In 1916, Albert Einstein (1879-1955) published the article "Die Grundlage der allgemeinen Relativitätstheorie" ("Die Grundlage der allgemeinen Relativitätstheorie"), successfully using the tools and methods of tensor analysis to express his general theory of relativity, and derived the famous gravitational field equation of general covariance (i.e., Einstein's equation):
where gμν is the Riemannian gauge tensor, Rμν is the condensation of the Riemannian curvature tensor Rμλνκ, T is the energy–tensor scalar of matter, and Tμν is the energy–tensor of matter. The famous American physicist John Archibald Wheeler (1911-2008) profoundly explained the essence of Einstein's equations - "matter tells how space-time bends, space-time tells matter how to move". From 1916 to 1917, when Hermann Weyl taught a course on the theory of relativity at the Federal Institute of Technology in Zurich, he tried to combine philosophical ideas, mathematical methods, and physical theories, and used his own ideas to clearly and rigorously expound the general theory of relativity. The first German edition of the work was published in 1918 (the first English editions of Space, Time, Matter were published in 1922) and by 1923 it was already in its fifth edition, and it was one of the earliest monographs on the theory of relativity, and one of the most influential.
In Space-Time-Matter, Hermann Weyl explores the ultimate philosophical questions about "space, time, and matter", and contains his profound mathematical philosophical ideas and their profound connections between geometry, physics (especially general relativity) and philosophy, so this book is a world masterpiece combining physics, philosophy and mathematics, and has important academic value in the history of modern science (including the history of mathematics and physics).
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← Swipe left and right to view Details → Weyl's book "Space-Time-Matter" combines philosophical ideas, mathematical methods, and physical theories, and uses his own ideas to clearly and rigorously expound the general theory of relativity. Published in 1918 under the title Raum, Zeit, Malerie, and reprinted five times in five years, it became a favorite of young people, and the great physicist W.K. Heisenberg and others learned from it.
The late famous mathematician Mr. Qi Minyou of the mainland believes: "Weyl was a mathematician of the whole mathematics, and knew a lot about the whole science, and he wrote "Space-Time-Matter" (Raum, Zeit, Materie) based on the general theory of relativity, which has not yet been translated into Chinese. In fact, Weyl's writings should all be translated into Chinese." (Mathematical Culture, Vol. 10, No. 4, p. 102, 2019) Therefore, the translation and publication of this book not only fulfills Mr. Qi's last wish, but also marks a special commemoration of the centenary of the publication of Hermann Weyl's works. Chen Huiyong, School of Mathematics and Statistics, Jiangxi Normal University, May 20, 2022
Introduction to Space-Time-Matter
Space and time are often thought of as forms of existence in the real world, and matter is its content. At a given time, a specific part of matter occupies a specific space. It is in the comprehensive concept of motion that these three basic concepts form a close relationship. Descartes defined the goal of the exact sciences as describing all that happens with these three basic concepts, thus calling them motion. Ever since the human mind first awakened from obscurity and was allowed to navigate freely, it has never ceased to perceive the profound and mysterious nature of time—the evolution of consciousness, the development and change of the world in time. This is an ultimate metaphysical question that philosophers have attempted to elucidate and address in any historical period. The Greeks took space as an extremely simple and definite scientific subject. In the ideas of classicism, the idea of pure science was born from this. Geometry became one of the most powerful expressions of supremacy that inspired the ideas of the time. Later, when the intellectual tyranny of the Church throughout the Middle Ages collapsed, and the tide of skepticism seemed to sweep away everything that seemed to be the most fixed, those who believed in the truth clung to geometry like a rock, and the highest ideal of every scientist was to make his science "more geometrical". Matter is considered to be a substance that participates in every change, and it is believed that every substance can be measured by a quantity, and the peculiar expression of this quantity as "matter" is the law of conservation of matter, which asserts that the quantity of matter is constant in every change. To date, this represents our knowledge of space and matter, which philosophers in many ways claim to be a priori knowledge, absolutely universal and necessary, which today seems to be a crumbling structure of knowledge. First of all, physicists represented by Faraday and Maxwell proposed the "electromagnetic field" as opposed to matter, as a different category of reality. Then, in the 19th century, mathematicians followed a different line of thought, secretly undermining trust in Euclid's geometric evidence. Now, in our time there has been a great transformation, which has swept away space, time, and matter, which hitherto have been considered the most solid pillars of the natural sciences, but only to make way for a broader and deeper vision of these things.
