Is there infinity?
Is there infinity?
Dan D. Farcaș
Motto: "Only two things are infinite: the Universe and human stupidity; but of the Universe I am not sure" (Albert Einstein)
I think I discovered the notion of infinity, at the age of the first questions about the great mysteries of reality, reading an astronomy book that talked about the vastness of the Universe. I suppose it was an older book, since in its pages neither space nor time had any beginning, nor end, nor any boundary. To this picture was added, of course, some of the mathematics and materialism learned at school.
The conclusions I reached, when I was not even 15 years old, I had written down in a notebook, recording an exposition, made a few days before, on this topic, to a colleague of the same age, during a walk. I still have the notebook, so I can reproduce below, unchanged, a few edifying fragments, interspersed with small explanations, so that the text is at least somewhat intelligible.
"Infinity, multiplied by infinity times the square, still remains infinite, whence it follows that infinity divided by infinity times itself gives infinity. There are four dimensions... Forward is as much as backward, to the right as to the left, up as much as down, the times past are equal to the times to come. And all are equal to infinity."
"If we admit that each molecule in the brain occupies [at a given moment] a well-established place, we have a very large but finite number of possibilities of placing them in different ways, and each combination is characteristic of a particular psyche" [thus defining a particular person, or consciousness]...
"There are an infinity of stars, an infinity of variants of worlds, buildings, lives, thoughts... Infinity divided by infinity equals infinity. Therefore, each of these possibilities exists an infinite number of times. The same thing has been spoken and will be spoken an infinite number of times. And there are worlds identical to ours"...
"There being a finite number of psychic possibilities [of combinations of molecules in the brain], in an infinity [of space and time], each [combination, therefore psychic] will exist an infinite number of times. It is said that we have lived an infinite number of times and will live as many more times. And not only that. We have lived the same life that we are living now and we will live it an infinite number of times. So we do not have to complain about death..."
In other words, I wanted to say that the reality we live here now also exists in another corner of the Universe and even an infinite number of times. And these identical copies existed in the same way in the past and will exist in the future, endlessly... Of course, the various variants, more or less different, of the present reality will also be infinite, all drowned in the multitude of variants that have nothing to do with our world.
Today I understand that I had thus, independently, come across an ancient paradox of infinity. But which is far from the only one.
Back then I was too young and too confident in the things in books or in the words of great people. We accept that the infinite does exist, once it is talked about, that we can play in our imaginations with infinite worlds, even if this game generates all sorts of weirdness. Distrust in the knowledge received from others and the need to check and rethink it with my own mind came to me a little later. It was only then that I asked myself whether the infinite really exists, since no one has experienced it, even through indirect consequences, but impossible to reject.
In fact, even a sharp-minded ordinary man, if asked if there is any "infinity", would answer that he can only put his hand on fire for the things he has experienced. You may have seen Webb or Hubble images of galaxies billions of light years away; and apparently there was something beyond them. But he saw only that, not infinity. In short, to our common sense, infinity does not really exist, since no one has ever seen or touched it.
In the meantime, I also learned that the Universe looks completely different than I thought as a teenager. In time, it (seems) has a beginning, in the famous "Big Bang", and perhaps it will have an end. As for space, here things are a bit more complicated. I'm trying to use an analogy. By "infinity" we usually mean a situation where no matter how far we go, no matter how far away a point is, we will be able to pass beyond, continuing on to an even further horizon. The surface of the Earth (which, simplifying slightly, we can consider two-dimensional) is obviously finite (510,072,000 km2), however, however we walk on it, wherever we end up, we will still be able to go further. The ancients still believed that it was possible to somehow reach the "end of the world"; today we know that such a thing is not possible. So, in a sense, the surface of the Earth is boundless, without any end (so "infinite"?).
