Lecture 14. Decline of Greek Mathematics

Figure 14.1 Ancient Greece

Summary of the Greek Achievements What are the accomplishments of Greekmathematicians?

• The Greeks are to be credited with making mathematics abstract Making mathe-matics abstract was the greatest contribution that Greeks made and it was of immea-surable significance and value. Think about this: a concrete algebraic equation couldbe applied to hundreds of different physical situations, which was indeed the secret ofthe power of mathematics.

• The Greeks insisted on deductive proof This was an extraordinary step. Of thehundreds of civilizations that did develop some crude arithmetic and geometry, onlythe Greeks had the idea of establishing conclusions by deductive reasoning and haddetermination to realize it. With the methods mankind has utilized in all other fields,many other civilizations did not think insisting deductive rigorous proof was necessary.Even learned such method from Greeks work later, many civilizations still decided notuse it. From this point of view, Greeks’ decision to require deductive proof was entirelyat odds, or almost irrational, at that time.

85

• Why Greeks insisted on deductive proof ? The Greeks wanted the truth, absolutetruth. They realized that a mathematical theorem, once proved, will never fail and itis an absolute truth. They believed this is the way to understand the universe, andthey saw that unquestionable methods of deductive reasoning was the only way forthem to obtain these truths.

The Greeks also realized that to secure truths they had so start from truths, and thatany unwarranted fact will never be assumed. As a result, the Greeks collected allbasic truths, called axioms, explicitly and regarded them as the starting points so thatfurther mathematical conclusions could be built upon these axioms.

The Greeks considered arithmetic, geometry, and astronomy to be the art of the mindand considered music for the soul.

Figure 14.2 The theatre of Epidauros, 4th century B.C.

• Two milestones: Euclid’s Elements and Apollonius’ Conic These two books con-tained 467+389 propositions, all derived from the 10 axioms in the Elements. Theydeveloped a new way for mathematicians to do their work.

• Creation of geometry and trigonometry Greek mathematics included plane geome-try, solid geometry, plane trigonometry and spherical trigonometry.

• Developed geometric algebra Eudoxus’ proportion theory resolved the first conflicton irrational numbers. It was almost shut down the real number theory, which awaitedonly the recognition of irrational numbers and translation into symbolic language.

• The beginning of the number theory.

• The method of exhaustion The beginning of the calculus.

86

• Applications Greek conception of nature, which identified mathematics with thereality of the physical world and saw in mathematics the ultimate truth about thestructure and design of the universe.

Summarize Greeks’ limitation Despite its marvellous achievements, Greek mathemat-ics has limitations:

• Greeks were unable to find the concept of irrational number, which separated arith-metic and algebra from geometry for thousand years.

• Greeks only concentrated on geometry by using rigorous mathematics. As a conse-quence, geometry became more and more complicated, particularly in the area of solidgeometry.

• Greeks put restrictions on curves, only accepted the figures that could be obtainedfrom line and circle.

• Greeks feared the infinite process so that they missed the limit process, although theywere able to approximate a circle by a polygon and they were able to use arbitraryfinitely small to describe the limit process.

Figure 14.3 Ancient Rome

Romans and mathematics It may not worth mentioning Roman mathematics. From atleast 200 B.C., the Romans were in close contact with the Greeks. However, during eleven

87

hundred years, there was no single Roman mathematician; this fact in itself already givesus an idea about what happened for Roman mathematics at the time.

By borrowing or learning from Greek sources, Romans did have a crude arithmetic, didhave some approximate geometric formulas, did have some improvement in the calender,and, from about 50 B.C. on, wrote their own technical books.

In Rome, the astrologers were called mathematicii, and astrology was condemned byRoman emperors. The emperor Dicletian (245-316) regarded geometry to be learned andapplied in public service; but felt that the “art of mathematics” —– that is, astrology —–was damnable and forbidden in its entirety. The distinction between the terms “mathemati-cian” and “geometer” lasted until well past the Renaissance. Even in the seventeenth andeighteenth centuries, “geometer” meant what we mean by “mathematician.”

Unlike the Ptolemys of Egypt, the Roman emperors did not support mathematics andin fact they did not understand pure science. This is a lesson that we shall learn fromthe history of mathematics that without highly theoretical work of mathematicians andscientists and without fully recognizing its usefulness, important practical developments areimpossible to achieve.

