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# (p. 811) Index

(p. 811) Index

Aabsolute objects, 541

abstract algebra, 422

abstract concepts, 305–6

abstract entities, 483–84

abstraction principles, 167, 170, 182

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 structures, 536

Ackermann, Wilhelm, 740–41

aggregates, 64–65

algebra

abstract, 422

arithmetical, 275–77

Boolean, 378

early, 35

and formalism, 237

fundamental theorem of, 597

and group theory, 636

Heyting, 378

Maseres on, 266

Playfair on, 265

and symbolic formalism, 263

usefulness of, 268–71

algebraic language, 267

algebraic reasoning, 266

algorithmically generated sequence, 322

alienated revolutionary nominalism, 520–23

alien epistemologists, 496–99

Amphinomus, 243

analogies, 644

analytic number theory, 9

analytic truth, 11

antecedent justification, 224

Apollonian theory of conic sections, 630

Apollonius, 245

application and applicability

and apriority, 29–49

of arithmetic, 137–38

canonical nonempirical, 632–41

metaphysical problem of, 511

of numerical concepts, 141

application function, 761

(p. 812)
apriority

and application, 29–49

of cognition, 44–48

and implicit definition, 67

of propositions, 55

and realism in ontology, 6

of space, 347

synthetic propositions, 51

and tautology, 66

arbitrary function, 104

Argand, J.R., 270n.47

Aristotle, 270, 644

division of mathematics, 238–39

and logically valid inferences, 672

on matter, 244n.13

statement of genetic ideal, 240

on viscosity, 346–47

arithmetic

applicability of, 137–38

application in physics, 631

cardinal, 629

first‐order, 768

foundational work in, 9

induction principle, 777

Kant on, 5

“logicist” analyses of, 81

nominalist analysis of, 64

order‐sensitivity in, 286

pure, 645

Quine on, 647n.29

standard inference, 508–9

standard laws of, 283–84

subject matter of, 21

arithmetical analysis, 600

arithmetic progression, 257

Avigad, Jeremy, 590n.1

Axiom of Countable Choice, 364

axioms, 687, 738, 739

in ancient geometry, 245

consequences of, 784

Gödel on, 306

in set theory, 801

Azzouni, Jodi, 775

B
Bar Rule, 609

Barwise, Jon, 774

*Basic Laws of Arithmetic*(Frege), 205

bipolarity, 88

Bonevac, Daniel, 484

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

bounding principles, 404

Brouwer, L.E.J.

and assumptions of actual infinite, 619

on logic, 334–35

reform of mathematics, 389

and Weyl, 601

Brouwerian sequence, 323–24

Burali‐Forti paradox, 591

Burgess, John

C
and Anderson‐Belnap tradition, 697–98

on difference between mathematical and scientific terms, 457n.38

and indispensability argument, 454–56

calculus, 30, 35nn.13–14, 39n.30, 104, 626, 628n.4, 773

*See also*deductive calculus; sequent calculuscanonical language, 425–27

canonical second‐order consequence, 781–808

Cantor‐Bendixson theorem, 612

Cantor's diagonal argument, 334

Cantor's diagonal construction, 591

Cantor's Paradox, 183

Cantor's theory of higher cardinals, 323

Cardano's Rule, 296

cardinal arithmetic, 629

cardinal number(s)

