For example, the surface of a sphere. It has a finite area, but as you move along it you will never reach the edge.
The question of whether the Universe is finite or infinite is still a mystery of our time, and there are mathematical models, taking into account both of these possibilities. Are there any infinite objects in the Universe? This question also arouses genuine interest among scientists.
In April of this year, philosophers, cosmologists and physicists gathered in Cambridge University as part of a conference on the philosophy of cosmology to discuss this topic.
Infinity that doesn't exist
People have been studying infinity and its relationship to reality for a long time.
The study of infinity began in the time of Aristotle. He clearly distinguished two types of infinity. One he named potential infinity, which was found in his descriptions of the world. This includes lists that have no end. These are, for example, ordinary numbers: one, two, three, four, five and so on to infinity, which cannot be reached. There are many such infinities in cosmology. Thus, the Universe is probably infinite in size or infinite in age, or may continue to exist indefinitely. These are all potential infinities that we cannot prove, we simply say that certain things are infinite
Most people accept that potential infinities exist, but no one knows for sure whether this is actually the case.
When you look at the Universe, your view is strictly limited because the Universe exists for a finite period of time, approximately 14 billion years. Light moves at a constant speed, postulated back in 1905 by Albert Einstein, so you can see no more than 14 billion light years away. You can't see infinity. It is very similar to how when you stand on a tower and look into the distance, you are able to see everything up to the horizon, but cannot look beyond it. But here there is an option to board a plane and fly to another place on the planet. In the case of the Universe, the scale is such that we cannot change the point of view, we are stuck in one place and can only see the Universe from that point and to a finite distance.
But even this 14 billion year limit that Ellis refers to more of a theory, how real facts. We know that the Universe is currently expanding, and if we move backwards in this case, we will eventually arrive at a point in time, the Big Bang, which we call the beginning of our Universe. However, generally accepted physical theories, Einstein's general theory of relativity and quantum physics, do not take this point into account. There is currently no theory to describe this case, other than a host of "supposed" theories.
cosmologist, University of Cape Town Some of these theories say that there never was a beginning, others say that there was. We try to make more or less reasonable assumptions. But we cannot conduct any experiments to prove this or that assumption, since there is no sufficient quantity energy.
Moment Big Bang is out of reach modern theories, however, there is a generally accepted model that explains the first moments after it. For example, cosmic inflation. Anthony Aguirre of the University of California, Santa Cruz, believes it could tell us something about the expansion of the Universe.
Inflation is the concept that, early on, the universe expanded by geometric progression, doubling in size hundreds of times over a short period of time. This theory leads to a lot of guesses, many of which turned out to be correct, and some of which can be tested in future experiments. This makes us believe in inflation, but it also has some very interesting side effects.
One of these side effects suggests that inflation may have continued at varying rates in various areas Universe. In some region, the rapid doubling in size will stop after some time, eventually forming a observable Universe like ours. In other regions, due to spatial changes, inflation can last forever.
Physicist, University of California, Santa Cruz We have infinite space-time, and not because we decided that space-time is infinite, but because we took into account the process that naturally leads to infinite space-time.
The theory also suggests that the expansion of space and time depends on point of view. According to Albert Einstein's general theory of relativity, time and space are inextricably linked, hence the term spacetime. If you want to mention space or time separately, you need to separate space-time mathematically.
Physicist, University of California, Santa Cruz It turns out that the answer to such questions as "is space finite or infinite?" may depend on how you define space and time separately. There is space-time, Einstein teaches us this. We can divide it into space and time in various ways. They are all valid and give the same results in all experiments, but they contain different meaning, and some values are more convenient than others to achieve certain goals.
If you have infinite spacetime, in which case you can break it up so that the universe can be finite and expanding. It can expand indefinitely and become infinitely large, but finite. Or this same space-time can be divided in such a way that space is infinite, resulting in an infinite, expanding Universe.
IN inflationary universe in places where inflation stops, its natural division occurs, in this case the Universe is close to homogeneity. A Universe arises, which is spatially infinite.
Inflation gives rise to homogeneous infinite universe, which can turn into something similar to ours. It's great that we can form assumptions about such a rich, multifaceted and interesting reality in which the Universe is infinite.
Actual infinity
The question of whether the universe is infinite concerns one type of Aristotelian infinity, a potential infinity that we can imagine but can never see. But there is another type of infinity according to Aristotle, actual infinity.
In this case, some object that we can measure is infinite.
Such virtual infinity could arise in a black hole, which is formed when massive object, for example, a star begins to collapse. Theoretically, this leads to an infinite mass density at one point. But do such infinities exist in the Universe?
