I. Rise
Reason:
The rise of Greek mathematics was in Athens, when people had a strong academic debate atmosphere and the academic atmosphere of rationalism was very prosperous. In addition, people believe in many religions and have freedom of thought, which can give full play to their imagination and help to separate science and mathematics from religious theology. As a result, there are many schools, and a hundred flowers blossom, resulting in the Ionian School and Pythagoras School represented by Thales.
Features: Starting from the initial concepts and axioms, the deductive system of demonstrating mathematics (or geometry) was born. Therefore, from the perspective of research methods, the Greeks paid attention to theory and were good at using formal logic, and the later Geometry is a typical representative.
1, Thales School (Ionian School)
Thales' epoch-making contribution in mathematics is to put forward the idea of proposition proof. It marks that people's understanding of objective things has risen from experience to theory, which is an unusual leap in the history of mathematics. The significance of introducing logical proof in mathematics lies in: ensuring the correctness of propositions; Reveal the internal relationship between theorems, make mathematics form a strict system and lay the foundation for further development; Make mathematical propositions fully convincing and persuasive.
The famous scholars of Ionian School are Anaquel Simander and Anaquel Simini. They had a great influence on Pythagoras later.
2. Pythagoras School
Pythagoras is another founder of demonstration mathematics. This school tries to explain everything with numbers, not only thinking that everything contains numbers, but also thinking that everything is numbers. They are famous for discovering Pythagorean Theorem (called Pythagorean Theorem in the West), thus discovering incommensurable metrics. One of the characteristics of this school is that arithmetic and geometry are closely linked.
However, due to the discovery of irrational numbers, the philosophical foundation of Pythagoras school that "everything is a number" was shaken, which led to the tragedy of the discovery of irrational numbers and the first mathematical crisis in the history of mathematics.
The first mathematical crisis tells us that reasoning and proof are reliable. From then on, Greece set out from the axiom of "self-evident", established a geometric system through deductive reasoning, and insisted on logical deductive reasoning to establish a complete axiom system, which made mathematics an abstract deductive science and laid the foundation of modern science.
3. Other schools of thought
There are many Greek schools, including Elijah School represented by Zhi Nuo, who studied the continuity, fluidity and infinity of the material world and founded "dialectics";
The atomism school represented by democritus put forward the viewpoint that "the material world is composed of a large number of indivisible atoms", and calculated the area and volume of some graphs from this viewpoint;
Plato school, represented by Plato, especially advocates geometry, and mainly studies irrational number theory, regular polyhedron and conic curve.
The Aristotelian school, represented by Aristotle, discussed some basic principles of mathematics. Its members, aldous Moss, wrote History of Arithmetic, History of Geometry and History of Astronomy, which became the earliest pioneers in the history of science.
The contributions of these schools in mathematics mainly include: three major drawing problems of geometry, namely, cubic multiplication, square change of circle, angle trisection, etc. At this time, some mathematical branches such as conic curve theory, cubic and quartic algebraic curves have also appeared. And the early concept of infinity. Aristotle is the founder of formal logic and a famous "syllogism". It laid a methodological foundation for the formation of Euclid's deductive geometry system.
Second, the heyday
Features: Alexandria was the heyday of ancient Greek mathematics, and its characteristic was that geometry broke away from philosophy and became a real deductive science independently. Axiomatic method has made quite good achievements in geometry and algebra has also made some achievements. Greek mathematics reached its peak, and there appeared outstanding mathematicians including Euclid, Archimedes and Apollonius.
1, Euclid
Euclid's Elements of Geometry is the crystallization of ancient Greek mathematical achievements, ideas, methods and spirit. It is the most widely distributed and widely used book in the whole history of science and has become the "Bible" of mathematics. Its great historical significance lies in that it is the earliest example of establishing deductive system by axiomatic method.
2. Archimedes, the God 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.
3.apollonius
His main contribution is the in-depth study of conic curve and the completion of the handed down book On conic curve. The tangent problem of his conic curve has become one of the driving forces for the development of calculus and played an important role in the development of mathematics in the17th century.
The achievements of Euclid, Archimedes and Apollonius marked the pinnacle of Greek geometry. With their limited skills, they have achieved most of the results obtained by using these skills.
Third, decline.
Special point: The later period of Alexander was the decline period of ancient Greek mathematics. The characteristic of this period is that geometry is mainly supplemented on the basis of Geometry Elements and other works, and it has made great achievements in algebra and trigonometry. Famous mathematicians include Helen, Ptolemy, Menelius, Seva, Diophantine, Pappus and Hipatia.
Helen's main contribution is to give the calculation formula of triangle area in metrology.
Ptolemy theorem is often selected and compiled in ancient and modern geometry books and works, and it is widely used;
Diophantine, a Greek mathematician, introduced symbols into algebra and made extensive and in-depth research on indefinite equations, making arithmetic and algebra independent disciplines, and was called "the father of algebra".
Papos's Compilations of Mathematics is a requiem of ancient Greek mathematics.
Hipatia annotated Diophantine's arithmetic and Apollonius's conic theory. She was the first female mathematician in history, but she was killed because she did not believe in Christianity. Her death also marked the decline of Greek mathematics.
The Greeks' pursuit of mathematics stems from their exploration and pursuit of nature. They deeply understand that mathematics is the key to understanding the universe, and mathematical laws are the essence of the layout of the universe. With the help of conjecture, the Greeks attached importance to abstraction and ignored concrete reality. For example, through typical proof, some imaginative and easily accepted definitions, postulates and axioms are selected and extended to the general, which greatly promotes the structural perfection and discipline development of mathematical science.
Although there are many achievements in Greek mathematics, there are also shortcomings and limitations. From the characteristics of various schools studying mathematics, we can sum up the following limitations:
The first limitation is that you can't grasp the concept of irrational numbers and passively escape:
They can't master irrational numbers, they are skeptical about irrational numbers, they passively escape, and there has been an irrational number tragedy in the history of mathematics. This also confuses the vision of future generations.
The second limitation is that it pays too much attention to geometry and ignores arithmetic and algebra:
Closely related to the first limitation, the Greeks could not grasp the concept of irrational numbers, which led them to pay more attention to geometry and concentrate on it, because geometric thought could make them avoid explicitly meeting the question of whether irrational numbers are numbers. This inevitably limits the development of arithmetic and algebra.
In a word, the achievements of Greek mathematics are brilliant. It has created great spiritual wealth for mankind, which is second to none in the world in terms of quantity and quality. More important than the concrete achievements made by Greek mathematicians, Greek mathematics produced the mathematical spirit, that is, the deductive reasoning method of mathematical proof. The abstraction of mathematics and the belief that nature is designed according to mathematical methods have played a vital role in the development of mathematics and even science. And a series of thoughts such as rationality, certainty, eternity and irresistible regularity produced by this spirit occupy an important position in the history of human cultural development.
These are all the stories about the history of mathematics: the rise and fall of ancient Greek mathematics compiled by geek mathematics.