Numbers

From Fish to Infinity

 

THE BEST INTRODUCTION to numbers I’ve ever seen—the clearest and funniest explanation of what they are and why we need them—appears in a Sesame Street video called 123 Count with Me. Humphrey, an amiable but dimwitted fellow with pink fur and a green nose, is working the lunch shift at the Furry Arms Hotel when he takes a call from a roomful of penguins. Humphrey listens carefully and then calls out their order to the kitchen: “Fish, fish, fish, fish, fish, fish.” This prompts Ernie to enlighten him about the virtues of the number six. 

 

 

Children learn from this that numbers are wonderful shortcuts. Instead of saying the word “fish” exactly as many times as there are penguins, Humphrey could use the more powerful concept of six.

As adults, however, we might notice a potential downside to numbers. Sure, they are great timesavers, but at a serious cost in abstraction. Six is more ethereal than six fish, precisely because it’s more general. It applies to six of anything: six plates, six penguins, six utterances of the word “fish.” It’s the ineffable thing they all have in common.

Viewed in this light, numbers start to seem a bit mysterious. They apparently

exist in some sort of Platonic realm, a level above reality. In that respect they are more like other lofty concepts (e.g., truth and justice), and less like the ordinary objects of daily life. Their philosophical status becomes even murkier upon further reflection. Where exactly do numbers come from? Did humanity invent them? Or discover them?

 

An additional subtlety is that numbers (and all mathematical ideas, for that

matter) have lives of their own. We can’t control them. Even though they exist in our minds, once we decide what we mean by them we have no say in how they behave. They obey certain laws and have certain properties, personalities, and ways of combining with one another, and there’s nothing we can do about it except watch and try to understand. In that sense they are eerily reminiscent of atoms and stars, the things of this world, which are likewise subject to laws beyond our control . . . except that those things exist outside our heads.

 

This dual aspect of numbers—as part heaven, part earth —is perhaps their most

paradoxical feature, and the feature that makes them so useful. It is what the

physicist Eugene Wigner had in mind when he wrote of “the unreasonable

effectiveness of mathematics in the natural sciences.”

 

In case it’s not clear what I mean about the lives of numbers and their

uncontrollable behavior, let’s go back to the Furry Arms. Suppose that beforeHumphrey puts in the penguins’ order, he suddenly gets a call on another linefrom a room occupied by the same number of penguins, all of them alsoclamoring for fish. After taking both calls, what should Humphrey yell out to the kitchen? If he hasn’t learned anything, he could shout “fish” once for eachpenguin. Or, using his numbers, he could tell the cook he needs six orders of fish for the first room and six more for the second room. But what he really needs is a new concept: addition. Once he’s mastered it, he’ll proudly say he needs six plus six (or, if he’s a showoff, twelve) fish.

 

The creative process here is the same as the one that gave us numbers in the

first place. Just as numbers are a shortcut for counting by ones, addition is a shortcut for counting by any amount. This is how mathematics grows. The right abstraction leads to new insight, and new power.

 

Before long, even Humphrey might realize he can keep counting forever. Yet despite this infinite vista, there are always constraints on our creativity.  We can decide what we mean by things like 6 and +, but once we do, the results of expressions like 6 + 6 are beyond our control. Logic leaves us no choice. In that sense, math always involves both invention and discovery: we invent the concepts but discover their consequences. As we’ll see in the coming chapters, in mathematics our freedom lies in the questions we ask—and in how we pursue them—but not in the answers awaiting us.