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The Numbers: What Are They?

Mar 28, 2008
There's a question asked by any number of bored sophomores - and hard-working, frustrated adult learners as well - a question many math teachers dread, and that a few of the best welcome: "What does this math stuff have to do with everyday life?"

The question may strike some teachers as rude and aggressive, but it actually has a strong basis. After all, mathematicians themselves stress the fact that theirs is a discipline based on purer and purer abstractions, so that even a problem that may seem the simplest imaginable - for example, the problem of what a number is - requires a high level of theorizing to be answered. In fact, we can come to understand how math works - and, in a roundabout way, what it's good for in everyday life - by investigating that very problem: what are numbers?

Your kindergarten teacher probably taught you what the number five is by showing you, say, five oranges. Looking at those five oranges in your mind, the concept "five" seems very clear indeed - why, it's having this many of something.

Expert mathematicians, however, caution us that this way of thinking is a trap. Most trained mathematicians will agree that you understand the idea of number most clearly when you understand numbers as functions in a rule-governed, logical system - not as tangible, standalone things that you experience in the way you experience, say, the bus, or the TV, or a plate of filet mignon. The smaller numbers fool us a bit - we can see five oranges, or nine of them, and so it's easy to imagine "five" and "nine" as real things that exist on their own, like cars or sticks of chewing gum. But larger numbers show the problem with such thinking.

As mathematician Timothy Gowers observes in Mathematics: A Very Short Introduction, "numbers do not have to be very large before we stop thinking of them as isolated objects and start to understand them through their properties, through how they relate to other numbers, through their role in a number system." After all, it's easy enough to imagine five oranges, or even nine of them (three rows of three, perhaps) - but how are you going to picture the number 148, let alone 148,000? How do you touch or experience numbers like these? You can't - just try to conjure up a mental picture of 148 oranges. But you can very well understand 148 as the number two numbers less than 150, or as the thing that, when multiplied by 10, makes 1480. You can understand it, in other words, by its relation to other numbers (it's one more than this one, one less than that one), and by the kinds of transformations it can undergo (we know 148 because it's the number that, multiplied by ten, becomes 1480). To quote from Gowers again, "think about the rules rather than the numbers themselves - [they] are tokens in a sort of game."

All well and good - but then, what does math get us? If even the simplest numbers - which represent the one part of mathematics that most people understand on some level and can work with - are best understood as functions in a highly complex rule-governed system that has its roots in pure logic and abstraction, and not in physical life, then, once again, what does math have to do with real life?

Quite a lot, of course. For an example, think of negative numbers - a part of the number system that confuses many students on their first encounter with it. We know what seven looks like, but where in nature do you find negative seven of something? Students (including the present author) get hung up on a seemingly-unanswerable question - does negative seven actually exist? In what sense?

Mathematicians deal with this question the simple way - they ignore it. After all, when the concept negative seven is applied and allowed to function in the rule-governed number system, it yields results we all, instantly recognize. Let's say that I write a check for an amount seven dollars more than I have in the bank (by accident, of course!)a mistake that can meaningfully be expressed as minus seven dollars, or negative seven. Using negative integers in this case allows us to quickly and clearly express what I owe my bank (not counting those onerous overdraft fees, of course). It seems that pure abstraction has its use, like everything else.
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