This book presents the author's research on automatic learning procedures for categorial grammars of natural languages. The research program spans a number of intertwined disciplines, including syntax, semantics, learnability theory, logic, and computer science. The theoretical framework employed is an extension of categorial grammar that has come to be called multimodal or type-logical grammar. The first part of the book presents an expository summary of how grammatical sentences of any language can be deduced with a specially designed logical calculus that treats syntactic categories as its formulae. Some such Universal Type Logic is posited to underlie the human language faculty, and all linguistic variation is captured by the different systems of semantic and syntactic categories which are assigned in the lexicons of different languages. The remainder of the book is devoted to the explicit formal development of computer algorithms which can learn the lexicons of type logical grammars from learning samples of annotated sentences. The annotations consist of semantic terms expressed in the lambda calculus, and may also include an unlabeled tree-structuring over the sentence.

The major features of the research include the following:

We show how the assumption of a universal linguistic component---the logic of language---is not incompatible with the conviction that every language needs a different system of syntactic and semantic categories for its proper description.

The supposedly universal linguistic categories descending from antiquity (noun, verb, etc.) are summarily discarded.

Languages are here modeled as consisting primarily of sentence trees labeled with semantic structures; a new mathematical class of such term-labeled tree languages is developed which cross-cuts the well-known Chomsky hierarchy and provides a formal restrictive condition on the nature of human languages.

The human language acquisition mechanism is postulated to be biased, such that it assumes all input language samples are drawn from the above "syntactically homogeneous" class; in this way, the universal features of human languages arise not just from the innate logic of language, but also from the innate biases which govern language learning.

This project represents the first complete explicit attempt to model the aquisition of human language since Steve Pinker's groundbreaking 1984 publication, "Language Learnability and Language Development."

The original title of this manuscript was Pages - A Voyage to Infinity. It's kinda like Walt Whitman's Leaves of Grass; a collection of poems with an underlying mystical theme. My dissertation is a kaleidoscope puzzle of images and thoughts and concepts and ideas taken from mysticism, science, logic, and mathematics. The end result, as the puzzle pieces are linked together, is a new portrait of Number. The reader is challenged to solve a conceptual picture puzzle using the chaotic scattering of puzzle pieces set forth in the thesis.

Some of the pieces challenge established ideas. Some of the pieces are decoys leading to dead ends. Some are background. Others are transition pieces. And, there are pieces that give the reader glimpses of me, the writ er of this thesis. So, exact ly what is infinit y? As the pieces of the puzzle are put together a new concept of infinity emerges, a concept that may be of interest to the mystics, the philosopher, the quantum physicist, and mathematicians who are open to a new window through which to view reality.

This book presents the author's research on automatic learning procedures for categorial grammars of natural languages. The research program spans a number of intertwined disciplines, including syntax, semantics, learnability theory, logic, and computer science. The theoretical framework employed is an extension of categorial grammar that has come to be called multimodal or type-logical grammar. The first part of the book presents an expository summary of how grammatical sentences of any language can be deduced with a specially designed logical calculus that treats syntactic categories as its formulae. Some such Universal Type Logic is posited to underlie the human language faculty, and all linguistic variation is captured by the different systems of semantic and syntactic categories which are assigned in the lexicons of different languages. The remainder of the book is devoted to the explicit formal development of computer algorithms which can learn the lexicons of type logical grammars from learning samples of annotated sentences. The annotations consist of semantic terms expressed in the lambda calculus, and may also include an unlabeled tree-structuring over the sentence.

The major features of the research include the following:

We show how the assumption of a universal linguistic component---the logic of language---is not incompatible with the conviction that every language needs a different system of syntactic and semantic categories for its proper description.

The supposedly universal linguistic categories descending from antiquity (noun, verb, etc.) are summarily discarded.

Languages are here modeled as consisting primarily of sentence trees labeled with semantic structures; a new mathematical class of such term-labeled tree languages is developed which cross-cuts the well-known Chomsky hierarchy and provides a formal restrictive condition on the nature of human languages.

The human language acquisition mechanism is postulated to be biased, such that it assumes all input language samples are drawn from the above "syntactically homogeneous" class; in this way, the universal features of human languages arise not just from the innate logic of language, but also from the innate biases which govern language learning.

This project represents the first complete explicit attempt to model the aquisition of human language since Steve Pinker's groundbreaking 1984 publication, "Language Learnability and Language Development."

The original title of this manuscript was Pages - A Voyage to Infinity. It's kinda like Walt Whitman's Leaves of Grass; a collection of poems with an underlying mystical theme. My dissertation is a kaleidoscope puzzle of images and thoughts and concepts and ideas taken from mysticism, science, logic, and mathematics. The end result, as the puzzle pieces are linked together, is a new portrait of Number. The reader is challenged to solve a conceptual picture puzzle using the chaotic scattering of puzzle pieces set forth in the thesis.

Some of the pieces challenge established ideas. Some of the pieces are decoys leading to dead ends. Some are background. Others are transition pieces. And, there are pieces that give the reader glimpses of me, the writ er of this thesis. So, exact ly what is infinit y? As the pieces of the puzzle are put together a new concept of infinity emerges, a concept that may be of interest to the mystics, the philosopher, the quantum physicist, and mathematicians who are open to a new window through which to view reality.

The relativity of time, space, and mass is covered first, giving some attention to the history of the two main divisions of relativity, the special and the general. Once special relativity and its mathematics are established, general relativity is covered, beginning with its relationship to Newton's laws and advancing through its revolutionary concepts as well as its mathematics.

This process is carried all the way to the level of tensor equations. The Mathematics of Relativity for the Rest of Us treats topics such as: The constant speed of light, the invariant laws of physics, the basis and meaning of the equation E = mc², the nature of curved four-dimensional space-time, the importance of non-Euclidean geometry, the gravitational bending of light, experimental confirmation of relativity, the philosophical and intellectual appeal of relativity, the nature of black holes, and the cosmologic significance of relativity -- both as concepts and as mathematical issues.

As a result the sufficiently attentive reader is set at ease with the reputedly incomprehensible but essential details about relativity. Even subjects such as "tensor calculus" and the "covariant partially differential field equations of general relativity" will be clear. For instance such a reader will know just what a "tensor" is, why the equations are "covariant," why they are "partially differential," why they are "field" equations, why relativity can be "general," and most importantly just what is meant by "relativity." Furthermore, if a reader is shown the fundamental equation of general relativity,

he or she will understand what every term of this equation means, why each is included, what obstacles Einstein and his colleagues overcame to derive each term, what impact this equation has on modern science, and why this equation revolutionized our understanding of our universe.

The Mathematics of Relativity for the Rest of Us also devotes a chapter to the relationship between relativity and quantum mechanics. It reveals the limitations of relativity and the direction of future work in this branch of science. The chapter concludes with the role of string theory in reconciling relativity and quantum mechanics.

There are four Appendices. The Appendix A contains units of measurements from the Système International d'Unitès along with some selected physical con stants. The Appendix B contains a listing of Christoffel symbols of the second kind associated with various coordinate systems. The Appendix C is a summary of use ful vector identities. The Appendix D contains solutions to selected exercises. The text has numerous illustrative worked examples and over 450 exercises.