addition of, 629–30

as attribute of property, 504–5

Cantor's theory of, 323

universal applicability of, 96

cardinal‐ordinal equivalence, 629

Carnap conditional, 176–77

Cartesian dualism, 461

categorical component, 553

categoricity, 803–4

category theory, 546–51

Cauchy completeness, 598

Cauchy proof, 597

causal attention, 329

causal laws, 648n.33

causal reasoning, 648n.33

causal theory, 484

Cayley, A., 261

choice sequences, 366

circle, 648

classical mechanics, 627n.15

coefficients, 254

cognitive verbs, 110

Cohen, Paul, 497n.35

coherence, 490

Collegio Romano, 247

combinatorial subset, 802nn.29–30

common difference, 257n.33

commutativity, 285–86

compactness theorem, 8

completeness axiom, 776

compositional semantic theories, 20

compound sentence, 686

computers, 748

conceivability test, 63–64

concept(s), 9, 10, 81

abstract, 305–6

construction of, 292

denoting, 154–59

of direction, 172

finite, 152

formation, 296

Hilbert on, 294–95

and language, 67

noncontentual role, 297

of set, 194

and thinkable predication, 200

conceptual set, 797

conic sections, 244n.12

connectedness, 346n.56

connotation, 60

consciousness, 329

consequence, 24, 652–53, 655, 659, 668, 672n.2, 783–84

*See also*logical consequence; canonical second‐order consequenceconservation laws, 637

consistency, 304–5

constitutive completeness, 294

constructible hierarchy, 603

constructional component, 241

content

in analytic sentences, 66

Kant on, 52

mathematical, 334

and paraphrase‐functions, 212

of thought, 67

within framework, 53

within linguistic scheme, 55

content‐hermeneutic nominalism, 523–25

contentual reasoning, 272

corollaries, 738

correctness, 664–65

correctness principle, 403

correspondence conception, 54–55

couples, 502–3

criterion of identity, 135

crystallography, 636

cumulative deductive progress, 702–4

cumulative theory of types, 603

Curry, Haskell, 17

D
Dedekind abstraction, 545

Dedekind completeness (continuity), 598

Dedekind infinity, 160–61

Dedekind sections, 597–98

Dedekind sets, 362

deduction

automated, 722

of conclusions from premises, 462

cumulative deductive progress, 702–4

of early mathematicians, 422

Frege's use of, 645

as producer of knowledge, 60–61

in sentences or formulas, 687

unrestricted transitivity of, 706

Deduction Theorem, 708

deductive validity, 653

definability theory, 603–6

definition by property, 244n.15

De Morgan, A., 271n.50

denotation, 60

denoting concepts, 154–59

derivations, 371

de Rouilhan, Philippe, 592

Descartes, René, 3, 4, 30, 43, 438

on ancient geometers, 281

on essence of material substance, 35–39

on extension, 35–39

and representational methods, 33–34

on space, 38–39

“wax argument,” 37

designative occurrence, 146

deviance, 345

Dewey, John, 340n.47

dialectical consequence, 747

dialectical valuation, 746–47

difference equations, 644

differential equations, 644–45

Dilution Elimination Theorem, 710

discharge of assumptions, 714

disjunctive weakening, 732–33

distinctness, 179

Distributivity, 719

Diversity of the Dissimilar, 571

Division Problem, 306–9

domain of discourse, 751

double negation elimination, 399

double valuation, 745–46

Duhem‐Quine problem, 700

(p. 816)
Dummett, Michael

E
coining of linguistic turn, 11

on Field, 646

on Hume's principle, 184–85

on induction, 777

on meaning, 682n.6

on radical conventionalism, 70

on sortal concepts, 179

on truth conditions, 680

electrons, 638

elementary formulas, 730

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

orientation to knowledge, 4

empiricist formalism, 299–302

enumerative induction, 63

epistemically reductive philosophy, 109

epistemic truth, 747

epistemology

abstraction principles, 170–79

of Brouwer, 319

and logical consequence, 659–60

and mathematical intuition, 331–32

of Mill, 72

normative and descriptive, 417–18

equivalence, 274

Erlanger Program, 262

existence theorems, 516–17

extendability, 540

external relations, 84

extramathematical linguistic context, 107

extramental reality, 39–40

F
fallacy of relevance, 721

Fermat, Pierre de, 30

fermions, 640

Feynman, Richard, 645

finite cardinal structures, 545n.10

finite concepts, 152

finite description, 322

finite lines, 40n.30

finite mathematics, 323

finite model property, 375

finitude, 768

first‐order abstraction, 190

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 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

