However, his philosophy and methodology play a more important role in his life activities. His philosophical thoughts had a great influence on the later development of philosophy and science. Descartes is generally regarded as the founder of modern western philosophy, and he was the first to establish a complete philosophical system. Philosophically, Descartes is a dualist and a rationalist. Descartes believes that human beings should be able to use mathematical methods-that is, rationality-for philosophical thinking. He thinks that reason is more reliable than sensory feeling. He gave an example: when we dream, we think we are in a real world, but in fact this is just an illusion. See Zhuang Zhou Dream Butterfly. He found four rules from logic, geometry and algebra: never admit that anything is true and take what I have no doubt as truth; Every problem must be divided into several simple parts to deal with; Thought must be from simple to complex; We should make a thorough inspection from time to time to ensure that nothing is missed. Descartes applied this method not only to philosophical thinking, but also to geometry, and founded analytic geometry.
Therefore, Descartes first thought that doubt is the starting point, and the knowledge of sensory perception can be doubted, so we can't trust our senses. Descartes emphasized that the purpose of science is to benefit mankind and make man the master and ruler of nature. He opposed scholasticism and theology and put forward a "systematic questioning method" that doubted everything. So he won't say "I see therefore I am" or "I listen therefore I am". From this, he realized a truth: what we can't doubt is "our doubt." Meaning: What we can't doubt is the "doubt itself" when we "doubt" this matter. Only in this way can we be sure that our "suspicion" is true, not a fake. Anyone who takes it for granted or takes it for granted is puzzled, so he introduces the famous philosophical proposition-"cogito ergosum". It emphasizes that the existence of an independent spiritual entity with thinking as its attribute is unquestionable, and demonstrates the existence of an independent material entity with universality as its attribute.
He thinks that the above two entities are finite entities, and their juxtaposition shows that he is a typical metaphysical or ontological dualist. Descartes regards this as the most basic starting point in metaphysics, from which it is concluded that "I" must be something independent of body and thinking. Descartes also tried to prove the existence of God from this starting point. Descartes thinks that we all have the concept of perfect entity. Because we can't get a perfect concept from an imperfect entity, we must have a perfect entity-God-to get this concept. In other words, God is the creator and ultimate cause of finite entities. Starting from these two points, Descartes continued to infer that since the perfect thing (God) exists, then we can be sure that the previous demon hypothesis cannot be established, because a perfect thing cannot allow such a demon to deceive people, so we can determine that "this world really exists" through constant doubts, and the mathematical logic after proof should be correct. In the real world, there are many characteristics that can be rationally perceived, that is, their mathematical characteristics (such as length, width and height). When our reason can clearly recognize a thing, then it must not be illusory, but it must be what we know. That is, Descartes applied the reasoning method and deductive method of geometry to philosophy, thought that the clear concept was truth, and put forward the "natural concept".
Descartes' natural philosophy is completely opposite to Aristotle's theory. He believes that all material things are machines governed by the same mechanical law, even the human body. At the same time, he also believes that there is a spiritual world outside the mechanical world. This dualistic view later became the fundamental thinking method of Europeans.
Although Descartes proved the existence of the real world, he believed that there are two different entities in the universe, namely, thinking (mind) and the external world (matter), both of which come from God, and God exists independently. He believes that only man has a soul, and man is a binary being, who can think and occupy space. Animals belong only to the material world.
Descartes emphasized that thought is beyond doubt, which had an important influence on European philosophy. I think, therefore, the controversy I caused lies in the so-called existence of God and animal monism (chimpanzee, octopus, parrot, dolphin, elephant, etc. Are proved to be intelligent), and the main idea of doubt is really helpful for research.
Methodology
Descartes wanted to introduce his research achievements in a book called The World, but when the book 1633 was almost finished, he learned that Galileo, the authority of the Italian church, was guilty of supporting Copernicus' Heliocentrism. Although Descartes was not persecuted by the Catholic authorities in Holland, he decided to be cautious and put the manuscript in a box because he defended Copernicus' theory in the book. However, in 1637, he published the most famous book "Methodology of Correct Thinking and Discovery of Scientific Truth", usually referred to as methodology.
