The development history of Greek mathematics can be divided into three periods. The first period, from Ionian school to Plato school, lasted from the middle of the 7th century BC to the 3rd century BC. The second period is the pre-Alexandria period, from Euclid to BC 146, Greece was trapped in Rome. The third period was the later period of Alexandria, which was ruled by Romans and ended when 64 1 Alexandria was occupied by Arabs.
From the decline of ancient Egypt and Babylon to the prosperity of Greek culture, there are not many mathematical historical materials left in this transitional period. However, the rise of Greek mathematics is closely related to the exposure of Greek businessmen to ancient oriental culture through travel.
Ionia is located on the west coast of Asia Minor, and it is easier to absorb the experience and culture accumulated by ancient countries such as Babylon and Egypt than other places in Greece. In Ionia, clan aristocratic politics was replaced by the rule of businessmen, who had strong activity and were conducive to the free and bold development of ideas. The struggle within the city-state helps to get rid of traditional beliefs. Greece has no special clergy and no dogma to follow, so there is a considerable degree of freedom of thought. This has greatly contributed to the separation of science and philosophy from religion.
Miletus is the largest city in Ionia, and it is also recognized as the hometown of Thales, the originator of Greek philosophy. As a businessman in his early years, he traveled to Babylon, Egypt and other places, and soon learned the knowledge handed down from ancient times and carried it forward. Later, the Ionian school of philosophy was founded, which 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. Thales also studied astronomy and mathematics. He predicted the solar eclipse, prompting a truce between Mittai (south of the Black Sea and Caspian Sea) and Lydia (west of Turkey). Most scholars believe that the solar eclipse occurred on May 28th, 585 BC. When he was in Egypt, he calculated the height of the pyramid by using the sun shadow and the proportional relationship, which surprised the Pharaoh greatly.
Thales' contribution to mathematics opened the proof of proposition, which marked that people's understanding of objective things rose from perceptual to rational, which was an unusual leap in the history of mathematics. The famous scholars of Ionian School are Anaquel Simander and Anaquel Simini. They had a great influence on Pythagoras later.
Pythagoras was born in Samos around 580 BC. In order to get rid of tyranny, he moved to Croton in the southern part of the Italian peninsula. Organize a secret society that integrates politics, religion, philosophy and mathematics. Later, it was destroyed in the political struggle and Pythagoras was killed, but his school continued to exist for two centuries.
The Pythagorean school tries to explain everything with numbers, not only that everything contains numbers, but also that everything is numbers. They are famous for discovering Pythagorean Theorem (called Pythagorean Theorem in the West), thus discovering incommensurable metrics.
Another feature of this school is the close connection between arithmetic and geometry. They found a formula that three positive integers represent the lengths of three sides of a right triangle, and noticed that continuous odd numbers and squares from 1 are not only arithmetic problems, but also related to geometry. They also found five regular polyhedrons.
There are significant differences between the Ionian school and the Pythagorean school. The former studies mathematics not only for philosophical interests, but also for practical purposes. The latter does not pay attention to practical application, and connects mathematics with religion, hoping to explore eternal truth through mathematics.
In the fifth century BC, Athens became the center of cultural gathering, and people advocated the spirit of openness. In an open discussion or debate, you must have knowledge of eloquence, rhetoric, philosophy and mathematics, so the Homo sapiens School came into being. They teach grammar, logic, mathematics, astronomy, rhetoric, eloquence and other subjects.
In mathematics, they put forward "three difficult problems": bisecting any angle; Double cube, find a cube so that its volume is twice that of the known cube; Turn a circle into a square and find a square so that its area is equal to the known circle. The difficulty of these problems is that only rulers (rulers without scales) and compasses are allowed to draw.
The interest of the Greeks lies not in actual drawing, but in solving these problems theoretically under the constraints of the rulers, which is an important step in the transition of geometry from practical application to system theory.
An Tifeng of this school put forward the "exhaustive method" to solve the problem of turning a circle into a square, which is the embryonic form of modern limit theory. First, a circle is inscribed with a square, and then the number of sides is doubled at a time, so that 8, 16, 32, … polygons are obtained. An Tifeng is convinced that the "difference" between the "last" polygon and the circle will be "exhausted". This provides an approximate method for finding the area of a circle, which coincides with China's thought of Liu Hui's separatist regime.
In the third century BC, Plato founded the school and the Academy in Athens. He attaches great importance to mathematics, but unilaterally emphasizes the role of mathematics in training intelligence and ignores its practical value. He advocates cultivating logical thinking ability through the study of geometry, because geometry can give people a strong intuitive impression and embody abstract logical laws in concrete graphics.
This school has trained many mathematicians. For example, eudoxus studied under Plato, and he founded the theory of proportion, which was the predecessor of Euclid. Plato's student Aristotle was also a great philosopher in ancient times and the founder of formal logic. His logical thought paved the way for arranging geometry in a strict logical system in the future.
