Numbers, Language, and the Human Mind

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Acknowledgments......Page 11
Introduction......Page 15
CHAPTER 2: WHA DOES IT MEAN TO BE A NUMBER?......Page 18
CHAPTER 4: THE LANGUAGE LEGACY......Page 19
CHAPTER 6: THE ORGANISATION OF OUR COGNITIVE NUMBER DOMAIN......Page 20
CHAPTER 8: NUMBERS IN LANGUAGE......Page 21
CHAPTER 1 Numbers and objects......Page 23
Cardinality assignments for sets of objects: ‘3 pens’......Page 32
Cardinal number assignments including measures: ‘a3 kg pumpkin’, ‘water of 3 ºC’......Page 37
Direct measurement: ‘a 3 kg pumpkin’......Page 38
Indirect measurement: ‘Bathwater with a temperature of 3 ºC’......Page 43
ORDINAL NUMBER ASSIGNMENTS: ‘THE THIRD RUNNER’......Page 47
NOMINAL NUMBER ASSIGNMENTS: ‘BUS #3’, ‘FOOTBALL PLAYER #3’......Page 51
OVERVIEW: HOW TO USE NUMBERS......Page 54
B. Ordinal number assignments......Page 55
C. Nominal number assignments......Page 56
CHAPTER 2 What does it mean to be a number?......Page 57
THE INTERSECTIVE ANALYSIS OF NUMBERS......Page 58
THE ITEMISING APPROACH TO NUMBERS......Page 66
THE RELATIONAL VIEW OF NUMBERS......Page 71
A CRITERIA-BASED VIEW OF NUMBERS......Page 74
Implications of our view: possible number sequences......Page 76
Are we getting arbitrary? The power of convention......Page 78
CHAPTER 3 Can words be numbers?......Page 82
The elements of a counting sequence are well distinguished......Page 84
Counting words form an infinite progression......Page 86
Is infiniteness necessary?......Page 91
COUNTING WORDS ARE NON-REFERENTIAL......Page 93
COUNTING WORDS ARE ACQUIRED DIFFERENTLY......Page 99
A CUT WITH OCCAM’S RAZOR......Page 103
COUNTING WORDS AS NUMERICAL TOOLS......Page 105
C. Counting sequences are acquired as verbal tools......Page 106
D. Counting sequences can be defined as numbers......Page 107
CHAPTER 4 The language legacy......Page 108
How to grasp the cardinality of sets: subitising and noisy magnitudes......Page 109
Before counting: babies know ‘How many’......Page 113
Quantitative and sequential concepts in other species......Page 116
Cardinality of sets......Page 117
Arithmetic operations on the cardinality of sets......Page 118
Serial order......Page 121
NUMBER SENSE AND BEYOND......Page 122
SYMBOLIC REFERENCE AND THE EVOLUTION OF LANGUAGE......Page 127
MATCHING PATTERNS IN SYMBOLIC REFERENCE AND NUMBER ASSIGNMENTS......Page 135
Convention......Page 138
Infinity......Page 141
A POSSIBLE SCENARIO FOR THE EMERGENCE OF NUMERICAL TOOLS......Page 145
LANGUAGE AND NUMBER AS HUMAN FACULTIES......Page 157
CHAPTER 5 Children’s route to number: from iconic representations to numerical thinking......Page 165
THE ACQUISITION OF A COUNTING SEQUENCE......Page 167
PRE-NUMERICAL PROPS FOR NUMBER ASSIGNMENTS......Page 171
GATEWAY TO NUMBER......Page 174
One–One Principle: match counting words and objects......Page 178
Stable-Order Principle: counting words have to be sequenced......Page 181
Order–Irrelevance Principle: counted objects can come in any order......Page 182
Cardinality Principle: use the last counting word to identify the cardinality of the set......Page 183
Abstraction Principle: anything counts......Page 186
Numerical-Tool Principle: counting words are just one conventional progression......Page 187
LANGUAGE AND THE EMERGENCE OF COUNTING AND CARDINALITY......Page 188
CHAPTER 6 The organisation of our cognitive number domain......Page 194
REPRESENTATION OF COUNTING WORDS AS NUMERICAL TOOLS......Page 195
CONCEPTS OF NUMERICAL QUANTITY......Page 196
ABSTRACT CARDINALITIES......Page 197
MEASURE CONCEPTS......Page 202
CONCEPTS OF NUMERICAL RANK......Page 216
Principles of ordinal number assignment......Page 219
CONCEPTS OF NUMERICAL LABEL......Page 223
THE ARCHITECTURE OF THE NUMBER DOMAIN......Page 228
CHAPTER 7 Non-verbal number systems......Page 233
ARABIC NUMERALS AS A NON-VERBAL NUMBER SEQUENCE......Page 236
Number forms......Page 238
The SNARC effect......Page 242
Distance effects......Page 243
Arabic numerals as signs for numerical concepts......Page 246
THE DEVELOPMENT OF NON-VERBAL NUMERALS: FROM ICONIC REPRESENTATIONS TO NUMERICAL TOOLS......Page 250
TWO ALTERNATIVE NUMBER SEQUENCES: THE CORRELATION BETWEEN COUNTING WORDS AND ARABIC NUMERALS......Page 255
‘Reading aloud’ non-verbal numerals......Page 256
The correlation of structures......Page 261
The Modular Model......Page 267
The Encoding Complex View......Page 270
The Triple-Code Model......Page 272
A model for the correlation of numerical tools......Page 273
CHAPTER 8 Numbers in language: the grammatical integration of numerical tools......Page 278
COUNTING WORDS AND THEIR REFERENTIAL COUSINS......Page 279
HOW TO EXPRESS NUMERICAL CONCEPTS: THE ORGANISATION OF NUMBER WORD CONSTRUCTIONS......Page 284
REFERENTIAL NUMBER WORDS ARE NOT OUTCASTS......Page 288
Grammatical parallels between numerical and non-numerical domains......Page 289
Conceptual parallels between numerical and non-numerical domains......Page 295
THE RETURN OF THE COUNTING WORDS......Page 302
Appendices......Page 309
CARDINAL, ORDINAL, AND NOMINAL NUMBER ASSIGNMENTS......Page 311
Direct measurement.......Page 312
Nominal number assignments......Page 313
Schema used for the definitions......Page 314
Definition of 0 and 1 via paradigm sets (Frege 1893: 41–42)......Page 315
Necessary and sufficient conditions for the set N of natural numbers (Dedekind 1887)......Page 316
Generation of the elements of C by inductive definition......Page 318
Sequential ordering of C by a relation (C;<)......Page 320
Derivation of actual counting words from the ‘idealised’ words in C by a function…......Page 322
Sequential ordering of A by a relation (A; <)......Page 323
Mapping from A to C......Page 324
Mapping from C to A......Page 325
The pre-numerical basis: early concepts of cardinality......Page 328
Concepts of numerical quantity: cardinality identified by numerical tools......Page 329
Concepts of direct and indirect measures......Page 330
Concepts of numerical rank: ordinality identified by numerical tools......Page 331
NOMINAL CONCEPTS: NUMERICAL LABEL......Page 332
Explanations of the elements used in the formulae......Page 333
Ordinals and superlatives: assignment of a rank within a progression......Page 334
‘#’-constructions and proper names: assignment of an identificational label within a set......Page 335
References......Page 336
Index......Page 354