“A black hole is not necessarily a solid object, it is a kind of surface in the universe,” explains Barrow, “Once you go inside, you will never come back, because for this you need to move faster speed light, otherwise gravity will be stronger. In a black hole, it is as if a giant cloud is collapsing, becoming denser and denser. Ultimately, a surface forms around it, which we call the horizon. If you are on the horizon of a very large black hole that is, say, a billion times larger bigger than the sun, then you will feel as if you are in a large room, nothing strange. But if you try to get out of there, you won't succeed. In the black hole itself, everything begins to move towards the center with unlimited density. However, this is not visible from the outside. These effects are isolated and cannot affect the outer Universe."
"Many years ago, Roger Penrose made a proposal known as cosmic censorship. It states that if singularities or infinities were to form in the Universe, and nothing could stop them, then they would always be within the horizons. The so-called" There cannot be naked" singularities; thus, there cannot be infinities that affect us on the outside. In in some cases the theory has been proven, but it is far from a general proof. This is a very difficult mathematical problem."
Another type of infinity that can exist is called infinitely small or infinitely divisible. With super-precise rulers and pencils, could we divide the segment into pieces that become smaller each time?
Ellis thinks the idea is ridiculous. "If you hold your fingers 10 cm apart and believe that there is a real line of dots between them, as in mathematics, then between your fingers there is an incalculable infinity of dots. This is absolutely unreasonable. I believe it is purely mathematical idea, which does not correspond to physics.
Richard Feynman once said that the only thing he would want to leave to future generations, if he had to leave one thing, would be the statement “Matter is made of atoms.” I think we have good reason to believe that a similar statement can be applied to spacetime, asserting its discrete nature. There is a very large amount between your fingers physical points, but it is finite and countable."
If space-time are indivisible parts, then there must be the smallest distance scale, the shortest length. Physical theories do support this idea by suggesting that there is nothing shorter than the so-called Planck length. It is approximately 10 -35 m (this is a number with 34 zeros after the decimal point). Modern methods do not allow us to get close to this number; even in theory, with very powerful instruments, we would never be able to measure anything less than the Planck length.
Space hot dog
Ellis made an important distinction. On the one hand, there is mathematical concept infinity (the line is infinitely divisible), on the other hand physical concept, which concerns real quantities and phenomena that may or may not exist in nature. But there is also a third type of infinity, probably the most familiar to us.
cosmologist, University of Cambridge We can distinguish mathematical infinities, physical infinities and transcendental infinities, which theologians or philosophers spoke about. Almost everyone on the street seems to be familiar with this transcendental infinity. This is a kind of cosmic everything. Like a hot dog in a diner - one with everything.
In many religions, absolutely everything lies in God or some cosmic force. This is something different than what physicists and mathematicians deal with. Consider the history of ideas in mathematics and physics, anyone can make one of the following statements: “I believe or do not believe in mathematical infinities,” “I believe or do not believe in physical infinities,” or “I believe or do not believe in any other type of infinity.” transcendental infinity."
You can choose any of the proposed points of view. And opinions really differ. Barrow and Aguirre work with mathematical infinities, but do not neglect physical infinities.
"I think it's natural to create theories that contain infinity," Aguirre says. “Yes, we are finite beings and can only comprehend a finite part of the Universe, but I see no reason to limit the entire Universe in principle.”
Ellis, on the other hand, does not believe that physical infinities exist and points out potential problems with using infinities in mathematical arguments related to physics. He refers to mathematician David Hilbert's famous thought experiment - Hilbert's Hotel, in which infinite number rooms and residence infinite number guests, so every room is occupied. When a new guest arrives, is it possible to accommodate him? Of course, this requires asking each guest to move to the next room and placing the new visitor in the first. This is possible because the n+1st room exists. What if an infinite number of guests come again? It’s also simple - just ask each guest from room n to move to room n*2. It turns out that the hotel is full and not full at the same time.
Because of paradoxes like these, Ellis believes we should be very careful when using infinities in a physical context.
cosmologist, University of Cape Town I'll clarify. Often when people talk about infinity, they really mean something very large quantities. Infinity in in this case is simply used as a code word. In this case, I think it is worth avoiding the word “infinity” and talking specifically about a large number. In other cases, people use infinity in its deep, paradoxical sense, like Hilbert's Hotel, for example. In my opinion, if an argument depends on such a paradoxical argument, then it is false and should be replaced by another.
Thus, scientists have not come to a consensus on whether infinities exist in the real world or not. Due to the lack of concrete scientific answers, it makes sense to turn to philosophers.
