Subject Index

A

Abelian groups, 202

Algebraic geometry, 121

Alikeness relation, 155–56

Analysis, 122

Analytic/synthetic distinction, 168

Antinominalism, 10, 134, 135

Antirealism, 4, 10, 12, 135–37

Approximation, 78

Arbitrariness, 120

Archimedean, 17, 30; axiom, 8, 15, 47, 79; condition, 67; in difference sequence (ds-Archimedean), 23; in standard differences (sd-Archimedean), 19; in standard sequences (ss-Archimedean), 18; intrinsically, 7, 17; ordered group, 26; property, 7

Archimedes’ law, 157, 158

Associativity, 5

Attributes: extensive, 3; intensive, 3

Automorphism, 25, 36–37; invariant under, 29

Axiomatization, 185

Axioms: empirical status of, 8, 53; nonnecessary, 68

B

Best explanation, argument from, 175

Boolean: algebra, 58, 202; algebras, ordered, 57; terms, 61

Borel field, 8

C

Categories, 216

Classificatory notions, 139

Coding, 120, 125–27, 132

Coherentism, 11

Collision, laws of, 102

Commensurability, 16

Comparative notions, 139

Comparison, higher-order, 218

Computability of measurement, 105

Concatenation, 2, 4, 189; structures, 17; structures, Dedekind complete, 24; structures, extensive, 22; structures, intrinsically Archimedean, 21

Confirmation, structuralist view of, 112

Congruence, 145

Conservation of momentum, law of, 102

Continuity axiom, 131n

Continuum, 16

Conventionalism, 4, 11, 182, 190

Conventions, 167–69

Coordinates, frame of, 120

Correspondence rules, 186, 197

Counting, 132

Covering condition, 149

D

Data: empirical strength of, 8, 61–62; identity, 61; inclusion, 61; observational, 54, 56, 69; ordinal, 61, 64

Datum, elementary, 61

Dedekind complete, 16

Demarcation, 113

Difference sequence, 23; Archimedean in (ds-Archimedean), 23

Dilations, 26

Dilemma, axiomatizer’s, 56

Dimension: complex, 171; simple, 171

Dimensional analysis, 198

Distribution: beta, 50; binomial, 50; moments of, 40; normal, 51; principle, 79, 80; uniform, 50

Divine Mind, 133

E

Electrodynamics, 172

Empirical: content, 8, 54; content, contributory, 56; equivalences, 66; laws, quantitative, 142; presuppositions, 170, 177

Empirical/conventional distinction, 168, 174

Empiricism, 12, 179, 195–98, 202–4, 206, 210, 216, 220; logical, 100, 109

Entities, theoretical, 175, 180

Equal: n-partition, 59; n-partitionable, 59; temperatures, law of, 153–54, 155n

Equipartionability, axioms of, 8

Error: mean squared, 83; principle, minimum, 84; single, 76; theory of, 39

Errors of measurement, 9

Expected utility, 191; theory, 185, 190

Experimental laws, 77

Extension, common, 204

F

Facts, purely logical, 57

Falsifiable theory, 70

Fixed-point equation, 215

Forces, fundamental, 176

Formalism, 7–8, 12, 182–84

Formalist view of measurement (see Formalism)