foundational work, 9

fourth‐order variables, 754

Frege, Gottlob, 3, 64, 78

account of number, 153

account of reasoning by mathematical induction, 138–40

challenges to formalism, 299–305

on combinatorial sets, 797

context principle, 90

contributions and errors of, 101

definition of numbers, 643n.23

dependence relations among definitions, 142–43

hierarchy of functions and senses, 149

Hilbert's criticism of, 251

interest in irreducible case, 297n.98

and mathematical objects, 511

on necessity, 84

and numbers as objects, 170–71

philosophy of math, 166–70

on quantity, 189–90

as realist in ontology, 11

on symbolical reasoning, 298

theory of thoughts, 149–52

Frege‐Russell definition of cardinal number, 135

Frege's constraint, 191

French, Steve, 639

Frend, William, 269

Friedman, Harvey, 703n.1

full comprehension axiom, 609

full mathematics, 325–28

function(s)

descriptive, 158

Frege's hierarchy of, 149

in full models, 766

material, 93

mathematical notion to structure of sentences, 82

predicative, 158

successor, 11

in theories, 14

value ranges of, 168

functors, 549

G
Galois, Evariste, 636

Galois group of an equation, 636

game formalism, 16

Gell‐Mann, Murray, 639

general cubic, 297

generality

of construction of infinite system, 153

and denoting concept, 154

of form, nature, and value, 274

general proof, 686

Gentzen, Gerhard

consistency proof for arithmetic, 601

natural deduction presentation, 370

negative translation theorem, 371

sequent calculus, 738

geometry

construction in, 244–46

constructive ideal in, 241

Maseres on, 266

nominalist analysis of, 64

Pasch on, 250–51

Plato on, 243

Playfair on, 265

Poncelet on, 265

problems in, 35

projective, 258–62

of space‐time, 640

symplectic, 637n.14

for two‐dimensional vector space, 640n.20

Gödel, Kurt, 3, 112, 512

challenge to formalism, 305–6

on combinatorial subsets, 802nn.29–30

negative translation theorem, 371

theorem of unprovability of consistency, 601

Goldbach's conjecture, 328

Goldfarb, Warren, 596–97

Goodman, Nelson, 647

Goodwin, William, 506n.43

Gottlieb, Dale, 484

gravitational field theory, 646

Gregory, D.F., 271n.50

H
hadrons, 639

Hahn, Hans, 249–50

Hallett, Michael, 597

Hamilton, W.R., 285–86

Hankel, H., 286

Hardy, G.H., 631n.7

Hebb, Donald, 496n.29

Heine‐Borel covering theorem, 300n.103

Heisenberg, Werner, 638

Henkin construction, 766n.5

Henkin interpretation, 783

Heyting, Arend

on mathematical objects, 19

and mathematical sentences, 380–81

Neighborhood Theorem, 366

Heyting algebra, 378

higher‐order abstractions, 167

higher‐order identity, 760

higher‐order logic, 170, 751–77

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

metatheory, 764–69

model theory, 757–64

plural interpretation, 804–7

plural quantification, 762–64

reconsidered, 781–808

higher‐order nonlogical constants, 754

Hilbert, David, 3, 376, 433, 601, 619

axioms for Euclidean geometry, 58

border between finitary and nonfinitary methods, 308

criticism of Dedekind and Frege, 251

on intuitionism, 382–83

metamathematics, 106

on proof, 282–83

Hilbert's Program, 299

Hippasus of Metapontum, 239n.5

Hippocrates, 257n.35

holism, 444n.15

basic idea of, 414–5

with empiricism, 413

and logic, 418–19

objections to, 419–23

and web of belief, 414–23

Holism‐Naturalism Indispensability Argument, 430–32

homogeneity, 593

Hyperarithmetic Comprehension Rule, 608

hypothetical component, 553

I
ideal propositions, 289n.80

ideal reasoning, 298

