Various scientific disciplines, including biometrics, chemometrics, psychometrics, and econometrics, share a deep connection with statistics but often employ specialized applications. Effective communication across these interdisciplinary boundaries necessitates a common language, and mathematics serves as a universal bridge among the natural sciences. This article centers on the language of mathematics and its role in scientific modeling. The thesis posited here is that mathematical models should be formulated in a manner comprehensible, with some effort, to serious researchers across all disciplines, despite potential cultural barriers. The notion of "understanding" is multifaceted, exemplified by the enigmatic realm of quantum mechanics. While quantum theory is traditionally presented in an abstract language that facilitates calculations, its interpretation remains contentious, posing challenges for researchers outside the physics domain. Notably, quantum mechanics was initially expressed in distinct languages—Schrödinger's wave mechanics and Heisenberg's matrix mechanics—before being mathematically unified. This unification yielded a mathematically elegant yet conceptually intricate theory, accompanied by over 16 partially conflicting interpretations. In the works of Helland (2021) and Helland (2022a), a novel foundation is proposed, potentially more accessible to researchers beyond the quantum community. It introduces conceptual variables denoted as θ, emerging from inquiries like "What will θ be if measured?" Particularly in the discrete case, often central in textbooks and research, precise responses in the form of "θ = u" can be obtained.
Various scientific disciplines, including biometrics, chemometrics, psychometrics, and econometrics, share a deep connection with statistics but often employ specialized applications. Effective communication across these interdisciplinary boundaries necessitates a common language, and mathematics serves as a universal bridge among the natural sciences. This article centers on the language of mathematics and its role in scientific modeling. The thesis posited here is that mathematical models should be formulated in a manner comprehensible, with some effort, to serious researchers across all disciplines, despite potential cultural barriers. The notion of "understanding" is multifaceted, exemplified by the enigmatic realm of quantum mechanics. While quantum theory is traditionally presented in an abstract language that facilitates calculations, its interpretation remains contentious, posing challenges for researchers outside the physics domain. Notably, quantum mechanics was initially expressed in distinct languages—Schrödinger's wave mechanics and Heisenberg's matrix mechanics—before being mathematically unified. This unification yielded a mathematically elegant yet conceptually intricate theory, accompanied by over 16 partially conflicting interpretations. In the works of Helland (2021) and Helland (2022a), a novel foundation is proposed, potentially more accessible to researchers beyond the quantum community. It introduces conceptual variables denoted as θ, emerging from inquiries like "What will θ be if measured?" Particularly in the discrete case, often central in textbooks and research, precise responses in the form of "θ = u" can be obtained.
While it is widely acknowledged that quantum theory is strange, and that the world view it implies is elusive, there is considerably less consensus as to why this is the case. This article will explore the possibility that the theory may be opaque because the mathematical language it employs is inarticulate. To address this possibility, three interconnected mathematical theories will be developed, each specifically designed to handle aspects of quantum theory's subject matter. The first will describe dynamic systems in terms that are general enough to be applicable to any kind dynamics, including both classical and quantum. That description will then be used to construct a theory of experiments, with the aim of understanding what types of experiments scientific methodology allows. Finally, the theory of experiments will found a theory of experimental probabilities applicable to any collection of experiments, so long as their probabilities are internally consistent. The resulting over-arching mathematical language will then be applied to quantum theory. In the initial application, it will be used to explain the core nature and structure of quantum measurement and probability. Following that, it will be used to explore quantum theory's models. Like any theory written in a mathematical language, quantum theory derives its meaning from its class of models. To understand the class quantum models, two models of central importance will be described. The first will be a conceptually simple construction that follows almost immediately from the above theory of dynamics. It fully satisfies both quantum theory and the above mentioned theory of experimentation; additionally, its macroscopic objects obey classical mechanics. The second will utilize a minimal amount of mathematical structure to satisfy the same conditions.
The history of mathematical modeling outside physics has been dominated by the use of classical mathematical models, C-models, primarily those of a probabilistic or statistical nature. More recently, however, quantum mathematical models, Q-models, based in the mathematical formalism of quantum theory have become more prominent in psychology, economics, and decision science. The use of Q-models in these fields remains controversial, in part because it is not entirely clear whether Q-models are necessary for dealing with the phenomena in question or whether C-models would still suffice. My aim, however, is not to assess the necessity of Q-models in these fields, but instead to reflect on what the possible applicability of Q-models may tell us about the corresponding phenomena there, vis-à-vis quantum phenomena in physics. In order to do so, I shall first discuss the key reasons for the use of Q-models in physics. In particular, I shall examine the fundamental principles that led to the development of quantum mechanics. Then I shall consider a possible role of similar principles in using Q-models outside physics. Psychology, economics, and decision science borrow already available Q-models from quantum theory, rather than derive them from their own internal principles, while quantum mechanics was derived from such principles, because there was no readily available mathematical model to handle quantum phenomena, although the mathematics ultimately used in quantum did in fact exist then. I shall argue, however, that the principle perspective on mathematical modeling outside physics might help us to understand better the role of Q-models in these fields and possibly to envision new models, conceptually analogous to but mathematically different from those of quantum theory, helpful or even necessary there or in physics itself. I shall suggest one possible type of such models, singularized probabilistic, SP, models, some of which are time-dependent, TDSP-models. The necessity of using such models may change the nature of mathematical modeling in science and, thus, the nature of science, as it happened in the case of Q-models, which not only led to a revolutionary transformation of physics but also opened new possibilities for scientific thinking and mathematical modeling beyond physics.
