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Principles and ideals of a 21st Century (II) math student


We do not claim to dictate the best way to progress in individual knowledge of mathematics. Our claim is far more modest. It comes down to recommending a few values ​​in the form of simple, yet effective principles and ideals to help students optimize their efforts to advance their knowledge of mathematics. Our goal is, in other words, to help the student achieve a minimum desirable mathematical knowledge for someone who wants, even superficially, to share the mathematical and scientific thinking of the Information and Knowledge Age. Students, who imagine that being regularly enrolled in colleges and "normally" pursuing their courses, are likely to be exposed only to past mathematical knowledge, which contributes little to an understanding of contemporary science. The motto "is very common"you They must first learn ancient mathematics, because it is basic, and without it one cannot understand anything about current mathematics.Exaggerating a little, we believe it is not so absurd to think that it is possible to acquire a more or less deformed mindset from Isaac Newton's time, which would not be all bad if the student at least became competent in Newtonian physics! Differential and Integral Calculus is a 17th Century knowledge. If the student does not become aware of Elie Cartan's work and ideas, developed around the 1920s, he will inevitably become a Mathematics teacher. , or a Bachelor of Mathematics, or a Master or Doctor, with a very similar, and probably less competent, mathematical mentality than nearly 400 years ago.

It's a real possibility that a 21st Century math student will end up "graduating" with a mentality roughly equivalent to that of the 17th Century, or the 18th Century, etc. The pressure to learn only topics from the first half at most 20th Century, is irresistible. We do not see how, in Brazil, the overwhelming majority of mathematics students can escape this reality. However, as from our point of view, a truer mathematical background is only possible through the self-taught route, we think there is a natural way out which is as we were explaining in the previous column. If a student can absorb knowledge of 20th Century mathematics in his mathematics course, he can be satisfied. This is because, in many cases, he is under pressure to devote his time to much older and anachronistic mathematical theories, until about 2500 years ago. As is the case with Euclidean geometry. It is very common in mathematics colleges to spend a great deal of time with this discipline without even bothering to present it as David Hilbert put it in the early 20th century.

Therefore, we already have two very simple criteria, derived from the first principle proposed to study mathematics. The student must verify that he is not wasting his vital time, which is still very short, with obsolete and useless knowledge for the rest of his life as Euclidean geometry before Hilbert. Or the Differential and Integral Calculus without Stokes' Theorem (even if only commented and illustrated through simple but interesting and meaningful applications and examples, emphasizing Elie Cartan's ideas about what is after all an integral and especially , the deep and extremely beautiful symmetry of Stokes's formula).

A Mathematics course without the Fundamentals discipline, in the correct sense of the term, that is, Fundamentals as Set Theory and Mathematical Logic, is a totally useless course for the student who dreams of raising his consciousness to the historical level of the Information Age and the Knowledge. Simply because in such a course the student, except by accident or miracle, will never know, for example, that It is not known if mathematics is consistent!

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