Infinity during humanity

Sasha Răuta

Infinity during humanity

The idea of ​​infinity and its concept have aroused people's interest since ancient times, since the times of the Greeks and Romans. They tried to define it in one way or another, from various points of view: in philosophy, mathematics, theology and even art.

In order to understand the topic more clearly, I will divide the article into several segments, by eras, from a historical point of view, starting with the first mention of infinity and continuing to the present.

Antiquity:

Among the first mentions of infinity can be found in 450 BC, in "Zeno's Paradoxes", formulated by the Greek philosopher of the same name, from southern Italy, a member of the philosophical school in Elea, founded by Parmenides, who was also his disciple. The paradoxes contain three arguments formulated to support his disciple's ideas. Unfortunately, there is no physical historical document, only a few references by other Greek philosophers, as his confessions were only transmitted orally. But even if there had been an object that was part of the paradox collection, it would have been very difficult to recover, given that its dating spans several millennia.

Thus, from the stories handed down by other Greek philosophers in the following centuries, three of Zeno's most important arguments stand out.

The first paradox: It tries to prove that crossing from one point to another is impossible, because the distance between one point and the middle of the road to another point can be halved to infinity, thus the road does not end, being endless. This reasoning is seen by many philosophers as a mistake. He formulated the argument as follows: "In order to travel this distance of one kilometer, a man first travels half a kilometer (that is, half the total distance), then half the remaining distance, then half the remaining distance, and so on, so that he will never reach the end, because this division could exist ad infinitum."

The second paradox: "Achilles and the Tortoise" – in this paradox, the Greek philosopher tries to prove that the one who runs faster will never overtake the one who runs slower. This theory will also be supported by Aristotle in „Physica” two centuries later. Suppose there is a race between the famous athlete Achilles and a slower rival, transformed by legend into a turtle. Suppose that Achilles has twice the speed of the tortoise, and the tortoise is 1 km ahead of Achilles (who starts at the start). You will most likely conclude that, after 2 km, Achilles will reach the turtle. Using the paradox presented earlier, Zeno tells us something else: when Achilles reaches 1 km, the tortoise has reached 1.5 km, and when Achilles reaches 1.5 km, the frog has reached 1.3 quarter km, and so on, so that Achilles will never be able to overtake the tortoise.

The third paradox: "The Arrow Paradox" tells us that "an arrow in motion between points A and B is at any given time neither at point A, because it has left there, nor at point B, because it has not yet arrived there. If you reduce the distance AB to the length of the arrow, it means that the arrow is, in fact, at rest." Since the conclusion is obviously false, it is a logical paradox.

Two centuries later, Aristotle tries to address these theories, but also the problem of infinity from a mathematical and philosophical point of view, concluding that infinity is divided into two subdivisions: a potentially infinite and an actual infinite. The philosopher says thus: "No distance is actually infinite, only that any such distance is potentially infinite" (it can be divided to infinity, but it is finite at the base). This statement is completely opposed to Zeno and is the reason that slowed down the search for infinity for a millennium.

Infinity in the Middle Ages:

This era, also called "Dark Ages", is characterized by the authority of central institutions, such as the church, over the population with a low literacy rate, low standard of living caused by the migrations of aggressive nomadic peoples and the appearance of the plague epidemic. The most significant element for the history of infinity in this age is the renunciation and prohibition of ancient values ​​regarding science and human development. People immersed themselves in faith and renounced reason, mathematics, philosophy and other subjects that were part of the arsenal of the cultured man. Thus, there was no need to explain infinity or approach it from a scientific point of view.

The only view tolerated was infinity from a theological perspective. In those times it was considered that not only God is infinite, omnipotent, omnipresent, but also the other elements related to him are attributed infinity: "All the attributes or works of God are infinite." Divinity had a close connection with the infinite, being defined by eternity and immortality. Indirectly, eternity meant infinity relative to time, and omnipresence meant infinity relative to space.