Einstein's theory of relativity takes our understanding of the structure of the universe one step further. It's like a wall separating us from the truth has come down. Now, we're seeing a much broader field and deeper knowledge that we don't even have any hunch about. It brings us closer to grasping the laws by which all matter occurs. This revolution was largely driven by the ideas of Albert Einstein. At present, it seems that some kind of conclusion has been drawn from these basic ideas; However, whether or not we are already confronted with a new state of affairs, we all feel the need to carefully analyze these new ideas, and it is impossible to retreat. The development of scientific thought may once again lead us beyond what we have achieved so far, but it is impossible to return to the narrow and constrained patterns of the past. Philosophy, mathematics, and physics each have their own strengths in the questions raised here. However, we will first focus on the mathematical and physical aspects of these issues. I will simply speak briefly about the meaning of philosophy, for the simple reason that no final result has yet been reached in this regard, and that personally, I cannot give such an answer to the epistemological questions involved, as my conscience allows. The ideas to be presented in this book are not the result of some speculative inquiry into the foundations of physical knowledge, but have been developed in the general process of dealing with specific physical problems that have arisen in the midst of the rapid development of science, which has so to say that it has broken through its old shell, because it has now become too narrow. This revision of the Basic Principles was made later, and only to the extent necessary for the newly formulated idea. As it stands, there is no other way for the sciences to proceed along this line in their own discretion, that is to say, the sciences should faithfully follow the path which is guided by the rational motives peculiar to their particular methods and special limitations. However, elucidating these questions from a philosophical point of view is still an important task, as it is very different from the questions that belong to many individual sciences. At this point, philosophers must tread with caution. If he always pays attention to the boundaries established by the difficulties inherent in these problems, he can direct the development of science, but in no way hinder the development of science, since the field of study of science is confined to the field of specific objects. Still, I'd like to start with some reflections of a philosophical nature. In the activities of daily life, human beings will find that their perceptual behavior is affected by substances. We classify them as "real" beings, and we generally accept their composition, shape, color, etc., and when they appear in our "general" perception, we exclude possible illusions, mirages, dreams, and hallucinations. These material things are submerged in a kind of undefined reality-like manifold, and are injected into it. These realities combine to form an eternally existing spatial world to which my body and I belong. Let us consider here only the objects of these entities, and not all the other different categories of things that we face as ordinary people, such as living beings, people, everyday objects, values, and entities such as the state, rights, language, and so on. Philosophical reflection probably begins with each and every one of us, when it first becomes suspicious of the naïve realist worldview that I have just briefly mentioned, and it is endowed with an abstract way of thinking. It is easy to see that a quality like "green" exists only when the sensory "green" is associated with the object given by perception, and that it is meaningless to attach itself as a thing to the material things that exist in itself. This recognition of the subjectivity of the nature of sensation was created by Galileo (as well as Descartes and Hobbes) in a form closely related to the basic principles of the constructive mathematical approach of modern physics that negates "properties". According to this principle, color is the "real" vibration of the ether (æther), i.e. the motion of an object. In the field of philosophy, Kant was the first to take a decisive next step towards the idea that not only the properties revealed by the senses, but also space and spatial features have no objective meaning in the absolute sense; In other words, this space is also just a form of our perception. In the field of physics, perhaps only the theory of relativity makes it clear that the two essences of space and time, which enter into our intuition, have no place in the world constructed by mathematical physics. Therefore, color is "real", not even other vibrations, but simply a series of functional values that occur in four independent parameters corresponding to three-dimensional space and one-dimensional time. As a general principle, this means that the real world, each of its components and its accompanying features, is and can only be an intentional object of conscious action. The direct data that I receive is the form of conscious experience that I receive. They are not made up of mere perceiving things, as many positivists have argued, but we can say, for example, that in a sense, an object is actually physically presented to the person associated with that feeling in a way that everyone knows, and yet, because of the uniqueness of this feeling, it cannot be described more fully. According to Brentano, I'm going to call this an "intentional object". For example, in the process of experiencing perception, I saw this chair. My attention was completely focused on the chair. I "have" this perception, but it is only when I turn this perception in turn as an intentional object of a new inner perception (an act of free reflection that enables me to do so) that I "know" something about it (and not just a chair), and determine exactly what I have just said. In this second act the intentional object is immanent, i.e. the act itself, which is a real component of my experience, whereas in the primary perceptual act, the object is transcendental, i.e. it is given in the experience of consciousness, but not the real component of consciousness. What is inner