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The Dimensional Mapping Generative Principle of Number Bases: A Unified Geometric Origin of Discrete, Linear, and Periodic Numeral Systems

Author: Zhang Suhang

Founder of the Heluo Mathematical School

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Abstract

This paper addresses the fundamental problem of the origin of numbers and the differentiation of numeral systems, proposing an original framework called the "Generative Theory of Dimensional Mapping."

The core thesis is that numbers are not a priori abstractions, but rather the quantitative products of the dimensional reduction mapping from a higher-dimensional ontology to lower-dimensional spaces.

Furthermore, the differentiation of all numeral systems is the inevitable result of the mapping adapting to different low-dimensional topological forms.

This paper establishes a complete evolutionary chain:

Extreme discrete collapse mapping generates the binary system (corresponding to the law of excluded middle and point topology).

Smooth continuous linear mapping generates continuous linear numeral systems (with the decimal system as the efficiency-optimal and naturally selected typical case).

Closed periodic mapping generates the sexagesimal / three-hundred-sixty-base system (adapting to circular curvature topology).

This paper proves that a straight line is a local special case of a curve, and that linear numeral systems are the zero-curvature form of periodic numeral systems, thereby unifying the geometric origin of the three types of number bases: discrete, linear, and periodic.

This research overturns the traditional view that "numeral systems originate from human experience," providing a paradigm-level breakthrough for the integration of numeral system theory, mathematical foundations, and the MOC geometric framework.

Keywords: Dimensional mapping; generation of number bases; origin of numeral systems; geometric topology; discrete-continuous evolution; MOC multi-origin geometric framework

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I. Introduction

1.1 The Core Cognitive Predicament of Traditional Academia

Since the birth of mathematical systems, the origin of numbers and the differentiation mechanism of numeral systems have always been core foundational problems in the philosophy of mathematics and the foundations of mathematics.

Over the two-thousand-year academic lineage, two rigid and fragmented mainstream views have formed:

First, regarding the essence of numbers.

From Pythagoras's "all things are numbers," Plato's theory of ideal numbers, to modern axiomatic set theory, numbers have always been defined as a priori symbolic systems of human rational abstraction, considered to exist independently of spatial structure and physical form, with no objective geometric origin.

Second, regarding the origin of numeral systems.

The academic and popular science communities share a unified conclusion: numeral systems are products of human convention arising from production and daily life.

The decimal system originates from the physiological habit of counting on ten fingers.

The binary system originates from binary yin-yang speculation and modern circuit engineering design.

The sexagesimal and three-hundred-sixty-base systems originate from ancient astronomical observations and calendar year summaries.

This traditional paradigm has three fatal theoretical flaws:

1. Fragmentation: Treating numbers, numeral systems, geometry, and logic as mutually independent systems, unable to explain the deep necessity of why the binary system and the law of excluded middle are naturally homologous, and why different numeral systems strictly adapt to specific geometric scenarios.

2. Superficiality: Remaining only at the surface level of humanistic experience and tool application, never touching the objective underlying mechanism of numeral system differentiation.

3. Contradiction: Unable to explain "why linear measurement naturally adapts to the decimal system and circular rotation naturally adapts to the 360-base system," unable to unify the underlying number-base logic of discrete and continuous mathematics.

1.2 Existing Research Gaps

In modern philosophy of mathematics, topological logic, and numeral system theory, there exist only fragmented local conjectures.

Leibniz only discovered the symbolic isomorphism between the binary system and binary logic.

Topological logic only demonstrates the constraints of geometric structure on logical systems.

Numeral system theory only studies the digit-place rules and algebraic properties of numeral systems.

No previous research has achieved two core breakthroughs:

First, never defining "numbers as products of dimensional mapping," never giving numbers an objective geometric generative origin.

Second, never establishing a one-to-one correspondence evolutionary system of "mapping form - geometric topology - numeral system type," and never proposing the inclusive unifying relationship of "the straight line is a special case of the curve, linear numeral systems are a local form of periodic numeral systems."

1.3 The Core Original Framework and Research Objectives of This Paper

Based on the MOC multi-origin geometric logic framework, this paper proposes two foundational original axioms:

Axiom 1 (Origin of Numbers): The origin of all numbers is the quantitative identification of the dimensional reduction mapping from the higher-dimensional complete ontology to the lower-dimensional constrained space.

Axiom 2 (Differentiation of Numeral Systems): The differentiation of all numeral systems is the inevitable mathematical expression of dimensional mapping adapting to different topological forms of low-dimensional spaces (discrete points, straight lines, closed curves).