Physicist, University of California, Santa Cruz I think it is worth combining the efforts of physicists and philosophers. In this case, physicists will reproach philosophers for not knowing science and not knowing what they are talking about. Philosophers look at physics from a different point of view, as an intellectual fund, compared to practical scientists. I think this kind of exchange of thinking would be incredibly valuable.
Dissimilar infinities
Infinity is one of those mathematical images that is difficult to imagine not only for non-specialists, but also for scientists. One famous mathematician, who taught geometry at the physics department of Moscow University, confidentially admitted to students that when he tries to imagine infinity, he feels his mind begin to cloud.
Nevertheless, mathematicians, physicists, and astrophysicists in their research have to deal with infinities, with infinitely large quantities and operate with them. Moreover, it turns out that infinities can be different, and they can even be compared with each other.
The simplest, most “elementary” infinity and at the same time the “smallest” is the infinity of numbers in the natural series. It can be obtained by adding one unit after another to one over and over again.
Since such an operation is not limited by anything and can be repeated for as long as desired, the result is an infinite set of integers - a “countable” set, as mathematicians call it. This infinity, convenient in many respects, plays the role of a kind of “measuring ruler”, a kind of standard for measuring other infinities. To do this, you need to try to simply number their elements. And see what comes of it...
Just? Why not? We know how to count from one and so on. But here a completely unexpected surprise awaits us. One of those that we encounter almost at every step when we deal with infinities. For example, let us “apply” our standard to the infinite set of all even numbers. Two is the least even number, let's number one, four - two, six - three, and so on, and so on... And we will be surprised to discover that not only are there enough numbers to designate all even numbers - this was to be expected - but there are also free numbers left .
It turns out that both infinities - countable and the infinity of all even numbers - are the same? How so? After all, of every two consecutive numbers in the natural series, only one is even. This means that there should be half as many such numbers as all integers! In other words, the set of all even numbers is only part of the set of all integers. And the corresponding infinities are the same, have, as mathematicians say, the same power.
But that doesn’t happen, it can’t be! The set of any object cannot be equal to its own part! Yes, indeed, it cannot, as long as we are dealing with finite formations. But infinities have their own laws - bizarre, of course, from an ordinary point of view - but nevertheless quite strict. By the way, that infinite sets can be equal to their own subsets, Galileo noted... Much to his surprise!
However, any discovery, as we already know, inevitably entails new questions. The one we are talking about is no exception. For example, the following question arises: are there infinite sets more “powerful” than countable ones? Here is a straight line segment. How many dots can fit on it? It is clear that there are countless of them. But how much exactly?
Let us once again resort to the help of our standard - a countable set. And in the end we will find that this time there are too few numbers in the natural series to number all the points of the segment we have chosen. In mathematics, a strict theorem is proven in this regard: no matter how many points of a segment we number, there will always be points for which there are not enough numbers in the natural series. Thus, we have discovered an infinity of a higher order than a countable set - an infinity called continuum. But the continuum is not the limit. In principle, one can construct infinities of arbitrarily high rank.
Let's return to the question of geometric properties ah Universe. You may have noticed that when discussing this problem, either the possible infinity of world space or its unlimitedness is mentioned. In the “ordinary” world, for which Euclidean geometry is valid, the same geometry that we study in school, these concepts are essentially equivalent, meaning the same thing. Although there are still some differences. Strictly speaking, infinity is a quantitative, “metric” property: infinity of length, area, volume. What about unlimited?..
“What do we want to express when we say that our space is infinite? – wrote Einstein, who had the fortunate ability to express the most abstract ideas with the help of visual images. - Nothing other than the fact that we can apply one thing to another equal bodies, say, cubes in any number, and at the same time we will never fill the space. This kind of construction will never end. There will always be room to add one more cube..."
This is what infinite space is. As for unlimitedness, this property is structural, as mathematicians say, topological. This circumstance was especially emphasized at one time outstanding mathematician Bernhard Riemann.
"When considering spatial constructions in the direction of the infinitely large,” he noted, “one should distinguish between the properties of unlimitedness and infinity: the first of them is the property of extension, the second is the metric property.”
In Euclidean space, any line that can be extended indefinitely is infinite. But we live in a curved world... In such a world, infinity and unlimitedness differ in an even more significant way. To the point - another unexpected paradox - that unlimited space can be both infinite, that is, without a boundary, an “edge,” or finite!
To somewhat soften this latest blow to common sense, let’s use an analogy. Analogies in science are not strict evidence, but they allow us to better understand the essence of certain complex phenomena.