identity‐isomorphism, 231–32

identity of structural indiscernibles, 544

identity‐predicate, 231–32

identity relation, 770

image, 78

impredicative definitions, 7

impredicative propositional function, 157

impredicative separation, 802

(p. 820)
incompleteness theorems

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

on set of arithmetic truths as noneffective, 9

on true nonprovable propositions, 70

indispensability arguments, 613–14

induction scheme, 609

inequations, 141

inertia, 415

inferentialism, 390

infinite systems, 152–54

intellect, 38

interpretational semantics, 676

interpretation function, 757

*In the Light of Logic*(Feferman), 599

intuition

in conceptualism, 61

decline of, 249–52

and deductive systems, 661–62

Descartes on, 252

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

formal logic and internal mathematics, 369–79

as house divided, 344–45

logical domains, 357–62

models and modality, 373–75

and naturalism, 473n.12

negative doctrines, 334–35

ontology, 333–34

phases of, 318–20

and philosophy, 318–51

reconsidered, 387–409

second act of, 330–31

(p. 821)
intuitionistic logic

connectives, 394–95

disagreements on meaning, 394–98

disagreements on preservation, 398–400

disagreements on truth, 400–402

epistemic argument for, 388–90

formal system, 336–39

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

inverse square law, 632

investigation, 255

irrational numbers, 34

irrational quantities, 240

isomorphism property, 663

isotopic spin, 638–39

iterative sets, 796–97

J
Jesuits, 246–47

K
Kant, Immanuel, 5, 32, 38, 40, 138, 263

critical epistemology, 280

doctrine of judgment as synthesis, 86n.12

on experience, 61

on inverse square law, 632

on logic, 85

philosophy of math, 44–49

on schematization, 141

and truth, 112

Kant/Frege conception, 56–58

Kepler, Johannes, 630

Klein, Felix, 262

knowledge, 3–4, 51–53, 55, 236

of cause, 237

geometrical, 243

of infinity of numbers, 152–54

Kant on, 52

limits on, 280

logical, 61

and logical consequence, 659–60

metalinguistic, 66

produced by deduction, 61

König's paradox, 591

L
language(s)

algebraic, 267

Berkeleyan conception of, 263–68

Boolos on, 806n.35

Brouwer on, 335

in communication, 20

consequence relation, 788

Dummett on, 342

equation as rule of, 110

Hilbert on, 291

linguistic rules, 67–68

(p. 822)
necessity and a priori knowledge in, 11

nonrepresentational role in mathematical reasoning, 237

philosophy of, 490

Laudan, Larry, 511

Lavine, Shaughan, 777

Law of Continuity, 259n.37

law of multiplicative commutativity, 285–86

Law of the Permanence of Algebraic Forms, 274n.56

learning, 626

least upper bound, 7

Leibniz's Law, 571

levels, 753–54

limitation of size, 193

limitative theorems, 8

lines, 245

linguistic anti‐realism, 320

linguistic context, 107

linguistic deficit, 656

linguistic rules, 67–68

lists, 65

logic

as analytic, 57

background of, 748

basic principles of, 4

Brouwer on, 334–35

deductive, 701–2

definitions of, 651

failure of classical, 749

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

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

requirements of, 699–702

Wittgenstein on philosophy of, 75–118

*See also*higher‐order logic; intuitionistic logic; reason and reasoninglogical 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

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)
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 equivalence, 88

