In addition to thanking readers for the effort of reading these small reflections that we bring each week on different aspects related to mathematics, which we write with the exciting idea of trying to make them closer and more understandable, we value the contributions that some of you leave in the comments. Many times they enrich the content of the article, and other times they shake us and make it quite clear that it is difficult, no matter how hard we try, to really convey the true essence of what we work, teach and study.We also perceive it almost daily in our classrooms with the doubts that students ask us. Most end up assuming the procedures, techniques, algorithms that we explain and apply them with some ease in solving exercises, but they do not understand what the purpose of this discipline is, or how it is organized, or how it is carried out. they deduce those procedures that they see work, but that in one way or another, they assume, period as if it were something revealed and inaccessible.

In the highly recommended book In honor of the human spirit. Mathematics today (Jean Dieudonné, Alianza Editorial, SA 1989), which also tries to offer an overview of our profession, the author warns that «it is very rare to obtain a valid answer from the interlocutor if he has not received the corresponding mathematical education, for at least the first two years of college. Even some leading scientists in other fields often have only aberrant notions about the work of mathematicians’.Of course, the author’s knowledge of the subject is beyond any doubt: Jean Dieudonné was one of the most active members of the Bourbaki group (a group that proposed to review all known mathematics around the 1940s and provide all its branches with extreme rigor), an expert in algebraic geometry and functional analysis.

In spite of everything, we are many enthusiasts and idealists who believe that the attempt is necessary (as I wrote these lines, the notice of the new RSME Bulletin appeared on the computer, in which the new editor of publications of the entity, Joaquín Pérez, indicates that “it is our duty not only to create new mathematics but to transmit those already created, ensuring the survival of the advancement of knowledge in this field”). In my case, my greatest motivation is to try to show perfection, within its limited limitations, of the, for me, the best way to get closer to understanding everything that surrounds us and to share the incomparable satisfaction that comes with ending up demonstrating rigorously something you’ve been thinking about for a long time. In particular, tell it to the youngest, those who can take over in that great company.In the highly recommended book In honor of the human spirit. Mathematics today (Jean Dieudonné, Alianza Editorial, SA 1989), which also tries to offer an overview of our profession, the author warns that «it is very rare to obtain a valid answer from the interlocutor if he has not received the corresponding mathematical education, for at least the first two years of college. Even some leading scientists in other fields often have only aberrant notions about the work of mathematicians.

Of course, the author’s knowledge of the subject is beyond any doubt: Jean Dieudonné was one of the most active members of the Bourbaki group (a group that proposed to review all known mathematics around the 1940s and provide all its branches with extreme rigor), an expert in algebraic geometry and functional analysis.In spite of everything, we are many enthusiasts and idealists who believe that the attempt is necessary (as I wrote these lines, the notice of the new RSME Bulletin appeared on the computer, in which the new editor of publications of the entity, Joaquín Pérez, indicates that “it is our duty not only to create new mathematics but to transmit those already created, ensuring the survival of the advancement of knowledge in this field”). In my case, my greatest motivation is to try to show perfection, within its limited limitations, of the, for me, the best way to get closer to understanding everything that surrounds us and to share the incomparable satisfaction that comes with ending up demonstrating rigorously something you’ve been thinking about for a long time. In particular, tell it to the youngest, those who can take over in that great company.

In short, the proof is reasoning that links a set of logical deductions between previously known and proven statements (or accepted for their indisputable validity; these, a few, are the axioms, absolute truths, such as, for example, that given a point and a length, it is possible to construct a circumference with the center point: it is indisputable because, among other reasons, I construct it physically). Until now, no civilization prior to the Greek in which this type of reasoning was made is known, therefore, we can consider the Greeks from the 6th century BC. C. as the precursors of logical-mathematical rigor. As Dieudonné indicates in the above-cited book, the earliest known written proofs appear in texts by the Greek philosophers Plato and Aristotle. In the «Meno» dialogue, Socrates wants to show a young man how to find a square whose area is double that of another given ABCD (that is, the doubling of a square, a step prior to the attempt to duplicate a cube, a problem that we know is unsolvable in general). The innocent young man’s first response is to double the size of the square (see image). Then, patiently, Socrates explains that this square would not be double the dice but quadruple. It is clear from the drawing that there are four squares ABCD in the purple square. However, a more analytical mind would demonstrate this from the well-known expression of the area of the square: if the side of the square ABCD is L, its area is L ^ 2. By doubling the side, that is, 2L, the area becomes (2L) ^ 2 = 4L ^ 2. Both (the graph and the formula) are verifications that the proposed object does not answer the question posed by the philosopher.

Next, Socrates tells him to build a square whose side is the diagonal of square ABCD. Note that said diagonal (both in purple) is the hypotenuse of the isosceles right triangle ADC. Therefore the Pythagorean theorem is involved (to see if the reader is able to demonstrate in this way that the red square is exactly twice the original and that this gives a general procedure, a proof). With the drawing (and origami, as a didactic resource), it is clear that if we fold the AA’B triangle on the side AB inwards, and repeat the same operation with the other three triangular flaps, we will have a single square, the original, which shows that the amount of paper we have is twice the square ABCD. Is this a proof just as rigorous as the Pythagorean theorem? In this case, yes, but the drawing is not always a rigorous demonstration, although, strictly speaking, that verifies the result once the construction is known. But, without that construction given, how do we find it from the statement alone?

As mathematics teachers (according to Dieudonné, most of us are just that: in order to call someone a mathematician, “I should have found and published at least the proof of a non-trivial theorem”) we are somewhat polyhedral in some respects, try to show ( or refute) if it is possible to make the previous drawing with a single stroke and without going through the same place twice (except for one point, that is, it is not possible to go through the same side twice, but through the same point), and already put, if it is possible to finish in the point that began. That is, in terms of graphs, is the above drawing an Eulerian or a Hamiltonian cycle?

The above is a constant dispute between mathematicians and students. Is a drawing worth demonstrating? It is clear that a good drawing can help you a lot when it comes to demonstrating something, but be careful !!: it is not always worth a demonstration. I give you another couple of examples for you to ponder (there would be much more to say on the subject of the demonstration, but this is a newspaper review, not a book, so, I am sorry I cannot delve further into the matter for today; in any case, fortunately almost no colleague reads these reviews. This is another constant in mathematicians: there is no time to read trivial and/or well-known things, it takes to work on the investigations that we are, it takes a lot of concentration And in the same way, it brings us to the fresh if what we are working on has a practical application or not. We are in our business and it is over, whether others like it, understand it or not, Science with capital letters that are indifferent. So to the reader who was saying that we discover things that “are not useful at this moment but can be applied later”, speaking again in terms of possibility, it is completely indifferent to us. That is why it costs us a lot to write ibir columns like this, among other things).

Let’s go with the examples.

I leave you another question that comes to mind. Proving this, would it be the same as proving that the sum of consecutive odd numbers is always a perfect square?An applied student would give us a demonstration by induction, for example. If someone made us a drawing like the one shown below, would it serve as a rigorous demonstration?And the second: we know that the surface of any triangle is half that of a rectangle. So with two equal triangles, like dice, can you always construct a rectangle? Of course, it is not worth breaking, cutting, bending, or doing any evil with those poor triangles. If this is not possible, then is the result false? As a final reflection, that of the writer and philosopher Bernard le Bovierde Fontenelle, in 1699. We tend to call things that we do not understand useless. It is a kind of revenge and since mathematics and physics are generally not understood, they are considered useless. Fontenelle is also credited with the following quote, on which more than one should reflect. Don’t take life too seriously; In any case, you will not get out of it alive.