Numbers: Their Tales, Types, and Treasures.

Chapter 1: Numbers and Counting

1.9.ABSTRACTION

Considering the counting procedure and its results, we come to understand a natural number as a property of a set, describing numerosity. For example, the number nine will appear as the cardinal number of any collection of nine objects or any group of nine people or any set with nine elements. With experience, one learns to associate a certain “size,” or “numerosity,” with the cardinal number nine. Through repetition and by force of habit, numbers can gain an abstract connotation independent of the concrete collection of objects. And finally, one is able to think of ninewithout picturing any particular group of nine objects. The (cardinal) number nine has finally achieved a meaning of its own. It describes what nine apples, nine people, or nine black dots on a sheet of paper have in common. It has become an abstract concept that represents a property of certain sets, namely all those sets for which counting ends with the number word nine. And the particular nature of the set, the type of objects in the collection, is completely irrelevant, as long as there are precisely nine elements in the set. Therefore, this concept of nine-ness of a set has become something that lives in our imagination and does not need a concrete realization any longer. A number seems to be a property of a collection in a similar sense that a color is a property of an object.

Forming abstractions is a natural process in our language. Consider, for example, the word table, which is the end result of a process of abstraction that starts with concrete objects. These concrete tables will differ in shape and material. But the abstract notion of a “table” makes no distinction and refers to all concrete tables simultaneously. The word table lets us think of an object that typically has a flat horizontal surface and is supported by one or more “legs.” Without further information, we do not know whether the word table refers to a dining-room table, a coffee table, a billiard table, or a workbench. The word table, therefore, contains less information than a reference to a particular table. Obviously, abstractions are created by reducing the amount of concrete information. Therefore, the process of forming abstractions can be seen as a process of simplification, and it is very handy, because we can now talk about tables in general without having to refer to any particular table.

The abstraction leading from special groups of objects to a number is a quite similar process. It is the process of removing information about the concrete nature of the counted objects. And as soon as we have successfully performed this abstraction, we have achieved a simplification. When we think of four, we do not need to think of four apples or four persons or four corners any longer. Four refers to all sets with four elements. We can work with numbers without having to think of their concrete realizations. And this is a prerequisite for successfully doing computations with numbers. We can perform computations, such as 9 + 4 = 13, without having in mind real manipulations with concrete objects. And while concrete realizations of computational tasks are still possible with smaller numbers like 9, 4, and 13, it becomes impractical, or even impossible, for larger numbers. When you think of the number 2,734, you will probably think of “many,” but it isn't always useful to imagine a particular collection of 2,734 objects. From a practical point of view, the transition from concrete realizations to an abstract number concept becomes absolutely necessary when dealing with really large numbers.

There is indeed some evidence from neuropsychology that numbers are represented in our brain in an abstract way. That means whenever we see the symbol 4 or hear the word four or see a collection of four dots, the same group of neurons in the same part of our brain gets activated. No matter how the number is presented to us—whether in verbal or nonverbal form—the end result of the information processing in our brain is always the same neuronal activity pattern. The fact that all the different sensual inputs invoke the same representation of “four” in our brain is the neuronal origin of our perception of the number four as an abstract “mathematical” object.

Thinking of numbers as abstract objects in the sense described above belongs to the first and most fundamental concepts of mathematics. When the abstraction described here is not learned, then a number is not perceived as a concept independent of the things that are counted. Numbers would be inseparable from concrete objects, and one could not understand that the four seasons and the four wheels of a car have something in common. The old language of Fiji Islanders, where ten boats would be bola and ten coconuts would be koro, provides examples of number words that cannot be detached from objects and did not acquire an abstract meaning. A vestige of this state of human development can still be found in modern languages—for example in Japanese, where different (although related) counting words are used for counting different types of things:

are just a few examples.