Descartes pointed out in Methodology that the method of studying problems is divided into four steps:
1. Never accept any truth I don't know, that is, I will try to avoid recklessness and prejudice. I can only be very clear and certain according to my own judgment, and there is no doubt about the truth. In other words, as long as you have no personal experience of the problem, no matter what authoritative conclusion you have, you can doubt it. This is the famous "doubt everything" theory. For example, Aristotle once concluded that women have two fewer teeth than men. But this is not the case.
2. The complex problems to be studied can be decomposed into several simple small problems as far as possible and solved separately one by one.
3. Arrange these small problems from simple to complex, and start with the problems that are easy to solve.
4. After all the problems are solved, check them together to see if they are complete and the problems are completely solved.
1960 years ago, the methods of western scientific research, from mechanics to human anatomy, were basically carried out in accordance with Descartes' talking method, which greatly promoted the rapid development of modern western science. But it also has some defects, such as the function of human body is only the synthesis of various parts of machinery, and the interaction between them is not well studied. It was not until the appearance of Apollo 1 moon landing program that scientists found that some complex problems could not be decomposed and had to be dealt with by complex methods, which led to the emergence of system engineering, and methodology was replaced by comprehensive methods for the first time. The emergence of systems engineering has greatly promoted many large-scale western traditional sciences, such as environmental science, meteorology, biology, artificial intelligence and so on.
"I think therefore I am"
Descartes' most famous thought. From the methodology.
Literally: "when I doubt the existence of everything, I don't have to doubt my own thoughts, because the only thing I can be sure of at this time is the existence of my own thoughts." Descartes believes that when I doubt everything, I can't doubt the existence of the "I" I am doubting. Because this "doubt" itself is an ideological activity. And this kind of thinking and questioning the essence of "I" is also an ideological activity. Note that "I" here does not refer to me who is integrated with body and mind, but to an independent mind.
Deep meaning: Descartes' philosophical proposition uses the so-called "suspicious method" to verify whether the source of "knowledge" is reliable. We can doubt everything around us, but there is only one thing we can't doubt, and that is: to doubt the existence of the "I" that we are doubting. In other words, we can't doubt our own doubts, because only in this way can we affirm our doubts. Descartes also proved the existence of God from his "I think therefore I am". Because the subject of "I" is beyond doubt, there is a higher "existence" to make "I" exist. In other words, because I exist, there must be an "existence" that makes me exist, and the "existence" that makes me exist must also be the "existence" that makes everything exist. Therefore, the "existence" that can make all things exist must be possible only by God.
This famous saying, regarded by Descartes as the starting point of his own philosophical system, was regarded as the general representative of extreme subjective idealism in Eastern Europe before17th century and China in 2 1 century, and was severely criticized. Many people even regard Descartes as "putting the cart before the horse" and "ridiculous" on the grounds that "existence must precede consciousness" and "there can be no thought without body". Descartes' doubts are not doubts about some specific things and principles, but absolute doubts about mankind, the world and God. From this absolute doubt, Descartes wants to lead to unquestionable philosophical principles. Descartes made a useful contribution to physics by virtue of genius intuition and strict mathematical reasoning.
Descartes has been paying attention to lens theory since he read Johannes Kepler's optical works from 16 19. He also participated in the study of the essence, reflection and refractive index of light and lens grinding from both theoretical and practical aspects. He believes that the theory of light is the most important part of the whole knowledge system. Descartes firmly believed that light travels "instantaneously". In his works "On Man" and "Principles of Philosophy", he fully expounded the concept of the essence of light. Descartes used his coordinate geometry to study optics, and put forward the theoretical demonstration of the law of refraction of light for the first time in Refractive Optics. Share the honor of discovering the law of refraction of light with Snell in the Netherlands. He thinks that light is the propagation of pressure in the ether. From the viewpoint of light emission theory, he calculated the reflection, refraction and total reflection of light on the interface between two media by using the model of tennis ball hitting cloth, and thus deduced the law of refraction for the first time under the assumption that the velocity component parallel to the interface is unchanged. But his hypothesis is wrong, and his deduction leads to the wrong conclusion that the speed of light increases when it enters the dense medium from the sparse medium. He also made an optical analysis of people's eyes, explained that the cause of vision impairment was the deformation of the lens, and designed a lens to correct vision.
