Platonism: Platonism.
Platonism is the most influential mathematical philosophy in the history of mathematics. It originated from Plato in ancient Greece, and since then there has been an explicit or implicit Platonism concept in the field of western mathematics. In the19th century, it almost occupied the dominant position in the field of mathematics. At the beginning of the 20th century, the debate among the three schools of mathematics foundation just subsided, and Platonism became one of the hot topics of discussion.
The basic view of Platonism is that the object of mathematics is mathematical concepts such as numbers, quantities and functions, and mathematical concepts exist objectively as abstract generality or "* * *". Plato believed that they existed in a special world of ideas, and later Platonists did not accept the "theory of ideas", but they also believed that mathematical concepts were a special objective existence independent of the real world and an eternal existence independent of time, space and human thinking. Mathematicians get a new concept, not creation, but a description of this objective existence; New achievements in mathematics are not inventions, but discoveries. Correspondingly, Platonism believes that the truth of mathematical theory is objectively determined by the existence independent of the real world, and this truth is understood by "mental" experience and some kind of "mathematical intuition". Only through intuition can people reach the "mathematical world" independent of the real world.
Modern Platonism is also called "realism" because mathematical concepts are considered to be real. Platonism has a considerable influence in modern western mathematics, and some great mathematicians, such as G. Cantor, Russell, Godel and bourbaki, basically hold this view. It is generally believed that this is not accidental, but a philosophical reflection on the simple belief that mathematics reflects the objective world and mathematics has objective truth. Because of this, Platonism has a certain positive effect on the historical development of mathematics: it urges mathematicians to take an objective and scientific position in their own research, and when some highly abstract mathematical theories are doubted by people because they can't find realistic prototypes, it may also give people some faith. Although this belief is blind, it may also lead to mistakes.
The mistake of Platonism is obvious: taking the form of reflection as the object of understanding; Regard abstraction as concrete objective existence; It is of course an objective truth to think that a form of thinking itself is objective. Studying truth without human practice will inevitably lead to fallacy. Platonism is objective idealism in philosophy.
Plato's Philosophy: Philosophy of Objective Idealism
Platonic love: spiritual love, the desire to abandon the body.
Aristotle (384 ~ 322 BC)
Aristotelian school
Ancient Greek philosopher, logician and scientist. He summed up the achievements of the development of ancient Greek philosophy since Thales, distinguished philosophy from other sciences for the first time, and initiated independent research on logic, ethics, politics and biology. His academic thoughts had a great influence on the development of western culture.
Aristotle was born in Stajila. He entered Plato Academy at the age of 18 and later became a teacher. In 342 BC, he was called the teacher of Prince Alexander by King Philip II of Macedonia. He left Macedonia in 339 BC. In 335 BC, Aristotle returned to Athens. Another forum was set up on the playground near the temple of Apollo outside Lv Keang. Therefore, his academy is called Lv Keang. His teaching activities are mostly carried out in the pedestrian area of the sports ground, and he discusses problems while walking, so he is also called Minstrel. Lv Keang established a completely different style of study from Plato's Academy. It pays more attention to reality, studies and raises questions, collects materials from various aspects and tries to explore. Great achievements have been made in many departments of philosophy and ancient knowledge. In 323 BC, the anti-Macedonian movement broke out in Athens, and Aristotle became the target of political attack. Like Socrates, he was accused of blasphemy. After taking refuge in Calci, he died the following year.
The map of Aristotle's theory described by Arabs (8 ~ 1 1 century)
Aristotle's works are divided into two categories: the first category is that he published them publicly for ordinary people to read before his death, using dialogue style. Most of these works have been lost, and only a few fragments have survived to this day. These works are elegant and imaginative, among which the most important ones are works about soul, justice and philosophy. The second kind of works is unpretentious and rigorous in reasoning. Probably Aristotle's syllabus, research notes or students' lecture notes have been preserved.
Aristotle's existing works mainly include: category, explanation, pre-analysis, post-analysis, demonstration and refutation of wise men, collectively referred to as instrumental theory, which mainly involves logical issues; Metaphysics mainly involves abstract general theoretical problems; Physics, the theory of heaven, the theory of life and death and the theory of soul mainly involve natural philosophy. Nikolai Kyle's ethics, grand ethics and oldham's ethics mainly involve ethical issues. In addition, there are political science, poetics, rhetoric and other works related to biology and economy.
Theoretical science is further divided into three categories: physics, mathematics and first philosophy or theology. Aristotle and his free and unfettered school focused on physics and the first philosophy.
