Tuesday Afternoon at U3A 14 June 2016

AMajorFlawinAncientGreekMathematics-wasPythagoraswrong?

Dr.RobertColomb

(RetiredProfessorComputerScience,withdegreesinMathematicsandPhilosophy)

This story, which is about numbers, the Greeks, and the crippling of the Pythagorean philosophy, began with an observation of a pattern of tiles on a bathroom floor. Before we get to that pattern, I will show you a couple of simple patterns, so as to establish some terminology, and also show you what to look for.

This is a photo of a tiled bathroom. It is common for tiles to be of different sizes. Here the grey tiles are bigger than the white tiles. What is interesting is where the two kinds of tiles meet. You can see that on the left the left edges of the grey tiles and the left edges of the white tiles meet exactly. Then if you move one tile to the right, the edges don’t meet. Keep moving right and eventually the left edges do meet again. You have to move across three of the white tiles and two of the grey tiles until the edges meet again. Keep going three more white tiles and two more grey tiles, and the edges match up a third time. No matter how far you go, every three white tiles and two grey tiles, the edges will match up again. This is because the white tiles are 2/3 as long as the grey. You could also say the grey tiles are 3/2 as long as the white. It is common to say that the grey and white tiles are in a ratio of 3 to 2.

Here is another example. In this case the white tiles are twice as wide as the grey, so they are in a ratio of 2 to 1. The designer has put a half tile on each side of the pattern to make it less obvious, but you can see that the left edge of the white tiles meets the grey tiles at the same relative place every time, which is every two grey tiles.

This kind of situation happens whenever the two kinds of tiles are in a ratio of p to q. Say p is 5 and q is 4, so the red tiles are 5/4 the size of the blue tiles, and the blue tiles are 4/5 the size of the red tiles. So starting from matching edges, the edges will match again every 5 of the blue tiles, which is the same as every 4 of the red tiles.

Here is another bathroom, where the tiles are more complex. Look at the white triangular tiles, and also the dark green triangular tiles, both of which are diagonal halves of square tiles. They are laid against orange . The ratio isn’t obvious. The pattern never repeats in the whole three metres of the floor, in either case.

When I contemplated this pattern, at first it seemed ugly. Then I said to myself: the owner is an architect. There must be beauty in this. After some thought, I realized that this pattern illustrates a profound mathematical truth, one that shattered the mathematical philosophy of the Pythagoreans in ancient Greece!

Unfortunately, we are going to need some more concepts before we can finish the story. First, numbers are based on the integers: 1, 2, 3, 4, 5 and so on. The discussion of ratios brings in fractions:

1 : 2 is either 1/2 or 2/1 2 : 3 is either 2/3 or 3/2 5 : 6 is either 5/6 or 6/5

As we can see from the first line, integers can be thought of as fractions: the integer n is the fraction n/1. Fractions, either proper or improper, thought of as a system of numbers, are called rational numbers. Rational numbers can be of any size. 739/113 is a rational number. It is possible to have a tile pattern where the tiles are in that ratio. So the pattern repeats after 739 smaller tiles and 113 of the larger tiles. The fact that the pattern between the triangular tiles and the long orange tiles does not repeat over a short range is aesthetically unusual, but not necessarily mathematically problematic.

All calculation is done with rational numbers. Addition

𝑎𝑏+𝑐𝑑=𝑑𝑎𝑑𝑏

+𝑐𝑏𝑑𝑏

=𝑑𝑎 + 𝑐𝑏𝑑𝑏

12+23=3×12×3

+2×22×3

=3 + 46

=76

Multiplication 𝑎𝑏

×𝑐𝑑=𝑎 × 𝑐𝑏 × 𝑑

12×23=26=13

Division 𝑎𝑏÷𝑐𝑑=𝑎𝑏×𝑑𝑐=𝑎×𝑑𝑏×𝑐

12÷23=12×32=34

These rules are easy to apply in many simple everyday calculations, but in general can get messy. Suppose we have to calculate 739/113 + 83/577!! Because of this, it has long been common in particular fields to use fractions with a fixed denominator. One dozen (12) is 1/12 of a gross (144). One gross is 1/12 of a great gross (1728). We can easily calculate a half-dozen, or a third of a dozen, or a quarter of a dozen. Potatoes and other similar things used to be sold in fractions of a hundredweight (112 pounds). A quarter (28 pounds) is a quarter of a hundredweight. The common retail unit of potatoes was 7 pounds, which is a quarter of a quarter.

