The relation of mathematics to physics. The simplest mathematical notion is that of the number of a class. This is the property common to the class and to any class that can be matched with it by pairing off the members, one from each class, so that all members of each class are paired off and none left over. In terms of the definition we can give meanings to the fundamental operations of addition and multiplication. Consider two classes with numbers a, b and no common member. The sum of a and b is the number of the class consisting of all members of the two classes taken together. The product of a and b is the number of all possible pairs taken one from each class. We cannot always give meanings to subtraction and division, because, for instance, we cannot find a class whose number is 2–3 or 7/5. But it is found to be a great convenience to extend the notion of number so as to include negative numbers, ratios of numbers irrespective of whether they are positive or negative, and even irrational numbers. When this is done we can define all the four fundamental operations of arithmetic, and the result of carrying them out will always be a number within the system. We need trouble no more about whether an operation is possible with a particular set of numbers, since we know that it is, once we have given sufficient generality to what we mean by a number.