The triangular relationship between man, mathematics and the universe

Let me begin with a piece of mathematics that every schoolchild is required to learn, the Pythagoras theorem: In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. With this, Pythagoras had laid the basis for a problem -- Fermat’s last theorem -- that confounded some of the greatest mathematical minds for over 300 years.

Pythagoras of Samos (570-490 BC) is a foundational figure in mathematics, and since there is no first-hand account of his life and work, he remains a mysterious and legendary persona. What is known for certain is that Pythagoras developed the idea of numerical logic and heralded the first great revolution in mathematics. It was he who discovered that numbers exist independent of the physical world and that their study was unaffected by prejudice or opinion.

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Pythagoras garnered his mathematical skills in the course of his travels across the ancient world, some believing he travelled as far as India. What is certain, though, is that after 20 years of travel, he had mastered all the mathematical rules extant in the known world. When he returned to Samos with the intent of founding a school devoted to the study of philosophy, he saw that the tyrant Polycrates had turned the once-liberal island into an intolerant society. Invited by Polycrates to join his court, Pythagoras declined and instead left the city and moved to a cave in a remote part of the island, where he could contemplate and teach without fear of persecution.

Soon, he was forced to flee to Croton, where he found a patron in Milo, the wealthiest man in Croton and a 12-time champion of the Olympic Games. He offered a part of his home for Pythagoras to establish a school for philosophy. Thus, a partnership was forged between a creative mind and a strong physical body.

In his new home, Pythagoras founded the Pythagorean Brotherhood, an egalitarian school dedicated to higher mathematical thinking, and thereafter, invented the term philosopher, and in so doing defined the aims of his school. Leon, Prince of Phlius, supposedly asked Pythagoras how he would describe himself. Pythagoras replied, ‘I am a philosopher’. Pythagoras went on to explain that life may well be compared with the Olympic Games and in the vast crowds that gather ‘Some are influenced by the love of wealth while others are blindly led on by the fever for power; but the finest type of man gives himself up to discovering the meaning and purpose of life itself. This is the man I call a philosopher.’

Pythagoras discovered that numerical perfection depended on a number’s divisors, and that the rarest numbers are those whose divisors add up exactly to the number itself and these are perfect numbers. The number 6 has the divisors 1, 2 and 3, hence it is perfect because 1+2+3 = 6. So it is with 28, because 1+2+4+7+14 = 28.

Besides studying the relationships between numbers, Pythagoras was interested in the relationship between numbers and nature. He recognised natural phenomena are governed by laws that can be described by mathematical equations. One of the first such relationships he discovered was that simple numerical ratios could generate harmony in music; and he applied this to the lyre, examining the properties of a single string. Fixing the string at a point exactly half-way along it, or a third, a quarter, or a fifth of the way along it, harmonious notes are produced. In essence, he had demonstrated the fundamental relationship between mathematics and science, and in an epiphany saw that numbers were hidden in everything, from the harmonies of music to the orbits of planets, exclaiming ‘Everything is Number’.

Of all the relationships between numbers and nature, the most important was expressed in the Pythagoras theorem; an equation true of all right-angled triangles, and therefore defines the right-angle itself. In turn, the right-angle defines the perpendicular; the relation of the vertical to the horizontal; and therefore, the relation between the three dimensions of our universe.

Though associated with Pythagoras, the substance of this theorem was used by the Chinese, Babylonians, perhaps Indians too, a thousand years before. However, they had no way of showing that it was universally true; that it was a law of mathematics. Pythagoras’s claim to the theorem is that he was the first who demonstrated this with mathematical proof.

Though easily understood today, mathematical proof as the search for knowledge that is absolute was unknown before Pythagoras. This method as the necessary condition to establish the truth has driven mathematics and science for the last 2,500 years. It was Pythagoras who revolutionised the eternal human quest for meaning. ‘Pythagoras – A Life’ by Peter Gorman tells this captivating story, one worth reading.