Representation and Invariance of Scientific Structures

Cover......Page 1
Title Page......Page 5
Contents......Page 9
Preface......Page 15
1.1 General Viewpoint......Page 19
The traditional sketch......Page 20
Models versus empirical interpretations of theories......Page 21
Intrinsic-versus-extrinsic characterization of theories......Page 23
Coordinating definitions and the hierarchy of theories......Page 25
Instrumental view of theories......Page 26
1.3 Plan of the Book......Page 28
1.4 How To Read This Book......Page 32
2.1 Meaning of Model in Science......Page 35
Quotations from the physical sciences......Page 36
Quotations from the social sciences......Page 37
Comments on quotations......Page 38
2.2 Theories with Standard Formalization......Page 42
Example: ordinal measurement......Page 43
Axiomatically built theories......Page 44
Difficulties of scientific formalization......Page 45
Useful example of formalization......Page 46
2.3 Theories Defined by Set-theoretical Predicates......Page 48
Example: theory of groups......Page 49
Meaning of ‘set-theoretical predicate’......Page 50
Basic structures......Page 51
Reservations about set theory......Page 52
Before Euclid......Page 53
Euclid......Page 54
Archimedes......Page 55
Euclid\'s Optics......Page 58
Ptolemy\'s Almagest......Page 59
Jordanus de Nemore......Page 61
Newton......Page 62
Modern geometry......Page 63
Hilbert and Frege......Page 65
Physics......Page 66
3.1 Kinds of Representation......Page 69
Definitions as representations......Page 71
3.2 Isomorphism of Models......Page 72
3.3 Representation Theorems......Page 75
Homomorphism of models......Page 76
Embedding of models......Page 80
Extensive measurement......Page 81
Difference measurement......Page 84
Bisection measurement......Page 85
Conjoint measurement......Page 87
Proofs of Theorems 2–4......Page 88
3.5 Machine Representation of Partial Recursive Functions......Page 92
Unlimited register machines (URM)......Page 94
Partial recursive functions over an arbitrary finite alphabet......Page 98
Aristotle......Page 99
Hume......Page 101
Kant......Page 104
James......Page 106
Special case of images......Page 110
Psychological views of imagery......Page 111
4.1 Invariance, Symmetry and Meaning......Page 115
Meaning......Page 120
Objective meaning in physics......Page 121
4.2 Invariance of Qualitative Visual Perceptions......Page 123
Oriented physical space......Page 124
4.3 Invariance in Theories of Measurement......Page 128
Second fundamental problem of measurement: invariance theorem......Page 130
Classification of scales of measurement......Page 132
4.4 Why the Fundamental Equations of Physical Theories Are Not Invariant......Page 138
Covariants......Page 140
4.5 Entropy as a Complete Invariant in Ergodic Theory......Page 141
Isomorphism of ergodic processes......Page 143
5 Representations of Probability......Page 147
Primitive notions......Page 148
Language of events......Page 150
Algebras of events......Page 151
Axioms of probability......Page 152
Discrete probability densities......Page 154
Conditional probability......Page 156
Independence......Page 162
Random variables......Page 164
Joint distributions......Page 171
Modal aspects of probability......Page 172
Probabilistic invariance......Page 173
5.2 Classical Definition of Probability......Page 175
Laplace......Page 177
Classical paradoxes......Page 181
Historical note on Laplace\'s principles 3–10......Page 184
5.3 Relative-frequency Theory for Infinite Random Sequences......Page 185
Von Mises......Page 189
Church......Page 191
5.4 Random Finite Sequences......Page 196
Kolmogorov complexity......Page 197
Universal probability......Page 200
Relative frequencies as estimates of probability......Page 201
Keynes......Page 202
Jeffreys......Page 203
Carnap\'s confirmation theory......Page 208
Hintikka\'s two-parameter theory......Page 216
Chuaqui......Page 218
5.6 Propensity Representations of Probability......Page 220
Propensity to decay......Page 221
Discrete qualitative densities......Page 228
Propensity to respond......Page 229
Propensity for heads......Page 232
Propensity for randomness in motion of three bodies......Page 236
Some further remarks on propensity......Page 238
5.7 Theory of Subjective Probability......Page 243
De Finetti\'s qualitative axioms......Page 244
General qualitative axioms......Page 248
Qualitative conditional probability......Page 252
Historical background on qualitative axioms......Page 256
De Finetti\'s representation theorem......Page 258
Defense of objective priors......Page 259
General issues......Page 260
Decisions and the measurement of subjective probability......Page 263
