Section II — A Greek Tragedy

1. Section II — A Greek Tragedy

2. History The story of the Greeks from 900 BCE to 300 BCE is one of the most fascinating tales of human history. This is something that is really worthwhile studying on your own. Not only were the Greeks extremely good at math, it seems they were extremely good at nearly everything. Lets take a very quick look.

3. Art Egyptian Persian Greek At its height, Greek art developed a rare understanding of the human form not seen before.

4. Greek theater was the originator of our own. Many Greek plays are very entertaining, even for an audience today. Far from boring, they are often filled with many things more shocking than what you find in movies today (Oedipus). The theater company at NMSU sometimes puts them on. Go see one for fun if they do it again. Literature

5. Politics As we all know, the Greeks (Athenians) originated the idea of a democracy. After the Greeks, the next major step toward democracy was the English Magna Carta in 1215. Then full democracy came again with the U.S. constitution.

6. War: Greeks vs Persians The Persian empire was the largest the world had ever seen, and was set to destroy the Greeks.

7. The Greeks repeatedly beat the Persians through superior skill, devilish trickery, and luck, though always outnumbered. • Marathon • Thermopylae (300 Spartans) • Salamis Later, Alexander the Great left Greece with 40,000 men and conquered the whole Persian empire of millions (and more) before he was 27 years old. After his death in 323 BCE, Alexander’s soldiers remained a formidable army in Persia until many were well into their 60’s.

8. Education Around the fall of Rome (450 AD) a system of education based on the classical Greek studies was introduced. This system was called the Seven Liberal Arts • The triumverate of Grammar, Logic, Rhetoric • The quadumverate of Arithmetic (number), Geometry (form), Harmonics (number in motion), Astronomy (form in motion) This has remained in effect for 1500 years. This “Liberal Arts” education is likely the primary reason you are asked to take a class such as Math 210.

9. Math and Philosophy Of the many wonderful Greek accomplishments, their efforts in Mathematics and Philosophy are likely the greatest. They made these subjects what they are, and had no rival in their accomplishments until the 1600’s. For some 2000 years humanity looked back for the answers rather than forward. We look at a chosen few aspects of their mathematics.

10. A secretive religious sect founded by Pythagoras about 550 BCE. Vegetarian, communal, equal rights for women, believed in reincarnation, … Complex, unusual beliefs that numbers (1, 2, 3, … ) were the only real thing in the universe, all else comes from their interaction. Their discovery of harmonics (sounds in ratio 2:1 or 4:3 etc) helped build their beliefs. The Pythagoreans

11. The Pythagoreans All this may sound crazy, but peoples beliefs are very serious things to them. The Pythagoreans did math because in a very direct way they felt numbers described the universe. Having found deep connections between number and music, they looked to connections between number and geometry. A basic idea is the measure, or length, of a line. They would consider one line to be 2 times the measure or another if it was twice as long, etc. This connected geometry to number, at least a little bit.

12. Common measure The Greeks sought to find the largest line that would fit an even number of times (1, 2, 3, 4, … ) into each of several other lines. This was called their common measure. Example: _____ _______________ _________________________ The red line is the common measure of the two black ones. It goes 3 times into the first line, 5 times into the second.

13. Common measure In modern terms, this means choosing a scale so that all lines have whole number lengths (3 and 5 in the example). Equivalently, this means picking one line to have length 1, then the lengths of the others are fractions (5/3 in the example). They were very happy with this as it let them treat geometry with numbers (meaning whole numbers or fractions).

14. The Greek Crisis The Greeks were able to prove there are lines that have no common measure! This was horrible for them as it completely altered the very basis of their view of the world. It meant that geometry was not built from numbers such as 1, 2, 3, … or at least not in any kind of way that they could understand. Lets look at this more closely …

15. 1 1 Take a right angle triangle with two sides of equal length. Let the short sides be length1. If there is a common measure, the long side (hypotenuse) has fractional length p/q. Pick p,q so that the fraction is in lowest terms, meaning nothing divides both the top and the bottom. By the Pythagorean theorem 12 + 12 = (p/q)2 so p2 = 2q2. As 2q2 is even, p2 must be even, so p must be even, so p2 must be a multiple of 4, so 2q2 must be a multiple of 4, so q2 must be even, so q must be even. So p, q are both even. But this means p/q is not in lowest terms as 2 divides p and q! p/q

16. The role of proofs The Greek devotion to proving things perhaps had no direct practical use. Yet it lead them to a deeper understanding of what they were doing. In this case it showed them their view of the world was either completely wrong, or in need of major revision. About 200 years later, Eudoxus found solutions to this problem of common measure. His work is very close to what modern people would call the real numbers.

