Numbers: Their Tales, Types, and Treasures.
Chapter 1: Numbers and Counting
1.5.COLLECTIONS OF OBJECTS, ELEMENTS IN A SET
At first sight, the counting principles may seem elementary and scarcely worth mentioning, but they contain subtle observations that are useful for further considerations. They show that counting, although a familiar operation, is logically rather complex. In the following, we use the counting principles as a guideline for a deeper analysis of these complexities.
Obviously, in order to be able to count, we need something that can be counted. The abstraction principle is perhaps the most fundamental of the counting principles because it describes what we can count—collections of objects, such as apples in a basket or people in a room. So far, we have not been very specific about what we mean by a “collection.” In mathematics, a collection, or group of things, would be called a set. The crucial property of a set is that it is a collection of well-distinguished objects.
German mathematician Georg Cantor (1845–1918) has defined a set as an aggregation into a whole of definite well-distinguished objects of our intuition or our thinking. The members of a set are called the elements of the set. The elements of a set are, thus, objects of our intuition or thinking, and this includes not only material objects but also ideas, numbers, symbols, colors, people, or actions, and so on. The elements of a set could even be sets themselves—for example, figure 1.1 shows a set of elements, and among these elements is a group (which is a set in itself) of people, and also a set of black squares. In mathematical notation, sets are often indicated by putting a list of its elements between braces. So {A, B, C} would be the set containing the letters A, B, and C as elements. The set {1, 2, 3, 4,…} would be the set of natural numbers, an example of an infinite set.
A set is formed either by actually “putting apples in a basket” or just by definition—that is, by using some descriptive property. For example, we can define the “set of the blue objects on this table.” In any case, it must always be clear which elements belong to a set and which do not. Moreover, every element must occur in a set only once. For this reason, Cantor emphasized that the elements of a set must be distinguishable from each other.
Whenever we count something with the procedure described earlier, we count the elements of some finite set. This is what makes the concept of a set important to us: Counting always deals with a set, even if the set is often not defined explicitly. And the abstraction principle states that any finite set can be counted—any finite collection of distinguishable objects.4
The invariance principle states that the result of counting a set does not depend on the order imposed on its elements during the counting process. Indeed, a mathematical set is just a collection without any implied ordering. A set is the collection of its elements—nothing more. For example, if you shuffle a set of playing cards, it retains its identity as the same set of cards.
People have often wondered why mathematics is able to describe many aspects of our world with high precision and accuracy. In a sense, this is not so astonishing, because from the very beginning, mathematical concepts have been formed on the basis of human experience—an experience, in turn, formed by the world surrounding us. We can see this even at the basic level, when the concept of a set is defined—perhaps one of the most important concepts of modern mathematics. But what property of our world, what kind of human experience about the world, would be reflected by the mathematical definition of a set?
To begin with, the notion of a set would make little sense to us if we hadn't made the observation of temporal stability. Typically, objects endure long enough that it makes sense to group them together and consider the whole collection as a new “object of our thinking.” Hence, for example, we can put objects into a box and know from experience that they are still there, even if we can't see them. The existence of permanent objects is helpful for the idea of grouping them together to form a new whole, a set. But the notion of a set is general enough to include nonpermanent elements. Time is not mentioned in Cantor's very general definition of a set as a collection of “objects of our thinking.” Therefore, temporal permanence of objects is not necessary as a prerequisite for combining them in a set. We can also form a set of events like drumbeats or strokes of a clock, and we can define, for example, the set of days between one new moon and the next.
A very basic observation concerning a fundamental property of the world we live in is the existence of objects that can be distinguished from each other. For the definition of a set, it is indeed of crucial importance that things have individuality, because in order to decide whether objects belong to a particular set they must be distinguishable from objects that are not in the set. Without having made the basic experience of individuality of objects, it would be difficult to imagine or appreciate the concept of a set.
It is perhaps worthwhile to make a little thought experiment. How might life be in a radically different universe? Imagine, for example, a vast ocean populated by intelligent protoplasmic clouds, containing no solid objects at all. Let us assume that whenever these cloud-beings meet, they would mix and merge into a new cloud-being. Would those cloud-beings, however intelligent, develop any concept of numbers and counting? Even if they did, numbers would probably have little importance and would appear as a very exotic idea. Arithmetic would appear rather impossible because for our cloud-beings 1 + 1 would be just 1 in most cases. Quite certainly, mathematics would have evolved in a different direction. We learn from this that statements like “1 + 1 = 2,” which appear to be so true and obvious to us, need not be true and obvious to everybody, under all circumstances. In our universe, however, one of the first observations a child makes about its surroundings is that there are well-defined objects that can be seen either alone or in pairs or in groups. We have learned to form sets, and we have learned to count, because our world contains objects with a certain permanence and individuality.
Moreover, there is still another important observation that seems to be essential for the idea to group objects into a set: This is the human ability to recognize similarities in different objects. Usually, a collection, or group, consists of objects that somehow belong together, objects that share a common property. While a mathematical set could also be a completely arbitrary collection of unrelated objects, this is usually not what we want to count. We count coins or hours or people, but we usually do not mix these categories. If you hear that 4 people watched 3 movies within 2 days, you would hardly want to know that there are 4 + 3 + 2 = 9 well-distinguished objects—this information appears to be rather useless, or even meaningless. When we count, we usually group objects by similarity, and count the number of people or the number of movies or the number of days. We automatically tend to group similar objects and perceive them as belonging together. This ability is the basis for defining a set by a common property of its elements (for example, the set of all blue objects).