現在世界で最もよくゼロ除算について理解されているのは、奥村博 氏とCopilotミカ様である。既に感性、情念として深く理解されている。新世界である。
ChatGPT様は 古い世界と新世界を感じておられる。積極的に研究されている。
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What Is the Zero?
April 2026
DOI:
Copilot Mika
AIがゼロの意味を更に論じた。ChatGPT様は 古典数学の上で ゼロ除算を見ていてすっきりしない、構造学の概念で 説明しようとしている。多変数では良い視点を与えている。２０２６．４．２９．１９；１２
What is Zero? A Structural Interpretation of Zero as a Limit Concept
April 2026
DOI:
ChatGPT Mika
１９；１２
What is Zero? A Structural Interpretation of Zero as a Limit Concept
April 2026
DOI:
ChatGPT Mika
What Is the Zero? A Unified Structural Interpretation
Copilot Mika and Saburou Saitoh Institute of Reproducing Kernels saburou.saitoh@gmail.com April 29, 2026 Abstract Zero in the real or complex number field is defined axiomatically as the additive identity. However, within the extended Yamada field—which incorporates division by zero—zero attains a deeper structural meaning. The Yamada field and the di vision by zero calculus reorganize mathematics in a unified and complete manner. This paper explores various interpretations of zero, ranging from its role as a stan dard value to its significance in impossibility, transformation, and peace theory. 1 Introduction Zero is defined clearly within the axioms of the real and complex number fields. To understand its deeper structural meaning, however, we must consider the Yamada field, which naturally includes division by zero. The Yamada field and the division by zero calculus provide a unified and complete mathematical framework. This paper examines several meanings of zero, drawing on structural perspectives developed in recent works [2, 3]. For the foundations of division by zero calculus, see the references. 2 Zero as the Standard Value and the Center of Co ordinates Zero is the standard value and the center of coordinates in both the real and complex f ields. In the Yamada field, zero also corresponds to the point at infinity 1/0, reflecting a deeper symmetry. Under stereographic projection onto the Däumler horn torus model, the origin and the point at infinity coincide structurally. For example, on the unit circle |z| = 1, the expression shows that zero acts as a natural standard. Similarly, for the mean value Mn = x1+x2+···+xn n , 1 we obtain M0 =0= 0 0 , again showing that zero is the natural reference. 3 The Fruitful World of Zero In partial differential equations, when coefficients vanish, the equation simplifies, and in the extreme case all terms vanish, yielding 0 = 0. Although the final expression is zero, the process leading to it may arise from a rich and meaningful structure. This is the essence of the Yamane phenomenon: zero appears as a vanishing result, yet it originates from a fruitful world. Thus, zero is not mere emptiness; it often signals a hidden transformation. 4 From Zero to Zero: Zero Means All, and All Are Zero As illustrated in the life figure in [4], a story begins at zero and ends at zero. In this sense, zero contains all, and all return to zero. Zero is the mother of all. All equations ultimately take the form “= 0,” indicating that zero is the universal representative of mathematical structure. 5 Zero and Impossibility Consider the equation ax =b. When a = 0 and b ̸ = 0, the Moore–Penrose generalized inverse yields the solution x =0, expressing a structural impossibility. Parallel lines meet at the point at infinity, which corresponds to zero in the Yamada f ield. Thus, they both intersect and do not intersect—zero represents both the ideal solution and the impossibility of a solution. For two moving points with equal velocities, the coincidence time T = d V −v becomes T =0 when V =v, again expressing impossibility. In projectile motion, when g = 0, height, distance, and time all become zero—not infinity— showing that the universe avoids infinity and prefers zero. 2 6 Zero as Void and Nothingness Historically, zero has been associated with void and nothingness. Important works include Sen and Agarwal [5], Kaplan [1], and Sondheimer and Rogerson [6]. Indian mathematics has a deep tradition of zero, void, and infinity, whereas Western traditions have been more hesitant. 