What We Cannot Know: Explorations at the Edge of Knowledge. Marcus Sautoy du

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СКАЧАТЬ the key to the ingredients of Greek chemistry: the elements of fire, earth, air and water. Each element, Plato believed, had its own three-dimensional mathematical shape.

      Fire was the shape of a triangular-based pyramid, or tetrahedron, made from 4 equilateral triangles. Earth was cube-shaped like my Vegas dice. Air was made from a shape called an octahedron, constructed from 8 equilateral triangles. It is a shape that looks like two square-based pyramids fused together along the square faces. Finally, water corresponded to the icosahedron, a shape made from 20 equilateral triangles. Plato believed that it was the geometrical interaction of these basic shapes that gave rise to the chemistry of the elements.

      The atomistic view of matter was not universally held across the ancient world. After all, there was no evidence for these indivisible bits. You couldn’t see them. Aristotle was one of those who did not believe in the idea of fundamental atoms. He thought that the elements were continuous in nature, that you could theoretically keep dividing my dice up into smaller and smaller pieces. He believed that fire, earth, air and water were elemental in the sense that they could not be divided into ‘bodies different in form’. If you kept dividing, you would still get water or air. If you take a glass of water then to the human eye, it appears to be a continuous structure which can theoretically be infinitely divided. If I take a piece of rubber then I can stretch it in a smooth manner making it appear continuous in nature. The stage was set for the battle between the continuous and discrete models of matter. The glissando versus the discrete notes of the musical scale. The cello versus the trumpet.

      Intriguingly, it was a discovery credited to the Pythagoreans that would threaten the atomistic view and which turned the tide for many years in favour of the belief that matter could be divided infinitely.

       NUMBERS AT THE EDGE

      From the atomistic point of view, if I draw two lines on the page then each line will be made from a certain number of these indivisible atoms and hence their lengths would be in a ratio of whole numbers corresponding to the number of atoms making up each line. But things didn’t turn out to be so orderly. In fact, it was Pythagoras’ own theorem about right-angled triangles that revealed that the world of geometry could give rise to lines whose relative lengths could not be captured by simple fractions.

      The dimensions of my dice already have hidden in them a challenge to this atomistic view of nature. Take two of the edges of the cube at 90 degrees to each other. They have equal length. Now consider the diagonal line across the face of the cube that completes the triangle made up from the two edges of equal length. How long is this diagonal line relative to the shorter sides?

      Pythagoras’ theorem about right-angled triangles says that the square of the length of the diagonal is equal to the sum of the squares of the lengths of the two shorter sides. If I set the length of the sides of my dice as 1, then Pythagoras’ theorem implies that the length of the diagonal across the face of my dice is a number which, when you square it, is equal to 2. So what is this number?

      The Babylonians had been fascinated by the challenge of calculating this length. Dating back to the Old Babylonian period (1800–1600 BC), a tablet housed at Yale University has an estimate for the distance. Written using the sexagesimal system, or base 60, they got the length to be:

      which in decimal notation comes out at 1.41421296296 …, where the 296 repeats itself infinitely often. It is true of all fractions that when written as decimals they repeat themselves after some point. Indeed, any decimal expansion which does repeat itself can always be written as a fraction. The Babylonian calculation is quite a feat. It is correct to six decimal places. But when you square that fraction it just misses being 2. What the ancient Greeks discovered is that however hard the Babylonian scribes tried, they would always find that their fractions just missed squaring to 2.

      It was one of Pythagoras’ followers, Hippasus, who is credited with the discovery that the Babylonians were doomed to failure. He proved that the length of this diagonal across the side of my dice could never be expressed as a fraction.

      Pythagoras’ theorem about right-angled triangles implied that this long side had length the square root of 2 times the length of the short sides. But Hippasus could prove that there was no fraction whose square was exactly 2. The proof uses one of the classic tools in the mathematician’s arsenal: proof by contradiction. Hippasus began by assuming there was a fraction whose square was 2. By some deft manipulation this always led to the contradictory statement that there was a number that was both odd and even. The only way to resolve this contradiction was to realize that the original assumption must have been false: there can be no fraction whose square is 2.

      His fellow Pythagoreans were reputed to be dismayed by the revelation that their beautiful right-angled triangles could produce such inharmonic lengths. The sect took a vow of silence, but when Hippasus let the discovery out of the bag, the story goes that he was drowned at sea for revealing such disharmony in the physical world. But these new numbers, called irrational numbers because they are not ratios of whole numbers, were not so easily silenced.

      Irrational lengths inside the cube.

      I certainly have the feeling that this length exists. I can see it on a ruler held up against the long side of the triangle. It is the distance between two opposite corners of any side of my dice. Yet try to write down the number as an infinite decimal and I can never capture it. It begins 1.414213562 … and then continues to infinity never repeating itself.

       IRRATIONAL EXUBERANCE

      The discovery by the ancient Greeks that there were lengths that couldn’t be expressed as simple ratios of whole numbers led the mathematicians of the time to create new mathematics, the mathematics of irrational numbers, in order to truly take the measure of the universe. Other basic lengths like π, the circumference of a circle of unit diameter, also turned out to be irrational and not captured by ratios of whole numbers. Although the ancient Greeks knew 2000 years ago about the irrationality of the square root of 2, it took till the eighteenth century for Swiss mathematician Johann Heinrich Lambert to prove that no fraction could capture π.

      Despite my aversion to things we cannot know, reading about numbers that can’t be captured using simple whole-number ratios or fractions was one of the defining moments that sparked my love affair with mathematics. The same year that my music teacher introduced me to the trumpet in the storeroom cupboard, my maths teacher introduced me to the proof of the irrationality of the square root of 2. The proof was contained in one of the books my maths teacher recommended to me to ignite the mathematical fire in me. It worked. I was amazed to find that you could prove with a finite logical argument that only the infinite could articulate what a length such as the diagonal across a square measured. If I couldn’t write out this length, the next best thing was knowing why I couldn’t know this number.

      Since reading that proof as a schoolkid I have learnt about alternative ways to explore these irrational numbers – so perhaps these are numbers we can know. There are infinite expressions with patterns that make the number less mysterious. For example,

      The discovery of these expressions pulls these irrational numbers into the known. A fraction is a number whose decimal expansion repeats itself from some point. Couldn’t I regard these expressions as a pattern, not too dissimilar from the repeating pattern of the decimal expansion of the fraction? СКАЧАТЬ