Compass directions, and angles generally, are usually based on fractions of 360 degrees. North is 0 degrees, east, one quarter of the circle, is 90 degrees, south, one half, is 180 degrees, west, three quarters of a circle, is 270 degrees. The ancient Babylonians used a system of representing numbers as fractions of 60. It is easy to divide 60 into halves, thirds, quarters, fifths, sixths, tenths, twelfths, fifteenths, twentieths and thirtieths. We, today, represent numbers as fractions with a denominator of powers of ten.

One half = 0.5 = 5/10 One quarter = 0.25 = 25/100 One tenth = 0.1 = 1/10

When we add one half and one quarter, we get

!!"+ !"

!""= !"

!""+ !"

!""= !"

!""= 0.75

The rules of arithmetic in our system are sufficiently simple that almost all children master them in primary school, even though they have been mechanised into calculators and spreadsheets and so on, so many adults have forgotten.

A cost of this simplicity is that many useful rational numbers are not able to be exactly represented. Take 1/3, for example. So what we do is use in our calculations a rational number which is near to 1/3 but not exactly. We can make this approximation as close as we need. For ordinary purposes, 0.33 = 33/100 is good enough. But we can easily make much more accurate approximates. A scientific calculator represents 1/3 as 0.333333 = 333333/1 million. An approximation 1/3 = 0.333333333333 = 333333333333/1 trillion is close enough to send a spacecraft to the moon. We now have enough to talk about the ancient Greeks, and what is mathematically interesting about the floor with diagonal tiles.

Pythagoras The ancient Greeks were latecomers to the civilizations of the eastern Mediterranean and Mesopotamia. The arts of calculation (arithmetic) and measurement (geometry) were highly developed, and had been for thousands of years. We marvel at the first Egyptian Pyramids, 2700 years BCE and 2000 years before the Greek classical period.

Step Pyramid of Djoser

But think of the measurements and calculations to work out how many stones of what size had to be cut and transported, how many workers would be needed, what tools and transport facilities, how much to raise to pay the workers, how much housing was needed, how much food to reserve, to make sure the food required was within the surpluses of the harvests, and so on.

Arithmetic was much more complicated in those days, but there was an educated class of people who could, and did, perform the necessary calculations. Geometry was the same. The earlier ancients knew about right angles, and could make tools like a 3-4-5 right triangle to measure them. What was peculiar about the Greeks is that when they became settled in cities, there arose a class of people who had the leisure to think, but no official duties to perform. They were neither priests, nor architects, nor functionaries in the bureaucracies that built and organised things. What they did became known as philosophy (love of wisdom), and they became known as philosophers. They thought about all kinds of things, but they often thought about calculations and measurements. Since they didn’t need to actually perform calculations or measurements, they were free to develop the theory of calculation and measurement, which we know now as arithmetic and geometry.

The first person thought of today as a philosopher was Thales, who lived in Miletus, in Western Anatolia, during the sixth century BCE. The island Samos is just offshore. Among many other achievements, he proved the earliest known theorem of geometry, still taught today.

Overlapping with Thales was Pythagoras, who was born in Samos about 570 BCE, and died in Metapontum, in southern Italy, about 495 BCE. His major activity seems to have been at Croton, in southern Italy, from about 530 BCE. Pythagoras was a highly influential figure, both in religion and mathematics. He founded a school of philosophy which lasted for more than 100 years. One of his later followers was Plato. It is hard to know what was actually done by Pythagoras, since he had many brilliant followers, and it was customary for the followers to attribute any discovery to Pythagoras himself. Our interest is limited to Pythagoras’ mathematical discoveries, and it doesn’t matter which of his followers may have actually made them.