is absolute, that is, what is inside is the form in which I possess it, and I can reduce its essence to axioms through the act of reflection. On the other hand, transcendental objects exist only phenomenally; They are presented in a variety of ways and in a variety of "levels" to present their own appearance. The same leaf, depending on its location and lighting conditions, seems to have one size or another, or one color or another. Neither of these ways of appearance can claim to present a leaf as it "is". Moreover, in every perception there is no doubt the proposition of the reality of the object present in it; In fact, the latter is a fixed and enduring factor in the general assertion of world reality. However, when we transition from a natural point of view to a philosophical attitude, a contemplative approach to perception, we no longer endorse this thesis. We simply confirm that there is something real in it that is "assumed". The significance of this hypothesis now becomes a problem that must be solved from the data of consciousness. In addition, a reasonable reason must also be found. I do not wish to imply that the view that the events of the world are merely a game of self-generated consciousness contains a higher degree of truth than naïve realism; Rather, all we are concerned with is to see clearly that if we are to understand the absolute meaning and the right to assume reality, we must place ourselves on the basis of consciousness. In the realm of logic, we have a similar example. The judgment I have announced affirms certain circumstances; The judgment considers them to be true. Here the philosophical question of the meaning and justification of this proposition of truth arises again; Here, the notion of objective truth is not denied, but becomes a question that must be absolutely grasped from what is given. "Pure consciousness" is the basis of a priori things in philosophy. Moreover, the philosophical inquiry of the proposition of truth must and must come to the conclusion that no act of perception, memory, etc., which exhibits the experience by which we grasp reality, can give us a conclusive right to attribute the object of perception to the existence and composition of a perception. Conversely, this right can always be denied by a right based on other ideas. Inexhaustible in content is the essence of the real thing; We can gain a deeper understanding of this content by constantly adding new experiences, some of which are in obvious contradiction, and harmonize them with each other. In this interpretation, real-world things are approximate concepts. This gives rise to the empirical character of all our knowledge of reality [Note: The detailed development of these ideas follows Husserl's line very closely, see his "Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie" (Jahrbuch f. Philos.u. phänomenol. Forschung, Bd. 1, Halle,1913)].
Time is the original form of stream of consciousness. No matter how vague and confused our minds are, in fact, the content of consciousness does not simply manifest itself as existence (such as concepts, numbers, etc.), but is filling the form of persistence with a different content. Therefore, instead of saying it is, people say that it is now, but now it is no longer the present. If we project ourselves outside of the stream of consciousness, and represent the content of the stream of consciousness as an object, the stream of consciousness becomes an event that takes place in time, with different phases in our stream of consciousness existing with each other in earlier and later relationships. Just as time is the form of the stream of consciousness, one can reasonably assert that space is the form of external material reality. All the characteristics of material things in external perceptual acts (e.g., color) are endowed with the separability of spatial extension, but it is only when we establish a single, connected real world from all experiences that the expansion of space, which is a component of each perception, becomes part of the same all-encompassing space. Therefore, space is the form of the external world. That is to say, anything material, without changing its contents, can likewise occupy a different place in space than it is now. This immediately gives us the nature of spatial homogeneity, which is at the root of the concept of congruence. Now, if the world of consciousness and the transcendental reality are completely different, or rather, if passive perceptual acts bridge the gap between them, the state of affairs will remain as I have just spoken, that is, on the one hand, a consciousness passes in an eternal present form, but there is no space; On the other hand, it is an extension of the real world in time, but it is eternal, and reality contains only a different appearance. Before all perception, we have the experience of effort and opposition with both active and passive experiences. For a person who leads an active natural life, the most important role of perception is to clearly indicate before his consciousness the clear point of attack of the action he is going to take, as well as the source of opposition to it. As the executor and executor of actions, I become a single individual with a spiritual reality, an individual with a spiritual reality that occupies a place in the material things of the external world, through which I can communicate with other similar individuals. Consciousness, not giving up its immanence, becomes a part of reality, becomes this particular person, that is, myself, born and will die. Moreover, as a result of this, consciousness unfolds its network in the form of time on top of reality. Change, movement, the passage of time, existence and disappearance, all exist in time itself; Just as my will, as a driving force, acts on the external world through and beyond my body, the external world is equally active (as the German word "Wirklichkeit" indicates, derived from the "wirken" = the representation of reality to act). Its phenomena are always linked by cause and effect. In fact, physics shows that cosmic time and physical forms cannot be separated from each other. The new solution to the problem of spatio-temporal fusion proposed by the theory of relativity has made people have a deeper understanding of the coordination of world actions.
