qiu chengtong’s mathematics lessons for middle school students: beauty and practicality often arise naturally on the way to exploring the mysteries of mathematics.
2024-09-27
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as a basic subject, mathematics plays an important role in human understanding of the world and changing the world. whether it is the birth of the theory of relativity and quantum mechanics, or the rapid development of modern medicine and artificial intelligence, mathematics plays an important role in it.
to cultivate top innovative talents, the importance of mathematics education is self-evident. in recent years, the well-known mathematician qiu chengtong has invested a lot of time and energy in cultivating mathematical talents in the basic education stage. recently, at the founding ceremony of the shanghai institute of mathematics and interdisciplinary studies, qiu shing-tung awarded medals to qiu shing-tung's classes from many middle schools across the country, including shanghai middle school and the private huayu middle school. at the licensing ceremony, he taught a special mathematics lesson to middle school students. this journal publishes part of the lecture content for the benefit of readers.
mathematics is a beautiful and practical science that has fascinated all mathematicians since its birth. beautiful and practical mathematics arises naturally in nature, which is very wonderful in itself. what's even more amazing is that people often pursue the practical value of mathematics, but they discover the beauty of mathematics in the process. for mathematicians, beauty and practicality often arise naturally on the way to exploring the mysteries of mathematics, which is a very interesting experience.
in fact, every scholar has different views on beauty. qiuzhen academy of tsinghua university once invited professor liu jude from the academy of fine arts to give a lecture on beauty in the eyes of artists. here, i will also tell you about beauty in the eyes of mathematicians and scientists.
no subject has stood the test of time like mathematics
in my opinion, the beauty in the world must be based on truth, and only then can it be called "beauty". professor liu said that beauty is epoch-making and transcends time and space. however, the only thing that can exist beyond time and space is truth. frankly speaking, i think there is only one truth - mathematics. no subject has a description of the world that has stood the test of time quite like mathematics. from ancient greek scholars to scientists such as newton and einstein, and to today, human beings’ observations of the world and the theories formed based on them are constantly changing; in the laboratory, as technology continues to develop and strive for excellence, we observations of the physical world are constantly yielding new results, and previous conclusions are constantly being overturned.
the theory of relativity and quantum mechanics gave scientists in the 20th century different views. they changed classical physics and gave us a deeper understanding of the universe. whether it is the structure of extremely small protons or the distant space, the phenomena we can now observe are unimaginable by our predecessors. therefore, in a sense, the truth of physics is constantly changing.
in this series of magnificent scientific developments, mathematicians have made huge contributions. in many important development situations of basic physics, mathematicians are often at the forefront, leading physicists forward, and then working together to understand the universe.
let's recall an extremely important development in science: one of newton's great contributions was the development of important applications of calculus in physics, thereby revolutionizing mathematics itself. in the early 19th century, mathematicians gauss and riemann developed a set of mathematical theories to gain a deeper understanding of electromagnetism. this theory was finally completed by maxwell, thus establishing the perfect maxwell equations. in the early 20th century, the german mathematician and physicist weyl studied maxwell's equations through geometric methods and made them part of the gauge field he proposed. weyl was also the first mathematician to propose gauge principles. these concepts become the basis of modern science. we also know that in 1926, the french geometer cartan developed the contact theory, which is today's non-commutative gauge field theory.
since then, a large number of mathematicians, including the chinese mathematician mr. chen shengshen, have been engaged in the research of non-commutative gauge fields. it can be said that all classical gauge fields were completed by geometers. however, the quantification of gauge fields in physics had to wait until the 1960s to be completed by several great physicists; by the 1970s, the standard model of theoretical physics became the most important work in basic science. in the above-mentioned work, extremely profound mathematical theories are used. these theories are actually beyond the ability of physicists back then to absorb.
physicists' understanding of the truth is constantly changing, but the correctness of the mathematical theories used has never been questioned. because they are based on some assumptions that are difficult to question. these assumptions are the simplest logical systems. these systems are also the basis of all human civilization.
from great truth to simplicity, it is a summary of the beauty of mathematics
mathematicians use strict logical systems to construct different mathematical systems to describe nature, and from them they can see the laws that govern nature. in this process, we see how nature builds its own structure, which is magnificent and incomparable to all other things.
the greatest simplicity is a summary of the beauty of mathematics. the simplest mathematics, starting from 1=1, to 1+1=2, 1+2=3... and keep deriving. in this way, people understand natural numbers and thus have mathematics. from the beginning of mathematics, from the time of calculating livestock and taxes, humans have realized that these abstract concepts are an exquisite induction and summary of things. this is closely related to beauty. mathematics abstracts reality into truth, and beauty is based on truth. at the same time, beauty guides humans to continuously discover the truth. without the pursuit of beauty, it would be difficult for humans to detect the existence of truth. the development of mathematics relies on human beings' pursuit of beauty, perceiving the truth and finding the truth.