Figure 14.4 Battle Scene with a

Roman Army Besieging a Large City, a painting

by Juan De La Corte(1597-1660).

Conquered by Romans The Romans conquered the land of Greece (168 B. C. onwards).However, Greek culture would in turn conquer Roman life.

88

After having secured control of central and northern Italy, Romans conquered the Greekcities in southern Italy and Sicily.

Mesopotamia conquered Greece in 64 B.C.

In 47 B.C., facing a desperate struggle with the Egyptians, Caesar’s 1defensive tacticscaused a major fire in the harbor of Alexandria. The library was struck by flames and thescrolls vanished. Two and a half centuries of book-collecting and half a million manuscripts,which represented the flower of ancient culture, were wiped out. Few scrolls survived thedisaster. Fortunately, an overflow of books that could no longer be placed in the overcrowdedlibrary was by this time stored in the temple of Serapis and these were not burned.

Figure 14.5 In 47 B.C., the fire spread to the harbour

and from there to the the Great Library of Alexandria.

Greek culture destroyed The Emperor Theodosius (ruled 379-395), who was thelast Roman emperor to rule over the full extent of the empire, divided the extensive empirebetween his two sons, Arcadius in the east (Greece, Egypt, and the Near East) and Honoriusin the west (Italy and western Europe).

The western part was conquered by the Goths in the fifth century A.D. and its subsequenthistory belongs to that of medieval Europe.

The eastern part preserved its independence until it was conquered by Turks in 1453.The eastern Roman Empire is known also as the Byzantine Empire. In Byzantine Empire,Greek culture and works were to some extent preserved.

1Gaius Julius Caesar (100 BC - 44 BC) was a Roman military and political leader. He played a criticalrole in the transformation of the Roman Republic into the Roman Empire.

89

Early Christianity is commonly defined as the Christianity of the three centuries betweenthe Crucifixion of Jesus (c. 30) and the First Council of Nicaea (325). It was the period whenthe religion spread in the Greek/Roman world. Despite cruel persecution by the Romans,Christianity spread and became so powerful that the emperor Constantine (272-337) hadto consign it a privileged position in the Roman Empire. The Christians were now able toeffect even greater destruction of Greek culture.

The Christians opposed pagan learning and ridiculed mathematics, astronomy, and phys-ical science (Christians were forbidden to contaminate themselves with Greek learning). In364, Emperor Flavius Jovianus orders the burning of the Library of Antioch. In 392, theemperor Theodosius proscribed the pagan religions and ordered that the Greek temples bedestroyed. In 397, Emperor Flavius Arcadius orders all the still erect Pagan Temples de-molished. Many of the Greek temples were converted to churches. Pagans were attackedand murdered throughout the empire.

In 448 Theodosius II ordered all non-christian books burned, and the Christians destroyedthe temple of Serapis, which still housed the only extensive collection of Greek works. In450 all the Temples of Aphrodisias (City of Goddess Aphrodite) are demolished and itsLibraries burned down. The city was renamed Stauroupolis (City of the Cross). In 486, more“underground” Pagan priests were discovered, arrested, burlesqued, tortured and executedin Alexandria, Egypt.

Figure 14.6 Hypatia, an Alexandrian mathematician of note,

refused to abandon the Greek religion. Christian fanatics seized her in

the streets of Alexandria and tore her to pieces.

90

Greek books were burned by the thousands and it is estimated that 300,000 manuscriptswere destroyed. Many other works written on parchment 2 were expunged by the Christiansso that they could use the parchments for their own writing. In 529 the Eastern Romanemperor Justinian closed all the Greek schools of philosophy, including Plato’s Academy.Many Greek scholars left the country and some settled in Persia.

The final blow to Alexandria was the conquest of Egypt by the Moslems (i.e. Muslim)in A.D.640. Books were destroyed on the ground. In fact, Omar, the Arab conqueror, said:

“ Either the books contain what is in the Koran, in which case we do not haveto read them, or they contain the opposite of what is in the Koran 3, in whichcase we must not read them.”

And so for six months the baths of Alexandria were heated by burning rolls of parchment.

The Greek mathematicians were wiped out, but the fruits of their work were not.

2Parchment is a thin material made from calfskin, sheepskin or goatskin, often split. Its most commonuse was as a material for writing on.

3Koran or Qur’an is the sacred text of Islam, considered by Muslims to contain the revelations of Godto Muhammad.

91