Numbers: Their Tales, Types, and Treasures.

Chapter 4: Discovering Properties of Numbers



The origin of an elaborate philosophy of numbers lies in ancient Greece, where one made the distinction between logistic (counting and calculating for practical purposes) and arithmetic (philosophical number theory). Arithmetic as a philosophically motivated number theory is intimately connected with the school of Pythagoreans, who based their way of life upon the quasi-religious worship of numbers. Little is known about Pythagoras himself. He lived mostly in the second half of the sixth century BCE and was born on the Greek island Samos, close to the coast of Asia Minor, just a few kilometers from cities of Ephesus and Miletus. It is said that Pythagoras fled from the tyranny of Polycrates on Samos, and after travels to Mesopotamia and Egypt he settled in Croton around 530 BCE. Croton is today's Crotone in southern Italy, which then belonged to the Greek sphere of influence and had a considerable Greek population. In Croton he founded an influential secret order, which had the typical characteristics of a religious sect—secret conspirative meetings, a time of probation for new members, strict rules for nutrition and clothing, ascetic lifestyle, and its own cosmology. Later, after becoming too influential, the Pythagoreans were persecuted and Pythagoras left Croton. He died in Metapont (also southern Italy) in the early fifth century. His school continued its activities in the Greek cities of southern Italy for about one hundred years. One group of Pythagoreans were the mathematikoi (the learners), who engaged in developing the scientific aspects of Pythagoras's philosophy, while the akousmatikoi (the listeners) focused on the religious aspects of his teachings. Due to continuing political persecution, the school of Pythagoras dissolved in the late fifth century BCE. In the first century BCE, the Pythagoreans were revived in Rome, and most of the information about the original Pythagoreans stems from that later time.

What makes the Pythagoreans interesting and special for the history of mathematics is that, to them, numbers were the key to an understanding of the cosmos. The Pythagorean philosopher Philolaus of Croton (ca. 470–385 BCE) writes in fragment 4, “Indeed, everything that is known has number, for nothing is either conceived or known without this.” And Aristotle (384–322 BCE), two generations later, even describes the Pythagorean doctrine as “All things are number.” Aristotle writes the following, often-cited passage about the Pythagoreans’ ideas in his bookMetaphysics:

The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this study, but also having been brought up in it they thought its principles were the principles of all things. Since of these principles numbers are by nature the first, and in numbers they seemed to see many resemblances to the things that exist and come into being—more than in fire and earth and water…since, again, they saw that the modifications and the ratios of the musical scales were expressible in numbers—since, then, all other things seemed in their whole nature to be modeled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number.2

This goes much further than the statement of Philolaos that everything has number. According to Aristotle, the Pythagoreans believed numbers to be the essence of all things. Numbers were not just an abstract construction of the human mind; for the Pythagoreans, numbers formed the basis, the principle, of all other things. And they came to this conclusion because they saw a variety of natural phenomena—from cosmic cycles to musical scales—that could be expressed through numbers and specific ratios of numbers.

Interestingly, Aristotle also states that the unit, the “One,” is not itself a number; instead it is the fundamental principle that creates number and thus plays a very special role, philosophically. We find this also in Euclid's Elements, book 7, which starts with the definitions

·        A unit is that by virtue of which each of the things that exist is called one.

·        A number is a multitude composed of units.3

The “One” is the basic unit of which all numbers consist, and, as Aristotle explains, the unit is not a number in the same sense as a measure is not the things measured. Moreover, there was no need for a “number one” in counting, because if there was only one item, then there was no need to count. Counting, and hence numbers, therefore had to start with “two.”

For the ancient Greek scholars, numbers were not just a useful tool. They regarded numbers as philosophical principles, as fundamental entities, as the essence of everything. Numbers had to be explored, as their properties would reveal the nature of all things. This quasi-religious state of mind was the driving force creating the Pythagorean tradition of systematically cultivating mathematics as a science. While the philosophical underpinning might appear obscure from today's perspective, the Pythagoreans’ rational approach to the exploration of numbers by strictly logical reasoning nevertheless marks the historical origin of modern mathematics.