What is space? You can interpret and describe it in different mathematical ways, but space does not actually exist, it is an abstract geometric category. Space is an abstract category that accommodates existence.

Euclidean geometric space is an infinite category based on the definition of infinite length of straight lines;

There are two kinds of non-Euclidean geometry, one is the limited category meaning or local limited category caused by extreme bending, like four-dimensional space-time; The other is a four-dimensional hyperbody, which is based on the premise definition of mathematical rationality, based on clear three-dimensional geometry, and is also equivalent to a finite line segment, so it is a limited space category;

The finite and infinite meaning of mathematical culture also has a compatible expression. In other words, the basic premise definition is conditional, changeable or compatible. For example, ancient squares can represent infinite space; But if it represents a finite size, then the outside is the outside. This is an ancient expression of mathematical compatibility.

That is to say, although we are all talking about space, we can express different concepts of space based on different premises.

Space consists of faces, just like a stack of paper (faces can be Euclidean planes or non-Euclidean surfaces); Surfaces are composed of straight lines, and the only straight line (which can be Euclidean or non-Euclidean), of course, there is another mathematical expression compatible with the definition of straight curves-point-to-point moving into straight lines, which is usually considered as non-Euclidean expression. It is different from the usual non-Euclidean geometric surface because it is a straight curve expression. Non-Euclidean geometry is based on the expression of specific curves. );

The so-called expression difference is only what line you are based on in the initial definition. We need to be careful here. Mathematics has never been so realistic about geometric shapes.

Then the basic unit of a line is the unit of length; The basic unit of surface is the square of area unit and length; The basic unit of space is unit of volume, that is, the cube of length. This is the basic idea of Euclid's mathematics. Area and volume are also the result of a mathematical definition and abstract simplified expression.

The meaning of this definition is to facilitate our geometric comparison. We can use a numerical result (one-dimensional explanation) or add a condition (two-dimensional expression) to express some different information of two-dimensional and three-dimensional geometric structures.

Euclidean geometry and non-Euclidean geometry (excluding hyperbody geometry) can still be expressed in a unified way in this respect.

For example, the area of 5 is greater than that of 4; Or the volume of 10 can hold more things than the volume of 9. Although we don't know its original shape.

That is, the advantage of this dimension reduction expression is simplicity, but the disadvantage is that all information can be expressed by dimension reduction.

This is a basic problem that is easily overlooked in the expression of dimension reduction.

We can only guess at other original information that can't be obtained after dimensionality reduction. And guess there will be right and wrong. This is the inevitable result of this method or expression of mathematics.

In order to express all the information of the original dimension after dimensionality reduction, it is necessary to clarify the dimensionality reduction method with mathematical logic significance and increase the description conditions after dimensionality reduction to make up for the loss of the original information brought by dimensionality reduction method.

For example, the projection method of mechanical drawing, there are two kinds of mathematical projection of three-dimensional cylinder: rectangle and circle. How can you guess or calculate the original shape of an object if you don't have this picture and only give you the projection results?

Three-dimensional projection of a three-dimensional object in a specific rectangular coordinate system provides three different kinds of information, so that three-dimensional geometric figures can be deduced mathematically. Of course, there are other auxiliary information.

Such as increasing perspective projection. If there is a hole in the middle of the cylinder and it cannot be seen directly, the information of this geometric shape should be represented by a dotted line. This is a mathematical method, a method of reducing dimension and expressing geometric original appearance.

However, if we only use the partial principle to obtain the two-dimensional information of the projection of a circle instead of the explicit projection based on this mathematical method, then we will guess many three-dimensional possibilities: cylinder, sphere, ellipsoid and so on. Of course, it is also possible to be right. For natural science, this may lead to the problem of blind people touching elephants; For the humanities, it will be very romantic and mysterious, and it will become divergent thinking.

For this three-dimensional mathematical projection method, if we restore the original appearance, we need information from three projection directions, both virtual and real information. That is to say, using two-dimensional method can actually contain six kinds of information, and accurately restore the original appearance of three dimensions.