Imagine an ordinary ball of finite radius. A spherical surface is a two-dimensional formation curved in three-dimensional space. Imagine some fantastic flat creature living on this surface and not even suspecting that there is some kind of third dimension. Traveling through its curved world in any direction, this creature will never come across any border. And in this sense, the surface of the ball is unlimited space. But since the radius of our ball is finite, its surface area is also finite. Thus, the unlimited and at the same time finite world appeared before us in all its reality. It turned out to be possible what at first glance seemed absolutely impossible.
The next step will require more from us greater strength imagination. We will talk about a three-dimensional ball that is located in four-dimensional space... Unfortunately, it’s hard to visualize similar situation It is no less difficult for us, beings of the three-dimensional world, than for an imaginary inhabitant of a spherical surface to imagine a two-dimensional sphere curved in three-dimensional space.
But in the theory of relativity our world looks exactly like this: it is curved in four-dimensional space, where, however, the role fourth dimension time plays out. According to Einstein, we live in four-dimensional “space-time.” At the same time great physicist believed that our curved world has a finite volume, it is, as it were, closed in on itself.
The history of studying the geometric properties of the Universe has taken another sharp turn. The classical Newtonian ideas of infinite and boundless space had to be abandoned. They played their role, but the world turned out to be more complicated.
Thus, another extremely important step was taken in understanding the hidden properties of our world. However, the mathematical, or more precisely, geometric, model of our Universe, built by the general theory of relativity, in itself could not yet be considered proof of the finiteness of real space. But Einstein himself considered this option the most reasonable.
However, this was not the end of the road. It was still very, very far away. New level, which resulted from the study of the geometric properties of our world, gave rise to a number of questions to which answers have not yet been found.
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The theory of relativity views space and time as unified education, the so-called “space-time”, in which time coordinates play as significant a role as spatial ones. Therefore, in the very general case we, from the point of view of the theory of relativity, can only talk about the finitude or infinity of this particular united “space - time”. But then we enter the so-called four-dimensional world, which has completely special geometric properties that differ most significantly from the geometric properties of the three-dimensional world in which we live.
And the infinity or finitude of four-dimensional “space-time” still says nothing or almost nothing about the spatial infinity of the Universe that interests us.
On the other hand, the four-dimensional “space-time” theory of relativity is not just a convenient mathematical apparatus. It fully reflects certain properties, dependencies and patterns of the real Universe. And therefore, when solving the problem of the infinity of space from the point of view of the theory of relativity, we are forced to take into account the properties of “space - time”. Back in the twenties of the current century, A. Friedman showed that within the framework of the theory of relativity, a separate formulation of the question of the spatial and temporal infinity of the Universe is not always possible, but only under certain conditions. These conditions are: homogeneity, that is, the uniform distribution of matter in the Universe, and isotropy, that is, the same properties in any direction. Only in the case of homogeneity and isotropy does a single “space-time” split into “homogeneous space” and universal “world time”.
But, as we have already noted, the real Universe is much more complex than homogeneous and isotropic models. This means that the four-dimensional sphere of the theory of relativity, corresponding to the real world in which we live, in the general case does not split into “space” and “time”. Therefore, even if with an increase in the accuracy of observations we can calculate the average density (and therefore the local curvature) for our Galaxy, for a cluster of galaxies, for the observable region of the Universe, this will not yet be a solution to the question of the spatial extent of the Universe as a whole.
It is interesting, by the way, to note that some regions of space may indeed turn out to be finite in the sense of closure. And not only the space of the Metagalaxy, but also any region in which there are sufficiently powerful masses that cause strong curvature, for example, the space of quasars. But, we repeat, this still does not say anything about the finitude or infinity of the Universe as a whole. In addition, the finiteness or infinity of space depends not only on its curvature, but also on some other properties.
Thus, when current state general theory relativity and astronomical observations, we cannot obtain a sufficiently complete answer to the question of the spatial infinity of the Universe.
They say that the famous composer and pianist F. Liszt provided one of his piano works with the following instructions for the performer: “fast”, “even faster”, “fast as possible”, “even faster”...
This story involuntarily comes to mind in connection with the study of the question of the infinity of the Universe. Already from what was said above, it is quite obvious that this problem is extremely complex.
And yet it is even immeasurably more complicated...
To explain means to reduce to what is known. A similar technique is used in almost every scientific research. And when we try to solve the question of the geometric properties of the Universe, we also strive to reduce these properties to familiar concepts.