He also used the refraction law of light to explain the rainbow phenomenon, and analyzed the color through the rotation speed of elemental particles.
In mechanics, Descartes developed Galileo's theory of motion relativity. For example, in the book "Principles of Philosophy", a vivid example of the watch wheel of a seaman's pocket watch in navigation is given to illustrate the reason why the reference system needs to be selected for motion and rest.
In the second chapter of Principles of Philosophy, Descartes first expressed the law of inertia in the form of the first law and the second law of nature: as long as an object starts to move, it will continue to move at the same speed and in the same straight line direction until it meets some obstacles or deviations caused by external reasons. Here, he emphasized the linearity of inertial motion that Galileo did not explicitly express.
In this chapter, he also clearly put forward the law of conservation of momentum for the first time: the total amount of matter and motion will never change. It laid the foundation for the law of conservation of energy.
Descartes discovered the original form of the principle of conservation of momentum (Descartes defined momentum as an absolute value, not a vector, so his principle of conservation of momentum was later proved to be wrong).
Descartes made a preliminary study on collision and centrifugal force, which created conditions for Huygens' success later. Descartes applied his mechanistic viewpoint to celestial bodies, developed the theory of cosmic evolution and formed his theory of the origin and structure of the universe. He believes that it is easier to understand things from the perspective of development than just from the existing form. He founded the vortex theory. He thinks there are huge eddies and stars around the sun.
He believes that the motion of celestial bodies comes from inertia and the pressure of some cosmic material vortex on celestial bodies, and there must be a celestial body in the center of various vortex sizes, so this hypothesis is used to explain the interaction between celestial bodies. Descartes' etheric vortex model of the origin of the sun relies on mechanics instead of theology for the first time to explain the formation process of celestial bodies, the sun, planets, satellites and comets. , a century earlier than Kant's nebula theory, is the most authoritative cosmology in17th century.
Descartes' theory of celestial evolution, vortex model and close interaction, like his whole ideological system, on the one hand, is characterized by rich physical thoughts and rigorous scientific methods, which played a role in opposing scholasticism, inspiring scientific thinking and promoting the progress of natural science at that time, and had a far-reaching impact on the thoughts of many natural scientists; On the other hand, it often stays in the intuitive and qualitative stage, rather than starting from quantitative experimental facts, so some concrete conclusions often have many defects, which have become the main opposites of Newtonian physics and caused extensive debates.
He thinks that there is a huge vortex around the sun, which drives the planets to keep running. The particles of matter are in a unified vortex, which distinguishes the three elements in motion: earth, air and fire. Earth forms planets, and fire forms the sun and stars. Descartes' vortex theory about the origin of the sun,
He also developed some theories, such as universe evolution theory and vortex theory. Although there are many defects in the specific theory, it still has an impact on future natural scientists. Descartes' most important contribution to mathematics is the creation of analytic geometry. In Descartes' time, algebra is still a relatively new discipline, and geometric thinking is still dominant in mathematicians' minds. Descartes devoted himself to the study of the relationship between algebra and geometry, and successfully linked algebra and geometry which were completely separated at that time. 1637, after the coordinate system was established, the analytic geometry was successfully established. His achievements laid the foundation for the establishment of calculus, which is an important cornerstone of modern mathematics. Analytic geometry is still one of the important mathematical methods.
