INFINITY – AN INTIMIDING CONCEPT

Valentine Simion

Motto: „No other problem has so shaken the spirit of man” – David Hilbert

Infinity was also a point of attraction for Greek mathematicians before Aristotle. But the discussions about the problem of infinity became very fierce, hence David Hilbert's outburst about the current infinity. For mathematicians, infinity comes in two different forms: potential infinity and actual infinity.

Potential infinity is the infinity with which we are familiar, and we feel it almost whenever we trace the series of natural numbers with our thoughts; we know that after the number we imagine for a moment comes another with one unit more. The current infinite, however, is completely different. It is a terrible dream, it laid many traps for mathematicians, and those it caught could not escape its spell.

I found in Oskar Becker's book, “Fundamentals of Mathematics", the following account of how infinite multitudes – currently two of the most vain defenders of this notion – imagined. Dedekin said in connection with the concept of an infinite set that he imagines this set as a closed bag that would contain certain determined things, but which would not be seen and about which nothing would be known, except that they exist and are determined.

Some time later, Cantor expressed his representation of the infinite crowd by pointing up his colossal stature and describing with a raised arm a grand gesture and then looking into space he said: "a crowd I represent to me as an abyss." Long before and Aristotle in Physics he dealt with this problem and wanted to put things in order because the Greek philosophers before him, using the notion of actual infinity, had reached paradoxes, that is, conclusions that were downright monstrous from a logical point of view. Aristotle showed that the notion of infinity cannot be placed alongside the other mathematical notions such as number, line, plane, etc.

Infinity exists in potentiality. It is not permissible, however, to take potential existence as is done, for example, when a block of marble is considered to be a statue in potentiality because it may once and once be a statue. The same will not happen with something infinite in potentiality: we must not assume that it will be infinite in the act, it is not allowed to consider the infinite as something concrete, determined, as for example, a man, a house, but as one speaks of a day, of a holiday, whose existence does not have the meaning of an entity, but always the meaning of something that appears and disappears. And even if it is always limited, it is still something different and always different. This is also the conception of infinity that Euclid used in his work: a natural extrapolation of our ancestral experience.

Aristotle's views on avoiding actual infinity in mathematical speculations have endured for over two millennia, as there are still many mathematicians today who deny it. In the last century Gauss did, expressing in 1831 his horror at the present infinity in the following form: "I object to the use of infinite magnitude as something definite, it is never admissible in mathematics. Infinity is only a way of speaking, its true meaning is only a limit to which certain ratios approach indefinitely, while others may increase without limit”.

And with all these protests, no matter how vehement they were and will be after Georg Cantor introduced set theory into mathematics, the current infinity still called transfinite was accepted by most mathematicians. And Hilbert declared enthusiastically: "no one will take us out of the paradise that Cantor created for us”.

Variations on the same theme

Space is a form of existence conceived as an infinite, continuous, three-dimensional expanse containing all that is. Many treatises, books, studies have been written about space trying to answer questions like: who created it, how was it born, how long will it last, how is it perceived by man, is it infinite or finite, does it change over time?

About the origin of space we have a first answer in the Book of Creation: space, like everything that exists, was made by the Creator. If we knew, we would also become creators, which is not a common sense thought.

Another secular and purportedly scientific answer, but impossible to objectively validate, is that Space came later Big-Bang(The Big Bang). In science there is great freedom to define and use in theoretical models many kinds of multidimensional spaces. For example, the Hilbert space, a vector space with an infinite number of dimensions, with no relation to the physical dimensions of the system. People with a lot of imagination, especially philosophers and theorists, and concerned with this problem have formulated answers that are still and not experimentally proven.

Among the first people interested in the problem were the philosophers of Greek antiquity: Plato, Aristotle, Zeno, Euclid and others. Then, after about 2000 years, there appears another pleiad of "curious" physicists and philosophers such as Newton, Einstein, Planck, Berkeley and many others. Everyone had an opinion, interpretation, explanation, for what we call Space. The only rational being on Earth, man, perceives space as having three dimensions. Only an immaterial, virtual plane can be thought of as having two dimensions. The thread, no matter how thin it is, still has three dimensions. Existence is three-dimensional; the rest is metaphysics or mathematics.

Again, an inescapable question is: how was Space born and when? These two problems put our reason at a great impasse. We are wandering in a forest of symbols and mathematical reasoning. "All knowledge floats on a bottomless sea", said Werner Heinsenberg. However, someone must have the last word. Otherwise every argument can be countered with another and we will never finish. Our knowledge can only be limited, while our ignorance is inevitably infinite. Cosmologists are the ones who face the infinite."They make frequent mistakes, but they never doubt”, said Lev Landau.

Very short

Infinity, a question that disturbed the human spirit like no other.

Infinity, an idea that has influenced human intelligence like no other, so stimulating and out of the ordinary.

Infinity, a notion that needs clarification like no other, if possible.

Infinity, the essence of reality, a continuous process of self-transcendence.

The infinite, approximate and real, fictional and factual.

Infinity, a room without floor, walls and ceiling.

Infinity cannot be traversed rationally.

Infinity, a monster that must be tamed by mathematicians.

What is mathematically clear is physically cloudy.

Is space continuous? Can it be divided to infinity? They say no. It divides up to the Planck length, equal to 1.616229 X 10 ^-35 meters. How did he get to this result? Only Planck knows.

Black holes - real infinities? They are said to have infinite density and curvature. However, if there were any scientific common sense, they seem to belong more to myth or the category of the fantastic than to the real universe; mathematical artifices, figments of the human mind. In physics, a rule would be needed according to which if you predict a singularity or something infinite, it means that the theory breaks down and another one is needed. "Your math is correct, but your physics is terrible" - Einstein about Lemațtre.

Questions seeking answers

Space stretches to infinity. How can space not be infinite?

What is the ultimate fate of the Universe?

What is the origin of all things?

Is knowledge given only by senses and reason?

How can an infinite space be created in a finite time?

What is the Universe evolving towards, expanding?

What caused it Big-Bangthe? We have evidence of the singularity Big-Bangthe?

Are galaxies really receding or is space expanding?

What is dark energy?

Are galaxies really receding or is space expanding?

Infinitive, adjective or noun?

Does infinity have degrees of comparison?

Everything is phenomenon and nothing is rational?

A fool can ask more than a thousand wise men can answer. And thus I have brilliantly confirmed the above saying. But the misunderstanding is fleeting and should not discourage us. "It's my job to convince you not to give up because you don't understand. You see, my physics students don't get it either. And that's because I don't understand it myself. Nobody understands” said Richard Feynman.

We don't understand great poetry either, or we each understand it differently. So is infinity. I dare say that poetry or art, to be more comprehensive, is the shadow of infinity on Earth. Stars go out and re-ignite in an infinite number of cycles.

Or, as the Poet said:

"A sun that would go out in the sky

The sun is shining again

Seeming forever to rise

Finally death grazes him

For all are born to die

And they die to be born".