David Peat

32 From Certainty to Uncertaintynever run out. No matter how many fractions there are, the “bag” ofintegers will never empty and so the next fraction on the list can alwaysbe matched with an integer. In other words, the number of the integersand the number of the fractions is the same.Does this sound like a bit of a cheat? For a layperson it may seemodd, yet mathematicians are convinced by the argument. This showsthat in mathematics things are not always obvious, so it may be agood idea to take the time to prove the certainty of mathematicalpropositions.What Is a Number?Let’s begin with the idea of “number” itself. We can all count. We allknow that 2 + 2 = 4. But what exactly is a number? How can we defineit? John and Jill made an important discovery about numbers andmathematics. Jill has suddenly realized that she can do the same thingwith apples as she did with candies. She can match each candy with anapple from the bowl. In this way she discovers that there are as manycandies in her bag as there are apples in the bowl. She rushes aroundcomparing everything in sight—apples and pears, candies and coins,dogs and cats, shoes and socks. In every case the method works. If shehappens to have 10 candies, then even if she can’t count past five sheknows when she has exactly the same number of apples, candies, coins,shoes, and so on. She has realized that a sort of mental bag exists thatwe could call “the number ten.” Into this bag can be fitted anything andeverything, provided there are only 10 of them. Shoes and candies andapples are totally different things, but when there are 10 of them theyhave something in common and that is their number.At the end of the nineteenth century, philosophers and mathematicianswere considering precisely this issue—the definition of “number.”It was the mathematician and philosopher Gottlob Frege who hiton Jill’s discovery and defined “number” just as she did, in terms ofclasses and sets. As Bertrand Russell put it in his Introduction to MathematicalPhilosophy, “The number of a class is the class of all thoseclasses that are similar to it.” That bit of verbiage stops us in our tracks