Arithmetic and Rational Numbers One of Pythagoras’ discoveries is the theory of harmony, in which the sounds produced by strings are related by ratios of integers:

The Pythagoreans developed a cosmology in which everything was related by ratios of integers. This included the so-called “music of the spheres,” in which the sun, moon, and planets were thought to emit imperceptible tones based on the ratios of their orbital periods. Rational numbers were the central explanatory principle in this cosmology.

Geometry and Pythagoras’ Theorem

The most famous of Pythagoras’ results in geometry is the theorem proving that in a right triangle, the square of the hypotenuse is the sum of the square of the two sides.

The theorem applies to any right triangle, so in particular it applies to a triangle whose two sides are length 1:

If the two sides are of length 1, the squares of their lengths are also each 1. The square of side C is the sum of the squares of the other sides, so is 2. Therefore the length C is a number whose square is 2. It is not obvious exactly how long C is. But the Pythagoreans were theoreticians, and not particularly concerned with making the calculation. Also, as we have seen, for practical purposes a good enough approximation suffices.

It is easy to make approximations to C. I’m not sure how the ancients worked, but the modern technique is essentially controlled guesswork. We can start with the following table, with numbers that are fractions of 100, square each, and see which is closest to 2:

Approximation Square 1.3 1.691.4 1.961.5 2.25

The nearest guess is 1.4, so we can refine the guess with fractions of 1000:

Approximation Square 1.40 1.961.41 1.98811.42 2.0164

The nearest of these guesses is 1.41, so we can further refine the guess with fractions of 10000:

Approximation Square 1.413 1.9965691.414 1.9993961.415 2.002225

And so on. We can continue until we get an approximation that is near enough for our purposes. A modern calculator or spreadsheet works with the approximation 1.414213562, a fraction of one billion.

However, one of the Pythagoreans, Hippasus of Metapontum, sometime in the late 4th century BCE, is credited with asking the question: whatever the length of C is, is that length odd or even? That is to say, is C the product of 2 and some other rational number, D? If so, C is even. If not, C is odd. The proof used is a proof by contradiction, or reductio ad absurdum. We start with an assumption. If the proof reaches a contradiction, then the assumption must be false. For example, consider the proposition that there is a largest integer. If there is a largest integer, let’s give it a name: N. Now we look at N. Since N is an integer, we can add 1 to it. But N+1 is larger than N. Since we have assumed that N is the largest number, we have a contradiction. We conclude that there can not be any largest number. Our assumption in the following is that C is a rational number.

Since all numbers are thought to be rational, C must be the ratio of two integers, A and B:

𝐶 =𝐴𝐵

The square of C is 2, so (!!)! = 2

and (!!)! = (!

!

!!)

so 𝐴! = 2𝐵! Now we can assume without loss of generality that A and B are not both even. If they were, we could divide both A and B by 2, preserving the ratio. For example

24=12

The right side of the equation is even, since it has a factor of 2. Therefore the left side must also be even, since the two are equal. Since 2 has no factors, we must have some integer D so that

𝐴 = 2𝐷 So A is even, and

𝐴! = (2𝐷)! = 4𝐷! = 2𝐵! We can divide through by 2, getting

2𝐷! = 𝐵! By the same argument as before, B is also even. We have concluded that A and B are both even. But our initial assumption was that at most one of A and B are even. This is a contradiction. Therefore, there can be no numbers A and B such that

𝐶 =𝐴𝐵

C, the square root of 2, is not a rational number! As you can imagine, this result was alarming to the Pythagoreans. If their whole cosmology was based on rational numbers, this is something that is not rational. Further, this cosmology included the unity of arithmetic and geometry. That is, every geometrical structure could be measured by a rational number. But Pythagoras’ own most famous theorem produced a structure that could not be measured by a rational number! One story is that this discovery, made during a sea voyage, so upset the gods that they brought forth a storm, causing Hippasus to drown at sea! In any case, Pythagoreanism began to decline about that time, and seems not to have been a strongly organised force by the time of Plato, who lived at the end of the 4th century BCE.

Returning to our original bathroom floor,

The fact that neither the white nor dark green triangles form, and can never form, no matter how long the tiles continue, a repeating pattern with the orange tile strips is in fact an illustration of a profound mathematical truth, one that brought about the collapse of the Pythagorean branch of ancient Greek mathematics.