Descartes not only put forward the main thinking method of analytic geometry, but also pointed out its development direction. In his book Geometry, Descartes combined logic, geometry and algebra methods, and outlined a new method of analytic geometry by discussing the drawing problem. Since then, number and shape have come together, and the number axis is the first contact between number and shape. It is proved to the world that geometric problems can be attributed to algebraic problems, and geometric properties can also be discovered and proved through algebraic transformation. Descartes introduced the concepts of coordinate system and line segment operation. He creatively transformed geometric figures into algebraic equations, thus solving geometric problems by algebraic methods. This is today's "analytic geometry" or "coordinate geometry".
The establishment of analytic geometry is an epoch-making turning point in the history of mathematics. The establishment of plane rectangular coordinate system is the basis of analytic geometry. The establishment of rectangular coordinate system bridges algebra and geometry, so that geometric concepts can be expressed in algebraic form, and geometric figures can also be expressed in algebraic form, so algebra and geometry are combined into one.
In addition, many mathematical symbols used now are first used by Descartes, including known numbers A, B, C, unknowns X, Y, Z and so on. And the representation of the index. He also discovered the relationship between the edges, vertices and surfaces of convex polyhedron, which was later called Euler-Descartes formula. He also discovered the Cartesian branch line, which is common in calculus.
Cartesian coordinate system
Mathematically, Cartesian coordinate system, also known as rectangular coordinate system, is an orthogonal coordinate system. The two-dimensional rectangular coordinate system consists of two mutually perpendicular number axes, and their zeros coincide. In the plane, the coordinates of any point are set according to the coordinates of the corresponding point on the number axis. In a plane, the correspondence between any point and coordinates is similar to the correspondence between points and coordinates on the number axis.
Using rectangular coordinates, geometric shapes can be clearly expressed by algebraic formulas. The rectangular coordinates of each point of a geometric shape must obey this algebraic formula.
The French mathematician rene descartes established the Cartesian coordinate system. 1637, Descartes published his masterpiece Methodology. This book is devoted to the study and discussion of western academic methods, providing many correct opinions and good suggestions, and making great contributions to the later development of western academic.
In order to show the advantages and effects of the new method and help him carry out scientific research, he added another book, Geometry, to the appendix of Methodology. The study of Cartesian coordinate system appears in the book Geometry.
Descartes' research on coordinate system combines algebra and Euclidean geometry, which has a key guiding force for later achievements in analytic geometry, calculus and cartography.
Anecdote: Spider Web Weaving and Establishment of Plane Rectangular Coordinate System
It is said that Descartes was seriously ill in bed one day. Nevertheless, he repeatedly thought about a problem: geometry is intuitive, while algebraic equations are abstract. Can geometry be combined with algebraic equations, that is, can geometry be used to represent equations? In order to achieve this goal, the key is how to link the points that make up the geometric figure with each group of "numbers" that satisfy the equation. He thought hard and tried to figure out how to connect "point" with "number". Suddenly, he saw a spider in the corner of the roof and pulled down the silk. After a while, the spider climbed up along the silk again, drawing left and right on the silk. The spider's "performance" made Descartes' thinking suddenly clear. He thinks that spiders can be regarded as a point. He can move up and down and left and right in the room. Can you determine every position of the spider with a set of numbers? He also thinks that two adjacent walls in the room pass three lines to the ground. If the angle on the ground is taken as the starting point and the three intersecting lines are taken as the three axes, then the position of any point in space can find three numbers in turn on these three axes. Conversely, any given set of three ordered numbers can also find the corresponding point p in space. Similarly, a set of numbers (x, y) can represent a point on the plane, and a point on the plane can also be represented by a set of two ordered numbers, which is the prototype of the coordinate system.
descartes' rule of signs
Descartes' symbolic law was first described by Descartes in his book La Géométrie, which is a method to determine the number of positive roots or negative roots of polynomials.
If univariate polynomials with real coefficients are arranged in descending order, the number of positive roots of polynomials is equal to the number of sign changes of adjacent non-zero coefficients, or a multiple of 2 less than it. For example, 5,3, 1 or 4,2,0. The number of negative roots is the number of times that the sign of the polynomial changes after changing the sign of all the coefficients of odd terms, or a multiple of 2 less than it.