During this period, there was the Elijah School represented by Zhi Nuo, who put forward four paradoxes, which greatly shocked the academic circles. These four paradoxes are:
Oral, a thing goes from a to b, and it can never be reached. Because if you want to go from A to B, you must walk halfway first, but if you want to walk halfway, you must walk halfway first, and so on, forever. The conclusion is that the movement of this object is hindered by the infinite division of the road and cannot move forward at all; Achilles (the hero who is good at running) chased the tortoise and said that Achilles would never catch up with the tortoise. Because when he chased the tortoise's starting point, the tortoise had climbed forward for a while. After he chased this paragraph, the tortoise climbed forward for a short time. If you repeat this forever, you will never catch up; The flying arrow is stationary, which means that the arrow is in a certain position all the time, so it does not move; On the playground, Zhi Nuo proved that time is half of it.
The atomism school represented by democritus believes that line segments, areas and solids are composed of many inseparable atoms. Calculating the area and volume is equivalent to collecting these atoms. This less strict reasoning method is an important clue for ancient mathematicians to discover new results.
After the 4th century BC, Greek mathematics gradually separated from philosophy and astronomy and became an independent discipline. The history of mathematics then entered a new stage-the period of elementary mathematics.
The characteristic of this period is that mathematics (mainly geometry) established its own theoretical system, which developed from empirical science based on experiments and observations to deductive science. Starting from several original propositions (axioms), a series of theorems are obtained through logical reasoning. This is the basic spirit of Greek mathematics.
In this period, elementary geometry and arithmetic elementary algebra have generally become independent disciplines. Compared with the analytic geometry and calculus that appeared in the17th century, the research content of this period can be summarized by "elementary mathematics", so it is called the elementary mathematics period.
Alexandria, Egypt is the hub of land and water transportation between east and west. After the operation of King Ptolemy, it has gradually become a new Greek cultural center, and Greece is now relegated to a secondary position. Geometry first germinated in Egypt, then transplanted to Ionia, then flourished in Italy and Athens, and finally returned to its birthplace. After such practice, I have reached the position of lush trees.
From the 4th century BC to the demise of ancient Greece in 146 BC, until Rome became the ruler of the Mediterranean region, Greek mathematics with Alexander as the center reached its heyday. There is a huge library and a strong academic atmosphere, and scholars from all over the world gather here for teaching and research. Among them, Euclid, Archimedes and Apollonius were the three early mathematicians of Alexander.
Euclid's Elements of Geometry is an epoch-making work. Its great historical significance lies in that it is the earliest example of establishing deductive system by axiomatic method. The mathematical knowledge accumulated in the past is fragmentary, which can be compared to bricks, wood and stone; Only with the help of logical methods can we organize this knowledge, classify and compare it, reveal its internal relations, and arrange it according to a strict system, thus building a magnificent building. The Elements of Geometry embodies this spirit and has a far-reaching influence on the development of mathematics.
Archimedes was a physicist and mathematician. He is good at combining abstract theory with concrete application of engineering technology and gaining insight into the essence of things in practice. Through rigorous argumentation, he turned empirical facts into theories. He explored and solved the problem of area and volume according to the principle of mechanics, which already included the preliminary idea of integral calculus. Apollonius's main contribution is the in-depth study of conic section.
In addition to these three mathematicians, Eratosthenes's geodesy and the "Plain Screen" named after him are also famous. Astronomer hipparchus made the "string table", which was the pioneer of trigonometry.
After 146 BC, Alexander scholars under Roman rule can still inherit the work of their predecessors and continue to invent and create. Helen (about AD 62), Menelaus (about AD 100), Pappus and others all made important contributions. Astronomer Ptolemy played the role of hipparchus and laid the foundation of trigonometry.
Late Greek scholars also made great achievements in arithmetic and algebra. The representative figures are Nicole Hoth (about AD 100) and Diophantine (about AD 250). The former came from jerash (now northern Jordan). The author of Introduction to Arithmetic, whose Arithmetic is about number theory, most of which can be classified as algebra. It is completely out of geometric form, unique in Greek mathematics, and has a great influence on later generations, second only to the Elements of Geometry.
In 325 AD, Constantine the Great of the Roman Empire began to use religion as a tool of rule and put all his knowledge under the control of Christian theology.
In 529 AD, the Emperor of the Eastern Roman Empire, Charles Tinny, ordered the closure of Plato's Academy and other schools in Athens and prohibited the teaching of mathematics. Many Greek scholars fled to Syria and Persia. Mathematical research has been dealt a heavy blow. In 64 1 year, Alexandria was occupied by Arabs, and the library was destroyed again, so Greek mathematics came to an end.
The natural philosophy of ancient Greece was first put forward by the philosophers in the slave city of Miletus. Their view of nature is simple and primitive, but it is of great significance: it breaks the myth and belief of the ancient Greeks about the origin of the world and replaces it with a purely rational explanation.
The earliest recorded natural philosopher was Thales of Miletus (about 625-546 BC). He believes that all objects headed by the earth are not produced by the power of God, but by the process of nature itself, and everything is produced by water. This natural philosophy originated from Thales, who believed that the world was a whole and the universe was the result of natural activities of matter. The most fruitful natural philosophy derived from this tradition is atomism.