According to some documents created by the priests of that period, one of the most famous quotes was from Saint John Chrysostom: "His judgments are unfathomable, His ways unfathomable, His peace overwhelms the whole mind, His gifts indescribable, His greatness has no bounds, His wisdom has no number, all are unfathomable... It is not about the being, but about the economies... But if the economies are unfathomable, how much more so is He Himself." Through these affirmations, and many others, during the Middle Ages, the Church left its mark on infinity, linking it to divinity through faith in God.

Infinity in the modern age:

The modern age can be considered as a period of time in which the acceleration of human progress was manifested, to improve the unfavorable situation of a few centuries ago, when faith dominated instead of reason. Progress has been manifested in all fields: politics, industry, society, economy, trade, transport, communications, science, medicine, technology and culture. Modernity is characterized by a mechanized and automated industry, gradually renouncing manual work.

A remarkable personality for the history of infinity in this period is the great Galileo Galilei, physicist, mathematician, philosopher and "the father of modern observational astronomy". In the 1600 millennium, the great mathematician manages to discover a new concept of infinity through a paradox starting from the mathematical and logical principle that the whole is greater than a portion of the whole. Galileo's paradox is a demonstration of one of the surprising properties of infinite sets.

In his final scientific paper, "Two New Sciences", Galileo Galilei made seemingly contradictory statements about positive integers. First, a square is an integer that is the square of an integer. Some numbers are square, while others are not; therefore, all numbers, including squares and non-squares, must be more numerous than just squares. And yet, for every number there is exactly one square; therefore, there cannot be more of the one than the other. This is an early use of the idea of ​​one-to-one correspondence in the context of infinite sets. Galileo concluded that the ideas of "less than," "equal to," and "greater than" apply to finite quantities, but not to infinite quantities. Through these statements, he managed to reawaken the desire of specialists to find an answer for infinity, and after his discovery, a new era of infinity began, which manifested itself in several scientific fields and changed man's vision of the impossible.

Infinity in the contemporary age:

The contemporary era is considered the period in which technology and science continue to develop throughout the planet and also the period that succeeds the one in which manual labor was replaced by automated and mechanized industry, known as the modern era.

The contemporary period represents the moment of great discoveries, and science and technology are in continuous expansion, because man continues to create inventions to ensure his livelihood, to find out his origin and to discover as many things as possible about the Universe, among which is the attempt to solve the infinite.

Throughout this era, but especially in the 19th century, there were prominent figures who addressed the problem of infinity. Foremost among the revolutionaries of contemporary science brave enough to discuss this controversial subject was Georg Ferdinand Ludwig. A German mathematician, the creator of a theory of sets that later became a fundamental theory of mathematics, Georg Cantor determined the importance of a one-to-one correspondence between two members of the set, defining an infinite ordered set, showing that the real numbers are larger than the natural numbers. In fact, the way to prove this theorem involved the existence of "infinite infinity". He found a framework where this restriction is not necessary and it is possible to define comparisons between infinite sets in a meaningful way, showing that by this definition some infinite sets are strictly larger than others. He also introduced the concepts of cardinal and ordinal numbers, along with their calculation rules.

In 1895, Georg Cantor coined the term "transfinite", wanting to avoid certain connotations of the word "infinite" when referring to these entities, which were not finite but not infinite in the traditional sense. Cantor was aware of the profound philosophical impact of his work and the controversies that would follow his discoveries and initiatives.

At first, Cantor's theory of transfinite numbers was considered counterintuitive and downright shocking. This led to opposition from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré, and later others such as Hermann Weyl, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devout Lutheran, believed that the theory had been communicated to him by God. Some Christian theologians saw Cantor's works as a challenge to the uniqueness of absolute infinity in the nature of God. Not all theologians opposed Cantor's theory, however; a relevant example is represented by Cardinal Johann Baptist Franzelin, who regarded the argument as a valid theory, following important clarifications made by the German mathematician.

