Numbers: Their Tales, Types, and Treasures.

Chapter 11: Numbers and Philosophy

11.1.NUMBERS—INVENTED OR DISCOVERED?

For several thousand years, numbers have been involved in a wide spectrum of research and have been the focus of research as well. Mathematicians have developed and refined our understanding of numbers and accumulated a vast amount of knowledge about them and their application. They have developed sophisticated procedures using numbers for a wide variety of purposes in many different contexts. Apart from natural numbers, mathematicians have introduced new types of numbers—for example, negative numbers, rational numbers, real numbers, and complex numbers. And, of course, they have kept thinking about the nature of numbers, that is, about “what numbers really are” and why they play such a formidable role in the universe.

We have already seen in chapter 1 that the number concept reflects some basic properties of our world; in particular, the possibility to group objects into sets of distinguishable elements. Evolution has provided us (and some animal species) with a rudimentary number sense, which is exact for small numbers and approximate for large numbers. Counting arbitrary sets requires a synthesis of these aspects and thus requires mental abilities specific to Homo sapiens. Numbers were invented in early human societies as people started to become settled, and therefore numbers belong to the first cultural achievements of humankind. Numbers seem to be a human creation—a tool for the human mind to create an adequate and useful mental representation of certain aspects of our world. And the process of simplification and information reduction leading to an abstract number concept appears to be even more of a mental construction, a function of the human brain that helps to organize thought processes in an economic manner.

Mathematicians, however, often think differently about the nature of numbers or other mathematical objects. When mathematicians get deeply immersed in research, they have the impression that they are dealing with entities that are not just a human creation but exist in a more objective sense. They believe that numbers have been discovered, not invented, and that their laws and properties can be explored in the same sense as a physicist would explore the properties of elementary particles. The only difference seems to be that numbers are objects that exist in a nonphysical, and also nonpsychological, manner, while elementary particles exist in the physical universe. But, like elementary particles, numbers seem to exist independently of the human mind. And where a physicist would use experiments and measuring devices, mathematicians use their intuition, logical thinking, and abstract reasoning to discover the beauty and truth in a previously unexplored terrain. The world where mathematicians conduct their research is an abstract world populated by mathematical objects and ideas. When they find unexpected relationships, patterns, and structures, then a new range of mathematical knowledge, a new region of that abstract world, becomes accessible. The mathematician would then feel like an explorer of a past century who had discovered a new, previously unknown region of the earth.

This view cannot be discarded easily. For example, when we played with square numbers in chapter 4, we “discovered” the amazing result that the sum of the first n odd numbers equals n × n. We found that this result must be true from the obvious, and geometrically intuitive, way in which the next square number is constructed from a given square number. This sense of truth is further confirmed by algebraic methods, which do not rely on geometric visualization at all. And therefore it is the general consent among mathematicians that this statement is indeed true for all natural numbers, n. Once convinced of its truth, one has the feeling that this statement expresses more than just a psychological conviction or a social convention. Indeed, the result is an inevitable conclusion of logical reasoning, and hence is independent of human belief or attitude.

This gives the impression that the result represents an objective truth that existed and was true even before it was formulated and proved. It evokes the idea that there is a metaphysical realm of numbers that exists independently of the physical universe. In other words, if the whole universe disappeared tomorrow, the eternal world of numbers would still exist.

We have just described two contradicting philosophical positions concerning numbers: One position holds that numbers have a mind-independent existence in a metaphysical world “out there.” The other position is that numbers exist “in here” as creations of the human mind, and are designed to help us in various tasks, like classifying and ordering sets of objects.