How did human perception of space develop, and why do we need measurements

The theory of relativity states that we live in four dimensions. String theory - which is ten. What are “dimensions” and how do they affect reality?

When I write texts at my desk, I can stretch my hand up to turn on the lamp, or down to open the drawer of the table and take out a pen. Stretching my hand forward, I touch a small and strange-looking figurine that my sister gave me for luck. Reaching back, I can clap a black cat stalking behind my back. On the right are the notes made during the research for the article, on the left - a bunch of things that need to be done (bills and correspondence). Up, down, forward, back, right, left - I control myself in my personal space of three-dimensional space. The invisible axes of this world impose on me the rectangular structure of my office, defined, like most western architecture, by three right angles drawn together.

Our architecture, education and dictionaries tell us about the three-dimensionality of space. The Oxford English Dictionary defines space in such a way: “a continuous area or expanse, free, accessible or unoccupied. Dimensions of height, depth and width, within which all things exist and move. " [ Ozhegov's dictionary says in a similar way: “Length, a place not limited by visible limits. The gap between what's Mr.., The place where something Mr. holds. ”/ approx. trans. ]. In the XVIII century, Immanuel Kant argued that three-dimensional Euclidean space is an a priori necessity, and we, saturated with computer-generated images and video games, constantly remind us of this representation in the form of an axiomatic rectangular coordinate system. From the point of view of the 21st century, this seems almost self-evident.

Yet the idea of ​​life in space, described by some mathematical structure, is a radical innovation of Western culture, which made it necessary to refute ancient beliefs about the nature of reality. Although the emergence of modern science is often described as a transition to a mechanized description of nature, probably its more important aspect — and unequivocally longer — was the transition to the concept of space as a geometric structure.

In the last century, the task of describing the geometry of space became the main project of theoretical physics, in which experts, starting with Albert Einstein, tried to describe all the fundamental interactions of nature as by-products of the form of space itself. Although at the local level we were taught to think of space as three-dimensional, the general theory of relativity describes a four-dimensional Universe, and string theory speaks of ten dimensions - or of 11, if we take its extended version as a basis, M-theory . There are variants of this theory with 26 dimensions, and recently mathematicians enthusiastically adopted a version describing 24 dimensions. But what are these "measurements"? And what does the presence of ten dimensions in space mean?

To arrive at a modern mathematical understanding of space, you first need to think of it as a kind of arena that matter can occupy. At the very least, space must be imagined as something extended. Such an idea, albeit obvious to us, would have seemed a heretical to Aristotle , whose concepts of representing the physical world prevailed in Western thinking in late antiquity and in the Middle Ages.

Strictly speaking, Aristotelian physics did not include the theory of space, but only the concept of space. Consider a cup of tea, standing on the table. For Aristotle, the cup was surrounded by air, which in itself represented a certain substance. In his picture of the world there was no such thing as empty space — there were only boundaries between substances — a cup and air. Or a table. For Aristotle, the space, if you want to call it that, was only an infinitely thin line between the cup and what surrounds it. The baz of the length of the space was not something in which there could be something else.

A century before Aristotle, Leucippus and Democritus proposed a theory of reality with a mode of observation strongly tied to space — an atomistic vision in which the material world consists of tiny particles, or atoms, moving in a vacuum. But Aristotle rejected atomism, stating that the very concept of emptiness is logically contradictory. He said that the definition of "nothing" can not exist. The draft refutation of Aristotle’s objections against emptiness and the concept of extended space will take centuries. Only when Galileo and Descartes made extended space one of the cornerstones of modern physics in the 17th century did this innovative approach acquire the right to exist. For both thinkers, as the American philosopher Edwin Burt said in 1924, “physical space was assumed to be identical with geometrical,” that is, three-dimensional Euclidean geometry, which is now taking place in schools.

Long before the physicists accepted the Euclidean point of view, the artists discovered geometrical concepts of space, and it is to them that we owe an outstanding leap in the development of our conceptual platform. In the late Middle Ages, under the influence of new ideas based on the works of Plato and Pythagoras, the intellectual rivals of Aristotle, in Europe began to spread the views according to which God created this world in accordance with the laws of Euclidean geometry. Therefore, if the artist wanted to capture his true appearance, he needed to emulate the work of the Creator in his representation. From the 14th to the 16th century, artists such as Giotto di Bondone , Paolo Uccello and Piero della Francesca developed techniques for using what was later called perspective — a style originally called the “geometric image.” Consciously studying the geometric principles, these artists gradually learned how to create images of objects of three-dimensional space. In the process, they reprogrammed European minds to see the space as Euclidean.

