What do we mean when we use the word mathematics? The first thing that most people think about is numbers and arithmetic. But there is also geometry, topology, calculus, and other subjects that are certainly part of what we call mathematics, and which have little resemblance to arithmetic. What is common to these subjects that include them as branches of mathematics?
To start to answer the question, let’s look at arithmetic, which most people identify with mathematics. Let’s ask a fundamental question about arithmetic: “Why do the numbers we are familiar with work so well in so many different situations?” Numbers are used in accounting, physics, chemistry, biology, sports, gambling and many other places. For example we use numbers to represent:
The same kinds of numbers are used in all of these examples. We don’t have special kinds of numbers for accounting and special kinds for keeping sports scores. This tells us that our system of numbers is very adaptable. How do these same kinds of numbers work in all these different situations? Consider a fable of sort:
On a planet in a galaxy far far way, lived a civilization of mostly intelligent beings, who loved entertainment in the form of games. These beings were very different from earth people and they didn’t need dwellings or have any concept of property ownership, nor was the notion of money known to them. But, anyway, they invented a game for their pleasure that used a set of objects named after some common things in their own culture, and some rules for play. So possessing this invention, groups of these beings would often sit around, socializing, and playing a session of their game, which they dearly loved.
When earth people and these beings contacted each other, as stories such as this require, each group studied the other’s culture in great detail. The earth people, to their amazement, discovered that the alien’s beloved game, if earth-type names were given to its objects, such as real estate, currency, ownership, and hotels, the game was identical to the earth game of Monopoly!
Some great mind from one of the cultures (it doesn’t matter which one) declared “This is a fun game no matter what identities we assign to the objects of the game. What makes the game fun is simply the rules we use to play the game.” And everyone lived happily forever after.
The fable is an undisguised commentary on what mathematics is:
A collection of objects and a collection of rules that apply to these objects.
The objects do not have tangible meaning in the real world, but the objects are required to obey the rules we have assigned to them.
While the objects do not have tangible meaning, we often assign a unique symbol to some of the objects in the collection. So in arithmetic the symbol 3 is an identifier for a single unique integer. Such a symbol is called a literal symbol. A literal is quite similar to the proper name of a person.
We can also assign identifiers that may refer to any object in our collection. We often use lower case letters, such as x for this kind of identifier. Such a symbol is called a variable symbol. A variable is similar to a pronoun in English.
Is this what mathematics is all about? Yes and, of course, much more. In following sections we will explore various topics in mathematics, but a common theme is that the mathematical objects discussed in these topics are abstract and are not required to have a meaning in the physical world.
But, what good is mathematics if its objects are just abstractions and don’t mean anything in the real world? Mathematics becomes useful in the real world when someone says something like this:
What we see happening above is:
So we jump from the real world to mathematics, manipulate the mathematical objects, and then jump back to the real world. We say that we are using mathematics to model some part of the real world. Because mathematics is blind to what real-world meaning we assign to its objects, a mathematical system such as arithmetic can be used in many different situations.
A mathematical model is a description in mathematical terms of some aspect of the real world.
We also should realize that mathematics can be useful without expecting it to model any real world situation. Mathematics left detached from the real world is often called pure mathematics. Pure mathematics is studied by many mathematicians, and we might say that one of its uses is to give mathematicians good jobs.