The properties of the Universe are, as it were, “measured” to those existing in at the moment abstract mathematical concepts about infinity. But are these ideas sufficient to describe the Universe as a whole? The trouble is that they were developed largely independently, and sometimes completely independently of the problems of studying the Universe, and in any case on the basis of research limited area space.
Thus, the solution to the question of the real infinity of the Universe turns into a kind of lottery, in which the probability of winning, i.e., a random coincidence, is at least sufficient large number properties of the real Universe with one of the formally derived standards of infinity is very insignificant.
The basis of modern physical ideas about the Universe is the so-called special theory of relativity. According to this theory, the spatial and temporal relationships between the various real objects around us are not absolute. Their character depends entirely on the state of motion of a given system. Thus, in a moving system, the pace of time slows down, and all length scales, i.e. the sizes of extended objects are reduced. And this reduction is stronger, the higher the speed of movement. As we approach the speed of light, which is the maximum possible speed in nature, everything linear scales decrease indefinitely.
But if at least some geometric properties of space depend on the nature of the movement of the reference system, that is, they are relative, we have the right to pose the question: aren’t the concepts of finitude and infinity also relative? After all, they are most closely related to geometry.
IN recent years The famous Soviet cosmologist A.L. Zelmapov studied this curious problem. He managed to discover a fact that, at first glance, was absolutely amazing. It turned out that space, which is finite in a fixed reference frame, at the same time can be infinite relative to a moving coordinate system.
Perhaps this conclusion will not seem so surprising if we remember about the reduction of scales in moving systems.
Popular presentation complex issues modern theoretical physics is made very difficult by the fact that in most cases they do not allow for visual explanations and analogies. Nevertheless, we will now try to give one analogy, but while using it, we will try not to forget that it is very approximate.
Imagine that a spaceship is rushing past the Earth at a speed equal to, say, two-thirds the speed of light - 200,000 km/sec. Then, according to the formulas of the theory of relativity, a reduction in all scales should be observed by half. This means that from the point of view of the astronauts on the ship, all segments on Earth will become half as long.
Now imagine that we have, although a very long, but still finite straight line, and we measure it using some unit of length scale, for example, a meter. For an observer located in spaceship, rushing at a speed approaching the speed of light, our reference meter will shrink to a point. And since there are countless points even on a finite straight line, then for an observer in a ship our straight line will become infinitely long. Approximately the same thing will happen with regard to the scale of areas and volumes. Consequently, finite regions of space can become infinite in a moving frame of reference.
We repeat once again - this is by no means a proof, but only a rather rough and far from complete analogy. But it gives some idea of the physical essence of the phenomenon of interest to us.
Let us now remember that in moving systems not only do scales decrease, but the flow of time also slows down. It follows from this that the duration of existence of some object, finite in relation to a fixed (static) coordinate system, may turn out to be infinitely long in a moving reference system.
Thus, from Zelmanov’s works it follows that the properties of “finitude” and “infinity” of space and time are relative.
Of course, all these at first glance rather “extravagant” results cannot be considered as the establishment of some universal geometric properties of the real Universe.
But thanks to them you can do extremely important conclusion. Even from the point of view of the theory of relativity, the concept of the infinity of the Universe is much more complex than it was previously imagined.
Now there is every reason to expect that if a theory more general than the theory of relativity is ever created, then within the framework of this theory the question of the infinity of the Universe will turn out to be even more complex.
One of the main provisions modern physics, her cornerstone is the requirement of the so-called invariance of physical statements with respect to transformations of the reference system.
Invariant—means “not changing.” To better imagine what this means, let's take some geometric invariants as an example. So circles with centers at the beginning of the system rectangular coordinates are rotation invariants. At any turn coordinate axes relative to the origin, such circles transform into themselves. Straight lines perpendicular to the “OY” axis are invariants of transformations of the coordinate system transfer along the “OX” axis.
But in our case we're talking about about invariance in more in a broad sense words: any statement only then has physical meaning, when it does not depend on the choice of reference system. In this case, the reference system should be understood not only as a coordinate system, but also as a method of description. No matter how the method of description changes, the physical content of the phenomena being studied must remain unchanged and invariant.
It is easy to see that this condition is not only purely physical, but also fundamental, philosophical meaning. It reflects the desire of science to clarify the real, true course of phenomena, and to exclude all distortions that can be introduced into this course by the process of scientific research itself.
As we have seen, from the works of A.L. Zelmanov it follows that neither infinity in space nor infinity in time satisfy the requirement of invariance. This means that the concepts of temporal and spatial infinity that we currently use do not fully reflect the real properties of the world around us. Therefore, apparently, the very formulation of the question of the infinity of the Universe as a whole (in space and time) with modern understanding infinity has no physical meaning.