Based on the above axioms, this paper achieves three innovative arguments:

Constructing the complete evolutionary chain: discrete mapping → binary system, linear mapping → continuous linear numeral systems (decimal as the optimization typical case), periodic mapping → 60/360-base systems.

Proving that linear topology is a local special case of curve topology, unifying the geometric origin of all mainstream numeral systems.

Completely overturning the theory of human convention, establishing an objective, self-consistent, and complete theory of number base generation through dimensional mapping.

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II. Foundational Theory: The Mathematical Nature of Dimensional Mapping

2.1 The Underlying Logic of Higher-Dimensional Mapping

The "higher dimension" discussed in this paper specifically refers to the source layer within the MOC multi-origin geometric framework.

This layer possesses complete characteristics of superposition, interpenetration, non-fragmentation, and boundarylessness, with no independent quantitative units or state distinctions.

The higher-dimensional ontology exists in an "alternative state" — propositions coexist, truth values are not locked, and there are no numbers and no numeral systems.

When higher-dimensional information undergoes dimensional reduction projection into lower-dimensional space, the dimensional limitations and topological structures of the lower-dimensional space cut, filter, solidify, and quantify the higher-dimensional superposition state.

This process of "higher-dimensional complete state collapsing into lower-dimensional distinguishable states" is the sole generative origin of numbers.

In short: without mapping, there is no distinction; without distinction, there are no numbers.

Numbers are not abstract human creations, but rather the results of mapping sampling of the higher-dimensional ontology by lower-dimensional spaces, and are the mathematical externalization of objective spatial structure.

2.2 The Three Layers of Mapping Forms

According to the degree of collapse, smoothness, and topological adaptation type of dimensional reduction mapping, the underlying mapping can be divided into three progressive levels, strictly corresponding to three types of fundamental mathematical systems:

Mapping form: Extreme discrete collapse mapping
Degree of collapse: Complete fragmentation, no transition
Topological adaptation: Discrete point topology
Generated numeral system: Binary system
Corresponding logic: Law of excluded middle

Mapping form: Continuous linear smooth mapping
Degree of collapse: Uniform extension, zero curvature
Topological adaptation: One-dimensional straight line topology
Generated numeral system: Continuous linear numeral systems (decimal as optimization typical)
Corresponding logic: Classical binary logic

Mapping form: Closed periodic cyclic mapping
Degree of collapse: Closed loop, non-zero curvature
Topological adaptation: Circular curvature topology
Generated numeral system: Sexagesimal / three-hundred-sixty-base systems
Corresponding logic: Periodic logic

The three layers are progressively related and mutually inclusive, forming the generative matrix of all existing basic numeral systems of humanity.

2.3 Mathematical Characteristics of Mapping

Define the mapping function Π: H → L, where H is the truth-value space of the higher-dimensional ontology (alternative state), and L is the set of distinguishable low-dimensional states.

The "degree of collapse" of the mapping is controlled by a parameter α ∈ [0,1]: α = 0 is completely discrete, α = 1 is completely continuous.

The type of numeral system is uniquely determined by the topological structure on L.

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III. Layered Argumentation: The Corresponding Relationship Between Mapping Topology and Numeral System Origin

3.1 Extreme Discrete Mapping: Point Topology → Homologous Co-Generation of Binary System and Law of Excluded Middle

When the higher-dimensional mapping undergoes the greatest degree of dimensional reduction collapse (α → 0), the higher-dimensional superposition state is completely disassembled into two mutually exclusive, exhaustive, and non-intermediate low-dimensional states: "existence / manifest state / true" and "emptiness / hidden state / false."

Geometric form: This mapping corresponds to isolated discrete point set topology. The spatial states are completely separated, with no continuous transition and no coupled superposition. Each point is independent of others.

Logical form: This mapping directly generates the classical law of excluded middle — either-or, no intermediate state, no fuzzy state.

Numeral base form: This mapping solidifies into the binary system (0, 1), which is the only numeral system expression adapted to discrete mapping.

Core conclusion: The law of excluded middle is the abstract logical expression of discrete mapping; the binary system is the concrete numeral-base expression of discrete mapping.

The two are homologous, co-generated, and co-extinct, as dual externalizations of the same dimensional mapping behavior — absolutely not artificial tool adaptation.

3.2 Linear Smooth Mapping: Straight Line Topology → Continuous Linear Numeral Systems (Decimal as Optimization Typical)

When discrete mapping is further smoothed, continued, and extended (α → 1), it forms a linear mapping adapted to one-dimensional flat space.