logical falsehoods, 700–701

logical grammar, 68

logically unrestricted quantifiers, 213–15

logical objects, 152

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 validity, 83

logicism, 11–13, 129–62, 412

assessment of, 206–16

definition of, 203–5

and formalism, 592

and naturalism, 462–63

neo‐Fregean program, 223–29

partial, 559

Platonist version of, 166–67

Poincaré versus logicists, 596–99

in twenty‐first century, 166–200

logics, 392–94

Lyndon Interpolation Theorem, 735

M
MacLaurin, C., 268–69

magnitude (quantity), 42, 189–91

definition of, 238–39

of geometrical figure, 256

irrational quantities, 240

mathematical notion of, 29–30

multidimensional, 285–86

manifold, 637n.15

mapping, 78

Markov's Principle, 377

Martin‐Löf, Per, 685n.8

mathematical cognition, 44–48

mathematical deviance, 345

mathematical induction, 138–40

mathematical notation, 645

mathematics

analytic principles of, 11

Aristotelian division of, 238–40

as “confirmed,” 14

definition of, 238

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

logic of, 772

meaning in, 9

mixed, 31n.3

nominalistic reconstructions of, 492–505

phenomenology of, 330–31

relative notions of, 8–9

representational methods, 32–35

standard, 626

truth in, 11, 41–42, 111, 112, 444n.15, 445

*See also*arithmetic; philosophy of math;*specific branches,*e.g., geometrymatter, 244n.13

