Preface to the foundations of Hilbert geometry

         Remember 1954 In the autumn , Yuan Meng studied biology at Nanjing No. 10 middle school in Jiangsu 、 Darwin's theory of evolution , Learned the truth of survival of the fittest .

         The basis of Hilbert geometry is the evolutionary result of the history of Mathematics , We believe in the future 50 Within a period of years , Hilbert's innovative ideas based on geometry will not be eliminated .

         So , We will cover the relevant chapters of the book （ The full text ） Published on our website .《 Basis of Hilbert geometry 》

The contents are as follows ：

         Preface to the fundamentals of geometry  

The great German mathematician Hilbert is his masterpiece “ Geometric basis ” The preface is as follows ： Preface to the Russian translation of the seventh edition in German • Foreword • Hilbert once had a student , Wrote a paper to prove Riemann conjecture , Even though There is an irretrievable mistake , Hilbert was still deeply attracted . In the second year , The student didn't know what was going on and died , Hilbert asked to make a speech at the funeral . On that day , Wind and rain rustle , The family members of the student are in deep sorrow . Hilbert began his speech , He first pointed out that ：“ Such a genius left us so early , It's really deplorable ！” Everyone feels the same , Cry more and more fiercely . Next , Hilbert said ：“ Although this student's Prove wrong , But if you follow this path , It should be possible to prove Riemann conjecture .” Next , Hilbert braved the rain and preached passionately ：“ in fact , Let's consider A single variable complex function ” Everyone fell . Preface to the Russian translation of the seventh edition in German In Hilbert's 《 Geometric basis 》 And it's in this problem Position in the history of development * ;- .K. Rashevski Geometry as physics When we study geometry , In limine —— Just like learning geometry in Middle School —— It is in our understanding that Unique thinking world , It is strangely both realistic and fantasy . in fact , We are about straight lines 、 Plane 、 Geometry （ Like a ball ） And so on Statement , It is only after giving them completely certain properties . However, something with the shape we study , Where on earth In what sense does Li He exist ？ Don't we all know , No matter how we grind （ for example ） The surface of a metal plate , Because of work There is an inevitable deviation from the action itself , We can never grind it into “ Ideal plane ” The shape of the . What's more, it is not only impossible to achieve the ideal Flat shape , And according to the atomic structure of matter , It is not even possible to approach it indefinitely ！ in fact , When we strengthen the accuracy required When the degree of , The metal plate will be broken down into separate atoms , So that generally speaking, its surface is meaningless . And what about straight lines ？ It may be thought that light travels along an ideal straight line ？ However, quantum mechanics tells us , light Line is the use of different media —— quantum —— And spread , As for the path taken by this quantum in motion , Generally, it doesn't make sense . that , What exactly do we study in geometry ？ Do you only study fantasy that is incompatible with the material world 、 The creation of our imagination Made ？ But from daily experience and technical experiments , We can firmly know , The laws and... Derived from these imaginary objects law , All submit to the physical nature with insurmountable power ; So that engineers who carry out new designs , When you suffer failure , You can doubt its Any assumptions , And never doubt the formula about the volume of corner column . These geometric images , It seems to be insignificant 、 Immaterial , At the same time, it portrays the material world with insurmountable power , also It seems that （ As idealist philosophy often says ） God created it in his own image , What on earth is it ？ The materialistic view of the universe helps us answer this question . Let's start with a rough example . Suppose there is a building in front of us A fence on the edge of a piece of land . If we want to calculate the area of this land , To draw up its plan , Then we will draw Draw a closed curve to replace the fence , And use the plane segments it separates to replace the land . This use of geometric concepts to secretly replace matter object , What is the essence ？ The problem is ： Whether we build walls with wood or stone , No matter how wide or high we build , No matter whether we move to the side so One centimeter , wait , In fact, this land has not changed because of it . Because all we care about is the land itself , As for what happened along its boundary Made something , It doesn't actually work , Try to put all this aside . therefore , We abandoned the wall as an object 、 The vast majority of properties that are not important to us in the current situation . The properties of the fence that are important to us —— It is related to its extensibility in length The nature of , It belongs to our consideration , These properties are exactly the geometric properties of curves . There are various examples of the same fact There are countless ： When we talk about ropes 、 The course of speeding shells is isochronous , Under a certain degree of accuracy , All we have to care about is It's their properties , That is what we call geometric curves . All in all , When we study geometric curves , We also studied the enclosure of the land , A certain length —— Compared with thickness —— The rope of , And the course of the speeding shells , Yet for all these phenomena , We do not retain the diversity of their nature in all aspects , Because of them Preface to the Russian translation of the seventh edition in German In It does not have the greatest accuracy , But just choose our important one-dimensional extensibility in the current situation , And it only has The degree of accuracy necessary in practice . So the common properties of these objects, which we call the properties of geometric curves, become prominent . such , If I They say that curves have no width , That's just a brief indication , The width of the fence does not actually affect the land it surrounds , The cross section of the rope The size can be omitted compared with its length , Just wait . All other geometric concepts and propositions have similar meanings . They all reflect the nature of material objects and the laws of the material world . they Of “ ideal ” Characteristics only indicate that non essential properties are discarded in the known connection of object properties （ abstract ）, In particular, they only use one Be considered for the degree of accuracy . This abstraction can be used to clearly reveal the common and hidden properties of objects , We call them extended Properties and studied in geometry . The reason why geometric laws are necessary in nature , Just because they are abstracted from nature . thus , Geometric truth that reflects material reality , In simplified and formulated shapes , Approximately reproduce the material reality . Due to Abandoned countless complex facts , Only then produced such convincing integrity and rationality of geometric theory . And if so , Very Naturally , You can't force geometry ［ For the time being, it is always limited to Euclid （Euclid） geometry ] It is suitable for studying the material world without limitation ： When the accuracy of this research exceeds a certain limit , Geometry because it approximately reflects the essence of reality , It doesn't work . In order to make it useful again , We must make it more accurate based on new experimental data , We must come back and pick up in the abstract Those things abandoned in the process . However, when we established geometry , Material reality , What are the more prominent aspects , Abandoned ？ First of all, matter is The movement carried out in a certain period of time . naturally , In order to avoid excessive abstraction in geometry , Make it close to material reality , We should be Reconsider the process of material movement , And that means , Geometry should be discussed in an organic whole combined with mechanics .” Pure ” Geometry disappeared . All of the above are not just theoretical discussions ,20 The historical development of internal medicine in the 21st century is just along this road . narrow sense relativity （1905） Combine the extension of space and time into an indivisible whole , And general relativity （1916） It's more about geometry and The universal theory of the distribution and movement of matter is unified in one discipline . therefore , From the point of view we have talked about geometry so far It seems , It is part of Physics , Therefore, it should grow and develop with physics based on experiments . However, there are other things in geometry 、 Mathematics , That is what we have deliberately ignored until now . And this aspect is currently for We are the most important , Because it is exactly what this book is about .

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