Special case: Note that if you know that a polynomial has only real roots, you can use this method to determine the number of positive roots. Because the repetition of zero roots is easy to calculate, the number of negative roots can also be found. You can then determine the symbols of all the roots.
Euler-Descartes formula
Euler-Descartes formula is a formula in geometry.
The content of the formula is: on any convex polyhedron, let v be the number of vertices, e be the number of edges and f be the number of faces, then v? E+F=2 .
This formula was first proved by the French mathematician Descartes around 1635, but no one knows it. Leonhard euler, a post-Swiss mathematician, independently proved this formula in 1750. 1860, Descartes' work was discovered, and then this formula was called Euler-Descartes formula.
Cartesian lobed line
Descartes is an algebraic curve, which was first proposed by Descartes in 1638.
The implicit equation of Cartesian contour is
The equation in polar coordinates is as follows
The name comes from the Latin word "leaf", which means "leaf".
Characteristics of the curve: y' can be obtained by using the derivative rule of implicit function;
Using the point oblique equation of a straight line, we can find the tangent equation at point ():
Horizontal tangent and vertical tangent: When the tangent of Cartesian line is horizontal. So:
At that time, the tangent of Cartesian line was vertical. So:
This can be explained by the symmetry of the curve. We can see that the curve has two horizontal tangents and two vertical tangents. The cartesian line is symmetrical about y=x, so if the horizontal tangent has a coordinate (), there must be a corresponding vertical tangent with a coordinate ().
Asymptote: The curve has an asymptote: x+y+a=0.
The slope of this asymptote is-1, and the x and y intercepts are-a.
The story of the heart line between Descartes and Christine.
Heart shape curve
There is no strict evidence that the heart line was invented by Descartes. The center line is an epicycloid with a sharp point. That is to say, when a circle rolls along another circle with the same radius, the trajectory of a point on the circle is the center line.
The center line is a kind of epicycloid, and its n is 2. It can also be expressed in polar coordinates: r= 1+cosθ. The circumference of such a heart line is 8, and the enclosed area is.
Heart line is also a kind of line.
The figure between Mandleberg sets is a heart-shaped line.
The English name "Cardioid" of the cardiac line was published by de Castillon in the Journal of Philosophy of the Royal Society in 174 1. It means "like a heart".
In Cartesian coordinate system, the parametric equation of the heart line is:
Where r is the radius of the circle. The cusp of the curve is located at (r, 0)
The equation in the polar coordinate system is:
ρ(θ)=2r( 1-cosθ)
Region:
Its equation in polar coordinates needs to be investigated, which is for reference only.
The Story of Mathematics tells the love story of mathematician Descartes. Descartes was born in France on 1596. When the Black Death broke out in continental Europe, he wandered to Sweden. He met Christina, a princess of 18 years old from a small principality in Sweden, and later became her math teacher. Getting along with each other every day makes them fall in love. The princess's father, the king, flew into a rage and ordered Descartes to be executed. Later, because of his daughter's intercession, he was exiled to France. Princess Christina was also under house arrest by her father. Descartes contracted the Black Death shortly after he returned to France. He writes to the princess every day. Because she was intercepted by the king, Christine never heard from Descartes again. Descartes died after sending the 13 letter to Christine. The letter 13 only contains a short formula: r=a( 1-sinθ). The king couldn't understand that they didn't always talk about love, so he gave this letter to Christine, who had been depressed. As soon as the princess saw it, she immediately understood the intention of her lover. She immediately began to draw a diagram of the equation. She was very happy when she saw the chart. She knows that her lover still loves her, and the figure of the equation is a heart. The princess established a polar coordinate system on paper, traced the points of the equation with a pen, saw the heart line represented by the equation, and understood Descartes' deep love for herself. This is also the famous "heart line".
After the death of the king, Christina ascended the throne and immediately sent people all over Europe to find her sweetheart. Unfortunately, she left before her, leaving her alone on earth. ...
It is said that this world-famous alternative love letter is still preserved in the Descartes Memorial Hall in Europe.
