliott, 419–22, 431, 510–11
social constructivism, 118
Sophie Charlotte (Queen), 41
Sorites Paradox, 405
sortal concept, 179, 181, 229
soundness, 653, 665, 668, 669
space‐time, 433, 640, 644, 648
spatiotemporal form, 44–45
Special Relativity, 648n.33
specious arithmetic, 30
Speusippus, 243, 244
spin, 638
spread, 325
standard arithmetical inference, 508–9
standard logical truth, 758
standard model, 758
standard semantics, 654, 757–59, 762–63, 766–71, 773–75, 777
standard‐validity, 773
Steiner, Mark, 472n.11, 494, 506, 511
Stelau, Klaus, 487n.8
Sternberg, Shlomo, 636, 638, 648n.33
Stone, M., 378
Strahm, Thomas, 618
Straight Proposal, 224–25, 226, 227, 228
stringency, 740–46, 748
structuralism, 21–23, 536–61
in category theory, 546–51
modal, 23, 551–60
reconsidered, 563–87
in set theory, 538–41
structures as sui generis universals, 541–46
structures, 794–95
subformula property, 371
Subject with No Object, A (Burgess and Rosen), 16, 483, 518, 519, 521, 533
subsets, 796, 800–802
substitutional quantifiers, 426–27
subsystems, 752
Subsystems of Second Order Arithmetic (Simpson), 610
Success by Default, 226–28
successor, 99, 140–41, 152
successor function, 11
supranatural world, 47
surrogates, 760
Sylvan‐Plumwood valuation, 745
symbolical ideal, 238
symbolic conception, 272
symbolic formalism, 263–99
symbolic reasoning, 272, 298
symbols, 43, 274, 295n.96
symmetry, 636–41
symmetry groups, 637–39, 648
symplectic manifold, 637n.15
Symplectic Techniques in Physics (Guillemin and Sternberg), 625
symplectric geometry, 637n.14
synonymy, 70–71, 478
syntax, 66, 68, 80
system, 152, 303, 637
System of Logic, A (Mill), 54, 61, 62, 63, 642
T
Tarski, Alfred, 3, 399
on antinomy, 491, 783
on axiomatic theories, 785n.6
(p. 832)
definition of consequence relation, 782, 794
definition of truth, 676, 680, 683
on logical consequence, 392, 674–76
on terminology, 773
on topological spaces, 378
Tarskian semantics, 20
tautology, 65–66, 84, 89, 730–31, 733
technique, 298
temporality. See time
Tennant, Neil, 12, 20, 406n.29, 737–40
tertium non datur (TND), 358–61, 366, 370, 378–83
theorem, 237, 242, 508, 700, 702, 738, 739
theorem of unprovability of consistency, 601
Theory of Forms, 626
theory of participation, 626
theory of symmetries, 636–41
theory of types, 148, 156, 592, 594, 603
thinkable predication, 200
third‐order quantifiers, 212
third‐order variables, 754
Thomae, Johannes, 301, 303
thought, 88, 149–54, 198, 677–79
thought‐contents, 67
“tonk” operator, 390, 667, 683
topology, 366–67, 378–79
topos, 378–79, 548
Tractatus Logico‐Philosophicus (Wittgenstein), 65, 75, 81, 84–87, 91, 95–98, 101–5, 679
“Transcendental Aesthetic” (Kant), 44
transcendental idealism, 31, 44, 53, 348, 349, 626
transcendental realism, 348, 350, 351
transformations, 636, 637n.14
transitivity, 734–40
transitivity of entailment, 733
Treatise of Algebra (Wallis), 255
triangle, 245, 247, 256, 565, 648
truth
analytic, 11
apriority of, 52
arithmetic, 11, 130
conditions, 679–81
definition of, 68, 112, 680, 683
epistemic, 747
Frege and Russell on, 82, 83
guarantee of, 675
and intuitionism, 340–41, 400–402
Leibniz on, 40–42
