What Are Numbers?

Numbers are the most natural place to start a survey of mathematics. Each time I've taught the Survey of Mathematics course, we've started with the basic (but not easy) question: “What is a number?” Like many things that we think of as obvious, it is hard to explain. Naming examples of numbers is easy, and it is nearly as easy to list things for which numbers are used—counting, measuring, ordering, and labeling or identifying. We can take these uses of numbers and get a quick definition: a number is a mathematical object used to count or measure (the other uses are related to ordering and uniqueness, two properties most of our number sets will have). This definition is admittedly squishy, but it has one key element that rings true. A number is a mathematical object.

It is easy to confuse the concept of “numbers” with the symbol used to represent them, numerals. To see the differences, consider an example: “5” and “five” are two ways of representing the same abstract mathematical object. “Five” is something beyond the word or symbol we use to describe it. If you look at a stack of five apples and a stack of five books, you can see that while the two stacks may have very little in common, they share the same fundamental quality of “fiveness.” Getting at what “fiveness” or “threeness” actually is takes us down the road to mathematical Platonism, a view of mathematics that has been with us for more than 2000 years.

Plato's philosophy of mathematics grew out of his attempts to understand the relationship between particular things and universal concepts. The world we live in is filled with particular things—this chair, that chair, big chairs, little chairs, and so forth. There is a quality all of the instances of particular chairs share—for lack of a better phrase we'll call it “chairness”—which presents a bit of a problem. It is not itself a chair and unlike all chairs we know it cannot be located in some place or at some time, but that does not mean that “chairness” is a figment of our imagination. Replace “chairness” with “fiveness” or “threeness” and we see how all this applies to numbers. Numbers are not particular things, they are universal concepts.

Initially driven by the need to count things, our understanding of numbers has grown considerably, pushed along by problems we've needed to solve. Counting problems led to the natural numbers, {1,2,3,4,...}, and then to the integers, problems in geometry led to the rational numbers and to the irrational numbers, which collectively give us the real number system. Problems in algebra and physics led to the complex number system. And mathematical explorations have led to more exotic number systems such as quaternions, and generalizations of numbers in abstract algebra like fields. Mathematicians have found these generalizations through their favorite game, making up axioms and seeing where those axioms lead.

To the Platonic way of thinking, these generalizations weren't invented, they were discovered. Numbers, universal concepts that they are, were always there, as were the generalizations like fields, humans just had to figure out how to use and to represent them. Humans didn't invent numbers, we invented numeral systems, but like anything in mathematics, numerals systems rely on notation, and it took a long time to find a notation that worked really well. Exploring how numeral systems evolved, and learning how ancient civilizations did mathematics, is a fascinating journey.