The magazine Notices of the American Mathematical Society published in October / 04 and November / 04 two articles by Allyn Jackson about Alexandre Grothendieck. He currently lives in southern France, isolated from the "normal" world. From eighteen to forty, Grothendieck had a brilliant performance in mathematical research. Its notable feature was the mental posture of seeking to view a problem from a natural point of view. For this, it is necessary to find out what is the natural context of the problem, that is, what is the “natural place where the problem lives”.
It is common in mathematics and physics to find problems separate from their 'natural place' or their 'birthplace'. The discovery of an interesting problem should not be confused with its originator. If a mathematician's imagination enters a natural place in mathematics, then it naturally sees the problems that naturally reside there. This is not a mere vanity of the mathematician. It is the best way, or the true way to understand mathematical objects and their relationships. It is the proper way to find mathematical truths.
For example, Pythagoras encountered the problem of what is the square root of two. He was not, in his day, like anything familiar to him. Nearly two and a half thousand years later, some mathematicians located the "natural place" of the square root of two. His name is "the complete ordered body of the real numbers”.
Grothendieck's life is inspiring for Brazilian students. These are often massacred in our schools with the pernicious notion that mathematics comes from everyday life. Perhaps Grothendieck's life can inspire the rescue of millions of Brazilian students who are forced to live together, and most succumb and incorporate it, with the idea that the path to mathematical abstraction passes through the "concrete." Concrete widely propagated in our schools is perhaps the most dangerous threat to Brazilian intelligence, because it kills at source the possibility of creating the minds of our children and youth.
It is because of this sad reality of the Brazilian school that the two articles by Allyn Jackson interested us in commenting here. To generalize means to restrict oneself to the essential. Therefore, it is natural for the thinking mind to seek the generalization of a concept, since it understands a concept when it is no longer masked. The essence of the concept is the concept itself, free of properties that are not necessary for its existence in its natural place.
When one preaches in school the belief that concrete comes first, there is a great deal of confusion in the student's mind because concrete is, at best, the concept full of properties that do not characterize it. On the contrary, concrete hides the concept of anything. To insist on concrete as a starting point for mathematical knowledge is to mutilate thinking ability, to prevent the student's mind from freeing itself from confusion.
“What makes the quality of the inventor's inventiveness and imagination is the quality of his attention when he hears the voice of things.. ” Grothendieck
We do not preach the idea that Brazilian students are new Grothendiecks. This is impossible even for mathematicians. However, we would like our students to have a better chance of getting rid of the darkness and mental confusion that plague our schools.
Hyman Bass of the University of Michigan stated that Grothendieck was a mathematician with a "cosmically general" view. Grothendieck's point of view has been, and continues to be, so deeply absorbed by mathematicians that it is very difficult for novices today to imagine that mathematics has not always been this way, comments Allyn Jackson.
Grothendieck had a knack for locating the natural place of a problem. Already in high school, he exercised his gift:What was unsatisfactory to me in our math books from gym and high school it was the absence of any serious definition of the notion of length (of a curve), of area (of a surface), of volume (of a solid). I promised myself to close this gap when I had a chance.”
When World War II ended in 1945, Alexandre Grothendieck was seventeen and went to live with his mother near Montpellier. He obtained a college scholarship and worked with his mother to harvest grapes when it was time. His mother also worked as a maid.
Grothendieck went to school less and less when he realized that the teachers basically repeated what was in the books. Jean Dieudonné stated that the University of Montpellier was one of the most backward in France in mathematics teaching. During his three years in Montpellier, Grothendieck devoted himself almost exclusively to plugging the hole he found in high school and high school books in terms of length, area and volume.
According to Allyn Jackson, he discovered the Theory of Measure and Lebesgue's notion of integral. Just as Einstein, who was still very young, developed ideas for himself in Statistical Physics that he later found to have already been discovered by Josiah Willard Gibbs.
In 1948, finishing his degree in Montpellier, Grothendieck went to Paris, the main mathematical center of France. André Magnier, an employee of the French education system, recalled in 1995, in a French magazine, how was the interview with Grothendieck for the granting of a scholarship to live in Paris. Magnier asked him what he had worked for in Montpellier. “Instead of a twenty-minute meeting, he explained for two hours how he had rebuilt“with the tools availables ”theories that took decades to build. He has shown extraordinary wit. ”
"Grothendieck gave the impression of being an extraordinary young man, but marked by suffering and deprivation." Magnier immediately recommended him for a scholarship in Paris.
Alexandre Grothendieck was born in Berlin on March 28, 1928. In 1966, he received the Fields Medal the most honorable mathematics award on planet Earth. He was one of the founders of IHES, Institute of Advanced Scientific Studies of France.
This institute was created by the French businessman, a doctor of physics named Léon Motchane, who envisioned the need for France to have an independent scientific research institution similar to the Princeton Institute for Advanced Studies in New Jersey, United States. It has become one of the best centers for mathematical research and theoretical physics on planet Earth.
We have already said that we do not claim that our students are new Grothendiecks, only that they can more easily break free from the darkness and confusion imposed on them in our fallen schools.
On the other hand, it is clear that some of our children are brilliant minds, but they will have nowhere to go if they want to study mathematics independently, enthusiastically, and entering the natural place located by Grothendieck.
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