Descartes and Christina did have a friendship in history. But Descartes came to Sweden at the invitation of Christina on 16491October 4, when Christina had become the queen of Christina. Descartes and Christina are mainly discussing philosophical issues rather than mathematics. It is recorded that Descartes can only discuss philosophy with Queen Christina at five o'clock in the morning because of her tight schedule. Descartes' real cause of death was pneumonia caused by cold weather and overwork, not the Black Death.
Analytic geometry
The Renaissance enabled European scholars to inherit the geometry of ancient Greece and accept the algebra introduced from the East. With the development of science and technology, describing sports with mathematical methods has become the central issue of concern. Descartes analyzed the advantages and disadvantages of geometry and algebra, and said that he would "seek another method that contains the advantages of these two sciences without their disadvantages".
In the first volume of Geometry (a part of methodology), he used the distance from a point on a plane to two fixed straight lines to determine the distance of the point, and used coordinates to describe the point in space. He further founded analytic geometry, showing that geometric problems can not only be reduced to algebraic form, but also be discovered and proved through algebraic transformation.
Descartes transformed geometric problems into algebraic problems and put forward a unified drawing method of geometric problems. To this end, he introduced the concepts of unit line segment, addition, subtraction, multiplication and division of line segments and square root, thus linking line segments with quantity. Through the relationship between line segments, he "found two ways to represent the same quantity, which will form an equation", and then drew according to the relationship between line segments represented by the solution of the equation.
In the second volume, when Descartes used this new method to solve the Pappus problem, he took a straight line as the baseline on the plane, set a starting point for it, and selected another straight line intersecting with it, which was equivalent to the X axis, the origin and the Y axis, respectively, to form an oblique coordinate system. Then the position of any point on the plane can be uniquely determined by (x, y). The Pappus problem becomes a quadratic indefinite equation with two unknowns. Descartes pointed out that the number of equations has nothing to do with the choice of coordinate system, so curves can be classified according to the number of equations.
The book Geometry puts forward the main ideas and methods of analytic geometry, which marks the birth of analytic geometry. Since then, mankind has entered the stage of variable mathematics.
In the third volume, Descartes pointed out that the roots of an equation can be as many as its times, and also put forward the famous Descartes sign rule: the maximum number of positive roots of an equation is equal to the number of times its coefficient changes sign; The maximum number of its negative roots (which he called false roots) is equal to the number of times the symbol remains unchanged. Descartes also improved the symbol system created by Vader, using a, b, c, … to represent known quantities and x, y, z, … to represent unknown quantities.
The appearance of analytic geometry has changed the trend of separation between algebra and geometry since ancient Greece, unified the "number" and "shape" which are opposite to each other, and combined geometric curves with algebraic equations. Descartes' creation laid the foundation for the creation of calculus, thus opening up a broad field of variable mathematics.
As Engels said: "The turning point in mathematics is Cartesian variables. With variables, motion enters mathematics, with variables, dialectics enters mathematics, with variables, differentiation and integration become necessary immediately. "Descartes' views and major psychological discoveries have a great influence on later psychology.
He is a famous representative of modern dualism and idealism theory. His great discovery of reflection and reflex arc provides an important basis for the assertion that animals are machines. And put forward the hypothesis of response stimulus.
However, Descartes' concept of reflection is mechanical. He emphasized the difference between humans and animals. Animals have no hearts, but humans have hearts. This inference is a typical manifestation of dualism. In addition, the theory of psychosomatic resonance is another typical expression of Descartes dualism in the relationship between body and mind. He believes that the human body is composed of material entities, and the human mind is composed of spiritual entities. Mind and human body can influence each other, cause and effect each other and interact with each other.
He believes that there are six kinds of primitive emotions: surprise, love, hate, desire, joy and sadness, and other emotions are branches or combinations of these six kinds of primitive emotions.
Although Descartes' dualistic psychology thought was wrong in theory, it played a very promoting and progressive role in the social background at that time. He used dualism to get rid of the absolute control of theology on science and guided people's thoughts to rational thinking and concrete research. Therefore, his contribution to psychology can not be ignored.