Source of this article: Mathematics Culture Quality Education Resource Library
From: Mathematics and Artificial Intelligence, Philosophical Garden
The geographical extent of ancient Greece, in addition to the present-day Greek peninsula, included the entire Aegean region and Macedonia and Thrace to the north, the Italian peninsula and Asia Minor. In the fifth and sixth centuries BC, especially after the Greek-Polish wars, Athens gained the leadership position of the Greek city-states, the economic life was highly prosperous, and the productive forces were significantly improved, on the basis of which a brilliant Greek culture was produced, which had a profound impact on future generations.
Map of Ancient Greece
The history of the development of Greek mathematics can be divided into three periods.
The first period lasted from the Ionian school to the Platonic school, about the middle of the 7th century BC to the 3rd century BC;
The second period is the early Alexander period, from euclid until the fall of Greece to Rome in 146 BC;
The third period, the Late Alexander period, was a period under Roman rule that ended with the occupation of Alexander by the Arabs in 641.
The Ionian school's transition from ancient Egypt and the decline of Babylon to the flourishing of Greek culture leaves little mathematical history. However, the rise of Greek mathematics was closely related to the travel and interaction of Greek merchants with the ancient culture of the East. Located on the west coast of Asia Minor, Ionia is more likely than the rest of Greece to absorb the experience and culture accumulated by ancient countries such as Babylon and Egypt. In Ionia, clan aristocratic politics was replaced by the rule of merchants, who were highly active and conducive to the development of freedom of thought and boldness. Struggles within the city-state helped to break away from traditional beliefs. In Greece there was no special priestly class and no dogmas that had to be observed, so there was a considerable degree of freedom of thought. This greatly contributes to the separation of science and philosophy from religion.
Millikin is the largest city in Ionia and the home of Thales. Thales is recognized as the originator of Greek philosophy. In his early years, he was a merchant who traveled to Babylon, Egypt and other places, and soon learned the knowledge handed down from ancient times and carried it forward. Later, he founded the Ionian school of philosophy, got rid of religion, sought truth from natural phenomena, and took water as the root of all things.
At that time, astronomy, mathematics and philosophy were inseparable, and Thales also studied astronomy and mathematics. He had predicted a solar eclipse that would prompt the States of Mitai (in present-day Black Sea and south of the Caspian Sea) and Lydia (present-day western Turkey) to stop the war. Most scholars believe that the eclipse occurred on May 28, 585 BC. When he was in Egypt, he used the shadow of the sun and the proportions to calculate the height of the pyramids, which surprised the patriarch. Thales's contribution to mathematics was the beginning of the proof of propositions, which marked the rise of people's understanding of objective things from sensibility to reason, which was an unusual leap in the history of mathematics. Notable scholars of the Ionian School also include Anaximander and Anaximini. They had a great influence on the later Pythagoras.
Pythagorean school
Pythagoras was born around 580 BC in Samos (present-day a small island in eastern Greece). In order to escape tyranny, he moved to Croton in the south of the Italian peninsula. There a secret society of political, religious, philosophical, and mathematical unity was organized. Later, he was destroyed in political struggles and Pythagoras was killed, but his school continued to exist for two centuries (about 500-300 BC). This school of thought attempts to explain everything in terms of numbers, not only by thinking that all things contain numbers, but that all things are numbers.
Pythagoras
They are famous for discovering the Pythagorean theorem (known in the West as the Pythagorean theorem), which led to the discovery of insignificance. Another feature of this school is that it closely links arithmetic and geometry. They found a formula that represented the length of the three sides of a right triangle with three positive integers, and noted that the odd numbers and the must be square numbers from 1 onwards, etc., were both arithmetic and geometric. They also found five regular polyhedra. In terms of astronomy, the first earth circle theory is believed that the sun, moon, and five stars are all spheres, floating in space. Pythagoras was also the father of music theory.
The Ionian School
This school is significantly different from the Pythagorean school. The former studied mathematics not purely for philosophical interest, but also for practical purposes. The latter, on the other hand, does not pay attention to practical applications, connects mathematics with religion, and wants to explore eternal truths through mathematics.
Homo sapiens school
In the 5th century BC, Athens became a center of humanity, and the spirit of openness was revered. In open discussion or debate, knowledge of eloquence, rhetoric, philosophy, and mathematics must be available, and the "sophist school" (or the school of dialectics and philosophers) came into being. They taught grammar, logic, mathematics, astronomy, rhetoric, eloquence, and more. Mathematically, they ask "three big questions":
(1) third-part arbitrary angle;
(2) Times the cube, that is, to make a cube so that its volume is twice that of the known cube;
(3) To turn a circle into a square, that is, to make a square so that its area is equal to a known circle.
The difficulty is that drawing is only allowed to use rulers (rulers without scales) and compasses. The interest of the Greeks was not in the actual formulation of graphics, but in solving these problems theoretically within the constraints of rulers. This is an important step in the transition of geometry from practical applications to systems theory. Antiphon of this school (c. 430 BC) proposed the use of the "exhaustion method" to solve the problem of turning circles into squares, which is the prototype of modern limit theory. First make a circle inside the square, and then double the number of sides each time, 8, 16, 32、...... The edges, and so on, Antiphon was convinced that the "difference" between the "last" polygon and the circle would be "exhausted." This provides an approximation of the area of the circle, which coincides with the Chinese idea of liu hui (c. 263 c. 263) in circumcision.