We have received yet another convincing evidence that the “theoretical” concepts of infinity, which the science of the Universe has used so far, are very, very limited in nature. Generally speaking, this could have been guessed before, since the real world is always much more complex than any “model” and we can only talk about a more or less accurate approximation to reality. But in this case, it was especially difficult to gauge, so to speak, by eye, how significant the approach achieved was.
Now at least the path to follow is emerging. Apparently, the task is, first of all, to develop the very concept of infinity (mathematical and physical) based on the study real properties Universe. In other words: “to try on” not the Universe to theoretical ideas about infinity, but, on the contrary, these theoretical ideas to the real world. Only this research method can lead science to significant advances in this area. No abstract logical reasoning or theoretical conclusions can replace facts obtained from observations.
It is probably necessary, first of all, to develop an invariant concept of infinity based on the study of the real properties of the Universe.
And, in general, apparently, there is no such universal mathematical or physical standard of infinity that could reflect all the properties of the real Universe. As knowledge develops, the number of types of infinity known to us will itself grow indefinitely. Therefore, most likely, the question of whether the Universe is infinite will never be given a simple “yes” or “no” answer.
At first glance, it may seem that in connection with this, studying the problem of the infinity of the Universe generally loses any meaning. However, firstly, this problem in one form or another confronts science at certain stages and has to be solved, and secondly, attempts to solve it lead to a number of fruitful discoveries along the way.
Finally, it must be emphasized that the problem of the infinity of the Universe is much broader than just the question of its spatial extent. First of all, we can talk not only about infinity “in breadth”, but, so to speak, also “in depth”. In other words, it is necessary to obtain an answer to the question of whether space is infinitely divisible, continuous, or whether there are some minimal elements in it.
Currently, this problem has already faced physicists. The question of the possibility of the so-called quantization of space (as well as time), i.e., the selection of certain “elementary” cells in it that are extremely small, is being seriously discussed.
We must also not forget about the infinite variety of properties of the Universe. After all, the Universe is, first of all, a process, the characteristic features of which are continuous movement and incessant transitions of matter from one state to another. Therefore, the infinity of the Universe is also an infinite variety of forms of movement, types of matter, physical processes, relationships and interactions, and even properties of specific objects.
Does infinity exist?
In connection with the problem of the infinity of the Universe, it appears at first glance unexpected question. Does the very concept of infinity have real meaning? Isn't it just conditional? mathematical construction, to which nothing corresponds at all in the real world? This point of view was held by some researchers in the past, and it still has supporters today.
But scientific data indicate that when studying the properties real world we are in any case faced with what can be called physical, or practical, infinity. For example, we encounter quantities so large (or so small) that, from a certain point of view, they are no different from infinity. These quantities lie behind quantitative limit, beyond which any further changes no longer have any noticeable impact on the essence of the process under consideration.
Thus, infinity undoubtedly exists objectively. Moreover, both in physics and in mathematics we are faced with the concept of infinity at almost every step. This is not an accident. Both of these sciences, especially physics, despite the apparent abstractness of many provisions, ultimately always start from reality. This means that nature, the Universe, actually has some properties that are reflected in the concept of infinity.
The totality of these properties can be called the real infinity of the Universe.
The theory of gravity created by Einstein gave a powerful impetus to the development of cosmology, which received a number of fundamentally important results related to the understanding of space-time and, above all, the problem of their infinity. Relativistic cosmology showed the extreme complexity of solving this problem, made a naive approach to it impossible and raised the question of the need deep analysis the very concept of “infinity”.
Before the advent of relativistic cosmology, the view of infinity was dominated by a rather naive approach - infinity was understood as something that has no end in any direction. This understanding, coming from ancient times, has remained unchanged for more than two thousand years. True, in mathematics, starting from the second half of the 19th century, the complexity and depth of the concept of infinity has become increasingly clear. But among non-mathematicians, a complacent attitude towards infinity continued to prevail, and the difficulties faced by mathematics were portrayed as some kind of “mathematical subtleties”. This naively complacent attitude towards the problem of infinity, manifested in the opinion that we know the content of the concept of infinity, has been preserved by some philosophers until recently.