The historian Samuel Edgerton describes in detail this remarkable and smooth transition to modern science in the book Heritage of Giotto's Geometry (1991), noting how the rejection of Aristotelian thinking about space was due in part to a long process that is a by-product people's observation of pictures made in perspective and their intuitive feeling that they “peek” into three-dimensional worlds on the other side of the wall. The unusual thing about this is that while the philosophers and predecessors of scientists were carefully trying to argue with the Aristotelian perception of space, the artists cut their way through this intellectual territory, appealing to sensations. Absolutely literally a perspective image was a kind of virtual reality, which, in the manner of modern VR games, was aimed at creating in the viewer the illusion of moving to geometrically consistent and psychologically convincing other worlds.

The illusory Euclidean space of the perspective image, gradually being laid back in the minds of the Europeans, was accepted by Descartes and Galileo as the space of the real world. It is worth noting that Galileo himself had the experience of dealing with the prospect. His ability to image depth became critical in his breakthrough images of the Moon, which showed mountains and valleys, and which indicated that the Moon was made of the same hard material as the Earth.

Having taken the space of perspective images, Galileo was able to show how objects such as cannonballs move according to the laws of mathematics. The space itself was an abstraction: unremarkable, inert, intangible emptiness, whose only known property was the Euclidean form. By the end of the 17th century, Isaac Newton expanded the vision of Galileo to embrace the entire Universe as a whole, and now this idea has turned into a potentially infinite three-dimensional vacuum - an extended voidless empty space, which lasts forever in all directions. The structure of reality thus evolved from a philosophical and theological issue into a geometrical proposition.

While artists used mathematical tools to develop new ways of creating images, Descartes, at the dawn of the scientific revolution, discovered a way to create images of mathematical relationships. In the process, he formalized the concept of measurement and introduced into our consciousness not only a new way to look at the world, but also a new method of doing science.

Today, almost everyone recognizes the fruits of Descartes' genius in the form of a rectangular coordinate system - the lattice on the plane marked by the x and y axes.


By definition, the plane of Cartesian coordinates is two-dimensional, since we need two coordinates to determine any point on it. Descartes discovered that on such a platform it is possible to connect geometric forms and equations. In this way, a circle of radius 1 can be described as an equation x ² + y ² = 1


A huge set of forms that we can draw on this plane can be described by equations - and such “analytical geometry” will soon become the basis for the mathematical analysis developed by Newton and Leibniz for analyzing the motion of physicists. One way to understand matan is to study curves. He, for example, allows us to formally determine the place where the curve has the greatest slope, or where it reaches a local maximum or minimum. When applied to the study of motion, matan gives us a way to analyze and predict where, for example, an object thrown into the air will reach its maximum height, or where a ball rolling down a curved slope will reach a certain speed. Since the invention of matan has become a vital tool for almost all areas of science.

Using the example of the last diagram, it is easy to see how a third dimension can be added. Using the x, y, and z axes, we can describe the surface of a sphere — for example, the surface of a beach sword. The equation for a sphere of radius 1 becomes x ² + y ² + z ² = 1


With the help of three axes, you can describe shapes in three-dimensional space. Again, each point is uniquely defined by three coordinates - it is a necessary condition for the triality that makes space three-dimensional.

But why dwell on this? What if you add the fourth dimension? Let's call it "p". Now I can write the equation for what I call a sphere located in four-dimensional space: x ² + y ² + z ² + p ² = 1. I can not draw it, but from a mathematical point of view, you can add an additional dimension. “Can” means that there is nothing logically contradictory in this action.

And I can continue to do this further by adding extra dimensions. The sphere in the five-dimensional space with the axes (x, y, z, p, q) I can determine by the equation: x ² + y ² + z ² + p ² + q ² = 1. And in six-dimensional: x ² + y ² + z ² + p ² + q ² + r ² = 1, and so on.

Maybe I can’t portray spheres from higher dimensions, but I can describe them symbolically, and one of the ways to understand the history of mathematics is a gradual awareness of what reasonable things we can go beyond. This is what Charles Lutwidge Dodgson, aka Lewis Carroll, had in mind in the novel Through the Mirror, and what Alice found there (1871), when the White Queen claimed that she was able to "believe in six impossible things before breakfast."