At the same time, I hope to use this great discipline as an example of the intermingling of philosophical, mathematical and physical ideas, a study that I love so much. To do this, the only way to do this is to systematically build theories from the basics and focus attention on the fundamentals from beginning to end. But I have not been able to meet these self-imposed demands: the mathematician has a higher status than the philosopher. In this way, we have a clear outline of the line of argumentation in the future. If we talk about it separately, then what do we have to say about time and the mathematical and conceptual grasp of time that is included in this introduction? Chapter 1 focuses on Euclidean space and its mathematical structure. The ideas that force us beyond the Euclidean system will be expounded in Chapter 2; This culminates in the general spatial concept of the metric continuum (the concept of Riemann space). On this basis, Chapter 3 will discuss the above-mentioned integration of space and time in the world. From this point on, the results of mechanics and physics will play an important role, since, as already pointed out, the essence of the problem makes us see the world as an active entity. The edifice built according to the ideas of Chapters 2 and 3 will lead us to Einstein's general theory of relativity at the end of Chapter 4, which physically contains a new theory of gravity, and an extension of the latter, which includes electromagnetic phenomena in addition to gravitation. The revolution caused in our conception of space and time will inevitably affect the concept of matter as well. Therefore, all the issues regarding substances that must be addressed will be dealt with appropriately in Chapters 3 and 4. The theory of relativity discussed in Chapters 3 and 4 raises the question of relativity not only for the point in time, but for the physical world as a whole. However, we found that once the solution to the two forms of this world – space and time – was found, the problem was solved. By choosing the coordinate system of space and time, we can also conceptually determine the physical reality of all parts of the world using a numerical approach. It's hard to get started. Since the mathematician approaches his concepts in a strict and formal way, he must remind himself from time to time that the origin of things is often hidden deeper than his methods can reach. In addition to the knowledge gained from individual sciences, there is also the task of understanding. Although philosophical views oscillate between different systems, we cannot abandon it unless we are to turn knowledge into a meaningless mess.
The reader needs the least amount of theoretical stock at the beginning. The special theory of relativity not only discusses the theory of relativity in detail, but even Maxwell's theory and analytic geometry are developed in its main contents. It's part of the overall plan. The establishment of tensor calculus, which alone makes it possible to fully express the physical knowledge in question, occupies a considerable space. Therefore, it is hoped that this book will enable physicists to become better acquainted with this mathematical tool, and at the same time, it will also serve as a textbook for students, winning their approval of new ideas.
This article is excerpted from Space-Time-Matter [ by Hermann Weyl; Translated by Chen Huiyong and Xia Jijin. Beijing: Science Press, 2024.3] The book "Introduction to Space-Time-Matter", "Chinese Translator's Preface", "From the Author's Preface to the First Edition", has been deleted and modified, and the title is added by the editor.
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书名原文: SPACE, TIME, MATTER
ISBN 978-7-03-073894-3
Editor in charge: Li Xin, Fan Peipei
This book is considered one of the greatest mathematicians of the 20th century, the famous work of German mathematician Hermann Weyl (1885-1955), "Space-Time-Matter" (Raum, Zeit, Materie), which is a classic work in the field of Riemannian geometry and general relativity. From 1916 to 1917, when Weyl taught a course on the theory of relativity at the ETH Zurich, he tried to combine philosophical ideas, mathematical methods, and physical theories, and to use his own ideas to articulate the general theory of relativity clearly and rigorously. Between 1917 and 1919, Weyl made great contributions to geometry and physics, one of the most important of which was his monograph Space-Time-Matter, which included: Euclidean space, its mathematical representation and its role in physics; the continuum of measures; the relativity of time and space; The general theory of relativity consists of four chapters. The first German edition of this book was published in 1918 and the first English edition in 1922, more than 100 years ago. This book is intended for undergraduate and graduate students, teachers and researchers majoring in mathematics and physics, and is a valuable historical document for readers who want to understand the history of Riemannian geometry, general relativity, and related fields.
Space-Time-Matter.