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- What needs to be corrected in the management of Russia first?
- Why did Blaise Pascal become a religious man at the end of his life?
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Infinity is a mathematical abstraction. In real life, we don't meet her anywhere. Even the universe is finite for us, since it is limited to the visible part of it. Its mass, for example, although it is an astronomical number, is also finite. Time has a beginning, which means that it is no longer a straight line, but at least a ray. Quantum physics tells us that matter is not divisible into an infinite number of infinitesimal parts. And our daily life even more clearly indicates the finiteness of everything around us.
A classic example of difficulties with the perception of infinity is Zeno's aporia, which appeared as early as the fifth century BC (see, for example, Wikipedia). The seemingly paradoxical arguments about motion, space and time just demonstrate problems with the perception of infinity, and in particular with the very important concept of infinitesimal.
Actually, classical physics (starting with Newtonian mechanics) appeared only after this “problem with infinities” was solved by means of differential calculus (including by Newton himself). In classical physics, any changes in anything can be written (and in most cases this is the only way) in the form of differential equations, that is, in terms of infinitesimals. So the apparatus of differential calculus is the most powerful tool that has shaped all classical physics (let me remind you that only quantum physics belongs to non-classical physics; that is, for example, general relativity, which is also part of classical physics).
With quantum physics, everything is more interesting. Things that always seemed infinitely divisible, such as energy, did not turn out to be so. I had to invent my own apparatus for describing quantum systems. But that's a different story. I will only say that they also use the concept of infinities, but this is no longer the main tool of the theory.
Let's consider another branch of mathematics — set theory. There is the Banach-Tarski Paradox, the essence of which is that if we assume infinite divisibility of an object, then we can make two exactly the same balls from one ball by dividing it. In practice, we will be limited at least by the size of the atoms.
So the concept of “infinity” is not as simple as it seems. This is a mathematical abstraction that still has no place anywhere in the real world.
Infinity cannot be obvious, since any volumes calculated and represented by us will clearly not be enough. This is not a question of mathematics or physics, since everything that exists is, of course, more of a philosophical topic, or rather a topic of understanding the boundaries of consciousness… Any manifested or perceived dynamic is limited by the time of existence, and therefore is far from the possibility of comprehending infinity. But this does not mean that it is incomprehensible, it is comprehensible in a static state of consciousness, in the same one – which was before our time – limited and form – limited process of perception of life, and will be again-after the perception of the existence of each “I”is completed. In other words: infinity is accessible beyond personal perception, but is accessible – for a state of consciousness free of form and time.
“Sometimes, when you look at the sky , there it is, Boundlessness!”.. A straight line runs from your nose to Alpha Centauri, where you can stop, look back one last time, and throw yourself completely into the abyss. And your home will be everywhere.
From the point of view of mathematics and physics, infinity is not as obvious as it seems to you. The most striking example is the Euler number, or the number “e”, which is also the solution of the second remarkable limit. Obviously, a unit will be a unit to any degree. Logic suggests that the number is slightly larger than one, but raised to a higher power, it should not differ much from this very unit. And here's a bummer – a little less than three turns out. Or here's another – divide any number by zero – infinity is obtained. And if it's zero? Here is the first wonderful limit to help, and it turns out that there is a difference between what zero is and what zero it is divided by. Also, multiply infinity by zero, or divide infinity by infinity. Are the answers to these questions obvious to you? Infinity and a lot are not synonymous! The number of natural numbers and integers is infinite. What happens if one of these sets is divided into another? Obviously, integers are twice as large as natural numbers. What if we take rational numbers and compare them with integers? They are infinitely more than integers. And if you also take irrational numbers? And the transcendental ones? Are there more irrational ones? Logically, transcendental numbers on the numerical scale are less common than simply irrational ones, but both are infinite! The last example that infinity isn't obvious is the last 4 digits of Pi? If you understand what infinity is, you will understand that they simply do not exist.
They don't believe it, but they don't want to take it into account. Or they attribute a singular number in infinity to our universe. This is due to the fact that if we recognize that the cosmos has limits, and infinity, as an independent quantity of space that permeates everything from particles to the whole universe, then the latter suggests the concept of God.
Obvious to whom? Believers in infinity? And on the basis of what?
They have already given quite detailed answers, both from the point of view of mathematics and physics. I would like to touch on this topic from the point of view of philosophy.
In my opinion, the problem with such concepts as infinity, singularity, continuity and the like lies in their symbolic nature. I.e., they are constructed purely symbolically, namely, a particle or a negative prefix is added to the symbol denoting something concrete and finite, such as” not “or”without”. In such cases, these particles are added precisely in order to reject the concreteness and definiteness of the original concept. For what? Where does the wind blow from?
And it's all about the idea of the Absolute or the absolute. The essence of this idea is precisely the rejection of any specifics. This is a purely abstract idea and therefore its expression requires artificially constructed concepts-symbols that deny their original concreteness and finiteness. The absolute basically claims to be infinite in everything and in all directions.
In general, the Absolute is also just a symbol. And it is precisely this symbolic nature that makes it possible to use this concept in all symbolic systems, i.e. in languages. Including math. But it is precisely this symbolic nature that does not allow us to interpret it in the form of images or objects. Imaginative or physical interpretation of something requires representation of this something in the form of connections with something else, concrete and finite, i.e. figuratively representable. In other words, such an interpretation requires mediation through the environment. But the Absolute by definition excludes any relativity to anything. Hence the paradox and impossibility of imaginative representation of all concepts based on the absolute.