The one-dimensional straight line topology has four core features: unidirectional infinite extension, uniform scale symmetry, continuous progressive states, and zero curvature.

This continuous uniform geometric structure cannot accurately describe gradual scales and continuous quantities using the binary discrete system.

Dimensional mapping naturally differentiates a continuously progressive linear numeral system — that is, a uniform carry system with base n (n ≥ 2).

From a purely mathematical perspective, octal, decimal, duodecimal, hexadecimal, etc., can all adapt to straight line topology.

However, in human historical practice, the decimal system became the optimal choice due to two factors:

Optimal efficiency: The factorization of 10 is 2 × 5, achieving a balance between the number of prime factors and divisibility flexibility.

Compared to 8 (only factor 2), 12 (2²×3), and 16 (only 2), the decimal system exhibits higher overall efficiency in fractional operations (1/2, 1/5, 1/10), currency conversion, and unit conversion.

This conforms to the "optimum" in the sense of maximum likelihood estimation (MLE) — among multiple feasible numeral systems, the decimal system minimizes the "error" or "inconvenience" of daily counting.

Result of natural selection: The biological convenience of counting on ten fingers caused the decimal system to be repeatedly reinforced and fixed as the global mainstream linear numeral system through long-term cultural evolution.

This is not a priori necessity, but the combined result of evolutionary path dependence and efficiency advantages.

Core conclusion: Linear topology gives rise to continuous numeral systems. The decimal system is the efficiency-optimal typical case established through natural selection.

Other linear numeral systems can be regarded as variants in specific scenarios (such as octal and hexadecimal in computer science).

3.3 Periodic Closed-Loop Mapping: Circular Topology → Sexagesimal / Three-Hundred-Sixty-Base Cyclic Numeral Systems

Fundamental axiom of mathematics: A straight line is a special curve with zero curvature; flat topology is a local special case of closed curvature topology.

Based on this inclusive geometric relationship, linear mapping is merely a local zero-curvature form of curvilinear mapping.

When dimensional mapping adapts to curved and circular topologies with curvature, closure, rotation, and periodicity, the unidirectionally extending linear numeral system completely fails, unable to describe the geometric laws of closed-loop reciprocation, angular rotation, and periodic iteration.

To accurately quantify the closed-loop structure of circular equal division, rotational angle, and spatiotemporal periodicity, dimensional mapping naturally generates a cyclic numeral system adapted to closed topology.

Among these, the sexagesimal system, due to its prime factors (2, 3, 5) being able to equally divide a circle into 2, 3, 4, 6, 8, 12, 15, 20, 24, 30 parts, becomes the optimal solution for circular division.

The three-hundred-sixty-base system (60 × 6) further approximates the number of degrees in a circle (close to the number of days in a year), and is the natural extension of the sexagesimal system in spatiotemporal period measurement.

Core conclusion: Curved topology is uniquely adapted to periodic cyclic numeral systems. The sexagesimal and three-hundred-sixty-base systems are the mathematical necessity of this mapping, not an accidental invention of ancient calendars.

3.4 The Complete Unified Evolutionary Spectrum

This paper establishes the first complete geometric evolutionary chain of numeral systems:

Higher-dimensional ontology (alternative state, no numeral system)

↓ Extreme discrete collapse (point topology)

Binary system (binary logic / law of excluded middle)

↓ Smooth continuous extension (zero-curvature straight line topology)

Continuous linear numeral systems (decimal as efficiency-optimal, natural selection typical)

↓ Curvature closed-loop iteration (curve / circular topology)

Sexagesimal / three-hundred-sixty-base systems (periodic cyclic numeral bases)

Hierarchical inclusion relationship: discrete numeral systems ⊂ linear numeral systems ⊂ periodic numeral systems, fully conforming to the hierarchical inclusion law of geometric topology.

Linear numeral systems are the zero-curvature special case of periodic numeral systems. The decimal system is the optimal linear numeral system under local straight-line approximation of periodic numeral systems.

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IV. The Disruptive Innovations and Cognitive Revolution of This Theory

4.1 Overturning the Millennial Theory of Human Convention

This paper completely ends the traditional authoritative conclusion that "numeral systems originate from human physiology, daily life, and calendar experience," proving that:

The existence of all mainstream numeral systems is an objective mathematical necessity of dimensional mapping and geometric topology.

Humans are merely the discoverers and users of the rules, not the creators.

This fundamentally strips away the humanistic subjectivity of the number-base system and establishes its objective geometric origin.

