Mathematical Foundations of Linguistics

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.