IV Greek Mathematics Greek mathematics is a larger body of scholarship than what came before. It, of course, had the benefit of the Mesopotamian and Egyptian advances in scientific and mathematical knowledge. The main mathematical subjects we will study from the Greek heritage are: The Greek search for explanations of physical reality The Greek use of logic and axioms Greek geometry Greek number theory Greek astronomy ". Pre-Socratic Greek thought. We begin in Ionia, on the coast of Asia Minor, in the City of Miletus. The first important Greek philosopher and thinker of whom we have record is Thales of Miletus ! (~625 to 547 B.C.E.) He is said to have worked to explain the events in the world on the basis of logical reasoning about reality, rather than resorting to supernatural reasons. He visited Egypt, and lived in Miletus, a city with overland contact to Mesopotamia which was ruled by Cyrus the Great of Persia from 550 onwards. Aechaemenids were ruling Persia from 650 B.C.E. forwards, and there was a road from Persia to Sardis in Ionia, dating from Assyrian times which was at least partly in place before Cyrus the Great completed his Royal Road. Thales thus was exposed to two deep and venerable cultures with somewhat differing mathematics and science. Reconciling differences in the values of ", to choose a small point, might lead one to try to come up with careful argument for one, the other, or something new. Miletus was a cosmopolitan city with Greek, Mesopotamian, Persian, Egyptian and Phoenician people in its institutions. Points of view on most subjects must have been varied. Discussions about what are the true causes of ! natural events and of mathematics were not the exclusive purview of Thales. Thales posited that the fundamental building block of all of reality is water. The various forms of water could account for liquids, solids and gases. His cosmology posited a large disc-shaped earth floating on water. When the water got rough enough it caused the earth to shake, causing earthquakes. Supposedly Thales sought after proofs of theorems in geometry. But we have no surviving writings of his. A student of Thales, Anaximander of Miletus (~610 to 546 B.C.E.), made an outline of geometry according to Herodotus, the Greek historian. He replaced Thales’ water as the fundamental unit of reality by “the boundless.” Worlds come out of the boundless, exist for a while, and then are absorbed back into the boundless. Anaximander was partial to Mesopotamian mathematics, whereas Thales seems to have favored Egyptian mathematics. After this time, with Cyrus the Great conquering Ionia, intellectual life suffered and innovative Greek thought began to migrate away from its Ionian birthplace. Cyrus was a tolerant ruler, but the men he assigned to local rule were often despotic. It was into this world that Pythagoras (~560 B.C.E. to 480 B.C.E.) was born. He was originally from Samos, he studied in Miletus, possibly with Anaximander and then traveled to Egypt. While he was in Egypt, the Persian king Cambyses conquered Egypt in 525 B.C.E. and Pythagoras went to Babylon for 7 years. During these times he learned the mysticism, mathematics and science of both cultures. He returned to Samos, but was exiled by the tyrant Polycrates. He emigrated and went to Italy, where there were Greek colonies. He settled in Croton, in southeastern Italy and founded a society of men and women who devoted their lives to philosophy, mathematics, mysticism and secrecy. All results discovered by the Pythagoreans were attributed to Pythagoras himself. The Pythagoreans believed in metempsychosis, that souls migrate to other bodies when we die, including the bodies of animals. As a result of this belief, they were strict vegetarians. They believed that number was the basis of reality, possibly due to such discoveries as the harmonics gotten by varying the length of the string on a musical instrument and the observation that the motions of the planets could be described using numbers. Constellations of stars looked geometric. Their cosmology posited a spherical earth, surrounded by heavenly spheres that rotated around the earth, containing the sun, moon, 5 visible planets, and one for the distant stars. There was a problem, due to the recognized retrograde (backwards) motions of the planets from time to time. So the planets were loosely attached to the rotating heavenly sphere. They then correlated the numbers in music with the distance of the spheres from the central earth, so that each sphere had a characteristic tone, the music of the spheres. It is interesting to notice that the Greeks seem to have been influenced