General relativity showed that space is inextricably linked with matter and in the general case is non-Euclidean. And for non-Euclidean – “curved” – space, the concepts of infinity and limitlessness, which have been implicitly identified since the time of famous reasoning Pythagorean Archytas. The ancient Greek philosopher Archytas gave the following visual image of such an understanding of infinity. If we throw a spear, and then go to the place where it fell, and throw the spear again, repeating this operation again and again, then we will not come across a boundary anywhere that would not allow us to continue throwing. The absence of such an obstacle demonstrates the possibility of endless movement in space, Archytas believed. The understanding of the infinity of space as the possibility of an unlimited addition of ever new units of distance is complemented by the interpretation of the infinity of time as an unlimited addition of segments of duration. Mathematically This understanding of infinity is served by the natural series of numbers. Hegel, and after him Engels, called such purely quantitative infinity “bad” infinity.
In reality, we may have a case where a three-dimensional space of positive curvature is finite (or, as is more often said, closed, closed) and at the same time limitless: a creature living in such a space, moving in it, will not encounter any boundaries and, however, will be able to establish its limb by determining the curvature.
Relativistic cosmology starts from Einstein's fundamental equation of gravity. It is solved under certain assumptions based on known empirical data, and the resulting solutions (“models of the universe”) are examined and compared with experience. The resulting models can be divided into two large groups: models of a homogeneous and isotropic universe and models of an anisotropic inhomogeneous universe. The first group is the most developed.
In 1922, the Soviet scientist A. A. Friedman put forward the hypothesis of an expanding universe. She was so unusual that even Einstein initially reacted negatively to her. Academician Ya.B. Zeldovich noted that Friedman's work provides a more impressive example of foresight than classic example Le Verrier's predictions. After all, Le Verrier used celestial mechanics, which had been brilliantly developed and confirmed even before his work. And Friedman’s work was the first (and many decades after the hypothesis was put forward, the only) correct use theory of relativity to cosmology).
The non-stationary nature of the universe predicted by Friedman was proven by establishing the red shift. In 1929, the American astronomer Hubble discovered that in the spectra distant galaxies spectral lines shifted towards the red end. This means that galaxies are “moving away” from us at a speed that is linearly dependent on distance. The recession of galaxies should not be imagined as some kind of ordinary movement in space that does not change with time, and we should look for special dynamic reasons for this movement. This is not the movement of objects in a constant space, but an effect caused by previously unknown to us properties of the space itself - the nonstationarity of its metric. The explanation of the recession of galaxies given by relativistic cosmology is, in principle, similar to the explanation relativistic effects shortening length and slowing down time.
Within the framework of Friedman's model, questions about the finitude and infinity of space and time, in a certain sense, become empirically verifiable. Friedman's non-stationary world can have both positive curvature (closed model) and negative curvature (open model). It can have one special time point - the beginning of time (the expanding universe), but it can also have infinitely many special points. In this case, none of them can be considered the beginning of time, and their presence simply means that in the universe, periods of expansion, starting from a certain moment, when the density of all types of matter was infinite, are replaced by periods of compression, when galaxies “run together” - red the displacement changes to violet -, the density again takes on an infinite value, and then expansion begins again, etc. (pulsating universe).
The finitude of time, which the model of the expanding universe speaks of by introducing the beginning of time, is not a conclusion about the unconditional finitude of time in general, but a hint of an approach to the boundaries of measure, an indication of the possibility of transition to qualitatively new types of relations, where a radical revision of the known ones may be necessary physical laws and the very concept of time.
The choice of one or another model of the universe depends on the average density of matter and fields in the universe. Comparison of actual density ρ with (critical density) allows you to select the specified options. When ρ > we have a space of positive curvature, that is, closed and finite (but limitless) and infinitely many temporary singular points: the universe will pulsate. At ρ< we have a space of negative curvature, that is, open and infinite, and one special point from which the expansion of the universe began. Empirical data lead to a decision in favor of the open model, but a final verdict cannot yet be made.
Some Russian philosophers, having become acquainted with this kind of scientific data, took the point of view of their rejection. Dialectical materialism, they reasoned, asserts the infinity of space and time, and everything that does not agree with this position is a manifestation of idealism. In a Soviet philosophy textbook of the mid-60s one can find the following statement: “One of the attempts to refute the idea of the infinity of the world is the idealistic theory of the “expanding universe” (as if there were materialistic and idealistic scientific theories. - author) ... This is reactionary, frankly the fideist theory does not stand up to criticism...” The above reasoning is an example of how not to fight idealism. Why should the theory of an expanding universe be considered idealistic, reactionary, outright fideist? At one time, idealists seized on it. But they, like the churchmen, grab at any scientific theory that breaks established ideas, and to fight them by denying what they “grasp” means actually helping them. This kind of “criticism” is evidence of the critic’s natural scientific illiteracy.
