Enigmatic Pythagoras | Sunday Saga #2

The Introducer of Mathematics

Pythagoras (ca. 572-497 BC) may have been the first person who was both an influential natural philosopher and a charismatic spiritual philosopher a scientist and a religious thinker. In fact, he is credited with introducing the words "philosophy," meaning love of wisdom, and "mathematics"-the learned disciplines. Even though none of Pythagoras's own writings have survived (if these writings ever existed, since much was communicated orally), we do have three detailed, if only partially reliable, biographies of Pythagoras from the third century. A fourth, anonymous one was preserved in the writings of the Byzantine patriarch and philosopher Photius (ca. AD 820-91). The main problem with attempting to assess Pythagoras's personal contributions lies in the fact that his followers and disciples-the Pythagoreans-invariably attribute all their ideas to him. Consequently, even Aristotle (384-322 BC) finds it difficult to identify which portions of the Pythagorean philosophy can safely be ascribed to Pythagoras himself, and he generally refers to "the Pythagoreans".

Pythagoras and Pythagoreans

Pythagoras and the early Pythagoreans were neither mathematicians nor scientists in the strict sense of these terms. Rather, a metaphysical philosophy of the meaning of numbers lay at the heart of their doctrines. To the Pythagoreans, numbers were both living entities and universal principles, permeating everything from the heavens to human ethics. In other words, numbers had two distinct, complementary aspects. On one hand, they had a tangible physical existence; on the other, they were abstract prescriptions on which everything was founded. For instance, the monad (the number 1) was understood both as the generator of all other numbers, an entity as real as water, air, and fire that participated in the structure of the physical world, and as an idea-the metaphysical unity at the source of all creation.

The Fascinating Numbers

The fact that someone would find numbers fascinating is perhaps not surprising in itself. After all, even the ordinary numbers encountered in everyday life have interesting properties. Take the number of days in a year-365. You can easily check that 365 is equal to the sums of three consecutive squares: 365=10² + 11² + 12². But this is not all; it is also equal to the sum of the next two squares (365=13² + 14²)! Or, examine the number of days in the lunar month -28. This number is the sum of all of its divisors (the numbers that divide it with no remainder): 28= 1+2+4+7+14. Numbers with this special property are called perfect numbers (the first four perfect numbers are 6, 28, 496, 8218). Note also that 28 is the sum of the cubes of the first two odd numbers: 28 = 1³ + 3³. Even a number as widely used in our decimal system as 100 has its own peculiarities 100 = 1³ + 2³ + 3³ + 4³

OK, so numbers can be intriguing. Still, one may wonder what was the origin of the Pythagorean doctrine of numbers? How did the idea arise that not only do all things possess number, but that all things are numbers? Since Pythagoras either wrote nothing down or his writings have been destroyed, it is not easy to answer this question. The picture that emerges from assembling the different clues suggests that the explanation of the obsession with numbers may be found in the preoccupation of the Pythagoreans with two apparently unrelated activities: experiments in music and observations of the heavens.

Connection among Numbers

To understand how those mysterious connections among numbers, the heavens, and music materialized, we have to start from the interesting observation that the Pythagoreans had a way of figuring numbers by means of pebbles or dots. For instance, they arranged the natural numbers 1, 2, 3, 4,...as collections of pebbles to form triangles (as in figure 1). In particular, the triangle constructed out of the first four integers (arranged in a triangle of ten pebbles) was called the Tetraktys (meaning quaternary, or "fourness"), and was taken by the Pythagoreans to symbolize perfection and the elements that comprise it. As the man count "1, 2, 3, 4," Pythagoras interrupts him, "Do you see? What you take for 4 is 10, a perfect triangle and our oath."

But that was only the beginning. The Tetraktys made an unexpected appearance even in the scientific approach to music. Pythagoras and the Pythagoreans are generally credited with the discovery that dividing a string by simple consecutive integers produces harmonious and consonant intervals a fact figuring in any performance by a string quartet. When two similar strings are plucked simultaneously, the resulting sound is pleasing when the lengths of the strings are in simple proportions.

And where do the heavens fit into all of this? Pythagoras and the Pythagoreans played a role in the history of astronomy that, while not critical, was not negligible either. They were among the first to maintain that the Earth was spherical in form (probably because of the perceived mathematico-aesthetic superiority of the sphere). They were also probably the first to state that the planets, the Sun, and the Moon have an independent motion of their own from west to east, in a direction opposite to the daily (apparent) rotation of the sphere of the fixed stars. These enthusiastic observers of the midnight sky could not have missed the most obvious properties of the stellar constellations-shape and number. Each constellation is recognized by the number of stars that compose it and by the geometrical figure that these stars form. But these two characteristics were precisely the essential ingredients of the Pythagorean doctrine of numbers, as exemplified by the Tetraktys.

The Beloved Pythagoras Theorem

The square numbers associated with the gnomons (The word gnomn originates from the the name of a Babylonian astronomical time measuring device, similar to sundial) may have also been precursors to the famous Pythagorean theorem. This celebrated mathematical statement holds that for any right triangle , a square drawn on the hypotenuse is equal in area to the sum of the squares drawn on the sides. The discovery of the theorem was "documented" humorously in a famous Frank and Ernest cartoon.

Was Pythagoras truly the first person to have formulated the well-known theorem attributed to him? Some of the early Greek historians certainly thought so. In a commentary on The Elements the massive treatise on geometry and theory of numbers written by Euclid (ca. 325-265 BC)-the Greek philosopher Proclus (ca. AD 411-85) wrote: "If we listen to those who wish to recount ancient history, we may find some who refer this theorem to Pythagoras, and say that he sacrificed an ox in honor of the discovery." However, Pythagorean triples can already be found in the Babylonian cuneiform tablet known as Plimton 322, which dates back roughly to the time of the dynasty of Hammurabi (ca. 1900 1600 BC). Furthermore, geometrical constructions based on the Pythagorean theorem were found in India, in relation to the building of altars. These constructions were clearly known to the author of the Satapatha Brahmana (the commentary on ancient Indian scriptural texts), which was probably written at least a few hundred years before Pythagoras. But whether Pythagoras was the originator of the theorem or not, there is no doubt that the recurring connections that were found to weave numbers, shapes, and the universe together took the Pythagoreans one step closer to a detailed metaphysic of order.

Pythagoras views on Mathematics

On the question of whether mathematics was discovered or invented, Pythagoras and the Pythagoreans had no doubt mathematics was real, immutable, omnipresent, and more sublime than anything that could conceivably emerge from the feeble human mind. The Pythagoreans literally embedded the universe into mathematics. In fact, to the Pythagoreans, God was not a mathematician-mathematics was God!

The importance of the Pythagorean philosophy lies not only in its actual, intrinsic value. By setting the stage, and to some extent the agenda, for the next generation of philosophers-Plato in particular-the Pythagoreans established a commanding position in Western thought.

In the next Sunday Saga post we are going to look Plato views on mathematics (keep following my blog😅). Till then share your ideas in the comments.