maxim of narrow analysis, 722–23

McCarty, David, 387

means, 239

mechanics, 637

mechanism, 9

metabasis, 258

metalinguistic knowledge, 66

metalogic, 70

metaphysical problem of applicability, 511

Method of Exhaustion, 257n.35

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

on math and logic, 52

mind‐body problem, 626

minimal anti‐nominalism, 485

mixed line, 245

mixed motion, 245

modal operators, 426

modal statements, 18–19

monads, 40–41

monism, 85

Morgenbesser, Sidney, 627

morphism, 549

Mourey, C.V., 270n.47

multidimensional quantities, 285–86

multiplicative commutativity, 285–86

N
Naive Comprehension Axiom, 132–33

naturalism

distinguished from realism, 54

forms of, 437–58

and logic, 450

of Maddy, 468–72

and mathematics, 453–54

reconsidered, 460–80

and science, 446–50

naturalized revolutionary nominalism, 518–20

natural number(s), 11

categorical axiomizations of, 766

embedding in complex plane, 9

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

negation, 685

negation‐completeness, 294

negative numbers, 307

negative translation theorem, 371

negative uniform continuity theorem, 345n.55

neo‐Fregean logic, 196–200

neo‐Fregean real analysis, 186–91

neo‐Fregean set theory, 192–96

Neoplatonism, 246

neutrons, 638–39

Noether, Emmy, 637

nominalism

alienated revolutionary, 520–23

content‐hermeneutic, 523–25

and higher‐order logic, 770–71

naturalized revolutionary, 518–20

philosophical view, 489–92

reconsidered, 515–34

reconstructions of mathematics, 492–505

and varieties, 515–18

Vineberg defense, 510–11

Yablo's figuralism, 528–34

noncontentual role, 297

noninterference, 789

nonlogical terms, 785

non‐standard models, 769

normal‐form deducibility, 719

notational definition, 503–4

Nuisance Principle, 181–82

number(s), 14

applicability of, 141

definition of, 168

Frege's definition of, 643n.23

Leibniz on, 41

original laws of, 288

numerical identity, 98

numerical property, 134

O
(p. 826)
object(s), 89, 98

absolute, 541

geometrical, 141

infinity of, 199

introducting new, 428–29

of intuition, 141

logical, 152

Plato on, 243

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 relativity, 434

ontology, 5–6, 30

abstraction principles, 170–79

and assertability, 349–50

and ontological relativity, 432–35

positing, 429

Resnik on, 565–76

Shapiro on, 576–86

open‐endedness, 616

open‐sentence tokens, 500

open‐sentence types, 500

open‐sentence variables, 18

order, 628n.4

ordered field, 768

ordered pair, 503

P
Pairs abstraction, 186

paraconsistent system, 696

parallelogram law, 270

parallels postulate, 63

Parity Principle, 181n.29

partial logicism, 559

participation, 626

Pascal, Blaise, 3

Pasch, M., 250–51

past/future distinction, 332n.30

Peirce, Charles, 632

Pererius, Benedictus, 246–47

perfectionism, 734–40

perfect validity, 701

pessimism, 280

phase space, 637n.15

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

rationalist, 37–38

realism in, 492

of Wittgenstein, 75–118

physical world, 5

Piccolomini, Alessandro, 247

pions, 639

planetary orbits, 630

Platonism/platonism

and concept of direction, 172–73

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

Poincaré phenomenon, 731–32

points, 245

Pólya, George, 602

polyhedron, 10

positing, 428–29

positivism, 466

possible, meaning of, 500–501

postulation, 293

Potter, Michael, 138

practical pursuits, 626

Pragmatic Indispensability Argument, 431–32

predicative functions, 158

predicativity, 590–621

as chapter of definability theory, 603–6

emerging of, 591–92

and indispensability arguments, 613–14

mathematical reach of, 610–12

outer mathematical bounds of, 612

Poincaré versus logicists and Cantorians, 596–99

provability in 1960s, 606–7

reducible systems, 607–10

rethinking of (1970–96), 614–15

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

prescriptive criticism, 729–30

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 Mechanics*(Hertz), 79

Proclus, 241–45

projectible predicates, 647

projective geometry, 258–62

proof(s), 10, 422

by cases, 714

cumulative deductive progress, 702–4

formalized, 109

Frege on, 302

Gauss on, 270–71

general, 686

(p. 828)
imaginary and complex numbers as means of, 270

inferences in, 251

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

proposition(s), 230–31

Aristotle on, 654

content of, 90

correspondence conception, 54

elementary, 103

Hilbert on, 289

ideal, 289n.80

as internally structured entities, 198

“necessary,” 84

Russell on, 156

systems of, 102–3

tautologies, 89

proxy function, 434

Pythagoras, 648

Q
quadratic equations, 254

quadrature, 256

quarks, 639

quaternions, 283

Quine phenomenon, 731–32

R
ramified analytic hierarchy, 604

Ramsey‐conditionals, 212

real abstraction, 190

real numbers

and Brouwer's Theorem, 365–69

categorical axiomizations of, 766

and completeness scheme, 776

Cut Abstraction, 187

as mental constructions, 19

neo‐Fregeanism, 167

reason and reasoning

algebraic, 266

Berkeleyan conception of, 263–68

causal, 648n.33

contentual, 272

ideal, 298

Leibniz on, 41

logic as instrument of, 699

on natural numbers, 138

relevance in, 696–725

rectangle, 256–57

reducible systems, 607–10

refinement, 326

Reflection Principle, 542

regimentation, 425–27

relevance

banning Dilution, 704–5

fallacy of, 721

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

relevance logic, 747

reliability thesis, 484

Renaissance, 246–49

Replacement Axiom, 540–41

representational methods, 32–35

representational semantics, 676

*Republic*(Plato), 626

restricted functional calculus, 752

reverse mathematics, 161

Richard paradox, 591

Riemann's Hypothesis, 380

Routley‐Routley valuation, 745

Russell, Bertrand, 11, 78

account of reasoning by induction, 159

analysis of arithmetic, 98

ancestral construction of, 99

on causality, 648

contributions and errors of, 101

and Hume's principle, 136

on intensional entities, 753

and mathematical reality, 112

on necessity, 84

notion of incomplete symbol, 90

(p. 830)
rejection of denoting concepts, 154–59

theory of types, 83

Russell‐Myhill paradox, 151

S
schematization, 141

scheme/content distinction, 54–55

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

revolutions in, 490

speculative and practical, 246–47

as ultimate arbiter of existence, 427–28

Scott, D., 378

second‐degree equations, 254

second‐order consequence, 781–808

second‐order validity, 772

self‐consciousness, 54

self‐evidence, 80

self‐identity, 183

semantic consequence, 757

semantics

canonical, 783

compositional theories, 20

and deductive system, 653

of informal mathematical languages, 774

Mill on, 54

multisorted first‐order, 760–62

Tarskian, 20

semantic tradition, 11

senses, 149–50

sentence(s)