Let's add another dimension to this picture. Just as the two-dimensional, self-enclosed surface of the Earth (the surface of a sphere) is "immersed" in our ordinary three-dimensional reality, I propose to imagine that the three-dimensional Universe that we observe around us could, analogously, be the surface of a hypersphere, "immersed" in a reality with four spatial dimensions. In this case, if we had a miraculous ship that could travel millions of light years in seconds, we would find that no matter how far we went in one direction we could still keep going. Eventually (as on Earth) we could have the surprise that, going forward, unfazed, we remember that we have returned to the point from which we left. And we wouldn't hit the edge of the Universe anywhere.
In other words, just as an observer from space could observe, from a distance, the surface of the Earth and the movements of people on it, a four-dimensional observer could look, "from the outside", at our Universe. With one observation: the surface of the Earth is in the shape of a sphere, although theoretically it could have another shape; by analogy, our Universe, viewed from the outside, does not necessarily have the shape of the surface of a hypersphere but, possibly, of another geometric (hyper)body. What kind of body? This is where cosmologists still brag. Maybe a (hyper)tor, maybe something more complex.
It seems that, at the present time, this is the view most accepted by the learned; that is, a Universe finite in size, closed (so without the possibility of escape from it), yet without borders. Of course, that's not how we imagined that "infinity" we were talking about at the beginning of the article.
And we could further complicate the picture by talking about multiverses, which in turn are part of some hypermultiverses (and so on to infinity?), or about an increasingly large (infinite?) number of dimensions of space and perhaps also of time. But such a discussion is beyond the scope of the present lines.
Going to college, I learned of course that for mathematicians infinity really exists, being a very real concept, without which nothing in modern mathematics could exist. The ancients still feared the concrete infinity. Today we casually say that two parallel lines meet at infinity. Euclid was much more cautious. A reformulation of his famous postulate about parallels says only that they will never meet, no matter how much we extend them.
And in the infinitesimal calculus, used - behold - for more than three centuries, at first it was only said that a number "tends to infinity" or that it is "as small as possible". But at the present time integrals from minus infinity to plus infinity are already used, without embarrassment, as if these values existed in reality. There is even a little mathematical monster, a functional that has the value zero over the entire interval between minus infinity and plus infinity, except for one point, where it has the value infinity. And his integral, throughout this field, is something finite.
Later I also encountered Banach or Hilbert spaces, spaces that are no longer shy to postulate in certain cases, from the very beginning, that they have an infinite number of dimensions. All these tools, like many other related ones, therefore use various poses of an infinity as concrete as possible, being at the same time of great utility in the science and technologies of our days.
The one who introduced the infinite into mathematics, as a real, almost tangible object, was, a century and a half ago, Georg Cantor, by creating "transfinite" set theory. In this world it is rigorously proved that even integers (or primes, etc.) are as many as all integers. So a part, no matter how special, is always perfectly equal to the whole. Bizarre operations with infinity, which I evoked in the adolescent thoughts reproduced at the beginning of this article, have their source in these paradoxical findings.
Cantor also proved that all imaginable fractions (a/b, the rational numbers) are exactly as many as the natural numbers (1, 2, 3,...), however bizarre it may seem at first glance. How many? Infinitely many, but not just any infinite, but one called "countable" and denoted by A0 (aleph zero). Then it was also shown that the number of points on a line (or segment, or all real numbers, etc.) add up to an infinity greater than the "countable" one, one that can be denoted by A1 (I pick one). And that the number of mathematical functions is an infinity and greater, therefore denoted by A2 (aleph two). After which (although no other examples were found) it was but a step to say that nothing would prevent us from speaking of a hierarchy of infinities: An for any (infinitely?) large n.
Cantor's ideas greatly annoyed the mathematicians of his time, who wanted to stay down to earth. Harassments followed, which made the far too sensitive teacher end up more and more often in a mental hospital, where he eventually died. But his legacy will conquer mathematics.
