A SERIOUS PROBLEM IN EDUCATION: THE DUALITY OF THE "INDEFINITE INTEGRAL"

1. Dear ladies and gentlemen. I am seeking collaboration with organizations involved in mathematics and logic education: from educational institutions to ministries and government agencies. The fact is, we have uncovered and exposed an entire niche in the education system—integral calculus—where basic and fundamental concepts are taught and defined completely incorrectly. A multitude of textbooks and reference books contain these tragically flawed definitions and constructions. An example of such incorrect definitions in the most fundamental areas of mathematics is presented in our international publication, available in various languages: https://zenodo.org/records/16789804 ( Discussion and questions are available here: https://www.youtube.com/@Kohan65 ). This 15-page document, understandable to NON-MATHEMATICIANS, describes the problem of the incorrect teaching of the indefinite integral, which is found in many textbooks in various countries. In these countries, a situation has been created where categorically different, mutually contradictory concepts coexist under the same term. At the same time, students are being indoctrinated at a state level with a "pseudo-understanding" that all these incompatible definitions can be simultaneously correct. This article of ours will be useful, interesting, and engaging to read for everyone involved in teaching or studying integral calculus, and for those university graduates who feel that they have some murky spots left in their understanding of the integral. Or if they have a mindset about integrals that says: "this is not for me." I also recommend that all engineering firms and design bureaus have their employees leaf through this article, just in case. It's understood that we would be very grateful for positive and meaningful feedback on the article, both from specialists on this matter and from people who feel that their understanding of the topic has significantly cleared up. It is important to note here that this is not about any kind of "flat-earthism" or "denial of science." In advanced countries, teaching is done correctly, but unfortunately, in many others, even highly developed ones, the education system and textbook publications have created this confusion. Dual, mutually contradictory definitions are contained in both textbooks and reference books, as well as in the teaching system as a whole. A debilitating situation is created for pupils and students: they are forced to consider both correct and incorrect definitions as correct at the same time. The most severe consequence for the mind is that the student becomes accustomed to the idea that it is "normal" to consider mutually incompatible things to be correct. It is also terrible that university students spend three or four years on integrals without really understanding what they are. This is a humiliating, disgraceful, agonizing, and wasteful pursuit. For many, this leads to the collapse of their studies. In our article, you will also learn many interesting things about how this "conspiracy theory" is structured. This problem is currently causing significant harm to your country in science and engineering. A significant number of the smartest students often drop their major in their second year because of this very problem. In this relatively small publication, you will be able to see our innovative approach. The fact is, authors and educators who teach incorrect definitions are unwilling to discuss this topic when presented with serious arguments. Our article uses the technique of a dispute with artificial intelligence, which draws knowledge from publicly available textbooks. The AI itself puts forward counterarguments, rather than the authors inventing them on behalf of opponents. These arguments are broken, one after another, by obvious scientific reality. The difference in such discussions is huge! Artificial intelligence cannot say: "I don't have time for this right now," "I have to leave," "I'm not interested in this," "we're on a different topic here." It cannot suppress information or switch to other topics. Today, artificial intelligence possesses greater knowledge and reason than most of the figures "stuck" with the wrong definitions. My co-author and I are interested in the following forms of collaboration: Remote consultations, clarifications, preparation of methodological and educational materials (all in written form). Our participation in the development and implementation of plans to correct this large-scale incorrect situation with flawed textbooks and teaching. 2. Today, in many countries around the world, in the field of integral calculus, which is fundamental to mathematics, science, and engineering, three sequential incorrect or vague definitions (antiderivative function, definite integral, and indefinite integral) are thrust upon the student. We are interested in collaboration, as well as sponsors, investors, and publishers, to present the foundations of integral calculus clearly, simply, correctly, and understandably from the very beginning for educational systems in different countries and languages. 3. In over 30 years of work in this field, S.K. has identified erroneous teaching in several other fundamentally important topics. We are also open to collaboration in this area of pedagogical reform at the state and country level on several other subjects. These include: logic, programming, English, computer science, physical education, health basics, and singing. 4. As of today, we are also looking for sponsors, patrons, and partners in a very unexpected new direction, for which we have great groundwork and opportunities to help people. We want to become co-founders (contributing knowledge and technology) of various international and national associations for people who have dropped out of higher education. We have accumulated very serious groundwork that can help these people both with legal issues and with self-education through fast and effective self-learning. Among people who have discontinued their education, especially due to poor performance in mathematics, there are many who are interested in our work, and here we can provide great synergy in the long term (self-education, reinstatement in studies, their own special path in modern education). Ukraine. Kyiv, 08.2025.