G

Galilean conditions, 125

Geometric algebra, 121, 128–29, 130

Geometry: analytic, 1, 119; Euclidean, 10; synthetic, 1, 119

Goedel-Bolker transformations, 113n

H

Hausdorff’s theorem, 42, 45

Heat theory, 172

Hilbert space, 202

Hilbert’s program, 181

Hölder’s theorem, 202

Holism, 11–12, 167–68, 170, 175, 179; epistemological, 174

Homogeneous, 21, 26; M-point, 26

Homomorphism, 10, 198

I

Ideal gas law, 90

Idealism, 182

Ideology, 120, 124, 137

Indifference class, 82

Inertia, 159

Injection, 207

Instrumentalism, 4, 11, 182

Integral: Borel, 142; Lebesgue, 142; Riemannian, 142

Intensional comparatives, 212

Interactionist, 198; approach, 13, 196; theories, 198

Invariance, 120, 120n

Invariant, 124

J, K

Jeffrey utilities, 71

Kepler’s laws, 133

Klein’s Erlanger program, 206

Kolmogorov axioms, 7

Koopman Archimedean: axiom, 66; condition, 59

Kuhn-loss, 109

L

Laws, purely universal, 54

Least squares estimates, 84

Length: ideal, 79; measurement of, 85, 99; true, 82

Löwenheim-Skolem theorem, 35, 35n, 55

M

Magnitudes: additive, 188; commensurable, 151; derived, 12, 187–88; extensive, 188; fundamental, 187, 188; intensive, 185–86, 191; multiplication of, 151; nonadditive, 188; nonextensive, 185, 188

Materialism, 12, 182, 193

Materialist view of measurement (see Materialism)

Mathematization, 181, 193

Matter: parity of, 163; quantity of, 160–62

Measurement: as assignment of numerals, 2; as assignment of numbers, 2; associative, 187; bisection, 185; chains of, 111; class, fundamental, 97; conjoint, 185; conservative conception of, 2; derived, 9, 100, 109, 187, 189; difference, 185; direct, 2; errors of, 111; extensive, 3, 189, 202; extraphysical, 189, 190–91; fundamental, 9, 67, 96, 107–9, 187–88; fundamental extensive, 39; graph, 111; intensive, 3; length, 96, 97; liberal conception of, 3; method of, 106; ontological grounding of, 183; physical, 191; received view of, 94; representational, 199–200; theory, interactionist, 221; unit, 190–92

Measures: empirical, 162, 163; sensible, 163

Measuring model, 101–4, 106; for mass, 102

Mechanics: law of, 9; synthetic, 10

Metricization, 182, 183, 185–87, 189

Minimization principle, 79, 80

Minimizing observational error, 86

Modal: logic, 215; reasoning, 216

Moments: independence of, 47; measurement of qualitative, 41–42; of a distribution, 8; qualitative ordering of, 44

Monotonic, 18; completely, 42; qualitatively completely, 46

Monotonicity, 5

Moral weight, 212

N

Nominalism, 10–11, 120n, 122, 132, 134–35, 144

Nonnecessary axiom, 54

Nonnumerical: formula, 126; invariant, 127–28; ontology, 120

O

Observation: language, 174; statements, 81

Observational axioms, 200

Ohm’s law, 93

Ontology, 120, 126, 137

Operation: associative, 18; commutative, 18; idempotent, 18; positive, 18

Operationalism, 4, 11–12, 182–84, 186–88; homocentric, 182; methodological, 182

Operationalist view of measurement (see Operationalism)

Order indiscernability, 207

Ordering: asymptotic, 25; dense, 16; weak, 59

P

Partition, 145, 153

PCS, 25

Physics: analytic, 119, 120n, 122, 132; synthetic, 119, 120n, 130–32

Platonism, 10, 120n, 135, 182

Positivism, 167, 181–83

Predictions, 70

Primigenial particles, 161

Principia, 163

Probability: de Finetti-style qualitative, 200; Dirac, 218; Kolmogorovian, 215; measure, Gleason, 202; of measurement, 7; qualitative, 207; relations, qualitative, 216; spaces, compatibility of, 205; spaces, Kolmogorovian, 206; structure, qualitative, 13

Projection map, 13, 207

Properties: extensive, 2; intrinsic, 176; perceptual, 4; primitive quantitative, 170

Proportionality, 127

Proportions, Eudoxian theory of, 152

Psychological magnitudes, 4, 90

Psychometrics, 4

Pythagoras’ theorem, 93

Q

Qualitative concepts, 139–40

Quality/quantity, 11, 139ff

Quantifiers, cardinality-comparison, 121n, 131n

Quantitative concepts, 139–40

Quantitiative law, 173

Quantities, 95; basic, 172; dimensionless, 172–73; extensive, 198, 199; fundamental, 204; intensive, 198, 199; nonsensible, 163; random, 39