17. Modern arrogance It is easy to say, “those silly Greeks just didn’t know about the square root of 2”. But saying there is a number whose square is 2 doesn’t ensure there is one. Correct modern treatments of real numbers were first done around 1860 by Dedekind. But these modern treatments lead to new troubles that are still not resolved! In a way, the approach of Eudoxus still has some logical advantages over what we use today.

18. The Pythagoreans faded, but left an influence on Plato, Aristotle, and Archimedes, we feel today. Greek mathematics flourished from 500 BCE to 300 BCE forming much of what was known for the next 2000 years. Fortunately much of this was collected in a single book called the Elements by Euclid. Euclid

19. The Elements A book written by Euclid around 300 BCE. A detailed account of geometry, numbers, and arithmetic, with everything meticulously proven. It consists of 13 chapters, called books. This book was used continuously for over 2000 years as the primary source of mathematics for much of the world! Lets take a closer look.

20. Book I Triangles, parallels, area Book II Geometric algebra Book III Circles Book IV Constructions of figures Book V Theory of proportions Book VI Similar figures Book VII Basic number theory Book VIII Continued proportions Book IX Number theory Book X Incommensurables Book XI Solid geometry Book XII Measurement of figures Book XIII Regular solids Contents of The Elements

21. History of the elements None of the original versions of the Elements survive. But the book was copied (by hand) in many different versions and languages over the next 2300 years. It took a long journey to come to us … Greek Roman Byzantine Islamic European Modern

22. From a Byzantine copy around 880 AD.

23. From a beautiful Victorian copy around 1850 AD. It was color coded to try to help people see thing better.

24. A modern version of Euclid. It looks exactly like what you think a text book on math should look like. The remarkable thing is that it is 2300 years old!

25. Travels of the Elements Empire of Alexander the Great 300 BCE

26. The Romans continued Greek learning and culture.

27. After the fall of Rome to barbarians around 450 AD, the center of learning and culture shifted to the Eastern Roman empire, called the Byzantine empire. In particular, to its capital of Constantinople, and the library at Alexandria (Egypt).

28. As the Islamic Empire overran the Byzantines, much of the legacy of the Greeks and Romans was translated into Arabic and the Islamic empire became the center of learning.

29. During the crusades from 1100 - 1250, the classical Greek learning held by Islam, as well as much Islamic learning, was transferred to Western Europe.

30. The knowledge coming back from the crusades found a home in the universities being founded in the west, such as Oxford, founded around 1200.

31. A final home The Elements became a fixture of the seven liberal arts at western universities up to the 20th century. The elements would have been something studied by most educated people, including much of the nobility, throughout this time. Lincoln, unhappy with his lack of understanding of proving an argument, took to studying the Elements as a young lawyer and congressman in his 40’s.

32. Remarks on Euclid Euclid became the standard source for learning arithmetic, geometry and reasoning for thousands of years. But the Elements is more than just a text to learn the basics of arithmetic and geometry. It contains such a deep and precise account of these matters, it is something modern scientists would do very well to read carefully. A well-educated person from the middle ages who had thoroughly studied Euclid may very well have had a better perspective on basic aspects of real numbers than the average rocket scientist at NASA!

33. Remarks on the Greeks Greek accomplishments generally, and in mathematics in particular, were simply astonishing. They far surpassed those before, and long after. A fundamental aspect that drove the Greek success was a desire to understand completely every aspect of a subject. In mathematics, this manifested itself as a desire to make precise proofs of everything they encountered. More than any specific factual discoveries, it is this Greek insistence on proof that has had the greatest effect on future generations, and on the Greeks themselves.

34. The Greek crisis again Recall the crisis of the Pythagoreans (the discovery of lines that have no common measure). The Greek desire for complete understanding forced them to face the fact that their world view was either incorrect, or at least seriously incomplete. They didn’t abandon their desire for understanding and proofs, rather they faced their troubles and eventually managed to deal with lines without common measure (Eudoxus). They adapted their view of the world through the teaching of Plato, etc.

35. The Greek crisis again This Pythagorean view that the reality of numbers underlies all in the world has gone through many modifications, but has never left mankind. Recalling the quadrivium … number harmonics geometry astronomy

36. These subjects are linked in the mysterious phrase ... “Music of the spheres” This means to convey some underlying order and pattern to the universe. It drove pre-Newtonian attempts to understand the motion of the planets. Kepler’s model of planetary motion Dated about 1600. The Greek crisis again

37. The somewhat unusual world view of Erwin Schrödinger is reminiscent of this phrase “music of the spheres”. Erwin was the fellow who more than anyone else is responsible for our modern understanding of quantum mechanics. In a very real way, he viewed matter as a similar kind of thing to musical sounds. The Greek crisis again

38. The Greek crisis again The Greeks were not the last to face such a crisis. But perhaps the most serious and fundamental crises are ones that have arisen only in the past 100 years. Later we will look at these modern crises. Hopefully we will deal with these with the same grace that the Greeks dealt with theirs.