Inexact measurement of belief: upper and lower probabilities......Page 266
Early statistical mechanics......Page 274
Quantum mechanics......Page 275
Pragmatism in physics......Page 279
Statistical practice......Page 280
6 Representations of Space and Time......Page 283
6.1 Geometric Preliminaries......Page 284
6.2 Classical Space-time......Page 287
Historical remarks......Page 290
6.3 Axioms for Special Relativity......Page 293
Historical remarks......Page 296
Later qualitative axiomatic approaches......Page 299
6.4 How to Decide if Visual Space is Euclidean......Page 300
The hierarchy of geometries......Page 305
6.5 The Nature of Visual Space: Experimental and Philosophical Answers......Page 306
6.6 Partial Axioms for the Foley and Wagner Experiments......Page 315
Contextual geometry......Page 318
Distance perception and motion......Page 319
Objects of visual space......Page 320
6.8 Finitism in Geometry......Page 321
Quantifier-free axioms and constructions......Page 323
Affine Axioms......Page 324
Theorems......Page 326
Analytic representation theorem......Page 327
Analytic invariance theorem......Page 328
Assumed mathematical concepts......Page 331
Space-time structure......Page 336
Primitive notions......Page 337
The axioms......Page 338
Two theorems—one on determinism......Page 341
Momentum and angular momentum......Page 343
Laws of conservation......Page 345
7.2 Representation Theorems for Hidden Variables in Quantum Mechanics......Page 350
Factorization......Page 351
Locality......Page 353
GHZ-type experiments......Page 356
Second-order Gaussian theorems......Page 360
7.3 Weak and Strong Reversibility of Causal Processes......Page 361
Weak reversibility......Page 362
Strong reversibility......Page 364
Ehrenfest model......Page 366
Deterministic systems......Page 367
8 Representations of Language......Page 371
8.1 Hierarchy of Formal Languages......Page 372
Types of grammars......Page 374
Normal forms......Page 375
Operations on languages......Page 377
Unsolvable problems......Page 378
Finite automata......Page 379
Languages accepted by finite automata......Page 382
Regular grammars and finite automata......Page 385
Pushdown automata and context-free languages......Page 389
Turing machines and linear bounded automata......Page 391
8.3 Stimulus-response Representation of Finite Automata......Page 392
Stimulus-response theory......Page 395
Representation of finite automata......Page 398
Response to criticisms......Page 405
Another misconception: restriction to finite automata......Page 412
Axioms for register learning models......Page 415
Role of hierarchies and more determinate reinforcement......Page 419
8.4 Representation of Linear Models of Learning by Stimulus-sampling Models......Page 421
Modification of general axioms......Page 422
Preliminary theorems......Page 423
Theorems involving the sequence ω_n......Page 429
Limit assumptions......Page 432
8.5 Robotic Machine Learning of Comprehension Grammars for Ten Languages......Page 437
Problem of denotation......Page 438
Background cognitive and perceptual assumptions......Page 439
Internal language......Page 442
General learning axioms......Page 443
Specialization of certain axioms and initial conditions......Page 447
The Corpora......Page 450
Empirical results......Page 451
Grammatical rules......Page 456
Related work and unsolved problems......Page 459
Some historical background......Page 460
Observing the brain\'s activity......Page 462
Methods of data analysis......Page 464
Three experimental results......Page 468
Criticisms of results and response......Page 471
Computation of extreme statistics......Page 474
Analysis of earlier studies......Page 476
Test of a timing hypothesis for the experiment with 100 sentences......Page 479
Censoring data in the visual-image experiment......Page 481
8.7 Epilogue: Representation and Reduction in Science......Page 483
Summary Table of Representation and Invariance Theorems by Chapter......Page 489
A......Page 493
B......Page 494
C......Page 495
D......Page 497
E......Page 498
F......Page 499
G......Page 501
H......Page 502
I......Page 503
K......Page 504
L......Page 506
M......Page 508
N......Page 510
P......Page 511
R......Page 512
S......Page 513
T......Page 518
V......Page 519
Z......Page 520
B......Page 521
F......Page 522
K......Page 523
M......Page 524
R......Page 525
W......Page 526
Z......Page 527
A......Page 529
B......Page 531
C......Page 532
D......Page 534
E......Page 535
G......Page 536
H......Page 537
I......Page 538
L......Page 539
M......Page 541
O......Page 544
P......Page 545
R......Page 547
S......Page 550
T......Page 552
V......Page 553
Z......Page 554
Dust Jacket......Page 556