logical, 674, 676, 701
and logical constants, 679–80
logical following of, 83
mathematical, 11, 41–42, 111, 112, 172, 444n.15, 445
“necessary,” 11, 62, 84
of sentences, 66, 71
standard logical, 758
Tarski's definition of, 676, 680, 683
Truth‐Logicism, 204–6, 208–10, 213–15
truth‐preservation, 735, 747, 749
truth‐tables, 88–91, 373
truth‐values, 6–7, 14, 17, 19, 68, 88–89, 193, 373, 445, 566, 676
Turing jump operator, 604
Turing machine, 377, 381
two‐dimensional quantities, 285–86
“two‐factor” theory of entailment, 735
U
Ullian, Joseph, 414
ultraholism, 464–65
unconditional anti‐nominalism, 454n.32
understanding, 39–41, 43, 53, 107, 111, 244, 266
Uniformity Principle, 357–59, 361, 366, 376, 381
unit segment, 33
unity, 141, 142
universal discharge requirement, 719
universality, 30, 84, 86
unrestricted Cut, 717
unrestricted transitivity of deduction, 706
upward Löwenheim‐Skolem theorem, 767, 784
use, 681n.6, 682, 684
useful fictions, 42
V
vagueness principles, 404
validity, 83, 657, 662, 701, 772–73
value, 274
van Dalen, Dirk, 368
variable‐assignment, 757, 758, 759, 761
variables
first‐order, 751, 763, 796
fourth‐order, 754
function, 752–53, 760
(p. 833)
higher‐order, 763, 770
open‐sentence, 18
Peacock on, 274
relation, 752–53, 758, 759, 760, 763
second‐order, 751, 770, 771, 804
variable‐sharing requirement, 735
vector space, 637, 640n.20, 647
verification correctness principle, 398
verificationism, 68, 71, 682
vicious circle principle, 83, 99, 156–57, 491, 591, 592, 597
Vienna Circle, 11, 51, 55, 67, 68, 80, 101–2, 103
Vieta, François, 30
Viète, F., 254, 280
Vineberg, Susan, 510–11
Virchow, Rudolf, 280
viscosity, 347
von Neumann ordinals, 137
W
Waismann, Friedrich, 262
Wallis, John, 237, 255–58, 268, 269, 281
Wang, Hao, 774
warranted assertability, 340n.47
Warren, J., 270n.47
wax argument, 37
web of belief, 15, 412–35, 440, 464
Web of Belief, The (Quine and Ullian), 414
weight, 630
Wessel, C., 270n.47
Weyl, Hermann, 282, 599–600, 610–11, 619, 638, 801
Whitehead, Alfred North, 3, 11, 12, 81, 82, 83, 103, 753
Wigner, Eugene, 631, 632, 640, 646
Williamson, Timothy, 213, 214, 474
Wittgenstein, Ludwig, 10, 11, 70
algorithms and calculation, 106, 108
analyticity, 65
on apriority, 55, 65–66
bibliographic essay, 113–18
early period, 81–100
evolution of thought, 81–113
and Frege, 80, 81, 91, 95, 97, 101, 102
and Gödel, 116
on language, 66, 71, 76, 78, 80, 86, 87, 90–93, 99, 103, 107–11, 679
later philosophy, 107–13
logical grammar, 68
on mathematical reality, 112
“middle” life and philosophy (1929–33), 100–107
new logics, 113–14
on philosophy of logic and math, 75–118
place in history of analytic philosophy, 114–16
recent trends in interpreting, 116–18
rule following, 106, 109, 114
and Russell, 80, 81, 91, 95, 97, 101, 102
Wolff, Christian, 31n.3, 45
Word and Object (Quine), 415, 463, 498
Wright, Crispin, 12, 404, 463
Y
Yablo, Stephen, 528–34
Z
Zeno's paradoxes, 346
Zermelo, E., 597–98, 613
Zermelo‐Fränkel set theory (ZFC), 184, 211, 214n.20, 491, 539–40, 601–3, 788n.8, 790, 799
Zermelo‐Russell contradiction. See Russell‐Zermelo paradox
ZF2 (second‐order Zermelo‐Fraenkel), 789–93, 795