R

Ramsey: embedding, 207, 208; representation formula, 200; sentence, 12, 197, 201, 214–15

Rational reconstruction, 12, 167, 170, 178–79

Ratios, identity of, 131

Realism, 10–12, 88, 122, 133, 135–37, 155–56, 167–70, 175, 179–80, 193ff, 204ff; scientific, 176

Realist view of measurement (see Realism)

Reductionism, 187

Reference, structuralist view of, 112

Relations: classificatory, 140, 146; comparative, 140, 147, 210; empirical, 12; equivalence, 145–47; nonnumerical, 150; numerical, 150; qualitative, 11, 143–44; quantitative, 11, 140, 143–44, 148, 170; stochastic order, 213

Representability: hypothesis, probabilistic, 54, 59; necessary conditions for, 53–56; nonnecessary conditions for, 54–55

Representation: quantitative, 199; random variable, 39–40; theorem, 5, 12, 13, 41, 47, 94, 185–86, 189

Representational: approach, 196; theory of measurement, 141, 143

Representationalism, 6–10, 12, 75–76, 90, 108, 198, 202, 214

Representationalist view of measurement (see Representationalism)

Robinson Crusoe, 81, 82

S

Scale: additive, 4–5; admissible, 124; Celsius, 173; classification, 6; constructible, 6; derivative, 172; fundamental, 172; interval, 3, 5, 32, 192; invariance of, 191; Kelvin, 190; nominal, 3, 5, 183, 189; of measurement, 2, 5, 10, 120; ordinal, 3, 5; origin of, 190; ratio, 3, 32; similar classes of, 173, 178; system, 172–73, 178; type, 27; utility, 192

Semigroup, commutative, 47; with identity, 44

Sequence, standard, 18, 23, 33

Simplicity, 12, 170

Skepticism, 11

Skolem-Löwenheim theorem (see Löwenheim-Skolem theorem)

Solvability: right restricted, 21–24; unrestricted, 33

Space and time, absolute, 163

Standard deviation, 8–9, 76

Stone’s representation theorem, 58

Structuralism, 9, 108–12

Structuralist view of measurement (see Structuralism)

Structure: Archimedean conjoint, 33; basic de Finetti, 59; closed extensive positive, 99; conjoint, 33; de Finetti, 8, 66; empirical, 96; extensive, 23; extensive Archimedean, 6; homogeneous, 29; intrinsically Archimedean, 28; Koopman Archimedean, 8; numerical, 96; ordered relational, 26; random extensive, with independent objects, 46; real unit, 31; relational, 5; weak extensive, 44

Subjectivism, 11

Symmetries, 25

T

Television violence, measuring, 210

Temperature: measurement of, 84–85, 131, 188; same, 153n

Terms: observation, 168; theoretical, 110, 168, 196–97

Theories: regular, 66; empirically equivalent, 62

Theory-laden observation, 174

Thermal expansion: coefficient of, 87; law of, 87–90; theory of, 85, 86

Time, absoluteness of, 178

Topologization, 185

Transformation: grammarians, 210; linear, 104; admissible, 4, 6; linear, 6

Transformations, monotone, 6

Transitivity, axiom of, 9

Translations, 26

U

Uniqueness, 26, 103, 103n, 104, 114, 200–1; N-point, 26; theorem, 6, 185–86, 189

Universals, quantitative, 176

Utile, 191

Utility measurement, 12, 113n

V

Vacuum, 157

Variance, 8; of error distribution, 83–87

Z

Zero: absolute, 190; arbitrary scale, 191; of utility scale, 192; point, 190

Zeroth: law of thermodynamics, 154, 156; style law, 155