Lecture 14. Decline of Greek Mathematics

Figure 14.1 Ancient Greece Summary of the Greek Achievements What are the accomplishments of Greek mathematicians? • 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 could be applied to hundreds of different physical situations, which was indeed the secret of the power of mathematics. • The Greeks insisted on deductive proof This was an extraordinary step. Of the hundreds of civilizations that did develop some crude arithmetic and geometry, only the Greeks had the idea of establishing conclusions by deductive reasoning and had determination 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 not use it. From this point of view, Greeks’ decision to require deductive proof was entirely at odds, or almost irrational, at that time.

85 • Why Greeks insisted on deductive proof ? The Greeks wanted the truth, absolute truth. They realized that a mathematical theorem, once proved, will never fail and it is an absolute truth. They believed this is the way to understand the universe, and they saw that unquestionable methods of deductive reasoning was the only way for them to obtain these truths. The Greeks also realized that to secure truths they had so start from truths, and that any unwarranted fact will never be assumed. As a result, the Greeks collected all basic truths, called axioms, explicitly and regarded them as the starting points so that further mathematical conclusions could be built upon these axioms. The Greeks considered arithmetic, geometry, and astronomy to be the art of the mind and 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. They developed 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 conflict on irrational numbers. It was almost shut down the real number theory, which awaited only 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 the reality of the physical world and saw in mathematics the ultimate truth about the structure 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 solid geometry. • Greeks put restrictions on curves, only accepted the figures that could be obtained from line and circle. • Greeks feared the infinite process so that they missed the limit process, although they were able to approximate a circle by a polygon and they were able to use arbitrary finitely small to describe the limit process.

Figure 14.3 Ancient Rome

Romans and mathematics It may not worth mentioning Roman mathematics. From at least 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 gives us an idea about what happened for Roman mathematics at the time.

By borrowing or learning from Greek sources, Romans did have a crude arithmetic, did have 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 by Roman emperors. The emperor Dicletian (245-316) regarded geometry to be learned and applied 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 and eighteenth centuries, “geometer” meant what we mean by “mathematician.”

Unlike the Ptolemys of Egypt, the Roman emperors did not support mathematics and in fact they did not understand pure science. This is a lesson that we shall learn from the history of mathematics that without highly theoretical work of mathematicians and scientists and without fully recognizing its usefulness, important practical developments are impossible 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 Greek cities 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 tactics caused a major fire in the harbor of Alexandria. The library was struck by flames and the scrolls 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 the disaster. Fortunately, an overflow of books that could no longer be placed in the overcrowded library 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 the last Roman emperor to rule over the full extent of the empire, divided the extensive empire between his two sons, Arcadius in the east (Greece, Egypt, and the Near East) and Honorius in the west (Italy and western Europe).

The western part was conquered by the Goths in the fifth century A.D. and its subsequent history 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 critical role in the transformation of the Roman Republic into the Roman Empire.

89 Early Christianity is commonly defined as the Christianity of the three centuries between the Crucifixion of Jesus (c. 30) and the First Council of Nicaea (325). It was the period when the 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) had to consign it a privileged position in the Roman Empire. The Christians were now able to effect 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). In 364, Emperor Flavius Jovianus orders the burning of the Library of Antioch. In 392, the emperor Theodosius proscribed the pagan religions and ordered that the Greek temples be destroyed. 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 attacked and murdered throughout the empire.

In 448 Theodosius II ordered all non-christian books burned, and the Christians destroyed the temple of Serapis, which still housed the only extensive collection of Greek works. In 450 all the Temples of Aphrodisias (City of Goddess Aphrodite) are demolished and its Libraries burned down. The city was renamed Stauroupolis (City of the Cross). In 486, more “underground” Pagan priests were discovered, arrested, burlesqued, tortured and executed in 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 manuscripts were destroyed. Many other works written on parchment 2 were expunged by the Christians so that they could use the parchments for their own writing. In 529 the Eastern Roman emperor 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 have to read them, or they contain the opposite of what is in the Koran 3, in which case 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 common use 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 God to Muhammad.

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