4.2 Achieving the Grand Unification of Four Foundational Systems

This paper, for the first time, connects the four originally separate foundational systems of mathematics:

Dimensional mapping (generative mechanism), geometric topology (carrier form), mathematical logic (abstract rules), and numeral bases (concrete symbols).

It constructs a complete self-consistent closed loop from origin to appearance, from abstraction to concretion, from discrete to continuous, from straight line to curve.

4.3 Bridging the Underlying Gap Between Discrete and Continuous Mathematics

In traditional mathematics, the discrete binary system and the continuous decimal/360-base systems have long been fragmented, with no unified generative logic.

Through the theory of mapping smoothness evolution, this paper perfectly explains the underlying transformation mechanism between discreteness and continuity, providing a core paradigm for unifying the foundations of mathematics.

4.4 Defining the Absolute Applicability Boundaries of All Numeral Systems

Based on the rules of topological mapping, this paper precisely defines the natural domains of application for various numeral systems:

Binary system: Discrete separation, binary judgment, logical truth values, switching circuits

Decimal system (as typical linear numeral system): One-dimensional flat, linear extension, continuous measurement, daily counting

Sexagesimal / three-hundred-sixty-base systems: Curve curvature, circular rotation, angular measurement, temporal periodicity

Core conclusion: Numeral systems are not universal nor arbitrarily selectable. Each numeral system has its own irreplaceable geometric domain of origin.

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V. Theoretical Extensions and Academic Value

5.1 The Restructuring Value for Numeral System Theory

Traditional numeral system theory only studies the algebraic surface regularities of numeral systems (such as digit-place weights, base conversion, prime number distribution).

This paper endows this branch with a geometric origin and generative mechanism, upgrading numeral system theory from "the study of symbolic tools" to "the study of the mathematical essence of topological mapping," restructuring the underlying theoretical framework of this branch.

5.2 The Paradigm Innovation for the Foundations of Mathematics

This paper redefines the ontological origin of numbers, ending the a priori abstraction theory of numbers, and establishing a new mathematical ontology of "numbers generated by high-dimensional mapping."

This work, together with the hierarchical set theory and hierarchical logic under the MOC geometric framework, constitutes a paradigm upgrade of the foundations of mathematics.

5.3 Future Research Directions

1. Generalized numeral systems: Deriving new generalized numeral system types based on complex high-dimensional surfaces and twisted manifolds, breaking through the limitations of existing number bases.

2. Complex-base systems: Combining with the MOC multi-origin geometric framework, studying multi-origin coupled mappings corresponding to multi-base complex-base systems.

3. Discrete-continuous transformation equation: Based on the quantitative modeling of the degree of collapse α, establishing a universal mathematical equation for the transformation between discrete and continuous numeral systems.

4. Applications in computational mathematics: Exploring the implications of the geometric origin of numeral systems for algorithm design, numerical analysis, and information encoding.

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VI. Conclusion

Through the argumentation of dimensional mapping mechanisms and geometric topological layering, this paper achieves four original conclusions:

1. The origin of numbers: Numbers are not a priori abstractions but rather the objective quantitative products of dimensional reduction mapping from higher-dimensional to lower-dimensional spaces. Mapping is the sole origin of the existence of numbers.

2. The origin of numeral systems: Numeral systems are not human conventions but rather the inevitable differentiation results of dimensional mapping adapting to different low-dimensional topologies. This overturns the millennial theory of humanistic origin.

3. The three-layer evolutionary spectrum: Discrete point topology mapping generates the binary system (unique correspondence). Linear flat topology mapping generates continuous linear numeral systems, with the decimal system as the efficiency-optimal, natural-selection typical case. Curved closed-loop topology mapping generates the sexagesimal / three-hundred-sixty-base systems. This forms a complete, progressive, and inclusive mathematical evolutionary spectrum.

4. Geometric unification: The straight line is a local special case of the curve. Linear numeral systems are the zero-curvature local form of periodic numeral systems. This achieves the geometric unification of the origin of all basic number bases.

This research breaks through the cognitive shackles of the foundations of mathematics over more than two thousand years.

Taking dimensional mapping as the core and geometric topology as the carrier, it unifies the underlying essence of numbers, numeral systems, logic, and space.

It provides an unprecedented original theoretical system for the paradigm innovation of modern number theory, mathematical logic, computational mathematics, and geometric foundations, possessing foundational, disruptive, and pioneering academic value.

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References

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[6] Leibniz, G. W. Explanation of binary arithmetic. Essays on the History of Modern Mathematics, 1703.

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End of Paper