Mathematically, I can describe a sphere in any number of dimensions I want. I just need to add new coordinate axes, what mathematicians call "degrees of freedom." They are usually denoted as x ₁ , x ₂ , x ₃ , x ₄ , x ₅ , x ₆ , and so on. Just as any point on the Cartesian plane can be described by two coordinates (x, y), any point of the 17-dimensional space can be described by a set of 17 coordinates (x ₁ , x ₂ , x ₃ , x ₄ , x ₅ , x ₆ , ... x ₁₅ , x ₁₆ , x ₁₇ ). Surfaces of the type of the sphere described above in multidimensional spaces are called manifolds .

From a mathematical point of view, “measurement” is just another coordinate axis, another degree of freedom, becoming a symbolic concept, not necessarily related to the material world. In the 1860s, the pioneer in the field of logic, Augustus de Morgan, whose work influenced Lewis Carroll, summed up this increasingly abstract field, noting that mathematics is purely a “science of symbols” and as such does not have to be associated with anything except myself. Mathematics, in a sense, is logic that moves freely in the fields of imagination.

Unlike mathematicians, who freely play on the fields of ideas, physicists are attached to nature, and, at least, in principle, depend on material things. But all these ideas lead us to a liberating opportunity - because if mathematics allows for more than three dimensions, and we think that mathematics turns out to be useful for describing the world, how do we know that physical space is limited to three dimensions? Although Galileo, Newton, and Kant took length, width, and height as axioms, could there not be more dimensions in our world?

Again, the idea of ​​the Universe with more than three dimensions penetrated the consciousness of society through the artistic environment, this time through literary reasoning, the most famous of which is the work of the mathematician Edwin Abbott Abbott “ Flatland ” (1884). This charming social satire tells the story of a modest Square living on a plane, to which one day a three-dimensional being Lord Sphere comes to us, leading him to the magnificent world of three-dimensional bodies. In this paradise of volumes, Square observes his three-dimensional version, Cube, and begins to dream of moving into the fourth, fifth, and sixth dimensions. Why not a hypercube? Or not a hyper-hypercube, he thinks?

Unfortunately, in Flatland, Squares are counted as lunatics and locked up in an insane asylum. One of the morality of history, in contrast to its more sugary adaptations and adaptations, is the danger of ignoring social foundations. A square, talking about other dimensions of space, tells about other changes of being - it becomes a mathematical oddity.

At the end of the 19th and the beginning of the 20th centuries, the mass of authors (HG Wells, mathematician and author of NF-novels Charles Hinton , who invented the word tesseract to designate a four-dimensional cube), artists (Salvador Dali) and mystics ( Peter Demianovich Uspensky [ Russian occultist, philosopher, the theosophist, tarologist, journalist and writer, mathematician by education / approx. transl. ] studied ideas related to the fourth dimension and what a meeting with him could become for a person.

Then, in 1905, the then unknown physicist Albert Einstein published a paper describing the real world as four-dimensional. In his "special theory of relativity," time was added to the three classical dimensions of space. In the mathematical formalism of relativity, all four dimensions are connected together - this is how the term “space-time” entered our lexicon. Such a union was not arbitrary. Einstein discovered that using this approach, one can create a powerful mathematical apparatus that surpasses Newton's physics and allows him to predict the behavior of electrically charged particles. Electromagnetism can be fully and accurately described only in the four-dimensional model of the world.

Relativity has become something much more than just another literary game, especially when Einstein expanded it from “special” to “general”. Multidimensional space has acquired a deep physical meaning.

In the picture of the world of Newton, matter moves through space in time under the influence of natural forces, in particular, gravity. Space, time, matter and forces are different categories of reality. With the SRT, Einstein demonstrated the unification of space and time, reducing the number of fundamental physical categories from four to three: space-time, matter and forces. GR makes the next step, interweaving gravity into the structure of space-time itself. From the four-dimensional point of view, gravity is just an artifact of the form of space.

To grasp this remarkable situation, let’s imagine its two-dimensional analogue. Imagine a trampoline drawn on the surface of a Cartesian plane. Now let's place a bowling ball on the grid. Around it, the surface will be stretched and distorted so that some points move away from each other more strongly. We distorted the inner measure of distance in space, made it uneven. GTR says that it is precisely such a distortion that heavy objects, such as the Sun, are subjected to space-time, and deviation from Cartesian perfection of space leads to the appearance of a phenomenon that we perceive as gravity.