Platonic School and other academic centers
Plato (c. 427 BC – 347 BC) founded a school in Athens and founded a school. He attaches great importance to mathematics, but one-sidedly emphasizes the role of mathematics in training intelligence, while ignoring its practical value. He advocated the cultivation of logical thinking ability through the study of geometry, because geometry can give people a strong intuitive impression and embody abstract logical laws in concrete figures. This school produced many mathematicians, such as Eudoxos, who studied at Plato, who founded the theory of proportions, which was the precursor of Euclid. Plato's student Aristotle was also a great philosopher of antiquity, the founder of formal logic. His logical ideas opened the way for the future organization of geometry in a rigorous logical system.
Plato
The Greek mathematical center of this period also had the Elia school represented by Zeno (about 496 BC to 430 BC), who proposed four paradoxes that greatly shocked the academic community. The four paradoxes are: (1) Dichotomy says that a thing can never reach from A to B. Because if you want to go from A to B, you must first pass through half of the road, but to pass through this half, you must first pass through half of the half, and so on, there is no end. The conclusion is that the movement of this thing is hindered by the infinite division of the road, and it is impossible to move forward at all. (2) Achilles (a good running hero) chased the turtle and said that Achilles chased the turtle and could never catch up. Because when he chased the turtle's starting point, the turtle had crawled forward for a while, and after he chased this paragraph, the turtle climbed forward a little more. This will never be repeated forever, and it will never catch up. (3) Flying arrow stationary says that every moment the arrow is always in a definite position, so it is immobile. (4) On the playground problem, Zeno argues that time and half of it are equal.
The school of atomism, represented by Democritus, believes that line segments, areas, and stereoscopic beings are made up of many indivisible atoms. Calculating the area and volume is equivalent to assembling these atoms. This less rigorous method of reasoning was an important clue to the discovery of new results by ancient mathematicians.
Greek mathematics after the 4th century BC gradually separated from philosophy and astronomy and became an independent discipline. The history of mathematics thus entered a new phase—the period of elementary mathematics. This period was characterized by the fact that mathematics (mainly geometry) had established its own theoretical system, transitioning from empirical science based on experiments and observations to deductive sciences. Starting from a few primitive propositions (axioms), a series of theorems are obtained through logical reasoning. This is the basic spirit of Greek mathematics. During this period , elementary geometry , arithmetic , and elementary algebra became largely separate subjects. Compared with analytic geometry and calculus that appeared in the 17th century, the research content of this period can be summarized by "elementary mathematics", so it is called the primary mathematics period.
The egyptian city of Alexandria, the hub of land and sea transportation between the east and the west, and through the Ptolemaic king (about 367 BC to 285 BC), gradually became the new Greek cultural center, and the Greek mainland had taken a back seat at this time. Geometry first sprouted in Egypt, then transplanted in Ionia, then flourished in Italy and Athens, and finally returned to its birthplace. After this cultivation, it has reached the state of lush forest.
From the 4th century BC to the fall of ancient Greece in 146 BC, when Rome became the ruler of the Mediterranean region, Greek mathematics centered on Alexander and reached its heyday. There is a huge library and a strong academic atmosphere, where scholars from all over the world gather to teach and research. The most accomplished of these were the three great mathematicians Euclid, Archimedes and Apollonius in the early Alexander period.
Euclid's Primitive Geometry is an epoch-making work. Its great historical significance lies in the fact that it is the earliest example of the establishment of a deductive system using axiom methods. The mathematical knowledge accumulated in the past is fragmentary and fragmentary, which can be compared to bricks and tiles; Only by means of logical methods, organizing this knowledge, classifying and comparing it, revealing the internal connections between them, and sorting them out in a strict system can we build a magnificent edifice. The Primitives of Geometry embodies this spirit, and it has had a profound impact on the development of mathematics as a whole. Archimedes was a physicist and mathematician who was good at combining abstract theories with the concrete applications of engineering techniques, and in practice he gained insight into the nature of things, and through rigorous argumentation, made experience in fact theory. He explored and solved the problem of area and volume according to the principles of mechanics, and already included the preliminary ideas of integrals. Apollonius' main contribution was the in-depth study of conic curves.
In addition to the three major mathematicians, Eratosthenes (c. 276 BC – 195 BC) was also famous for his geodetic surveys and the "prime sieve" that he was known for. The astronomer Hipparchus (2nd century BC) made the "string table", which was the precursor to trigonometry.
After the late Alexander period 146 BC, Alexander scholars under Roman rule were still able to inherit the work of their predecessors and continue to invent. Helen (c. 62 AD), Mennalaus (c. 100 AD), Papus, and others all made important contributions. The astronomer C. Ptolemy (c. 85–165) organized the work of Hipparchus and laid the foundations of trigonometry.
Late Greek scholars were also accomplished in arithmetic and algebra, represented by Nicomachos (c. 100 AD) and Diophantus (c. 250). The former were people from the Jerash (present-day northern Jordan) region. He is the author of "Introduction to Arithmetic", the latter's "Arithmetic" is about the theory of numbers, and most of the content can be classified into the scope of algebra. It was completely divorced from the geometric form, unique in Greek mathematics, and had a great influence on later generations, second only to the Geometric Primitives.
In 325, Constantine the Great of the Roman Empire began to use religion as an instrument of rule, bringing all scholarship under the control of Christian theology. In 529, the Eastern Roman Emperor Charles Tenni ordered the closure of Plato's Academy and other schools in Athens, strictly prohibiting the teaching of mathematics. Many Greek scholars fled to places like Syria and Persia. Mathematical research has been hit hard. In 641, Alexander was occupied by the Arabs, the library was destroyed again, and Greek mathematics came to an end.