Language and Mathematics: An Interdisciplinary Guide

by Marcel Danesi

Overview

This book explores the many disciplinary and theoretical links between language, linguistics, and mathematics. It examines trends in linguistics, such as structuralism, conceptual metaphor theory, and other relevant theories,to show that language and mathematics have a similar structure, but differential functions, even though one without the other would not exist.

Product Details

ISBN-13:	9781614515548
Publisher:	De Gruyter
Publication date:	06/06/2016
Series:	Language Intersections Series
Pages:	344
Product dimensions:	6.10(w) x 9.06(h) x 0.04(d)
Age Range:	18 Years

About the Author

Marcel Danesi, University of Toronto, Canada.

Table of Contents

List of figures viii

Preface x

1 Common Ground 1

1.1 Logic 6

1.1.1 Formalism in linguistics and mathematics 8

1.1.2 Syntax 18

1.1.3 Formal analysis 24

1.1.4 The structure of logic 32

1.2 Computation 36

1.2.1 Modeling formal theories 40

1.2.2 Cognitive science 46

1.2.3 Creativity 50

1.3 Quantification 52

1.3.1 Compression 53

1.3.2 Probability 55

1.4 Neuroscience 56

1.4.1 Neural structure 57

1.4.2 Blending 62

1.5 Common ground 64

2 Logic 66

2.1 Formal mathematics 69

2.1.1 Lógos and mythos 70

2.1.2 Proof 72

2.1.3 Consistency, completeness, and decidability 81

2.1.4 Non-Euclidean logic 85

2.1.5 Cantorian logic 88

2.1.6 Logic and imagination 91

2.2 Set theory 96

2.2.1 Diagrams 98

2.2.2 Mathematical knowledge 101

2.3 Formal linguistics 103

2.3.1 Transformational-generative grammar 104

2.3.2 Grammar rules 108

2.3.3 Types of grammar 110

2.3.4 Formal semantics 114

2.4 Cognitive linguistics 118

2.4.1 Conceptual metaphors 119

2.4.2 Challenge to formalism 123

2.5 Formalism, logic, and meaning 125

2.5.1 A Gödelian critique 127

2.5.2 Connecting formalism and cognitivism 128

2.5.3 Overview 129

3 Computation 132

3.1 Algorithms and models 134

3.1.1 Artificial intelligence 138

3.1.2 Knowledge representation 139

3.1.3 Programs 144

3.2 Computability theory 147

3.2.1 The Traveling Salesman Problem 147

3.2.2 Computability 153

3.3 Computational linguistics 159

3.3.1 Machine Translation 160

3.3.2 Knowledge networks 163

3.3.3 Theoretical paradigms 167

3.3.4 Text theory 172

3.4 Natural Language Processing 174

3.4.1 Aspects of NLP 175

3.4.2 Modeling language 178

3.5 Computation and psychological realism 179

3.5.1 Learning and consciousness 180

3.5.2 Overview 184

4 Quantification 193

4.1 Statistics and probability 195

4.1.1 Basic notions 197

4.1.2 Statistical tests 200

4.2 Studying properties quantitatively 202

4.2.1 Benford's Law 203

4.2.2 The birthday and coin-tossing problems 206

4.2.3 The Principle of Least Effort 209

4.2.4 Efficiency and economy 216

4.3 Corpus linguistics 219

4.3.1 Stylometric analysis 219

4.3.2 Other techniques 221

4.3.3 The statistics on metaphor 222

4.4 Probabilistic analysis 224

4.4.1 The Monty Hall Problem 226

4.4.2 The Prosecutor's Fallacy 227

4.4.3 Bayesian inference 228

4.4.4 General implications 230

4.5 Quantifying change in language 237

4.5.1 Lexicostatisties and glottochronology 237

4.5.2 Economy of change 245

4.6 Overview 248

5 Neuroscience 255

5.1 Neuroscientific orientations 256

5.1.1 Computational neuroscience 257

5.1.2 Connectionism 262

5.1.3 Modularity 264

5.1.4 Research on metaphor 266

5.2 Math cognition 268

5.2.1 Defining math cognition 270

5.2.2 Charles Peirce 272

5.2.3 Graphs and math cognition 274

5.2.4 Neuroscientific findings 276

5.3 Mathematics and language 284

5.3.1 Mathematics and figurative cognition 285

5.3.2 Blending theory 287

5.4 Concluding remarks 294

Bibliography 297

Index 327