Download Lecture 14. Decline of Greek Mathematics

Lecture 14. Decline of Greek Mathematics
Figure 14.1
Ancient Greece
Summary of the Greek Achievements
mathematicians?
What are the accomplishments of Greek
• The Greeks are to be credited with making mathematics abstract
Making mathematics abstract was the greatest contribution that Greeks made and it was of immeasurable 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.
This was an extraordinary step. Of the
• The Greeks insisted on deductive proof
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.
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• 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.
These two books con• Two milestones: Euclid’s Elements and Apollonius’ Conic
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 geometry, 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.
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• 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
ics has limitations:
Despite its marvellous achievements, Greek mathemat-
• Greeks were unable to find the concept of irrational number, which separated arithmetic and algebra from geometry for thousand years.
• Greeks only concentrated on geometry by using rigorous mathematics. As a consequence, 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
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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 “mathematician” 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.
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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 1 defensive 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.
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Gaius 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.
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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 physical 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 demolished. 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.
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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.
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Parchment is a thin material made from calfskin, sheepskin or goatskin, often split. Its most common
use was as a material for writing on.
3
Koran or Qur’an is the sacred text of Islam, considered by Muslims to contain the revelations of God
to Muhammad.
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