by the Egyptian number representations more than the better Babylonian place-value system. The Greeks had a base ten number system and used alphabetic characters for the numbers. We only have evidence for this sort of representation of numbers from about 450 B.C.E. so that the Pythagoreans may have thought of number in terms of geometry, arrays of points. They dealt with ratios of two integers, and did not develop an arithmetic of fractions. Number for the Pythagoreans meant what we call rational numbers, the ratio of 2 positive integers. The discovery of 2 and the realization that it is not a rational number was devastating for them. It involved proving that 2 can not ever be expressed as a quotient of 2 integers, exactly. This sort of very careful mathematical argument is new to the world with Greek society. Previously mathematics was in the service of architects and ! practical people, so that such matters as differing values for ", which were each close enough to serve for all practical purposes!, caused no grief. The horror which the discovery of an irrational (not a ratio) number caused is still with us today in the use of the words “rational” and “irrational” to describe certain human behaviors. The Pythagoreans associated certain sequenc! es of integers with geometric figures. They defined the triangular, square and pentagonal numbers, and proved various relationships among them. The triangular numbers are T1 = 1, T2 = 1 + 2 = 3, T3 = 1 + 2 + 3 = 6, n(n + 1) and in general T = 1 + 2 + 3 + " " " + n = . These are the numbers of n 2 points which can be arranged in equilateral triangular arrays of points. ! 2 The square numbers are S1 = 1, S2 = 4, S3 = 9, and Sn = n . These are the num! bers of points which can be put into square arrays of points. We call (n x n ) n squared because of this association of the Pythagoreans. Similarly the pe!nt agonal numbers are P1 = 1, P2 = 1 + 4 = 5, P3 = 1 + 4 + 7 = 12, Pn = 1 + 4 + " " " + (3n # 2) n(3n " 1) = 2 ! They defined the idea of prime number, perfect number, amicable pairs of numbers and worked on “Pythagorean” triples of numbers which had been listed in Babylonian tabl!e s 1,000 years before Pythagoras. A prime number is a multiplicative atomic building block of numbers. It is any number which is divisible only by 1 and itself. 1 is not considered to be a prime number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and there are 25 prime numbers less than 100. We go to worksheet # 4 The Sieve of Eratosthenes, at this point. A perfect number is a number which is exactly equal to the sum of all of its proper divisors. Proper divisors are any divisors except the number itself. So 6, 10 and 28 are perfect numbers. The Pythagoreans knew these and also 496 and 8,128 Amicable numbers are a pair of numbers, each of which is exactly equal to the sum of the proper divisors of the other. The Pythagoreans only knew of the pair 220 and 284. The Pythagoreans are the originators of the subject of Number Theory, a branch of current mathematics. A prominent Pythagorean, Archytas of Tarentum (~438 B.C.E.to 347 B.C.E.) also from Italy established a curriculum of studies known as the quadrivium which consisted of Geometry, Number Theory, Music and Astronomy. Both Plato and Eudoxus were students of his. Homework III 1. Adapt the proof by Aristotle that 2 is an irrational number to prove that 11 is also an irrational number. n(n + 1) 2. (Math credit) Prove tha!t the nth triangular number equals 2 ! 3. Show, using a Pythagorean argument involving arrays of points, that the nth square number is a sum of two triangular numbers. ! 4.Show that 496 is a perfect number. 5. Show that 220 and 284 are amicable numbers. 6. (Math credit) Prove that the nth pentagonal number is equal to the nth square number plus the (n – 1)st triangular number. One remaining voice emanates from the Persian-dominated Ionian city of Ephesus, Heraclitus (~500 B.C.E.). He was an acerbic and brilliant philosopher who criticized the great minds of his time, including Homer and Pythagoras. He believed that diversity made up a cosmic unity, and is quoted as stating that “You cannot step twice into the same river, for fresh waters are ever flowing in upon you.” The unity that is diversity is a huge tension. He is frequently summed up, a little too quickly, as believing that everything is change. In keeping with his belief in a dynamic tension, he posits that the basis of all reality is fire. We now consider the school of the Eleatics, from Elea in southwest Italy. Parmenides (~515 B.C.E. to 450 B.C.E.) is the first eminent thinker of whom we have documentation from this school.