The first philosophy is metaphysics. Aristotle pointed out that the research object of philosophy is "existence as existence", that is, universal existence, and other disciplines are cut out from this whole for special research. Studying universality means studying those "things that belong to existence in themselves". This is the most primitive and primitive reason. On the one hand, Aristotle emphatically criticized Plato's idealism, and at the same time pointed out the shortcomings of democritus's materialist atomism. When Aristotle criticized idealism, he summarized this theory into a famous formula, which is "one among many". Very accurately hit the idealism of idealism.
In Aristotle's philosophy, the issue of entity occupies a central position, because entity and "being as being" are almost the same. Entity is the origin related to everything. Aristotle showed confusion and confusion on this issue. This is because at that time, philosophical thought was still in the pioneering stage from concrete to abstract, from individual to general. He can't really solve the problem of transition from individual to general, from perceptual to rational, thus falling into contradiction and confusion, vacillating between materialism and idealism, dialectics and metaphysics.
Physics is Aristotle's natural philosophy. This includes not only the general principle of the movement of things, but also two "sensible entities", namely celestial bodies and creatures. Aristotle believes that all objects in nature are made of matter, and there must be movement and change. He believes that space movement is accompanied by other kinds of movement. Space and position are universal conditions for all kinds of movements. Aristotle does not regard position as the space occupied by the extensiveness of an object, but in turn regards it as the boundary around the object. The extension of the universe is limited, and there is no void outside the universe. Aristotle defined time as measuring motion in sequence. He thinks that since movement is eternal, so is time. As for the materials that make up objects, Aristotle thinks that there are five kinds, and everything in nature has a purpose.
Aristotle is a systematizer of the geocentric theory. In his view, the universe is a sphere centered on the stationary earth. The outer layer of the earth is surrounded by 47 or 55 hollow spherical skies, and the outermost layer is starry sky. The movement of nature is a process from potential to reality, and it is a process in which matter is constantly shaped by form. In this process, different levels of creatures have been produced. In physics, Aristotle also has some puzzles, and it is difficult to justify himself.
Logic has no place in Aristotle's knowledge classification. In Minstrel's view, logic is not knowledge itself, but a tool and means to acquire knowledge.
Aristotle is the founder of formal logic. He believes that the research object of logic is language, that is, logos. But it only pays attention to the form of language and not to the content of language. Words are the most basic components of language, and each word is a judgment. He made a special study on the definition and put forward essential definitions, such as the definition of species difference, the definition of words, the definition of reasons and so on. , and discusses the possible mistakes in the definition, and puts forward some rules that are still used in logic textbooks. Divide propositions into simple propositions and compound propositions. The relationship between proposition and transposition is an important theory in Aristotle's logic. He didn't specifically discuss the relationship of difference, but actually applied it in syllogism. Aristotle studied reasoning and thought that reasoning was a logical form to draw inevitable conclusions through premises. The theory of outspoken syllogism is an important part of it. His syllogism is a relatively complete deductive reasoning theory and a primary axiomatic system. In addition, he also put forward the theory of modal syllogism, which is actually an axiomatic system, and some contents need further study. The theory of thinking law is the basis of Aristotle's logic. Aristotle put forward and established his theory of logical thinking law from ontology and logic. Law of excluded middle is another law of thinking. Aristotle studies scientific proof, and his proof premise must be true and inevitable; The initial proposition of proof must be direct. He also discussed direct proof and indirect proof, and thought that direct proof was better than indirect proof. At the same time, he also studied various fallacies and methods to refute them.
Aristotle's logic theory also includes induction and scientific methodology. Induction is to find universality from the special and generality from the individual through analogy, which is the source of all scientific proofs.
As a philosopher of slave owners, Aristotle's social and political stance is clearly reflected in Practical Science. There are three branches of practical science, namely ethics, financial management and political science. In these three areas, slaves were excluded. Aristotle called slaves "tools of life" and "submissive property". Although they are human beings, in essence, they should belong to their masters, obey them and engage in manual labor.
Ethics Aristotle's ethical theory is the inheritance and development of the happiness theory of ancient Greece from Solon to democritus, and its center is about happiness and the conditions to realize it. Aristotle, in his main ethical work, Nicokell Ethics, integrated the achievements of predecessors' ethical thoughts, and used the method of combining experience and reason to deeply explore all aspects of human moral behavior and various provisions of moral relations in slavery society, and established the first complete theoretical system of happiness ethics in the history of western ethical thoughts. The starting point of his ethical theory system is the slave owner's theory of human nature.