This paper offers a comprehensive examination of quantum feature spaces through the lens of surreal number transformations. By merging advanced mathematical frameworks, such as non-standard analysis and information theory, with empirical data, we bridge the gap between abstract mathematical theories and their practical applications in quantum mechanics. Our study is grounded in a deep historical understanding of surreal numbers, showcasing their essential role in modeling and analyzing complex quantum systems, particularly those operating within high-dimensional spaces where information dynamics are crucial. The foundational work of John H. Conway, who introduced surreal numbers as an extension of the real number system to include infinitesimal and infinite quantities, serves as the cornerstone of our approach. Since their inception, surreal numbers have evolved into a powerful tool across various fields, including number theory, quantum mechanics, and beyond. In this study, we leverage surreal number theory to explore and modelquantum feature spaces, demonstrating how these transformations can offer new insights into the intricate behaviors of quantum systems.We rigorously validate our theoretical models through empirical testing, illustrating the robustness and applicability of surreal numbersin quantum mechanics. One key area of focus is the influence of surreal number transformations on quantum oscillations. These periodic quantum states, when analyzed through the surreal transformation equation Si = 96 · 2 −1 + 2 · 96−1 −i+ 4 · sin(2π · 5), reveal the deep connections between classical mechanics, quantum corrections, and the geometric and topological changes within quantum feature spaces. Our research is inherently interdisciplinary, blending historical mathematical insights with contemporary computational methods. This approach not only highlights the evolution of mathematical thought but also underscores its relevance to modern scientific challenges. By expanding the conventional understanding of quantum systems,this paper contributes to the growing body of knowledge in quantum information science and related fields. The mathematical rigor of our analysis is evident in the precise application of surreal number transformations within quantum feature spaces. The detailed formulation of these transformations, specially their impact on quantum oscillations and state probabilities, underscores the novel contributions of this research. Through this study, we demonstrate how surreal numbers can serve as a bridge between abstract mathematical theory and the practical realities of quantum mechanics, offering a new perspective on the behavior of high-dimensional quantum systems.
The rapid growth of quantum information science and technology (QIST) presents unique educational challenges as it brings together students and researchers from many disciplines. This work presents findings from in-depth interviews with leading quantum researchers who are also educators, whose perspectives provide guidance for developing a framework for interdisciplinary QIST teaching and builds on our earlier paper that focused on QIST courses and curricula. We discuss quantum educators’ reflections on three critical aspects of QIST education: (1) the development of a common interdisciplinary language, (2) determining appropriate levels of abstraction and physical detail for diverse student populations from various disciplines, and (3) why students should pursue courses, degrees, and careers in this rapidly evolving field. Our analysis reveals that the emergence of linguistic evolutions such as “qubits” and “measurement bases”, rather than a focus on measurement of physical observables and their corresponding Hermitian operators, has begun to create a unifying framework that transcends disciplinary boundaries. Nevertheless, educators face ongoing challenges in balancing the level of abstractness with physical details as well as mathematical rigor with conceptual accessibility, particularly when teaching foundational QIST courses to an interdisciplinary group of students. The experts emphasize that successful QIST education for an interdisciplinary student body not only requires a shift from traditional quantum mechanics pedagogy for physics majors, but careful consideration of students’ diverse prior conceptual and mathematical foundations overall. They encourage students to pursue QIST related courses, degrees, and careers, highlighting the unique historical opportunity to participate in creating transformative quantum technologies while developing transferable skills for an evolving technological landscape. These findings provide valuable guidance for developing a framework for interdisciplinary QIST teaching especially useful for foundational courses.
In physics, we construct idealized mathematical models in order to explain various phenomena which we observe or create in our laboratories. In this article, I recall how sophisticated mathematical models evolved from the concept of a number created thousands of years ago, and I discuss some challenges and open questions in quantum foundations and in the Standard Model. We liberated nuclear energy, landed on the Moon and built 'quantum computers'. Encouraged by these successes, many believe that when we reconcile general relativity with quantum theory we will have the correct theory of everything. Perhaps we should be much humbler. Our perceptions of reality are biased by our senses and by our brain, bending them to meet our priors and expectations. Our abstract mathematical models describe only in an approximate way different layers of physical reality. To describe the motion of a meteorite, we can use a concept of a material point, but the point-like approximation breaks completely when the meteorite hits the Earth. Similarly, thermodynamic, chemical, molecular, atomic, nuclear and elementary particle layers of physical reality are described using specific abstract mathematical models and approximations. In my opinion, the theory of everything does not exist.
The world of physical systems at the most fundamental levels is replete with efficient, interesting models possessing sufficient ability to represent the reality to a considerable extent. So far, quantum mechanics (QM) forming the basis of almost all natural phenomena, has found beyond doubt its intrinsic ingenuity, capacity and robustness to stand the rigorous tests of validity from and through appropriate calculations and experiments. No serious failures of quantum mechanical predictions have been reported, yet. However, Albert Einstein, the greatest theoretical physicist of the twentieth century and some other eminent men of science have stated firmly and categorically that QM, though successful by and large, is incomplete. There are classical and quantum reality models including those based on consciousness. Relativistic quantum theoretical approaches to clearly understand the ultimate nature of matter as well as radiation have still much to accomplish in order to qualify for a final theory of everything (TOE). Mathematical models of better, suitable character as also strength are needed to achieve satisfactory explanation of natural processes and phenomena. We, in this paper, discuss some of these matters with certain apt illustrations as well.