How about the dimensionality reduction of four-dimensional space-time and four-dimensional hyperbody in these two four-dimensional mathematical directions? How many kinds of 3D projection information do you need to recover all its original information? In this respect, mathematics is still using similar three-dimensional projection method, but there is no clear constraint expression for the time being. We still get multiple three-dimensional or two-dimensional projections based on some mathematical method to "logically reason" four-dimensional information. Is the four-dimensional information of this reasoning comprehensive, unique and correct? This is not as mature as three-dimensional drawing, and it is still being explored, and it is still a mathematical problem. That is, is our mathematical way of dimensionality reduction the only expression of reversible information?

In at least four-dimensional space-time, two kinds of dimensionality reduction results are usually obtained. The rotation direction based on imaginary number I is different, even if the rotation angle is the same, two dimension reduction results will be obtained counterclockwise and clockwise. What is the basis for the choice and interpretation of these two results? Now they are usually used together, interpreted separately or together. Is there a problem with this interpretation? When imaginary numbers are represented by real numbers, some people just take one-way rotation to get a result, which is mathematically problematic. However, is there a problem with simultaneous interpretation? For example, the interpretation of the negative value of the time cone leads to the grandfather paradox. This is a mathematical problem that the author thinks about. I also want to get a satisfactory mathematical and logical answer, but I haven't found it yet.

However, the results of three-dimensional dimension reduction in the cell space of four-dimensional hyperbodies are not unique, but multiple and even infinite. For example, the rotating gyro is actually one of the three-dimensional dimensionality reduction of the four-dimensional hypersphere, so is the rotating gyro all the properties of the four-dimensional hypersphere? No, this is just one or part of the properties of four-dimensional hypersphere.

This is the difference between dimensionality reduction of a four-dimensional system to a three-dimensional system and dimensionality reduction of a three-dimensional system to two dimensions. However, we usually adopt the same dimensionality reduction method and simply assume that the dimensionality reduction results of the two systems are the same.

In physics, the uncertainty of the four-dimensional space-time method lies in that we have verified the total space-time in a hundred years, only verified some phenomena that the four-dimensional space-time has been reduced to three dimensions locally and individually, but we are unable or afraid to characterize the nature of the total space-time, such as the debate about whether God rolls dice. At the same time, there are still different theoretical and physical hypotheses about the same total space-time, which can also explain the observed phenomena. This also illustrates this mathematical problem. We try to explain the overall mechanism of the four-dimensional system by using three-dimensional phenomena, which are uncertain because of the loss of information in the process of dimensionality reduction.

There are many ways of projection, not just mathematical projection.

For example, the projection method used in oil painting is point perspective; The projection mode of traditional Chinese painting landscape painting is scattered perspective (diffuse perspective). This is also a projection method. Based on the two-dimensional expression of traditional painting, we have to guess what the back of the painting looks like and fill in some three-dimensional information that the painting itself does not provide.

Shadow is also the simplest and most direct projection.

This projection method is equivalent to analyzing the original information of high-dimensional objects through shadows. If you have seen the game of hand shadow (the projection effect of point light source from 3D to 2D), you will know that if the information obtained by this dimension reduction method is not enough (including the mathematical logic and comprehensive establishment of the method itself), it is difficult to understand all the original information, even if there is no necessary other auxiliary information, it is not feasible.

Objects in Euclid's three-dimensional space involve "light and shade"; Non-Euclidean four-dimensional space-time involves real problems, while virtual space-time is still in Euclidean three-dimensional system, which expresses the bending effect of the system based on one more influencing factor besides the straight three-dimensional system. Non-Euclidean multi-dimensional space (super-body cell space larger than or equal to four dimensions) also involves the problem of reality and falsehood, but this cell space is not in Euclidean three-dimensional system, or it cannot be explained compatibly.

In order to solve the problem of direct visibility and direct invisibility of three-dimensional objects, the two-dimensional projection of three-dimensional objects adopts "light and shade" classification. The mathematical projection mode in the above drawing method is represented by dotted lines for geometric structures that cannot be directly seen; Will this solve all the problems?

For the extremely complex hidden structure, in order to express it accurately, mathematics adds a kind of two-dimensional information, that is, sectional view. Only in this way can we avoid the complex structural problems hidden in the process of 3D restoration.

And math problems. For the extremely complex overall structure, mathematics adds decomposition diagram.

This is the two-dimensional diagram we are using now. On the whole, four prescribed projection drawing methods (solid line, dotted line, section and decomposition) are used to solve the three-dimensional restoration of a complex system. And this does not include details, such as accuracy, slope, height, material and other auxiliary information. This method is based on pure Euclidean geometry.

