Mathematics is a language, it is a way of naming, of describing things or situations.

We use mathematical language when we say that there are **4** people playing mus.

A child who does not yet know how to use it can only say: "Joseph, John, Thomas and another man are playing mus".

We give the GPS position of our favorite bar using mathematical language as: **34.123456 54.02134**.

The alternative is something like "the bar next to the big tree on the way out of town".

Like all language there are words and rules for using them, combining them, putting them together.

The words are the concepts: number (4), equation (x^{2}=9),... and the norms are the rules: addition, subtraction, multiplication, square root, etc.

To discover the truth in the world is to look with the eyes of Jesus Christ, it is to be conformed to him. Our life should be to become more and more like Jesus Christ: to try to look at things and people as He sees them and to behave as He would behave in every situation. He, as God, sees **the whole** truth. As we discover more (partial) truths, it is as if we get more pieces of the puzzle that shows the whole truth.

God, as Jesus Christ taught us, is "the Truth, the Way and the Life". Here we are interested in the "Truth" aspect.

In another article we spoke of the fact that there is only one Truth. This is well known especially to mathematicians, because together with Philosophy, they are the only exact sciences, all the rest are approximations to the Truth and, therefore, from various (multiple) directions and insecure, because they do not even know how far they are from the truth. That is, all the laws of Physics, Chemistry, tomorrow can be replaced by another more exact and general law. There are cases where it is already known that the laws used are not the best (electromagnetism), but of the best (Maxwell's laws) scientists say that they do not know how to use them. (In the "life" sciences - medicine, biology, psychology,... - it is much more difficult to get closer to the truth).

Well, mathematics leads us to God because it unveils the truth that is in all things. It is as if they were removing veils, or curtains, that cover the sancta-sanctorum, the center of the temple where God is.

The difference with the rest of the sciences is that everything it tells us are **truths**, always partial, but always true. And, as we advance, improving mathematical knowledge, we are adding truths but never refuting, correcting, rectifying what was said before. That is to say, whatever mathematics advances, two plus two will always be four, in any place, circumstance, in all eternity.

On the other hand, **the rest of sciences are always intrinsically false**, they can only give us **acceptable** **solutions** but never assure us that they give us the best solution (see this other article on what science can tell us and what it cannot).

Another difference is that the statements of mathematics are **universal**, they serve for everything, on the other hand the sciences use some laws for some things, others for others, and so on.

For these reasons, all the other sciences use mathematics, but the laws of the other sciences never help it.

Let us see this unveiling of the truth with the example of numbers.

First mathematicians only had the "natural" numbers: one, two, three,... and with them they described many realities of the world: "4 people playing mus", but they could not describe other realities: how to tell the centurion of the galley that 5 oarsmen were missing to complete the number of oarsmen? They had to use the number 5 and a word of spoken or written text ("missing"). There was no way to describe this reality in purely mathematical language. So they discovered the "integers", which are the "natural" numbers plus all the negatives (and the zero): the -1, the -2, the -3, and so on.

If we represent them on a line, first we had only the natural numbers.

The integers are "better"; because they serve for the naturals and for the negatives.

Integers allow us to dis-cover the existence of negative numbers, which are the essence of some situations.

The essence of my relationship with the bank is -345. (That I owe him 345 reais, wow). The essence of this year's "profit and loss account" of my company is -4,567 (loss of 4,567) (that is the essence, the most important thing).

Sometimes we have to find the number whose square is worth a positive number (which is the same as writing x^{2}= a (a being the positive number) or it is the same as saying that we are looking for the square root of a positive number. For this, positive numbers are enough:

But there are times when the essence of a reality (a particular truth) is described as the square root of a negative number. This is how they discovered the "complex" numbers, with their letter "i", which is the square root of -1.

Complex numbers have two parts: the "real" and the "imaginary" (i). We speak of a complex number but it is represented by two numbers, the second followed by an "i".

With them we can also add up the number of apples on a table by simply considering that the imaginary part is zero.

This small complication (that complex numbers are pairs of numbers) is largely compensated because thanks to them we can understand, name, some realities that God created and that until they were discovered we did not know how to name, understand, manipulate (or do it in a very expensive, complicated way).

God is simple and Omnipotent, that is why He prefers to use more numbers to describe, to name, a reality, than to use complicated formulas.

In this sense, God does not use complex numbers, but the so-called "**quaternions**" (which are like complex numbers but with three "imaginary" parts instead of one). With them, all the calculations of motions in space (rotations plus translations), are much simpler than with the numbers mentioned so far. They are used in videogames, flight simulators, etc. That is to say, they are currently indispensable.

Note that quaternions are the most general numbers we know, and the rest (natural, integer, complex, etc.) are not revealed to be false, but remain as particular cases of quaternions. |

In this effort to search for the truth, to uncover the deepest essence of everything (and at the same time the simplest, as mentioned above), the work of the best mathematician of the last 200 years is exemplary: Alexander Grothendieck (link to a website of his followers with a lot of information).

His main achievement has been to unify, to provide a "superior" point of view, more general, of a lot of mathematical specialties until then dispersed and isolated.

As with numbers, his discoveries turn those specialties, not into falsehoods, but into particular cases.

Apart from simplifying, another advantage of these generalizations is that with them we can name, address, situations that could not be addressed with previous ideas or of which we did not even know they existed.

Following the puzzle metaphor mentioned earlier, Grothendieck would be someone who would give us puzzle pieces that we were missing in order to put together various groups of pieces that we already had fitted together. And perhaps also to know where in the overall picture each group of the previous ones goes.

It is not famous, perhaps for a combination of reasons:

Grothendieck, with his discoveries, inevitably jeopardized the prestige of many famed mathematicians.

He was quite "anti-establishment", critical of those in charge, to the point of rejecting the most important world prize in mathematics (for people under 40 years of age) because he rejected the armamentism of the country that awarded it (USSR), abandoning a research institute after discovering that it was financed in part by the war ministry, and criticizing the misuse of science.

He was a man who seems to have lived continuously looking upwards (where God is with his simplicity and generalizations), both in his work as a mathematician and in his personal life: with a lifestyle unconcerned with earthly details (he ate only milk, cheese and bananas; he slept on a board, he did not accept to be anesthetized in operations, austere house and clothes,... (He lived in a village in the French Pyrenees, so probably the milk and cheese he drank were of the best quality).

He said there were two ways of trying to solve a problem (like cracking a nut):

The violent way, which consists of hitting it with a hammer (with the risk of accidents).

The way he used: "soaking" the problem (the nut) until it is so soft that the shell can be separated from the fruit "like the rind of a ripe avocado".

Applied to mathematical problems, this means that he did not try to solve the problems "at all costs", directly, accepting all the limitations that were necessary to reach the solution. He, by patiently going around the problem, listening to what the essence of things was telling him, was looking at the problem in a different way, and finding a solution indirectly, in a slower way, but which then allowed him to solve the problem in a very simple way.

As you can see, on 6/14/2024 I've added many articles. |

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