Numbers: Their Tales, Types, and Treasures.

Chapter 11: Numbers and Philosophy



The philosophical position that mathematical objects (such as numbers, triangles, equations, etc.) exist by themselves in some “realm of mathematics,” which is outside the world of physical objects, and also outside our mind, is called Platonism, named after the famous Greek philosopher Plato (428/427–348/347 BCE). In his “theory of forms,” Plato claimed that ideas possess a more fundamental kind of reality than material objects. Ideas (also called “forms”) are nonmaterial and abstract, and they exist in a metaphysical world of ideas. Material objects that are perceived through our senses are just “shadows” or “instances” of their ideal forms—their true essence. Human beings are like a caveman who sits with his back toward the entrance of the cave and can observe only the shadows of the outside reality on the wall in front of him. Consequently, real insight can only be gained through the study of ideas that are not directly accessible through our senses but are accessible through reason.

Until the twentieth century, this was indeed the common belief concerning the nature of numbers. Mathematicians considered numbers to be “real” objects in an immaterial realm of abstract ideas existing independently of human beings. While modern mathematicians usually do not go so far as to declare the material world as unreal, many of them would still uphold the Platonic view of the reality of mathematical objects. For example, as French mathematician Charles Hermite (1822–1901) stated: “I believe that numbers and functions of analysis are not the arbitrary result of our minds; I think that they exist outside of us, with the same character of necessity as the things of objective reality, and we meet them or discover them, and study them, as do the physicists, the chemists and the zoologists.”1

Elsewhere, he wrote, “There exists, if I am not mistaken, an entire world, which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation.”2

In the book A Mathematician's Apology (1940), the well-known British mathematician Godfrey Harold Hardy (1877–1947) expressed his belief as follows: “I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations,’ are simply our notes of our observations.”3