Aristotle criticized the mysticism and asceticism in Plato's ethical thought, affirmed the happiness and happiness of secular material life, and believed that happiness was the natural result of human virtue activities. However, he opposes attributing happiness only to happiness, especially to sensualism that attributes happiness only to perceptual happiness.
Aristotle attached great importance to moral education, and he especially emphasized the training of moral behavior. His ethical thought reflects the interests and moral requirements of middle slave owners and is the peak of ancient Greek ethical thought. However, he did not finally get rid of Plato's influence in theory, and both positive and negative aspects of his ethical thought had a far-reaching impact on the ethical theory of later generations.
Personal financial virtue must be cultivated and realized in the family. Family is the natural collective to which an individual originally belongs. Aristotle believes that the family consists of two elements, namely, people; Property and livestock. Slaves are the most important and indispensable family property. He admired agriculture, believing that it was the most honest industry, and it was of great help to cultivate people's heroism. Unlike business, people who engage in business will only become weak. Aristotle's economic thought was valued by Marx.
Aristotle in political science thinks that "man is essentially a political animal"; The city-state is a * * body developed by family collection, and the country is an inseparable whole, with the purpose of living a good life. Man is not a god. Without such political recognition, man will degenerate worse than social animals. As for the advantages and disadvantages of the political system, Aristotle believes that it lies not in the system itself, but in whether the rulers seek self-interest or public welfare. It is believed that the virtue of reason is the highest virtue, and the purpose of the country is not to develop military superiority, but to improve science and culture. Only the political unity of a literate country can last. Aristotle attached importance to the rule of law and thought that law should be consistent with the form of the country. Lawmakers should first care about the education of young people, and the highest purpose of law and discipline is to cultivate virtue. Education should not be vulgarized. The basic subjects of education are culture, physical education, music and painting.
Creative science Aristotle believes that creative science should be closely linked with practice. Creation is also a practical activity. It is a rational virtue to have the specialty of creation and be good at creation. Creative practice is different from ethical practice. The practical purpose of ethics lies only in practice itself, while the purpose and value of creative practice lies in products, and the purpose and value of poetry creation lies in poetry. Creation and knowledge-seeking activities are also different, and the object of knowledge is eternal, which is the truth as truth. The object of creative activities is changeable. If the object remains unchanged, creative activities cannot be carried out.
In Aristotle's view, practical technology serves real life, and imitation art gives people delicate enjoyment. A work of art, because it imitates something better and nobler than usual, becomes a means of moral cultivation. Art must idealize every object according to its unique attributes. The result of this imitation, although the imitated object is not better, nobler, even equal or worse than the ordinary object, is beautiful.
According to his own philosophical principles, Aristotle summed up the artistic experience of ancient Greece and wrote his main work Poetics, which represents his aesthetic thoughts. When solving the relationship between art and reality, he insisted on the principle of reflection theory. Representative theory is the core of Aristotle's aesthetics and philosophy of art. This theory affirms that art is an imitation of objective nature and life, and works of art should be "similar" to nature and life. Aristotle was the first philosopher in history to distinguish the aesthetic education function and moral education function of art theoretically, and he was also the first philosopher to study the category of "tragedy" from the aesthetic height. Most importantly, he pointed out that tragedy arouses the audience's pleasure through the purification of "pity" and "fear". The classical explanation of tragic effect has far-reaching historical influence and has been discussed many times in history. The basic aesthetic theory put forward by him has always been valued by various aesthetic schools in later generations.
The influence of Aristotle's philosophy had a great influence on later western philosophy. Many words in modern western philosophy and even in other parts of the world come from Aristotle. Because of Aristotle's philosophical confusion and complexity, his influence is multifaceted and even contradictory. In medieval Europe, orthodox scholasticism and monastic philosophy grasped the dead object in Aristotle's theory and demonstrated religious theology. Its opponents use the materialistic factors in Aristotle's philosophy to oppose the immortality of the soul and spread the double truth. Since the Renaissance, European humanists have tried to study Aristotle's original works from the original text and opposed the distortion of Aristotle by scholastic philosophers. In the modern west, neo-Thomas doctrine further distorts Aristotle's philosophy and defends Catholicism. The founders of Marxism-Leninism attached great importance to the study of Aristotle. While affirming Aristotle's great contribution to human culture, it also profoundly exposed his philosophical confusion and vacillation.