In Newton's physics, gravity comes from nowhere, and for Einstein, it naturally arises from the internal geometry of a four-dimensional manifold. Where diversity is most stretched, or moving away from Cartesian regularity, gravity is felt more strongly. This is sometimes referred to as “rubber film physics.” There are huge cosmic forces that hold the planets in orbits around stars, and the stars in orbits within the galaxies are nothing more than a side effect of the distorted space. Gravity is literally geometry in action.

If the transition to four-dimensional space helps to explain gravity, will there be any scientific advantage in five-dimensional space? “Why not try it?” Asked Theodor Franz Edward Kaluza , a young Polish mathematician, in 1919 , thinking that if Einstein included gravity in spacetime, then perhaps an additional dimension can treat electromagnetism in a similar way as an artifact space-time geometry. Therefore, Kaluza added an extra dimension to Einstein's equations, and, to his delight, found that in five dimensions both of these forces are perfectly artifacts of a geometric model.

Mathematics magically converges, but in this case the problem was that the extra dimension did not correlate with any particular physical property. In GR, the fourth dimension was time; in Kaluza’s theory, it was not something that can be seen, felt or indicated. It simply was in mathematics. Even Einstein was disappointed in such an ephemeral innovation. What is it?He asked; where is it?

In 1926, the Swedish physicist Oscar Kleingave an answer to this question, very similar to the passage from the work about Wonderland. He offered to imagine an ant living on a very long and thin section of the hose. The hose can run forward and backward, without even noticing a tiny circular change under your feet. This measurement can be seen only by ant physicists using powerful ant microscopes. According to Klein, each point of our four-dimensional space-time has a small additional circle in a space of this kind, which is too small for us to see. Since it is many times smaller than an atom, it is not surprising that we have not yet found it. Only physicists with very powerful particle accelerators can hope to get to such a tiny scale.

When physicists moved away from the initial shock, Klein's idea won them over, and during the 1940s, this theory was developed in great mathematical detail and transferred to a quantum context. Unfortunately, the infinitely small scale of the new dimension does not show how its existence can be confirmed experimentally. Klein calculated that the diameter of the tiny circle is about 10 ^-30 cm. For comparison, the diameter of the hydrogen atom is 10 ^-8 cm, so we are talking about something 20 orders of magnitude smaller than the smallest of atoms. Even today, we are not at all close to being able to see something on such a miniature scale. So this idea went out of fashion.

Kalutsu was so easy not to scare. He believed in his fifth dimension and in the power of mathematical theory, so he decided to conduct his own experiment. He chose a topic like swimming. He could not swim, so he read everything he found, according to the theory of swimming, and when he decided that he fully mastered the principles of behavior on the water, went with the family to the sea, rushed into the waves, and suddenly swam. From his point of view, the swimming experiment confirmed the truthfulness of his theory, and although he did not live to see the triumph of his favorite fifth dimension, in the 1960s, string theory specialists revived the idea of ​​a space with higher dimensions.

By the 1960s, physicists had discovered two additional forces of nature operating on a subatomic scale. They have been called weak nuclear interactions and strong nuclear interactions, and they are responsible for some types of radioactivity and for the containment of quarks, which form the protons and neutrons that make up the atomic nuclei. At the end of the 1960s, physicists began to study the new topic of string theory (stating that particles look like tiny rubber bands vibrating in space), and the ideas of Kaluza and Klein came to the surface again. Theorists gradually began to think whether it would be possible to describe two subatomic forces in terms of the space-time geometry.

It turns out that in order to cover both these forces, it is necessary to add five more dimensions to our mathematical description. There is no particular reason for five of them; and again, none of these extra dimensions are directly related to our sensations. They are only in mathematics. And this leads us to 10 dimensions of string theory. And here are four large-scale measurements of space-time (described by GR), plus six additional “compact” measurements (one for electromagnetism and five for nuclear forces), curled up into a damn complicated, shriveled geometric structure.

Physicists and mathematicians are making great efforts to understand all the possible forms that this miniature space is capable of accepting, and which, if any of these many alternatives, are realized in the real world. Technically, these forms are known as Calabi-Yau varieties , and they can exist in any number of higher dimensions. These exotic and complex creatures, these extraordinary forms, constitute abstract systematics in multidimensional space; their two-dimensional section (the best that we can do to visualize their appearance) resembles the crystal structures of viruses; they seem almost alive .