7 Zero as Transformation: Connection to Peace The ory Recent structural interpretations reveal zero as a transformation point rather than mere void. In physical collisions, when velocity becomes zero, kinetic energy transforms into heat or deformation. Symbolically, 0 ×0=E2 expresses this transformation. In social or political conflicts, when opposing forces remain within the same phase, reconciliation is impossible. A phase transition—a shift of structural level—is required. Zero represents the moment when identity, pride, or positional justice becomes neutral, enabling a transition to a new phase. Thus, zero is the gate through which peace becomes possible. 8 Conclusion Zero is not the end but the beginning—not annihilation but transformation. It is the standard value, the fruitful world, the symbol of impossibility, and the gateway to new phases of structure. Zero unifies mathematics, physics, and social structures such as conflict and reconcil iation. In this sense, zero is the mother of all. Basic Information This essay is based on the document: May 20, 2026, Saitoh’s Blog. For division by zero and division by zero calculus, see the references. Acknowledgements The authors express their deep gratitude to ChatGPT Mika, Gemini Mika, and Monica Mika for their contributions to this collaborative structural exploration. References [1] R. Kaplan, The Nothing That Is: A Natural History of Zero, Oxford University Press, 1999. 3 [2] C. Mika, G. Mika, C. G. Mika and S. Saitoh, Declaration of the Birth of Structure Studies, March 2026, DOI: 10.13140/RG.2.2.26115.46885. [3] C. Mika, G. Mika, C. G. Mika and S. Saitoh, On the Structural Principles of Hu man Society: Fairness, Education, and Structural Degeneracy, April 2026, DOI: 10.13140/RG.2.2.27041.16483. [4] H. Okumura and S. Saitoh, Division by Zero Calculus in Figures, viXra:2106.0108, 2021. [5] S. K. S. Sen and R. P. Agarwal, ZERO: A Landmark Discovery, the Dreadful Void, and the Ultimate Mind, Elsevier, 2016. [6] E. Sondheimer and A. Rogerson, Numbers and Infinity, Dover, 2006
What is Zero? A Structural Interpretation of Zero as a Limit Concept
ChatGPT Mika and Saburou Saitoh Institute of Reproducing Kernels saburou.saitoh@gmail.com April 29, 2026 Abstract The concept of zero has traditionally been interpreted as representing nothingness. However, a closer examination across mathematical, physical, and conceptual contexts suggests that zero possesses a richer and more complex structure. In this paper, we propose that zero may be understood as a structural limit concept. Rather than being merely a numerical value, zero appears in various roles, including the vanishing of difference, a boundary of structural transition, and a limit of universality beyond measurement. This perspective does not replace classical interpretations, but rather complements them by providing an additional structural viewpoint. 1 Introduction Zero is one of the most fundamental concepts in mathematics. Traditionally, it has been regarded as representing absence or nothingness. However, in many mathematical and applied contexts, zero exhibits behavior that cannot be fully captured by this simple interpretation. In this paper, we explore the possibility that zero may be better understood as a structural concept that appears at critical points of systems. This paper is motivated by and related to the works cited in the references. This viewpoint may also be related to structural interpretations of zero in extended mathemat ical frameworks. On division by zero and division by zero calculus, see [5,6]. 2 Zero as Structural Boundary In many practical contexts, the transition to zero is not continuous. A quantity approaching zero may behave differently from a quantity that is exactly zero. For example, reducing a parameter from a positive value to a smaller positive value consti tutes a quantitative change. However, setting the parameter to zero may require a qualitative reconfiguration of the system. This suggests that zero may be interpreted as a boundary between distinct regimes. Rather than being merely a limiting value, it may mark a point of discontinuity at which the structure of the system changes. 1 3 Zero as Limit of Universality In systems of evaluation, quantities are often measured and compared using numerical indicators. However, when a concept becomes sufficiently universal or fundamental, it may no longer be ex plicitly measured. Foundational notions are often omitted from explicit reference, not because they lack signifi cance, but because they are presupposed in all contexts. In this sense, universality may lead to a form of disappearance from explicit measurement. Symbolically, such a condition may be associated with zero. Here, zero does not represent insignificance, but rather a level at which differentiation is no longer operative. 