Story of Maths - Part 1.3: Greek mathematics: Pythagoras, Plato. (37:10-47:40)

C. Greek mathematics By 330BC, the Greeks had advanced their imperial reach into old Mesopotamia. Just like the Babylonians before them, the Greeks were passionate about mathematics. The Greeks were clever colonists, who took the best from the civilisations they invaded to advance their own power and influence; but they were soon making contributions themselves. Their greatest innovation was a shift in the mind: they gave humanity the power of proof.Somehow they decided that they had to have a deductive system for their mathematics and the typical deductive system was to begin with certain axioms, which you assume are true. And then, using logical methods and very careful steps, from these axioms you prove theorems and from those theorems you prove more theorems, and it just snowballs. Proof is what gives mathematics its strength.

Pythagoras Pythagoras of Samos is credited, rightly or wrongly, with beginning the transformation from mathematics as a tool for accounting to the analytic subject we recognise today. Pythagoras is a controversial figure. Because he left no mathematical writings, many have questioned whether he indeed solved any of the theorems attributed to him. He founded a school in Samos in the sixth century BC, but his teachings were considered suspect and the Pythagoreans a bizarre sect. There is good evidence that there were schools of Pythagoreans, and they may have looked more like sects than what we associate with philosophical schools, because they didn't just share knowledge, they also shared a way of life. There may have been communal living and they all seemed to have been involved in the politics of their cities. One feature that makes them unusual in the ancient world is that they included women. What's known as Pythagoras' theorem states that if you take any right-angled triangle, build squares on all the sides, then the area of the largest square is equal to the sum of the squares on the two smaller sides. But Pythagoras made another great discovery, that has to do with music and the discovery of the harmonic series. The story goes that, walking past a blacksmith's one day, Pythagoras heard anvils being struck, and noticed how the notes being produced sounded in perfect harmony. He believed that there must be some rational explanation to make sense of why the notes sounded so appealing. The answer was mathematics. Experimenting with a stringed instrument, Pythagoras discovered that the intervals between harmonious musical notes were always represented as whole-number ratios. According to legend, Pythagoras was so excited by this discovery that he concluded the whole universe was built from numbers. But he and his followers were in for a rather unsettling challenge to their world view, and it came about as a result of the theorem which bears Pythagoras' name. Legend has it, one of his followers, a mathematician called Hippasus, set out to find the length of the diagonal for a rightangled triangle with two sides measuring one unit. Pythagoras' theorem implied that the length of the diagonal was a number whose square was two. The Pythagoreans assumed that the answer would be a fraction, but when Hippasus tried to express it in this way, no matter how he tried, he couldn't capture it. Eventually he realised his mistake. It was the assumption that the value was a fraction at all which was wrong. The value of the square root of two was the number that the Babylonians etched into the Yale tablet. However, they didn't recognise the special character of this number. But Hippasus did. It was an irrational number. But these irrational numbers didn't fit the Pythagorean world view. Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy, but Hippasus let slip the discovery and was promptly drowned for his attempts to broadcast their research.

Plato Schools of philosophy and science started to flourish all over Greece, building on these foundations. The most famous of these was the Academy. Plato founded this school in Athens in 387 BC. Although we think of him today as a philosopher, he was one of mathematics' most important patrons. Plato was enraptured by the Pythagorean world view and considered mathematics the bedrock of knowledge. In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking the secrets of the universe, a view still held by scientists today. Indeed, the importance Plato attached to geometry is encapsulated in the sign that was mounted above the Academy, "Let no-one ignorant of geometry enter here." Plato proposed that the universe could be crystallised into five regular symmetrical shapes. These shapes, which we now call the Platonic solids, were composed of regular polygons, assembled to create three-dimensional symmetrical objects. The tetrahedron represented fire. The icosahedron, made from 20 triangles, represented water. The stable cube was Earth. The eight-faced octahedron was air. And the fifth Platonic solid, the dodecahedron, made out of 12 pentagons, was reserved for the shape that captured Plato's view of the universe. Plato's theory would have a seismic influence and continued to inspire mathematicians and astronomers for over 1,500 years.