{

What is a Number?

Philosophy of Mathematics

In philosophy and maths we like our definitions to give necessary and sufficient conditions.

This means the definition of an X separates the world into “things that are Xs” and “things that aren’t”, with no gaps or overlap.

What are the necessary and sufficient conditions for something to be a number?

Introduction

The conditions that make a definition can attach to:

What the thing is. For example, whether it’s physical or not, its properties and so on.

What the thing does: For example, how it interacts with other things of the same kind.

In the nineteenth century the emphasis shifted from the first to the second.

Perhaps this is the influence of Kant; asking what a number “truly is” in the first sense looks like metaphysics.

We ask instead: “If something is a number, what does that mean it can do?”

Note that some philosophers, including Russell, continued to look for “is” definitions. But they get quite abstract and certainly seem far from our intuition: http://people.umass.edu/klement/imp/imp.html#chapter2 A Change of Emphasis

Numbers can act on things that are not numbers – for example, by counting. But this is potentially quite complicated, since there are a lot of different things that aren’t numbers.

Numbers can also act on other numbers. This seems more tractable because we’re dealing with a narrow range of possibilities.

What can Numbers do?

The basic arithmetic operations: Adding Subtracting Multiplying Dividing Powers (maybe)

What can Numbers do?

The natural numbers, written , are the simple counting numbers 1, 2, 3, 4, …

Which arithmetic operations can we do with the natural numbers?

The Natural Numbers

The integers, written , are the positive and negative whole numbers, plus zero.

Note that the integers include all the natural numbers.

How does this change the way arithmetic works?

The Integers

The rational numbers, written , are all the fractions you can make with positive and negative whole numbers, plus zero.

We don’t allow fractions to have “0” on the bottom.

Note that the rational numbers include all the integers, which are the fractions with “1” on the bottom.

How does this change the way arithmetic works?

The Rational Numbers

We said the objects we were working with are numbers, but now we turn it around, and say we can put any objects in our sets, and make up any operations on them. If they behave in a suitably number-like way, maybe we’ll be tempted to call them numbers.

This is “abstract” in that we don’t care what the things are any more; we claim that the structure of how they relate to each other is enough.

We’ll now develop a number system of our own, to prove the point.

Abstract Algebra

For our purposes, a group is a set of numbers with an arithmetic operation associated with it.

The group concept is actually a lot more general than this.

We’ll write elements of the group as small letters and the operation as •.

For a set of numbers and its operation to form a group, the following axioms must hold:

1. Closure: a•b must be an element of the set we started with.

2. Identity: there must be an element – call it x – such that a•x = x•a = a for every a in the set.

3. Inverses: for each a in the set there must be some other element, call it b, so that a•b = b•a = x.

4. Associativity: for all elements a, b and c we must have (a•b) •c = a• (b•c).

We’ll now check each of these for the arithmetic operations on the number systems we’ve met so far.

Groups

Our number systems really have two main arithmetic operations: addition and multiplication.

We’ve looked at whether these are groups or not, but not at how they interact.

For our purposes, a ring is a set of numbers with two operations, + and ×, that obey the following axioms:

1. Addition is a group.2. Commutativity: a + b = b + a.3. Associativity: a × (b × c) = (a × b) × c4. Identity: there is a u such that a × u = u

× a = a5. Distributivity: a × (b + c) = (a × b) + (a

× c)

Rings

We can see that a ring can nearly be defined by the following axioms:

Addition is a group and is commutative Multiplication is a group if you ignore zero Together, they obey the distributive

property. This is neat, but doesn’t always work.

For example, the integers are a ring, but multiplication of integers is not a group because we don’t have inverses.

When we can do this, the object is called a “field”. Fields are especially nice.

The rational numbers form a field.

Fields

The kind of numbers we’ve been playing with aren’t just artificial constructions.

If we take the integers from 0 to n and define addition and multiplication on them in the modular way, we always get a ring.

If n + 1 is a power of a prime number, the result is a field.

These finite fields are important in cryptography and other areas of technology as well as more “pure” maths.

Modular Arithmetic

Although the rational numbers form a field, they still aren’t the “nicest” number system we can come up with.

We can square any rational number and get another rational number. This number is always positive.

But if we take the square root of a positive rational number we do not usually get a rational number!

Here’s a proof: http://www.math.utah.edu/~pa/math/q1.html The real numbers include all the rational numbers and

a lot of others, including all those missing square roots. The result is a complete field. Constructing or even “imagining” these numbers is quite

difficult, and wasn’t done until the nineteenth century. They’re one of the most commonly-used number systems

in general use. You can learn a bit more about them more on the courses on

Calculus and Infinity.

Real Numbers

One arithmetic operation we haven’t talked about yet is raising a number to a power.

Even the real numbers don’t behave nicely under this operation.

For example, there’s no solution to the equation x2 + 1 = 0.

This posed a problem for mathematicians all the way back in sixteenth century Italy. They solved the problem by “pretending” there were square roots of negative numbers while working out problems, and making sure they “got rid of them” before they got to the solution.

Complex Numbers

In his Ars Magna of 1545, Gerolamo Cardano presents the following problem: “To divide 10 in two parts, the product of which is 40”

In modern terms, call the two parts A and B. Then we have A + B = 10 and AB = 40.

From the first expression, A = 10 – B. Substituting into the secon: (10—B)B = 40 Expanding: 10B – B2 – 40 = 0

This is a quadratic equation, which can be solved by a method known to the ancient Babylonians. The solutions are and .

The problem is that square roots of negative numbers seem to make no sense at all.

And they crop up here even though there are no negative number or square roots in the original problem!

Cardano’s Problem

This idea was picked up and developed in fits and starts throughout the seventeenth century; by the middle of the eighteenth century it was more respectable.

The Complex Numbers were born: the real number system extended by adding the square roots of negative numbers.

Although you can begin by adding just one such number, , you end up having to add a huge amount of new numbers to get the complex numbers to satisfy the axioms of a complete field.

At the very end of the eighteenth century, Caspar Wessel found a way to think of the complex numbers geometrically, but this work was recognised and accepted only very slowly.History

Today the complex numbers are the basic number system used in much of physics and engineering.

As with hyperbolic geometry finding an application in relativity, we can ask: does this make these number more “respectable”, or more “real”?

In 1843 William Rowan Hamilton invented an even more complicated number system. Instead of just adding one square root of -1 he adds three different ones, and provides rules for the resulting arithmetic.

This number system is called the quaternions. For a century it was considered an obscure piece of pure-mathematical trickery, but today quaternions appear in a range of applications from aeronautics to computer games.

In the same year, John T Graves discovered an even bigger number system that became known as the octonions. These, too, have found a few applications in radio engineering and theoretical physics.

Later Developments

Our intention in this session has been to look at a historical change in the nineteenth century: the development of “abstract algebra” and a greatly widened sense of what numbers and arithmetic are.

In the twentieth century this shift was made more dramatic by category theory, in which objects (such as numbers) are radically de-emphasized in favour of maps involving them, which include arithmetic operations.

In philosophy, though, this has been less decisive. Many people still want an answer to the “is” question, and feel the more abstract mathematical approach just avoids the issue.

So what is a number?