Numbers: Their Tales, Types, and Treasures.

Chapter 4: Discovering Properties of Numbers



While everyone has an innate number sense, and while it is not too difficult to learn the syntax for counting, arithmetic usually poses problems of a new dimension. Learning to calculate is such an ordeal that it sometimes creates hostility toward mathematics and sympathy with those who fail. Stanislas Dehaene has formulated this as follows:

Mental arithmetic poses serious problems for the human brain. Nothing ever prepared it for the task of memorizing dozens of intermingled multiplication facts, or of flawlessly executing the ten or fifteen steps of a two-digit subtraction. An innate sense of approximate numerical quantities may well be embedded in our genes; but when faced with exact symbolic calculation, we lack proper resources. Our brain has to tinker with alternate circuits in order to make up for the lack of a cerebral organ specifically designed for calculation. This tinkering takes a heavy toll. Loss of speed, increased concentration, and frequent errors illuminate the shakiness of the mechanisms that our brain contrives in order to “incorporate” arithmetic.1

While there are numerical prodigies who succeed very well with complicated arithmetic tasks, such as extracting the square root of a five-digit number mentally or multiplying two very large numbers mentally, most people fail miserably. Dr. Arthur Benjamin (1961–), a mathematics professor at Harvey Mudd College in California, for example, often demonstrates his mathematical talents for national audiences. Yet most mathematicians do not have this ability. Usually, mathematicians don't think of themselves as being particularly good at arithmetic, and the prospect of having to mentally evaluate 891 × 46 is not very appealing to them either. They are aware of the fact that the average human brain is ill-equipped for this kind of task.

Without doubt, the ability to perform exact computations is important in a developed society. Therefore, the insight that these tasks are rather difficult could well be considered a motivation and a historical reason for the emergence of “real mathematics.” In a time before the advent of computers, one had to look for ways to achieve a deeper understanding that might relieve the trouble of tedious computational tasks. Instead of trying to perfect the mind to do error-free calculations, which is rather impossible, mathematicians looked for interesting properties of numbers and relationships between numbers. They preferred to play with numbers in search of logical structure and repeating patterns that might prove helpful. Finding meaning in the world of numbers makes life with them easier (or at least more entertaining).

Apart from the purely practical reasons for a deeper involvement with number properties, there is another reason that is more philosophical in nature. The human ability to create abstractions turns numbers into mathematical objects with a meaning of their own, which is independent of any concrete realization and thus applies to all kinds of different situations. The simple computation 5 + 3 = 8 might refer to apples or days or persons. But the statement “5 + 3 = 8” can very well stand for itself. It need not refer to anything outside the world of numbers. It seems self-evident, objectively true, and independent of the human state of mind. Surely, it must have been true before there were any human beings—and will it not be true even after the extinction of the human race? It appears that in proven statements about natural numbers lies an a priori truth, a certainty and absoluteness, that is completely independent of human experience. If there exists an eternal and undisputable truth at all, isn't it here, in the arithmetic of natural numbers, where we come as close to it as we will ever get?

As soon as people started to think philosophically about meaning and truth, about illusion and reality, they also started to think about mathematics and the nature of numbers. Are numbers just an instrument for counting things, or is there more to them? It is perhaps a common human trait to suspect a deeper meaning and hidden truth under the surface. So the relationship between humans and numbers is shaped not only by the need to count, measure, and calculate for practical purposes, but also by the desire to understand numbers and their meaning from a more theoretical and philosophical point of view.