give an intuitive example. many painters like to paint bamboo because bamboo is elegant, tough, and full of character, reflecting the spiritual pursuits of chinese intellectuals. there are many ways for painters to depict the character of bamboo. for mathematicians, when they first see bamboo, they see a straight line. like painters, they will add a lot of content to this straight line.
for example, the construction of straight lines is an interesting thing for mathematicians. on this line, we first label the natural numbers, that is, integers, starting from 1, 2, and 3, which are the basic mathematical structures; then, in order to enrich its structure, we also construct fractions, such as 1/2, 3/ 4 and so on, are drawn densely on this straight line.
what’s next? the greeks constructed irrational numbers. they used vertical lines to construct a triangle with two sides of length 1 and a hypotenuse of length √2 - this was the discovery of the greek pythagoreans. √2 is an irrational number. after successfully constructing irrational numbers, we added a large class of numbers to the straight line, and the numbers on the straight line were denser. but that wasn't enough, we started constructing numbers using compasses and rulers, but in any case, we couldn't fill the line.
it took almost another 1,500 years before we completely filled in the straight line and turned this bamboo into a complete solid line. in order to achieve this goal, mathematicians spent a lot of effort before finally having a thorough understanding of straight lines. just like a painter who wastes time describing the meaning of bamboo, mathematicians also use a lot of abstract number concepts to construct straight lines.
in the 15th century, mathematicians began to introduce imaginary numbers for this straight line, which changed our focus from a straight line to a two-dimensional space - the recognition of two-dimensional space is a very important thing in human history. after the emergence of imaginary numbers, we have a clearer understanding of many wave equations and various phenomena of waves.
the waves described by the painter are actually closely related to imaginary numbers. however, we are currently unable to vividly draw dynamic waves because our understanding of imaginary numbers is not clear enough. imaginary numbers are the most important numbers in the study of dynamical systems and are also used in the study of quantum mechanics.
from the introduction of positive integers such as 1, 2, 3, etc., to a straight line, to imaginary numbers, to a complete explanation of two-dimensional space, and then to three-dimensional space - this process is actually completed slowly and gradually through the development of mathematics. and this methodical advancement involves, in addition to rigorous mathematical reasoning, mathematicians’ pursuit of beauty. the goal that mathematicians hope to achieve is: the phenomena that people see and the world in their eyes should be as complete as possible. if there are some spaces that cannot yet be described, it is not ideal and must be understood more thoroughly, which requires drawing a perfect picture from a mathematical perspective. to a mathematician, this image looks like a picture. adding up the points from the integers results in a straight line - this is satisfying. but this alone is not enough. adding imaginary numbers can describe two-dimensional space; two-dimensional space is still not enough, so we add them all the way...
mathematicians rely on the guidance of beauty to find mathematical truths
some people may say that bamboo is obviously not a line, so why are mathematicians so stupid to regard it as a line? this is correct. the surface of bamboo is a cylinder. straight lines are just the first step in describing the characteristics of bamboo. this step does not fully comply with physical phenomena.
let's move on to describing bamboo. find a small circle with a fixed radius, which is perpendicular to the straight line, and pull the center of the circle out along the straight line to get the cylinder we expect. the best mathematical way to describe this cylinder is with complex numbers. after adding the structure of complex numbers, the cylinder is called a riemann surface in mathematics. the riemann surface is widely used in describing two-dimensional space and in modern physics, and it is really powerful.
when the radius of the above small circle becomes very small, the cylinder becomes a straight line. this is a way that high-dimensional space can be used to describe the low-dimensional reality. when the radius of the circle changes with the position of the center of the circle, the cylinder can become a bamboo.
geometers can use the above point of view when looking at bamboo. but galileo, the leader of the western scientific revolution, probably wouldn't see it this way, because he would study the various physical properties of bamboo, such as elasticity, structural properties, and other issues. these problems were more perfectly solved after the emergence of newtonian mechanics and calculus, and mathematicians such as fermat, euler, lagrange, etc. have been involved.
in modern physics, we can replace straight lines with three- or four-dimensional flat spaces, while circles can be replaced with more complex geometries. an important geometry is the calabi-yau space, from which various physical phenomena can be described.
schematic diagram of calabi-yau space
therefore, a mathematician looks at a piece of green bamboo differently from a painter or artist. mathematicians will logically reason and deepen their understanding, and then describe it. now, we still feel that it is not enough to reason about three-dimensional space. in the 19th century, the discovery of quaternions began. soon after, octonions were discovered, thus entering high-dimensional space. high-dimensional space can express more phenomena in life. high-dimensional space is an important issue. when dozens, thousands, or tens of thousands of particles roll around, a high-dimensional space is formed, and now artificial intelligence needs to reach tens of millions of dimensional spaces. all phenomena in high-dimensional space are very beautiful, and there are many truths in them, that is, the existence of mathematics.