In the past hundred years, we have used non-Euclidean geometry to describe the four-dimensional system in two-dimensional and three-dimensional ways, and the complexity of similar recovery information is being reflected. It is not a theory of relativity that has solved all the problems in the total space-time, but a bunch of problems that cannot be explained by the theory of relativity, leading to different theoretical physics hypotheses. This means that we may still lose information in the process of restoring the four-dimensional space-time system contained in our total space-time.

This involves a kind of space problem. The mathematical drawing method of Euclid geometry involves "light and shade", and darkness is the existence of facts. But the space-time of non-Euclidean method involves two possibilities: one is the influence effect of existence (four-dimensional space-time); One is that it may not exist (four-dimensional cell space). As for using imaginary numbers as real numbers, mathematics does not support it. However, in the usual use, the actual and false inclusions are going on.

In the two-dimensional system, there is a mixed mathematical basis of both real coordinate system and virtual coordinate system. Because I define it on the basis of two-dimensional premise. However, as far as interpretation is concerned, these two systems are not the same. Now there is also confusion in interpretation. I is based on a non-real number field, which can indicate non-existence, or potential influence, or invisibility. And real numbers can indicate existence, or direct influence, or visibility. The original explanation is different.

In the three dimensions, the combination of reality and reality will be different. For example, two real axes and an imaginary axis, or one real axis and two imaginary axes. There is not much specific research in this field, and it is usually a mixture of good and evil. Especially in the physical hypothesis, in order to unify mathematics and physics, mixing is more common. Ignoring mathematics to explain different problems.

In the fourth dimension, the combination of reality and reality is more complicated. Now the four-dimensional space-time of physics is three real axes and an imaginary axis. Four-dimensional hyperbody is a mathematical expression of the meaning of assuming that all four are real numbers.

To multi-dimensional, this mathematical differentiation is even more serious. The multi-dimensional hyperbody of hyperbody geometry is based on the expression of multiple (more than 3) real number axes; The five-dimensional space-time in physics is based on the assumption that the four-dimensional space-time is straight (it has been proved that the four-dimensional space-time is not straight now) and that a non-Euclidean system with a circle is also different from the four-dimensional space-time; Euclidean geometry has no concept of four-dimensional space; The meaningful research on the mathematical direction of imaginary numbers is quaternion and octahedron. That is, a system based on a real axis and other imaginary axes. These are incompatible mathematical systems.

The multidimensional nature of mathematical culture obliterates the mathematical fact that the three-dimensional projection of multi-dimensional hyperbodies is not unique. Assuming that it is unique and confused with four-dimensional space-time, we can travel through the so-called mathematical space at will.

On the premise of unclear understanding of mathematics, mathematical culture tends to be mixed with reality; Theoretical physics hypothesis is easy to confuse the explanation of reality and excess; Mathematical culture, based on the definition of non-European premise, develops independently, and many contents are incompatible.

Real number space, described by analytic geometry method of real number; Virtual space is described by the analytic geometry of imaginary numbers.

Then the non-Euclidean geometry method based on imaginary analytic geometry method faces two problems:

Based on the premise definition of non-Euclidean four-dimensional space-time, if there are four or more influencing factors, the bending effect of flat plate system will be caused. Four-dimensional space-time is the theory of relativity, five-dimensional space-time assumes that four-dimensional space-time is straight, and five-dimensional space-time adds a circle. A theoretical research direction of astronomy is thinking along this line of thought. This was proposed by Einstein's teacher after the special theory of relativity. This is different from the premise definition of non-Euclidean hyperbody dimension and cannot be expressed compatibly.

Although we already know that four-dimensional space-time is curved, and confirmed this effect. But on a large scale, assuming that this curvature is very small and can be ignored, it will produce the mathematical fitting idea of physics, a five-dimensional space-time

This is also different from the author's denial of the five-dimensional space of mathematical metaphysics based on different premise definitions

Other four-dimensional mathematical ideas, five-dimensional and multi-dimensional ideas are faced with mathematical and physical verification problems

Based on the premise definition of non-Euclidean hyperbody, if there are four or more influencing factors, it cannot be expressed intuitively in three-dimensional static state.

Non-European thinking and different premise definitions usually lead to different logical results. We must pay attention to the premise definition of every non-European way of thinking. If you are confused, you will usually fall into mathematical metaphysics.