analytic, 66

apriority of, 67

compound, 686

context of, 107

deduction composed of, 687

in formal languages, 752

and function, 82

and inference, 672n.2

internal relations between, 90

in linguistic framework, 70

logicality of, 91

with plural quantifiers, 763

quantificational forms of, 92

understanding of, 111

as verification, 682

(p. 831)
set theory, 8–9

axiomatic, 601

and canonical second‐order consequence, 797–800

categoricity, 803–4

connection to “correct reasoning,” 651

and full mathematics, 325–28

logical versus iterative sets, 796–97

paradoxes of, 491

and Principle of Permanence, 288

in second‐order language, 771

structuralism in, 538–41

subsets, 800–804

shape, 41

simple hierarchy, 158

simple theory of types, 594

simply infinite system, 152–53

Skolemite relativism, 774

social constructivism, 118

Sophie Charlotte (Queen), 41

Sorites Paradox, 405

spatiotemporal form, 44–45

Special Relativity, 648n.33

specious arithmetic, 30

spin, 638

spread, 325

standard arithmetical inference, 508–9

standard logical truth, 758

standard model, 758

standard‐validity, 773

Stelau, Klaus, 487n.8

Stone, M., 378

Strahm, Thomas, 618

structuralism, 21–23, 536–61

in category theory, 546–51

reconsidered, 563–87

in set theory, 538–41

structures as sui generis universals, 541–46

structures, 794–95

subformula property, 371

substitutional quantifiers, 426–27

subsystems, 752

*Subsystems of Second Order Arithmetic*(Simpson), 610

Success by Default, 226–28

successor function, 11

supranatural world, 47

surrogates, 760

Sylvan‐Plumwood valuation, 745

symbolical ideal, 238

symbolic conception, 272

symbolic formalism, 263–99

symmetry, 636–41

symplectic manifold, 637n.15

*Symplectic Techniques in Physics*(Guillemin and Sternberg), 625

symplectric geometry, 637n.14

T
Tarski, Alfred, 3, 399

on axiomatic theories, 785n.6

(p. 832)
on terminology, 773

on topological spaces, 378

Tarskian semantics, 20

technique, 298

theorem of unprovability of consistency, 601

Theory of Forms, 626

theory of participation, 626

theory of symmetries, 636–41

thinkable predication, 200

third‐order quantifiers, 212

third‐order variables, 754

thought‐contents, 67

“Transcendental Aesthetic” (Kant), 44

transitivity, 734–40

transitivity of entailment, 733

*Treatise of Algebra*(Wallis), 255

truth

analytic, 11

apriority of, 52

conditions, 679–81

epistemic, 747

guarantee of, 675

Leibniz on, 40–42

and logical constants, 679–80

logical following of, 83

standard logical, 758

Turing jump operator, 604

two‐dimensional quantities, 285–86

“two‐factor” theory of entailment, 735

U
Ullian, Joseph, 414

ultraholism, 464–65

unconditional anti‐nominalism, 454n.32

unit segment, 33

universal discharge requirement, 719

unrestricted Cut, 717

unrestricted transitivity of deduction, 706

useful fictions, 42

V
vagueness principles, 404

value, 274

van Dalen, Dirk, 368

variable‐sharing requirement, 735

verification correctness principle, 398

Vieta, François, 30

Vineberg, Susan, 510–11

Virchow, Rudolf, 280

viscosity, 347

von Neumann ordinals, 137

W
Waismann, Friedrich, 262

Wang, Hao, 774

warranted assertability, 340n.47

Warren, J., 270n.47

wax argument, 37

*Web of Belief, The*(Quine and Ullian), 414

weight, 630

Wessel, C., 270n.47

Wittgenstein, Ludwig, 10, 11, 70

Y
analyticity, 65

bibliographic essay, 113–18

early period, 81–100

evolution of thought, 81–113

and Gödel, 116

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

Yablo, Stephen, 528–34

Z
Zeno's paradoxes, 346