There are many versions of string theory equations describing ten-dimensional space, but in the 1990s, mathematician Edward Witten from the Institute of Advanced Studies at Princeton (old Einstein's lair) showed that everything can be simplified a little if you go to an 11-dimensional perspective. He called his new theory "M-theory", and mysteriously refused to explain what the letter "M" means. It is usually said that it means “membrane”, but besides this, there were also proposals such as “matrix”, “master”, “mystical” and “monstrous”.

So far, we have no evidence of these extra dimensions - we are still in a state of floating physicists who dream of inaccessible miniature landscapes - but string theory has had a powerful influence on mathematics itself. A recently developed version of this theory, with 24 dimensions, showed an unexpected relationship between several basic branches of mathematics, which means that even if string theory is not useful in physics, it will become a useful source of purely theoretical ideas. In mathematics, the 24-dimensional space is special - magic things happen there, for example, it is possible to pack the spheres in a particularly elegant way - although it is unlikely that there are 24 dimensions in the real world. Regarding the world in which we live and that we love, most string theory experts believe that 10 or 11 measurements will be enough.

Attention is worth another event of string theory. In 1999, Lisa Randall (the first woman to get her post at Harvard in the field of theoretical physics) and Raman Sandrum (American of Indian origin, an expert in theoretical particle physics) suggestedthat an additional dimension can exist on a cosmological scale, on scales described by the theory of relativity. According to their theory of "branes" (bran is a contraction from the membrane) - what we call our Universe can be located in a much larger five-dimensional space, in something like a superuniverse. In this superspace, our Universe can be one of a number of universes existing together, each of which represents a four-dimensional bubble in the wider arena of five-dimensional space.

It is difficult to say whether we can ever confirm the theory of Randall and Sandrum. However, between this idea and the dawn of modern astronomy, there are already some analogies. 500 years ago, the Europeans considered it impossible to imagine other physical "worlds" other than our own, but now we know that the Universe is filled with billions of other planets moving in orbits around billions of other stars. Who knows, maybe someday our descendants will be able to find evidence for the existence of billions of other universes, each of which has its own unique equations for spacetime.

The project of understanding the geometric structure of space is one of the characteristic achievements of science, but it may happen that physicists have reached the end of this path. It turns out that Aristotle in some sense was right - the idea of ​​extended space really has logical problems. Despite all the extraordinary successes of the theory of relativity, we know that its description of space cannot be final, because it refuses at the quantum level. Over the past half century, physicists have unsuccessfully tried to combine their understanding of space on a cosmological scale with what they are observing on a quantum scale, and it seems more and more that such a synthesis may require radically new physics.

After the development of UTO, Einstein spent most of his life trying to “express all the laws of nature from the dynamics of space and time, relegating physics to pure geometry,” as Robbert Dijkgraaf, director of the Institute of Advanced Studies at Princeton, said recently. "For Einstein, space-time was the natural foundation of an infinite hierarchy of scientific objects." Like Newton, the picture of the world of Einstein puts space at the head of existence, makes it the scene where everything happens. But on a tiny scale, dominated by quantum properties, the laws of physics show that this space to which we are accustomed may not be.

Some theoretical physicists are beginning to express the idea that space can be some kind of emerging phenomenon, resulting from something more fundamental, as temperature arises on a macroscopic scale as a result of the movement of molecules. As Dijkgraaf says: “The current point of view considers space-time not as a point of reference, but as a final finish line, a natural structure arising from the complexity of quantum information.”

The leading proponent of new ways of representing space is cosmologist Sean Carroll from Caltech, who saidrecently that classical space is not a “fundamental part of the architecture of reality”, and proving that we incorrectly assign such a special status to its four, or 10, or 11 dimensions. If Dijkgraaf gives an analogy with temperature, then Carroll suggests that we consider "humidity", a phenomenon that manifests itself because a multitude of water molecules come together. The individual water molecules are not wet, and the moisture property appears only when you collect many of them in one place. In the same way, he says, space comes from more basic things at the quantum level.

Carroll writes that, from a quantum point of view, the Universe "appears in the mathematical world with a number of dimensions of the order of 10 ^{10 100}"- a dozen with googol zeros, or 10,000 and another trillion trillion trillion trillion trillion trillion trillion trillion zeros. It is difficult to imagine such a huge amount, in comparison with which the number of particles in the Universe is absolutely insignificant. And yet, each of them is a separate dimension in mathematical space, described by quantum equations; each is a new “degree of freedom” available to the Universe.

Even Descartes would have been amazed at where his reasoning had led us, and what astonishing complexity was hidden in such a simple word as “measurement.”