LHL 10: Mathematical Foundations of Linguistics

Mathematical Foundations of Linguistics

H. Mark Hubey
Montclair State University

Only a few decades ago, only mathematicians, physicists and engineers took calculus courses, and calculus was tailored for them using examples from physics. This made it difficult for students from the life sciences including biology, economics, and psychology to learn mathematics. Recently books using examples from the life sciences and economics have become more popular for such students. Such a math book does not exist for linguists. Even the computational linguistics books (Formal Language Theory) are written for mathematicians and computer scientists.

This book is for linguists. It is intended to teach the required math for a student to be a scientific linguist and to make linguistics a science on par with economics, and computer science.

There are many concepts that are central to the sciences. Most students never see these in one place and if they do, they have to wait until graduate school to obtain them in the often-dreaded "quantitative" courses. As a result sometimes it takes years or even decades before learners are able to integrate what they have learned into a whole, if ever. We have little time and much to do.

In addition to all of these problems we are now awash in data and information. It is now that the general public should be made aware of the solution to all of these problems. The answer is obviously "knowledge compression". Knowledge is structured information; it is a system not merely a collection of interesting facts.

What this book does, and what all other math books do is teach people the tools with which they can structure and thus compress information and knowledge around them. It has also been said that mathematics is the science of patterns; it is exactly by finding such patterns that we compress knowledge. We can say that mathematics is the science of knowledge compression or information compression.

This book provides the basic tools for mathematics (even including a short and intuitive explanation of differential and integral calculus). The broad areas of linguistics, probability theory, speech synthesis, speech recognition, computational linguistics (formal languages and machines), historical linguistics require mathematics of counting/combinatorics, Bayesian theory, correlation-regression analysis, stochastic processes, differential equations, vectors/tensors. These in turn are based on set theory, logic, measurement theory, graph theory, algebra, Boolean algebra, harmonic analysis etc.

The mathematical fields introduced here are all common ideas from one which one can branch off into more advanced study in any of these fields thus this book brings together ideas from many disparate fields of mathematics which would not normally be put together into a single course. This is what makes this a book especially written for linguists.

Table of Contents:

1. Generic Building Blocks
Layering
Numerals, Multiplication, Constants and Variables
Summation--Gauss
Zeno's Paradox and Euler
Continuous Products
Decision Trees, Prisoner's Dilemma
CPM/PERT Methods
2. Symbolic Computation, Iteration and Recursion
Algorithmic Definition of Integers
Parallel and Serial Choices
Multiplicative vs. Additive Intelligence
Strong vs. Long Chain Trade-off
Recursion/Iteration and Solution of a Nonlinear Equation
Programming Charts
Learning Iteration
Frequency vs. Wavelength
3. Basic Counting and Reasoning Principles
Product, Series
Logical-AND Rule
4. Hazards of Doing Science
Dimensionless Numbers
Mass vs. Surface Area
5. Normalization
Grade Normalization, Boxing Normalization
Extensive vs. Intensive Variables
Gymnastics & Diving
Boyle's Law and Charles's Law
Color Space & Vectors
Torque
Brain and Body Mass
6. Accuracy and Precision
Significant Digits
Paleontology
7. Reliability and Validity
Ratio Scale
Distance
Hamming Distance
Phonological Distance -- Distinctive Features
Vowels and Consonants- Ordinal Cube
What's a Bird?
Interval Scale
Temperature Scale
Ordinal Scale
Likert Scale
Nominal Scale
Sets, and Categorization
8. Sets: An Introduction
Languages
Cardinality, Empty Set
Union, Intersection, Partition, Power Set, Complement, Difference
Characteristic Bitstrings (functions)
9. Graphs: An Introduction
Subgraphs, Unions & Intersections of Graphs
Graph Representation: Incidence, Adjacency, Degree, Paths, Digraphs
Hypercubes, Complete Graphs, Bipartite Graphs
Multiple Comparisons of Historical Linguistics
Representation: Incidence and Adjacency
Matrices
Euler Circuits
10. Objects & Spaces
Cartesian Products, Vectors, Matrices, Tensors
Matrix Multiplication
Zero-One Matrices, Toeplitz matrices
Markov Matrices, Leontieff Matrices, Phonotactics Matrices
Rotation Matrices of Computer Graphics
Sonority Scale and Vectors
Venn Diagrams (Set Independence?)
11. Algebra: How many kinds are there?
Arithmetic
Language Capability
Substitutes and complements
Intelligence Theory and Testing
12. Boolean algebra
Infinity Arithmetic
Electrical Circuits and Infinity
Parallel Circuits vs. Series Circuits
XOR, EQ
Representation of Integers
Hamming Distance and XOR
Phoneme Maps
13. Propositional Logic
Implication
Hempel's Raven paradox
Paradoxes of Logic
Rules of Inference
Fallacies
Integers: Division Algorithm
Divisibility
Fundamental Theorem of Arithmetic
gcd, and lcm
Mod, Div, and All that (methods of proof)
Congruence Mod m
Pseudo-Random Number Generators
Caesar Cipher, ROT13, Comparative Method
Fuzzy Logic
Appendix: Axiomatizations of Logic
14. Quantification
Syllogisms
Continuous Products & Continuous Sums
Predicates
Quantification of Two Variables
Mathematical Induction
Time-Space Super-Liar Paradox
15. Relations
Reflexive, Symmetric, Anti-symmetric, Transitive relations
Representation of Relations
Set-theoretic representation
Matrix-representation
Graph-theoretic representation
16. Boolean Matrices and Relations
Composition of Relations--associative operation
Powers of Matrices of Relations
Equivalence Relation
Inverses
Operators and Operands (see Section 26: Operator Theory)
17. Partially Ordered Sets: partitions
Hasse Diagrams -- prerequisite structure of this book
Lattices, Subsets
18. Functions, Graphs, Vectors
One-to-One Functions
Onto Functions
One-to-one correspondence
Function Inverse
Graph Isomorphism
Minimal Spanning Trees
Family Trees
Cladistics
Genetic Tree of Indo-European Languages and Isoglosses
Vector Functions
Distances on Vectors, Weighted Distances
Intelligence Measurement
Systems of Equations -- Algebraic Modeling
19. Asymptotic Analysis and Limits
Big-O notation
20. Fuzzy Logic
Axioms
Invariants of Logic
Continuous Logics
Generalized Idempotent and Continuous Max-Min Operators
21. Counting Principles
Pigeonhole Principle
Sound Changes
Permutations
Combinations
Words, Subsets, Sentences, Constrained Sentences
Queues, Books, Phonotactics, Length constraints
Distributing Objects to Containers
Vervet Languages
Distribution of Meanings
Pascal's Identity, Vandermonde's Identity, Binomial Theorem
Inclusion-Exclusion Theorem
False Cognacy Problem
22. Induction, Recursion, Summation
23. Recurrence, Iteration, Counting
Linear homogeneous first-order difference equation
Fibonacci Series
False Cognacy Problem
Coupled Difference Equations
Bitstrings and Polynomials
Polya's Method of Counting
24. Formal Language Theory
Real Human Languages
Finite State Automata and Regular Languages
Context-Free Languages
Context-Sensitive Languages and Natural Languages
25. Simple Calculus
Rates
Integration from Summation
26. Probability Fundamentals
Addition Theorem
Multiplication Theorem
Independence And Conditional Probability
27. Discrete Probability Theory from Counting
28. Bayes Theorem
29. Operator Theory (see Chapter 17)
Linear Operators
Commutativity
Integration and Differentiation
30. Statistics
Histogram
Correlation-Regression
31. Expectation Operator & Density Functions
Expectation and Moments
32. Discrete Probability Functions (Mass Functions)
Uniform
Geometric - Bernoulli
Binomial
Hypergeometric
Poisson
Birthday Problem
33. Continuous Probability Density Functions
Uniform
Exponential
Gamma Density and Chi-Square Density
Gaussian Density and the Central Limit Theorem
34. Joint and Marginal Density Functions
35. Stochastic Processes
Stationarity
Markov Processes
Chapman-Kolmogorov Equations
Speech Recognition
Random Walk
36. Harmonic Analysis
Delta function
Fourier Series and Fourier Transform
37. Differential Equations, Green's Function and the Convolution Integral
Complete solution of the First Order Linear DE
Carbon Dating
Menzerath's Law
Altmann's Law
Damped Harmonic Oscillator
38. Time and Ensemble Moments - Stationarity and Ergodicity
Stationarity
Ensemble Correlation Functions
Time Averages and Ergodicity
39. Characteristic Functions, Moments and Cumulants
40. Stochastic Response of Linear Systems
Example of Word Production in a Language
Stochastic Excitation of the DHO
41. Fokker-Planck- Kolmogorov Methods
Generalization of the Random Walk
Replacement of a General Process by a Markov Process
Appendices; Calculation of some integrals; References.

ISBN 9783895866418. LINCOM Handbooks in Linguistics 10. 260pp.1999.