Ed Gerck

Planalto Research · Mathematics and Physics

INTRODUCTION QM stands for quantum mechanics, CS for computer science, CC for cellphones/computers, and Mathematics seems to have somehow missed all three. Now is the time to catch up. We have come to a stand off. Students are ending up neurologically sick -- feeling math-averse (neurophobia and math trauma). It is not a matter of selection. Only group/rote work [1] seems to infuse Mathematics, as taught today, even in highly-qualified students. Paradoxically, one finds that the more intellectually qualified, the more averse! And yet current Mathematics still tries to "instruct" it to the students, and other disciplines, in the US and the world. This failure has been denied [1], and the “fault” has been put on the victims -- the students. Paradigm shifts are needed: first, QM. According to QM, only integer numbers should be necessary to build all one can see in nature. Otherwise, all of Physics would be contradicted. This would also contradict all of CS and all of CC. Computers, for example, only work with integers, and are error-free. Since this contradiction is not even imaginable, QM is ontologically correct. Therefore, all sciences must reflect it, according to a "holographic principle" (HP) in nature, including Mathematics. The HP is often referred to as "the micro as in the macro" or vice-versa. These two words macro and micro are commonly viewed as antonyms, meaning that they are the opposite of each other. Macro means on a large scale. Micro means on a very small scale. Both are important, though often complementary, views of nature. The view from the macro, while taking into account the micro, is called "universality" in Physics. The QM main principle was given by Niels Bohr as, "all states at once". This needs to be understood as different from the following possibilities: "Copenhagen interpretation", or a "collapse" of the quantum function upon measurement, or waves, as a picture of QM, or wave-particle duality, or the Heisenberg principle, or a probabilistic view of QM. In opening the "black box" of QM as viewed by Bohr, QM does not represent Natur (defined by the philosopher Kant), but Wirklichkeit (ditto). It is not how nature is, but seems to work. A story. A known analogy is the Plato's Cave". The shadows one can see on the wall are Wirklichkeit, and the open reality outside is Natur. After watching Wirklichkeit for some time, people can make an idea what Natur looks like, even though imprisoned in the Cave. QM is, ontologically, how one can describe Wirklichkeit -- which is the subject of Physics. Natur may not have QM behavior, and continuity may exist in Natur. Mathematics is not concerned with Wirklichkeit only, but can include Natur. Although, as no one can see Natur, one is led to treat any "pure" Mathematics as a speculation. Albeit, Mathematics must agree with Physics in the realm of Wirklichkeit. This does not happen today, and creates a clash. The inconsistency is seen as an opening. What IS mathematics? Today's calculus teaching can be seen as relying on outdated ideas, going back before QM, computer science (CS), and cellphones/computers (CC) were discovered. Three major paradigm shifts seem to have been missed. But these three paradigm shifts represent how science is done, in our seemingly endless task in going from Wirklichkeit to Natur. One recognizes then, that science does not behave continuously, but by jumps -- that break from the past, and open a new future -- in going from Wirklichkeit to Natur, and back -- does the envisioned Wirklichkeit reproduce the actual Wirklichkeit? This has been called a "paradigm shift" (PS) by T. S. Kuhn in Structure of Scientific Revolutions [2]. David Hilbert [3], at the turn of the XX century, proposed twenty-three problems intended to guide research in the dawning century, claims otherwise, that “History teaches the continuity of the development of science.” Michael Harris [4], a mathematician, writes that he, "would still be glad to lift the veil, but we no longer believe in continuity. And we may no longer be sure that it’s enough to lift a veil to make our goals clear to ourselves, much less to outsiders." The outdated ideas currently in Mathematics date back to the time of Newton, Leibnitz, and Cauchy, before the 3 PSs mentioned. They include (aka Fictions, or incorrect models): microscopic continuity, infinitesimals, hyper-reals, Cauchy epsilon-deltas, and Cauchy accumulation points. Even in 4-year universities one finds classes, such as at Caltech, MIT, CSU, and abroad, teaching Fictions today [1]. This failure can be denied [1], and the “fault” can be put on the victims -- the students. But, there is no feeling of inferiority. PS [2] combats this, American style -- by innovation. A technical innovation can reduce this gap by a PS [2] -- already heralded and error-free, as one can see in the history of sciences (contradicting David Hilbert [3]). This reduces risk by following an experimental model that works, even though no one has explained it mathematically, without Fictions. By jump-starting to QM one has a solution, which we mark as a first paradigm shift. Where Fictions interfere with known Physics, they must be mercilessly deprecated -- although, and yet, current Mathematics still tries to "instruct" it to the students, in the US and the world. The other paradigm shifts, stand for computer science (CS), and cellphones/computers (CC). We now introduce a coherent, holographic principle (HP), a universe, and understanding yet to be discovered, as two new paradigm shifts -- and gain a cosmic perspective: who is the Creator of all this marvelous scheme? This final PS leads to a comment. One does not need to learn anything, confirms https://www.researchgate.net/profile/Robert-Fuchs One just has to observe. Mathematics seems, thus, to be discovered — as a hologram and coherent with other sciences, and not, somehow, invented by mere humans. ACTION PLAN This new RG project will follow this -- in QM, and reforming mathematics by a PS [2]. It is future-proof, as special relativity is, and promises to be as revolutionary to several disciplines. The main innovation is to obtain the same traditional formulas, with added results. But only using integers, which the Mayans did in astronomy, Per Martin-Löf in CS, and CC use -- avoiding Fictions, and any difficulty to teach them [1]. AG embraced this method successfully at top public high-schools in La Jolla and San Diego, Calif., avoiding group/rote work, using its futility [1] to the students' advantage. To counter-exemplify, if one calculates a derivative using Fictions, one would soon pass the matter limits in molecular size, atoms and even quarks. This makes results unphysical and untrustworthy -- as students know today. To exemplify, CC can calculate this text as it is stored in magnetic domains, and subject to many read/write errors. Coprocessors do not calculate in “reals”. Encryption defends e-commerce. All these methods use only integers, and are error-free. To further reduce business risk, facing expected skeptical opposition, we plan to add more technical personnel, attracted by the timely opportunity, combating fear, uncertainty, and doubt (FUD), and the Dunning-Kruger effect, through several PSs [2]. We envision a series of books, talks, and online material, emulating today the goals of the School Mathematics Study Group of 1958, when Sputnik stunned America, and learning from opposing FUD. The books include some introductory self-teaching books and others, more advanced. REFERENCES [1] https://edsource.org/2022/high-calculus-failure-rates-thwart-students-across-csu/664771 [2] T. S. Kuhn, Structure of Scientific Revolutions, 1962. [3] David Hilbert, Paris International Congress of Mathematicians (ICM), 1900. [4 I] Michael Harris, “Mathematics Without Apologies”, Princeton University Press, ISBN 978-0-691-1-17583-6, 2017.

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We are revisiting the Scientific Method. It can be better understood using TCS (theoretical computer science) and univalent mathematics theory. It is not a guess. It is not testing many probable hypothesis.