He was seen by many philosophers, mathematicians and theologians of the time as a charlatan, a rebel "corrupter of youth" and a madman.
Although he was harshly criticized, Cantor continued to search for the infinite, gradually giving up his passion for mathematics due to the death of his son and taking up the philosophical side of the infinite, until he lost his mind. Thus, he spent the last year of his life alone in a sanatorium.
The ideas and theories of this genius continue to generate significant interest in this field today.

In the 20th century, more precisely in 1931, another great representative of the subject of the infinite stands out: Kurt Gödel. He was a logician, mathematician and philosopher, and along with Aristotle and Gottlob Frege, he is considered one of the most important logicians in history. His works had a profound impact on scientific and philosophical thought.
Immediately after completing his doctorate in Vienna, the young philosopher published the two incompleteness theorems. These theorems claim that any mathematical system (that is consistent and strong enough) includes true statements that cannot be proved. In other words, mathematics contains problems that cannot be solved.
Thus, through his conceptions, Gödel captures the impossibility of the impossible in mathematics, being, by itself, a paradoxical matter, in which even if a new idea is discovered, another uncertainty will automatically be created regarding the new discovery, and so on.

Conclusion:

The boundless was and continues to haunt us today. From ancient times it was logically approached and suspected because we were not advanced enough. In the Middle Ages, an attempt was made to explain it through religion and faith, because we did not want to recognize other methods. In Modernity we came close to it, or so we thought, but we created more problems by defining it. Finally, in the contemporary era, we tried our best to prove the infinite and its existence, being more advanced than ever, but we came to the conclusion that the infinite cannot be understood by finite beings but only guessed.

Personal reflections:

Infinity is a difficult subject to approach from any perspective without going crazy, that's why I proposed to add my mark and impression on it through literature, more precisely through a story of its own nature, and not a theory.

"He was standing alone at a pool of water, one slow and warm, the other cold and disturbing.

Lost in the concert of the stream and the deafening chaos, he was witnessing the creation. Next door, the universe lingered, looking for a sign from fate to exist, and nothing was waiting for it. The world was not, the celestial was floating somewhere around, but nothing noticed it.

A blind sight, the vast darkness polarly dominated the realm, like a curtain hiding the actors of a play, the edge of which swallowed up all light. Rivers roared unseen against the black background. He was used to nothingness with the unknown, he didn't know what it was, just as he didn't know himself. Why look if there was nothing to see? Why should he think if his mind was empty? Rhetorical questions, because he knew there was no one else with him.

He was enveloped by the sound of the waters that flowed invisibly and constantly. Curious, follow the sound of one of them for some time. He had already passed a wide loop and another crossing of waters; slowly the air had grown heavier and warmer. Suddenly, the nothing fell into the hot river, as if nearing the bottom, drowned in the white light and came out on the other side. Although he had no eyes or physical form, he could now see and burn. Not wanting to be alone, he looked for a friend among the galaxies. He joined the stars lost in the sky in tormented formations and invited the sun to dance. Charmed by the mystery of nothingness, the stars fled the sky with him, from the warm world to the empty one. To the hot river he was heading, to cover the road back home. It was drowning again in the black sky and scattering the stardust far and wide. Nothing was baptized from two waters, the banks of which end in loops, and in the middle they meet."

Bibliographic references:

David Foster Wallace “Everything and More”

John D. Barrow, The book of infinity

Valentin Curtef, Zeno and the concept of limit, November 13, 2009

article 8 of 102 of the "Philosophical Anecdotes" series

Father Arsenie Boca - great guide of souls from the 20th century, Teognost Publishing House, Cluj-Napoca, 2002, p. 86-87)

Safta Alexandru, "The art of infinity: history, mathematics, the impossible"

Cristina Cioaba, "Infinity Within"

"About Aristotle's infinity and that of Galileo Galilei, the visionary who achieved the impossible" - Moldova sovereign magazine

“Georg Cantor” – Enciclopedia Britannica

PHILOSOPHY MAGAZINE 1-2/2008 AFTER GÖDEL MIRCEA MALIŢA