The same applies to the concept of “Being”, which means the existence of the world by itself, i.e., without reference to anything. This means that the same applies to matter as a form of Being and all physical reality. They are all based on the absolutization of the concept of “exist”, which initially was extremely specific and relative. Existence presupposed a place and time that were thought of as being between other objects and as an event between other events. Not points on the axes. The introduction of absolute existence made it possible to introduce the same absolute concept of inertial motion as motion itself, which in turn gave the right to use reference systems. Further, differential calculus, infinities, singularities, etc. That's how in small steps we did not notice how we came to the physical vacuum and physical space.
It is the absolutized existence itself that gives the illusion that existence must be extended in all directions without end or end. I remember when I was 5 years old, I tormented my father with a similar question: “What is beyond the edge of the earth? After all, there must be something there!”
Mathematics, as a symbolic language, is allowed to use symbols that hide absoluteness. But physics, in the pursuit of a visual interpretation of such symbols, is sliding deeper and deeper into fiction. But there's no stopping it. Today, even the Nobel Prize has been awarded for beautiful interpretations of mathematical studies of black holes.
It's kind of brain-racking and totally unusual.
The Mobius strip has one side. And bottles of Klein.
This also breaks the brain.
People don't like brain-breaking stuff.
Give a clear definition of infinity and then it will become clear whether it exists or not. It is clear that there are infinitely many natural numbers, or, for example, simple ones. There are a finite number of people on earth. If we are talking about the size of the universe, then it is finite in 3 dimensions, even the size can be estimated if the speed of light is multiplied by the age. About the infinity of human stupidity is an open question, this is to Einstein from jokes and stories, but what is “belief in infinity” I do not understand at all. For mathematical or physical terms, the word “believe” only means “we don't know yet, but all the data points to..”
Infinity itself cannot be “believed” or “disbelieved”: this is a mathematical concept that is quite specifically used in matanalysis, functional analysis, set theory, etc., and the infinities themselves can be of different types. Although in the XIX-XX centuries there were many discussions related to the understanding of the concept of infinity in mathematics: paradoxes of set theory, the concept of continuum, transfinite numbers, constructivism, etc. But in general, infinity is no longer a matter of faith, but a part of mathematical tools.
One can “believe or disbelieve” whether infinite quantities exist in the physical world. The answer to this question primarily depends on what kind of infinity is meant: current or potential.
No one particularly objects to potential infinity (i.e., the situation of a potentially unlimited increase in some quantity). For example, the number of seconds that have elapsed since the big bang increases indefinitely, and there is no reason to believe that time will ever stop flowing in the universe. And in this sense, a situation where time lasts potentially indefinitely is quite logically meaningful, physically permissible, and no one really argues with this.
Problems arise with actual infinity, i.e., with the situation when a physical quantity is supposed to be virtually infinite at some particular moment. A classic example is the problem of singularities in black holes or in early cosmology, when a number of quantities (such as the curvature of space-time, pressure or density of matter) included in the solutions of Einstein's equations go to infinity in a certain region of space-time. Another example: the question of infinite divisibility of space-time. To do this, it is necessary that it actually contains infinitely many degrees of freedom, allowing for detailing on any arbitrarily small scale.
Physicists in general are not inclined to believe in the existence of actually infinite quantities. More precisely, this is not so much a problem of faith or disbelief as a linguistic question, if you look at physics from the positivist point of view. The statement “something is infinite here and now“is unverifiable and unobservable. It is impossible to state it meaningfully, because it lies outside the scope of any empirical experience, which always deals with finite quantities and finite standards. Everything that we can say meaningfully from the point of view of physics will, by definition, be grasped by us in terms of certain specific finite quantities. And everything that does not fit into this possibility will simply lie outside the scope of our perception, physical methods and measurement procedures. Of course, theoretically, we can imagine a situation where an attempt to measure a certain quantity over and over again leads to an off-scale increase in the available instruments-but even in this case, it will only be possible to state that the value is immeasurable by our means. But we cannot reliably guarantee that it is actually (i.e., actually) infinite at any step. This means that it is methodologically unjustified to talk about the actual infinity of physical quantities.
Again, I emphasize that this is from the point of view of positivism. From the point of view of metaphysics, the question of actual infinities is less trivial. But most physicists, as a rule, are positivists (alas or fortunately-the question is rhetorical).
The most common approach to the problem of singularities in physics is that the appearance of infinite quantities signals the limits of applicability of a physical theory (and then it should be replaced by a more advanced theory – for example, the theory of quantum gravity should replace general relativity), or the need to modify the theory through the renormalization procedure (as is the case in quantum field theory).
The problem of infinite divisibility of space-time has long been pushed into the background, since it is believed that there is a fundamental limit to divisibility (the Planck limit), smaller than which space-time loses its classical meaning and turns into a foam of quantum fluctuations.
One more problem remains: the possible actual infinity of the universe. Of course, from the positivist point of view, there are no problems here again: at any given time, the observable region of the Universe is finite. It can potentially expand indefinitely, but we cannot fix its actual infinity at any step. Well, from the point of view of metaphysics, the attempt to think of the universe as a factual infinity is very interesting in itself, and here the question is open.