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Back to the infinitely small…


Does the idea of ​​ever smaller quantities make as close to zero as we want? This is the famous, and difficult, problem of knowing how “out of nowhere something comes…”.

Perhaps the reader's intuition coincides with ours: "It is impossible for something to come out of nothing ...". If this intuition is physically correct, then there is no symmetry in the process of fractioning any amount of anything.

For example, imagine a time t positive. Next, let's imagine the sequence of times. t > t/2 > t/3 > t/4 >… > t/no >…> 0, representing a fractional of time tending to zero. Or even a quantity m of matter and the sequence m > m/2 > m/3 > m/4 >… > m/no >…> 0 representing a fractionation of a zero amount of matter. A being living in infinite time could accomplish the sequence m/no above approaching infinitely zero.

However, this same being could not perform the “inverse” process: for it would be impossible to take anything out of nothing. Immediately comes a question, unfortunately also very difficult: why the hell a process can not be reversed? That is, why can't it also happen the other way around?

In the June 2004 Scientific American Brazilian (Number 25), the famous physicist Gabriele Veneziano teaches us a little about String Theory that emerged from a mathematical model he proposed in 1968 to describe subatomic particles. The fundamental entity of the universe would no longer be a particle resembling a "dot" but rather a "line", still very small, but "larger than a dot." It turns out that a quantum string cannot be broken! According to the way it vibrates, a particle exists corresponding to that vibration. Like a guitar string that can give us many different notes depending on how it vibrates when tightened in a certain way. A quantum chord does not lose weight, so it cannot be split into pieces of weight tending to zero.

If one cannot reach a size smaller than that of the quantum chord, then the dimensions of reality are not just Albert Einstein's four proposals: length, width, height and time. According to string theory, there are seven more spatial dimensions.

Amazement, an electron, for example, is a quantum string whose ends move in the three spatial dimensions we can perceive, but stand still in the other seven!

String theory therefore proposes at least eleven dimensions for the universe. And there is no way to get infinitely small quantities. So the idea that we could go back in time, at least imaginatively, to the instant of time, is false! That is, also for the amount "time" there is no way out of zero because instant zero does not exist!

There must then be a minimum amount of time, as well as space, matter, etc., in fact there are already mathematical models for this kind of universe. In mathematics we are free from these limitations. The number zero is number like any other, although it has special properties. In the numerical line the symmetry between the infinitely large and the infinitely small is, say, perfect, and both exist!

Existence in mathematics is different from physical existence. Although there are also models of universes that “carry out any mathematical theory”. In the Scientific American Brazilian reader can find articles about the quantized universe, that is, about universe where space, matter and time do not exist below certain quantities, and about the models of "mathematical universes".

In mathematics, the existence of an object is given as long as it can be defined or axiomatized. The axioms prescribe the characteristics of initial objects that need not be defined. From these initial objects we can define others. But when this happens, we need to show that the definition makes sense, that is, that the defined object "exists."

Let's take an example. Axiomatically, let us accept that 1 and 0 exist, as does the addition of objects that produces new objects. So, there must be objects 1 + 1, 1 + 1 + 1,…, etc., and we define: 2 = 1 + 1! Thus, we define object 2 and show that it exists because it is 1 + 1.

We cannot define "anything" in mathematics and expect this "thing" to exist. For example, let's try to define the “three-sided square circle”. How do we show that it exists? If we try, we will face problems of intractable difficulty. We will not go there.

Having the intuition of the object that is worth defining (other than the “square circle” type) is the mathematician's job. Part of the intellectual pleasure of the mathematician comes from the "intuitive visualization" of certain objects, even if they have not yet been defined, or are already defined, but their existence needs demonstration. More intellectual pleasure the mathematician gets when he "sees" the demonstration of the existence of a certain object.

Mathematical existence, therefore, is merely a matter of coherence with a system of axioms and a compatible logical system of deductions. However, this does not mean that mathematics exists only as a mental elucubation. The History of Science clearly shows that mathematical ideas have always been of fundamental scientific importance.

Interestingly, physics has been a source of deep mathematical ideas. Everything indicates that it will continue to be and increasingly intensely. String theory is an example of mathematical theory awaiting empirical evidence of its physical reality and, at the same time, an example of a set of ideas in physics driving mathematical research, especially in the areas of topology and differential geometry.

It seems that by 2010 the Theory of Supersymmetry in Physics may or may not be corroborated, and this will finally bring clues about the validity of the mathematical theory of quantum chords for physical existence.

The mathematical existence of quantum chords is already a consolidated fact. Even if it has to be filed a few years from now. It has the same existence as the statement "the sum of the internal angles of a triangle is 180 degrees." The same can be said about the statements "the sum is greater than 180 degrees" and "the sum is less than 180 degrees".

Could the reader "visualize" three triangles with these characteristics? Think of triangles drawn on a sheet of paper, triangles on the peel of an orange, and triangles on the surface of a speaker.

There is nothing wrong with the mathematical existence of these three geometries, and likewise there is nothing wrong with the mathematical existence of Quantum String Theory.

If some physical theory is replaced by String Theory or not, both will continue to have mathematical existence as much as the three geometries above.

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