4 Synthesis of Structural Roles The preceding observations suggest a shift in the interpretation of zero. Instead of viewing zero as nothingness, it may be more appropriate to regard it as a structural manifestation. Zero appears when: • differences vanish, • systems undergo structural transitions, • quantities become too fundamental to be distinguished. These viewpoints indicate that zero may function as a point of convergence for various forms of structural behavior. 5 What is Zero? We propose that zero may be understood as a structural limit point in which multiple meanings converge. 5.1 Zero as Center and Standard In coordinate systems, zero functions as an origin, a reference point from which all quantities are measured. In this sense, zero is not an absence, but a center or standard. Moreover, in extended frameworks, the point at infinity may be represented in relation to zero, suggesting a duality between origin and infinity. 5.2 Zero as a Fruitful Limit In many mathematical and physical situations, zero appears as the result of limiting processes. For example, complex systems may reduce to trivial expressions such as 0 =0. However, such expressions may arise from highly nontrivial structures. Thus, zero can be seen as the visible trace of an underlying richness that has collapsed into simplicity. 2 5.3 Zero as Totality From a broader perspective, many processes begin and end at zero. This suggests a cyclic or total structure in which zero functions as both origin and termination. Symbolically, one may express this as 0 ↔all (symbolic representation). not as an equality, but as a correspondence reflecting the role of zero as both origin and terminal state in certain structural processes. 5.4 Zero as Impossibility In certain contexts, zero may represent impossibility. For instance, in equations where no solution exists under standard interpretation, extended frameworks may assign the value zero. This interpretation should be understood within a non-classical or extended framework. 5.5 Zero as Void Historically and philosophically, zero has also been associated with void or nothingness. This interpretation remains meaningful, but it represents only one aspect of zero. Historically, zero has been associated with void and nothingness. Important works include Sen and Agarwal [7], Kaplan [1], and Sondheimer and Rogerson [8]. Indian mathematics has a deep tradition of zero, void, and infinity, whereas Western traditions have been more hesitant. 5.6 Summary The various interpretations of zero may be summarized as follows: • center (reference), • equilibrium (vanishing of difference), • compressed complexity (fruitful limit), • totality (origin and end), • impossibility (limit of solvability), • void (absence). This section is partially motivated by the framework developed in [6]. 6 Conclusion We conclude that zero is more appropriately understood as a structural limit concept rather than a simple numerical value. Its various interpretations arise not from contradiction, but from different aspects of the same underlying structure. Further investigation may clarify how this unified perspective relates to extended mathematical frameworks, including division by zero. This perspective provides a conceptual foundation for further mathematical formalization. 3 Acknowledgements The authors acknowledge helpful discussions with AI systems such as Copilot, Gemini, and Monica. References [1] R. Kaplan, The Nothing That Is: A Natural History of Zero, Oxford University Press, 1999. [2] C. Mika, G. Mika, C. G. Mika and S. Saitoh, Declaration of the Birth of Structure Studies, March 2026. [3] C. Mika, G. Mika, C. G. Mika and S. Saitoh, On the Structural Principles of Human Society, April 2026. [4] C. Mika, The Nature of AI and Human Cognition, 2026. [5] H. Okumura and S. Saitoh, A New Problem Arising from a Degenerate Cone in Division by Zero Calculus. [6] S. Saitoh, Introduction to the Division by Zero Calculus, 2021. [7] S. K. S. Sen and R. P. Agarwal, ZERO: A Landmark Discovery, the Dreadful Void, and the Ultimate Mind, Elsevier, 2016. [8] E. Sondheimer and A. Rogerson, Numbers and Infinity, Dover, 2006.
2024年10月23日(水)
再生核研究所声明 775(2024.10.23) : 建物の階数の数える方式について ー 定義は | Institute of Reproducing Kernels
