# Chapter 1. Geometry, Numbers and Sets

Ask someone what mathematics is about, and most of the time in this day and age something to do with numbers will be mentioned straight away (unless it's just "something I hated in high school until I was allowed to give it up"). This is a long way short of the whole story: Webster's (1997) dictionary sums this up, defining mathematics as

1) the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.

There are numbers, for sure, and we will find them very useful tools --- but mathematics is also "the science ... of space configurations and their structure." This branch of mathematics is traditionally known as geometry, and was rigourously studied some two thousand years before the same exacting standards came to be applied to the study of numbers. The Oxford English Dictionary, with its typically traditional and didactic voice, recognizes this historical precedence, defining mathematics as

1. Originally: (a collective term for) geometry, arithmetic, and certain physical sciences involving geometrical reasoning, such as astronomy and optics; spec. the disciplines of the quadrivium collectively. In later use: the science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis; mathematical operations or calculations.

As this definition suggests, the traditional sciences such as astronomy and optics found geometry to be an invaluable tool, as have many other sciences, including music (understanding the relationships between notes of different pitch), mechanics (understanding the path of a projectile), chemistry and biochemistry (understanding the structure of different atoms and molecules) and physics (one of whose crowning glories, Einstein's General Theory of Relativity, is an entirely geometric theory). Geometry has contributed much and benefitted enormously in its interaction with these different fields. The purpose of this book is to describe the way in which a new interaction is developing between the study of geometry and the study of language --- in particular, the study of meaning in the new and practical field of natural language processing. To begin this story, it is important to consider the question, "What is geometry?" Why on earth should the study of space configurations and their structure be beneficial when dealing with meaning in language?

As well as introducing geometry, this chapter discusses some of the relationships between geometric objects, other mathematical concepts such as numbers and sets, and the way mathematical concepts have been used to describe and model the world around us. Some of this chapter (in particular, the introductions to numbers and set theory) are partly for later reference. Any readers with little mathematical background who find these sections difficult are urged not to get bogged down. Much of the material in the rest of the book can be understood, albeit less formally, without going into every bit of mathematical detail. Also, some of the complicated-looking ideas and symbols introduced in this chapter may become much more familiar through being used in practical linguistic situations. Readers who follow the gist but not the details in this chapter will probably get more knowledge and much more enjoyment from this book by pressing on to further chapters and referring back to different sections in Chapter 1 as necessary.

This health-warning aside, the next section which introduces geometry itself is a must-read, especially for those who have a good grounding in formal methods generally but are unaware of geometry and its story.

## Sections

#### 1. What is Geometry?

This section explains how pervasive spatial awareness is to the way we approach the world, and how much we can predict just by considering the relationships between objects. Examples include packing your luggage, steering a car, and the phases of the moon. The phases of the moon and the time of moonrise and moonset can be predicted by reasoning about the relative positions of three spheres.

This section also tells the story of Euclid of Alexandria who compiled the Elements of Geometry, the most widely used textbook in the history of the world.

#### 2. Sets, Relations and Mappings

Introductory material on set theory, introducing the idea of sets and the notions of intersection, union and complementation. Also describes the relationship between geometric containment and logical implication, Aristotle's fundamental link between geometry and meaning.

#### 3. Numbers and Ratios, Mysticism and Music

Why are numbers important? How do they describe the shapes of sounds and crystals, the fundamental elements of which the universe is made, and the orbits of the planets? The five Platonic solids: Plato taught that four of these were thought to correspond to the four elements of matter (earth, air, fire and water) ... and then a fifth was discovered by one of his pupils!

Some of these correspondences between mathematical patterns and the physical world are well-founded, others are now seen as spurious, history's reminder that we must be very careful when applying mathematical models to real-world situations.

#### 4. Numbers and Measurement

The main use of numbers in this book is to measure and compare different observations. This process often involves choosing a fundamental length (a unit) and a reference or zero point. These choices are often more or less arbitrary, and vary for scientific and often historical reasons. This section also discusses positional notation (introduced by the Babylonians some 3000 years before Hindu-Arabic numerals made it to Western Europe). The symbol "3" can refer to 3 units, 3 tens, or 3 hundreds, depending on its position - one of the first examples of systematic ambiguity we encounter, where symbols mean different things in different contexts.

#### 5. Different Kinds of Numbers and Measurements

Some measurements are continuous - however small an interval you consider, there are always smaller divisionds you could make. Others are discrete - the number of books on my desk is a whole number, never a fraction. This difference corresponds to the difference between digital and analogue desriptions. Most measurements in this book are discrete - we deal primarily with text, which has already been digitised or 'quantised' into a finite alphabet of characters for representation in a computer. However, many of the mathematical models used in the book are based upon continuous mathematics.

#### 6. Numbers in everyday language

Numbers are used to count all sorts of different objects. While numbers are thought of as absolute and unchanging, they are among the most ambiguous words we can use. Every street in the world may have a house with the number 9 - so long as there are no two houses with the same number in the same street or 'context,' everything is fine. In mathematics, numbers may be absolute - in language, we have to rely on a host of robust human conventions, and we are so used to this that we don't even notice. Small wonder that it is hard to get computers to respond appropriately to human language!
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