We still have a good way to go before we finish examining the first truths of Zermelo-Fraenkel Set Theory and thus the first truths of mathematics. We still need to introduce the Infinity Axiom, the Choice Axiom, and the Replacement Axiom Scheme. However, to finish 2001 and start the second year of the knowledge age, we would like to make some suggestions for perhaps helping the mathematics student, or lay people who like mathematics, to expand their mathematical awareness in year that begins.
From our point of view, the best way to progress in mathematics is to cultivate some simple but noble and very clear principles and ideals. What are these principles and ideals? The first principle is: "do not confuse mathematics as objective knowledge, fundamental to the scientific knowledge, and cultural heritage of all humanity, with the activities developed in our official schools." We are not suggesting that the student abandon his school and social life, this is meaningless. But "normal" school life has little to do with such ideals as curiosity to know how science really works and what the true role of mathematics is in scientific knowledge. It is possible to reconcile normal school life with the cultivation of certain ideals but it is important to realize that these are very different things. Our recommendation is for those who enjoy, admire and want to make progress in mathematics for noble reasons, that is, because they have deep questions about human life that await a clear and objective answer (we are not promising any kind of mathematical solution, as if magic, to the fundamental problems of human life. We only want modestly to support those who seek mathematical knowledge to at least think more rigorously and more efficiently in their search for the path of scientific knowledge, a meaning for life. The first recommendation is that the student try to be self-taught and seek instruction through good books and good texts. In the Age of Knowledge, which is now beginning its sophomore year, there are books galore, and a huge amount of text available on the Internet, mostly in English, for almost every area of Contemporary Mathematics, which gives the student complete freedom to study. develop his self-taught work in his effort to rise to the historical height of his time.
How to tell if a book or text is good? This is one of the problems the self-taught student has to face. We only know one way: the patient's research and organization of available information on good quality mathematics. We see no other solution than patiently learning to recognize good books through certain details such as critical reviews by other authors, citation in books already recognized as quality, and the presence in their content of certain themes and information that indicate the seriousness and competence of the authors. For example, we recommend caution with a book on Mathematical Foundations that does not in any way contain Set Theory topics, Mathematical Logic topics, and does not mention, for example, ideas from mathematicians such as Georg Cantor and Kurt Gödel.
All areas of mathematics have progressed prodigiously in recent decades, around the world, especially in the United States, Europe, and Japan. Students need to be aware that their studies are leading to even small and superficial knowledge. of Contemporary Mathematics, unless you have no interest in understanding a bit how Contemporary Science works, for example, Physics.
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