Among other things, its detractors have noted that accepting the existence of a real infinity leads to a number of paradoxes. It was recalled in this regard that, as early as almost two and a half millennia ago, Zenon Eleatul he enunciated, for example, a famous paradox which stated that Achilles, the swift of foot, cannot catch up with a tortoise by running. Why? Because by the time Achilles reaches the point T0, where the turtle was initially, it is no longer there, having reached a new position T1, a little further. When Achilles reaches T1, the reptile has already moved to point T2. This process can be repeated ad infinitum, with the same result, so apparently Achilles has no chance. Common sense tells us that, of course, such a thing does not happen in reality. For a long time the paradox puzzled scholars. Zeno's trick was that he boiled down a finite situation into an infinite (real) number of steps. And this infinite number can never be completed. Today's mathematicians have tools for such situations; they will recognize, for example, that we are in the presence of a banal development in a series that converges to the moment of encounter.
Here I come to another teaching, which is revealed in the faculty of mathematics and perhaps in philosophy, but not elsewhere. I learned that disputes over the existence or non-existence of real infinity, like other antinomies in logic (which also exist in the world of the finite), revealed that the foundations of mathematicians are much shakier than they think. What followed was an unimaginable fact for an ordinary man. It happened that, around the beginning of the 20th century, logicians and mathematicians split into different schools, each promoting methods and truths disputed by the others.
Representatives of one of the schools - the logicians, among them Gottlob Frege, Bertrand Russell, Alfred N. Whitehead etc., they hoped that a universal logic, free of paradoxes, sufficient for itself would still be possible, and that mathematics could be constructed exclusively through the tools of this logic. Intuitionists, through L. E. J. Brouwer, Arend Heyting, Jules Henri Poincaré and others, found that the solution lies in prohibiting the use of some concepts and principles, thus trying to eliminate the reasonings made beyond the boundaries that intuition can control. The formalists, by David Hilbert, John von Neumann etc., nurtured the conviction that, through sufficiently rigorous formal theories, constructions without antinomies could still be obtained. None of these schools managed to carry out their program to the end, the dispute between them remains smoldering to this day.
I remember that in one of the last classes I attended, before graduating from college, a professor gave us the following parting advice: "You know by now how controversial the foundations of mathematics are; you learned that at one point mathematicians split into various schools, which were unable to agree on some elementary principles. If you ever become a high school teacher, please persevere, do not tell the students any of this. Their spirit is still too plaguing; for them mathematics must remain one and the same."
Many years have passed since then. For over six decades I have used, with more or less success, mathematics and computers to solve a wide variety of practical problems. This practice has taught me to respect some elementary wisdoms, available to everyone, but often ignored. I would mention two of them.
The first idea is that reality is much more complex than we might imagine. To know it, man has three highly effective tools, but none of them perfect: the word, logic and mathematics. With their help we build models of reality (for example theories) never perfectly adequate (proof of paradoxes, etc.), but satisfactory for solving the problems we face. Many confuse these models with reality, imagining that what is valid in the model is also true in reality. It's a mistake that can lead to deeply flawed conclusions.
The second idea derives from the first. We often hear that "it has been shown that in reality X must exist" or that "Y cannot exist". It is good not to forget that any demonstration is carried out only inside a model, a functionally imperfect copy of reality. The conclusions of the proof are true within the model, but in reality they may or may not be true. The model is an extremely useful guide, but not an infallible one.
In particular, the infinity of mathematics has its charms, but it is only part of a pattern. Beyond a certain point, its use can become a coherent and fascinating game, but gratuitous and unrelated to reality. So even though infinity is almost a banality in mathematics, this fact does not imply any certainty about reality. In the real world, infinity could only be that of "tending toward" or "beyond something, however far," etc.; just as something real could be.
A third elemental wisdom, which I would like to remind the reader, in conclusion, would be that we must not forget that there are also problems, regarding reality, to which we not only do not have an answer at the present time, but for which the solutions will not become clear until hundreds of years from now, not sooner (perhaps there are even problems that completely exceed the capacity of the minds of homo sapiens). So let's be humble and patient. The existence of real infinity could be such a problem.