Ancient atomists believed that the basic elements of the universe were infinite, indestructible and indivisible atoms. These atoms are different in shape and size, but their structures are the same. They are always combined, separated and recombined in different ways because of their inherent motion. Therefore, every object or organic matter in the universe is the product of accidental aggregation of atoms.
Democritus (about 460-370 BC), who lived in Alfatilla on the Thrace coast, systematized the atomism of ancient Greece. He believes that in this universe, in the final analysis, there are only vanity and indivisible atoms; Atoms are constantly moving in the void, colliding with each other to form a vortex. Vortex is divided into inner and outer parts due to different weights. The atoms in the middle part produce the earth, while the atoms in the outer part produce the sky, air and fire. This thought reflects the final result of materialism tendency in early Greek thought.
The basic concept of atomism is that particles move in infinite void, which has become an extremely effective hypothesis in modern science. However, in history, atomism, as an atheism, has long been criticized and suppressed. It was not until modern times that it regained its luster.
Aristotle, the most famous philosopher in ancient Greece, also made his own contribution to the development of natural philosophy.
Aristotle studied under Plato, but his philosophy was different from Plato's. Plato respected ideas, while Aristotle attached great importance to experience. (Of course, his conceptual framework as the basis of understanding experience is completely different from modern science. He has a strong interest in ecology, physics and astronomy, and made a lot of observations with arrogant enthusiasm, leaving a lot of valuable information. His world outlook is teleological, which can be regarded as a compromise between Plato's idealism and transcendentalism and atomism. His philosophy pays attention to the initial state and final state, but ignores the process. He believes that matter and spirit are equally important, eternal and indispensable, and form is the cause of everything and the motive force with purpose. Changes in nature are regarded as the transition from form to completely obvious final state.
Aristotle's natural philosophy was introduced into Europe from the Islamic world in the Middle Ages. His thoughts were regarded as authority by scholastics at that time because his thoughts were combined with some dogmas of the Christian church and Europeans at that time were eager to learn the culture of ancient Greece.
Another famous natural philosopher in ancient Greece was Archimedes. He is a mathematician and a natural philosopher. It was he who liberated physics from the framework of natural philosophy and made it an independent experimental science. He is different from Aristotle. Although Aristotle attached importance to experience, he didn't do any experiments, and he ignored the application of mathematics. Archimedes not only did a lot of experiments, but also proved the lever theorem and buoyancy theorem with mathematics as a tool very early. His understanding of the role of mathematics in physics is similar to that of modern scientists.
The splendid civilization of ancient Greece declined with the constant invasion of foreign countries, and its civilization achievements were not inherited by Europe, which was still in a wild state at that time, but were preserved by its eastern Islamic world. After that, physics entered an extremely slow or even stagnant development period.
The ancient civilization of the west originated from the Aegean world, one of the earliest cradles of civilization in the world (Crete, Aegean islands and western Asia Minor in the Asian continent were called the Aegean world in ancient times). The ancient Greek civilization, starting from the Aegean civilization, is one of the earliest and most influential ancient civilizations in Europe.
Aegean civilization, as the predecessor of ancient Greek civilization, first appeared in the bronze age of slave city-states in Crete around 2000 BC. Then it extended to Mycenae in 1600 years ago, and died out in about 1 100 BC. Since then, the development level of Greek society has regressed to the stage of military democracy in the late period of clan system, and it is in a stage of both backwardness and reform and development. Westerners used to call it the dark age, that is, Homer's age, named after Homer's epic, which is the main document of this period.
After the Homer era, which lasted for 300 years, the ancient Greek civilization was reborn. Sparta and Athens built cities one after another, and the Greek city-state era began around 800 BC. "polis" originated from the ancient Greek word Polis (Pohris). The polis is a typical political and social organization of the ancient Greeks and a great political initiative. In ancient times, the city-state of an independent sovereign country with the city as the center must have three elements: city, state institution or state machine, citizen commune or citizen assembly.
In 43 1 BC, the Peloponnesian War broke out. This war, which lasted for 27 years and was triggered by the contradiction between the city-States, was the starting point for the decline of the prosperous ancient Greek civilization. In 336 BC, Alexander established an empire through military conquest. As a result, the city-state went to the empire. After Alexander's death, the huge empire he built collapsed rapidly, and civilization entered a new era-Hellenistic era. As a result, Greek civilization spread outward from the Greek world in northern West Asia, interacting with ancient Egypt and the two river civilizations.
In 28 1 BC, the basically stable separatist situation in several countries ended the chaotic period of central power struggle after Alexander's death. However, wars and internal disputes led to the rapid decline of Hellenistic countries. When Rome expanded in the Balkans in 200 BC, the long-term disunity and internal weakness of Greece provided an excellent opportunity for Rome's expansion. Finally, in 30 BC, with Egypt, the last region of the Hellenistic East, being incorporated into the Roman Empire, the Hellenistic era ended.