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

What is “axiomatic study of mathematics”? How do you get to know and to know mathematical truths accurately? The student who never asks himself these questions will never truly begin his study of mathematics, at least that fascinating mathematics we inherited from people like David Hilbert. We are ready for the second principle. Do not be deceived! If you are not capable of literacy for yourself, and correctly in mathematics, no one will do it for you. Your intellectual autonomy is your only hope of truly progressing in mathematics! You are at the same time free to think and doomed to distrust every statement that comes to you, until you make it yourself or become convinced that you could make it.

So check out everything you are learning in math. Both what is written in the books, and what is told by teachers, or by their peers. Don't let "your own thoughts" and your "intuition" slip away, as they can mislead or mislead you much more easily than it seems.

In 1931, the mathematician Kurt Gödel in Vienna, Austria, demonstrated that if mathematics, based on Russell and Whitehead's Principia Mathematica, is consistent, then there are mathematical truths that are not demonstrable. We imagine it embarrassing for a math student, whether he is a future math teacher, or a future bachelor of mathematics, to graduate without such fundamental knowledge. Here is one more simple criterion, derived from the first two principles: It is interesting to know if your efforts are not wasting away with obsolete mathematics; It is interesting to see for yourself whether your studies are contemplating a minimum of knowledge of Mathematical Logic, for example, notions of Gödel's Incompleteness Theorems, and the information that no one knows whether Mathematics is consistent or not!

By 1974, mathematician Gregory Chaitin endeavored to show that "most mathematical truths are indemonstrable." The student of conscious mathematics would not allow himself to be ignorant of this fact. If Gödel discovered that "there are indemonstrable mathematical truths," it was only 40 years later that it was interesting to ask whether the number of indemonstrable mathematical truths was not just an irrelevant number. But Chaitin has drawn attention to the fact that this number is not irrelevant, on the contrary, the number of demonstrable truths is that it is "small"!

We who write here have gone through all undergraduate and graduate school ignoring this fact. One excuse is that in the 1970s or 1980s scientific information was not circulating as widely, effectively and rapidly. Today it is very easy to know Chaitin's ideas, just visit his website on the Internet. This idea of ​​Chaitin is serious and has great repercussions on the way we study mathematics. If we take this information to its ultimate consequences, we will have to seriously consider the possibility of profoundly changing our current study and research strategies. We will not delve into this discussion here because our aim now is merely to suggest a few principles and ideals that, if followed by the student of mathematics, are effective in raising your consciousness to the historical height of the 21st century. For example, it is the case with those who write this column that it is very easy to go through one college, in fact many, ignoring fundamental information that is already part of mathematics.

Not surprisingly, there are many other examples. They are even dramatic. In the 1970s, while attending our first college, several spectacular revolutions took place in mathematics, of which we had absolute ignorance!

Students regularly enrolled at the Brazilian Faculties of Mathematics in the 1970s and 1980s were deeply occupied with their "ancient mathematical topics" and could not at that time be aware of the spectacular revolutions that took place in various areas of mathematics. It was the case of those who write this column. Examples: the discovery that nonlinear phenomena are not an irrelevant part of science, quite the contrary, they are almost the totality of nature!

Surprisingly, the discovery that the functions of the first and second degree already reveal, if studied from a certain point of view, the fascinating world of Chaos, the emergence of Order within Chaos, the self-organization of Chaos, the bifurcation present universally. in the phenomena of Nature, the Feigenbaum constant, Fractal Geometry (called by Mandelbrot "the true geometry of Nature"), and much more that we could not relate here. In other words, the fascinating and very serious discovery that nature's unpredictability and irregularity are already present in the behavior of “ridiculously simple, one-variable” functions!

Our suggestions are somewhat intended to prevent “our waste of time” from being repeated by current students who like mathematics and suspect that it is very important. No one here is claiming that it is easy to attend a mathematics faculty while at the same time becoming aware of the main facts that occur every day revolutionizing and advancing mathematics. But there are some facts that the mathematics student will never accept being ignorant of. For this he must follow certain noble principles and ideals that will guide him along a path that contains the main landscapes that need to be seen. How third principle We suggest the following: seek to distinguish books and texts, which didactically disseminate mathematical knowledge, from books written for specialists in a certain field.

The student of mathematics should look for books and texts that honestly competently disclose mathematical knowledge to a beginner. Unfortunately there are books that, instead of helping, hinder and can even cause irreparable damage to student education. A book is not error free just because it is a published book. The student should be aware that even good books can contain serious errors. Thus the prudent student will always seek to find out about a particular subject, whatever it may be, through several books, say at least 3! This is an arbitrary number, “a kick”, but it is much better than a single source and even better than just two reference sources.

There are certain "signs" that suggest the poor quality of a book or its inefficiency for student progress. For example, we recently came into contact with a “new to the market” spreadsheet that had no index. No reason has been given by the publisher for placing another Calculus book on the market, which is odd as there are already hundreds of good Calculus books available for purchase. The student should look for a good reason to study a book. If the book has no index, then it may have been marketed to the publisher's great disregard by readers, since the index is of great importance to any reading and study strategy.

One more example: whenever we examine a book of calculus of various variables we look for the demonstration of Stokes' theorem; if it seems to us to be well explained, intelligible and correct, then we will have a great attraction for this book. Another example comes from Algebra: A book on Group Theory that does not present Sylow's Theorems in its full form, including the group action demonstration strategy discovered by Wielandt, does not deserve credibility from the 21st Century student. The importance of these theorems and the beauty of Wielandt's demonstration are unmissable for a spirit of the Information and Knowledge Age.

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