The most consistent with the real state of affairs in resolving the issue of the infinity of space-time seems to be the position defended at one time by the Estonian scientist G.I. Naan. Here we are dealing with a consistently unconventional path, since the existence of some philosophical standard of infinity, with which specific data from physics, astronomy, and mathematics must be compared, is resolutely denied. The task of philosophy, according to Naan, is not to give a final solution to the problem of infinity in addition to natural science, which cannot give such a solution, but to examine the very origin of our concepts of infinity and indicate the path along which we should go understanding more and more new scientific data. Today we should not talk about comprehensive solution problems of infinity, but about improvement methodological tools her decisions. Following this path, we will approach an ever more complete elucidation of what infinity is. Now we do not have such a concept. And even if we assume for a moment that we somehow became aware of the specific understanding of infinity that our descendants will possess, it would not help us much. By figuratively Naana, we could use this concept not in to a greater extent than a primitive savage who found a jet plane in the forest.
So, the most important epistemological conclusion is that when solving our problem, we cannot assume the concepts of finite and infinite that we have today as unquestionable standards, with which we only need to compare changing scientific data, recognizing those that do not contradict our standards, and discarding those that contradict . On the contrary, it is necessary, based on these data, to clarify the concepts themselves, considering each new step in the development of natural science as another step along the path of this deepening and clarification. Therefore, it is clear that the results obtained by relativistic cosmology cannot be considered as providing today a final solution to the problem of the infinity of space-time.
And one more fundamental remark. The data of relativistic cosmology are relevant to the question of the finitude or infinity of the Metagalaxy, that is, the universe. In philosophy we are talking about the infinity of the Universe, or the world as a whole.
Brief summary of the work
Space without infinity
And, indeed, if the Universe is not infinite...
Could this be?
It turns out it can.
And not even in the sense that it occupies part of the space. The universe may occupy all of space, but this space does not have places in mathematics designated by the sign ∞ (infinity).
To understand this, we only have three steps to take.
First, let's depict such a space in general outlines, and then begin to draw all the details.
So, step one.
One-dimensional space.
In everyday understanding, it appears to us as something like a number line.
On the straight line, we mark the origin of the reference – point O and from it in one direction with a plus sign (+), in the other with a minus sign (-), at equal intervals, called a unit of measurement, we make markings +1, +2, +3, ... ,+ ∞ and, accordingly, -1, -2, -3, …, - ∞. That is, on both sides there are ∞ signs - this is a one-dimensional infinite space.
Here we ask our question: “Can there be a one-dimensional space that does not contain ∞?”
It turns out it can.
In the initial sketch we will give only those examples that will be necessary and sufficient for us to understand the essence and further logical description of the next steps. At the same time, we will try to avoid introducing any new definitions.
Let's draw a circle.
This is also a one-dimensional space.
But no matter how you mark out such a space, if we take a certain finite value as a unit of measurement, then the sign ∞ cannot be placed anywhere in such a space.
This circle is local example one-dimensional space that does not contain the sign ∞.
Step two.
Two-dimensional space.
Let us draw two mutually perpendicular lines on the plane. Let's mark them in exactly the same way as the straight line in the first step, taking the intersection point as the starting point for each. Thus we define two-dimensional infinite space.
Here again we ask our question: “Can there exist a two-dimensional space that does not contain ∞?”
It turns out that it can too.
Pick up the globe.
No matter how you mark its surface, you won’t be able to place the ∞ sign anywhere.
This sphere is a local example of a two-dimensional space that does not contain ∞.
Let's move on to the third step.
Through the point of intersection of two mutually perpendicular lines we draw a third line perpendicular to the first two. Let's mark it in exactly the same way as in the first two steps. We get a three-dimensional infinite space, or more precisely, a way to display it - a Cartesian coordinate system.
We ask the initial question: “Can there exist a space that does not contain the sign ∞?”
It turns out it can.
Local example, similar to examples in the first two steps, it will not be possible to give here.
These local examples were given only in order to obtain a method for displaying such a space in a Cartesian coordinate system, which will allow us to determine a method for calculating an ideally defined space - a space that does not contain the ∞ sign, in the global sense.
Let's move on to the method of displaying an ideally defined space in a Cartesian coordinate system.
Let's return to one-dimensional space.
How can you display a circle on a line?
Let's mark any point on the circle and take it as the origin, denoting it exactly the same as on the straight line - O (with a zero value). From point O we measure half a circle in any direction and designate this mark as point M (that is, OM - half a circle in any direction). From point O in one direction with a sign (+), in the other with a minus sign (-), with exactly the same identical in......