this is the world in the eyes of mathematicians and their pursuit of beauty. facing the thousands of worlds in front of us and so many ungraspable phenomena, we rely on the guidance of beauty to find the mathematical truth. a straight line is drawn from a bamboo to a two-dimensional space, and gradually enters a high-dimensional world. the connections are continuously enriched and this important subject is developed. it is full of the spirit of mathematics, from simple to complex, and then uses a simple principle to describe the most complex phenomena of nature, and finally gets infinitely closer to the truth.
whether it is ancient greece or the renaissance, this spirit has always been consistent. the painting art of the renaissance was inseparable from mathematics, which promoted the development of geometry - which also shows that beauty and mathematics have always been inseparable.
modern mathematics lays the theoretical foundation for artificial intelligence
mathematics has numerous applications in modern technology. for example, geometers deal with how to effectively express the beautiful features of curved surfaces; the distribution of prime numbers and the beautiful theory of integer solutions on elliptic curves have become extremely important tools for modern security systems; the effective description of waves, and the location of the fourier transform and momentum duality has produced fundamental changes for modern science, even computational science.
taking geometry as an example, there are not only fascinating theories that play a huge role in modern engineering practice.
modern technology requires a lot of knowledge about thin films, so how to accurately depict two-dimensional curved surfaces is an indispensable knowledge in engineering. the study of two-dimensional surfaces can be traced back to the great scientist euler, who lived in the same era as newton. he used calculus to explain geometry and created the variational method to calculate some important geometric figures. riemann and his teacher gauss are undoubtedly the two founders of modern geometry. gauss is the father of modern geometry, and the real founder is riemann. in the mid-19th century, he proposed the theory of riemannian geometry and conformal geometry, which not only played a key role in theoretical physics, but also played a key role in computer graphics, geometric modeling and medical images are widely used.
the theory developed by my student gu xianfeng and i using the method of riemann surfaces later developed into an important branch of imaging science - computational conformal geometry.
the core context of computational conformal geometry mathematics is to prove the existence, uniqueness, regularity, and well-posedness of the solution to single-valued theorems, especially how to extend it to discrete surfaces; the core of computer science is how to design algorithms and calculate single-valued theorems. value theorem. in computers, smooth surfaces are represented as discrete surfaces, theories in modern topology and differential geometry are extended to discrete situations, and computers are used to realize abstract geometric concepts, which can lead to good engineering practice.
conformal geometry is the study of invariants under conformal transformations. the so-called conformal mapping is a mapping that keeps the angle unchanged. for example, we map a three-dimensional human face curved surface to a two-dimensional flat disk, and draw two intersecting curves on the human face. the curves on the surface are mapped to the plane curve, but the intersection angle remains unchanged. since the conformal transformation is unique, it is easy to perform face comparison.
currently, artificial intelligence and data science technologies have been widely used in different fields such as clinical diagnosis, surgical guidance, and risk prediction. it can be said that modern mathematics has laid a theoretical foundation for artificial intelligence and pointed out the development direction for artificial intelligence to break through bottlenecks. on the other hand, artificial intelligence also poses challenges to mathematics and promotes the development of mathematics.
taking stock of the major changes in science and technology in the 20th century, they are based on mankind's in-depth understanding of the structure of matter. the theory of relativity and quantum mechanics are the basis of these studies, and mathematicians have made in-depth contributions to these studies. since the successful construction of the standard model of high-energy physics in the 1970s, which unified three different fields of physics, physicists' greatest wish has been to put gravity in the standard model. this integration requires a breakthrough in extremely creative concepts. i believe it will have a profound impact on the technological breakthrough we are looking forward to - quantum computing. how to construct quantum geometry will be an important milestone and a combination of truth and beauty.
the dispersion, gathering, rise and fall of all things, the structure of the heaven, earth and universe, and the context of human affairs and social economy are all related to basic mathematics. mathematics can provide truth and beauty. the goodness required by china for five thousand years, the benevolence and righteousness mentioned by confucius and mencius, can all be found in truth and beauty, that is to say, they can be called out from the sea of mathematics. therefore, basic mathematics is the basis for building a country and the bridge between eastern and western cultures. if chinese culture can be passed down and endure for thousands of years, we must pay attention to basic mathematics.
author: qiu chengtong
text: qiu chengtong (dean of qiuzhen college of tsinghua university and the first chinese winner of the fields medal). this article is a speech given by professor qiu chengtong at the shanghai institute of mathematics and interdisciplinary studies on august 17. part of the content is from "mathematical humanities". the author has authorized publication , no reproduction is allowed without permission) picture: the title picture comes from visual china, and the pictures in the article are provided by qiu chengtong editor: chu shuting editor: jiang peng.
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