Yucong Duan

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWP-SC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Throughout my early explorations of the world, I was captivated by patterns, relationships, and the underlying principles that govern natural phenomena. As I delved deeper into mathematics, I sought not only to understand existing concepts but also to explore new hypotheses and conjectures. To facilitate this exploration, I adopted a systematic approach known as the New DIKWP Semantic Framework, which stands for Data, Information, Knowledge, Wisdom, and Philosophy (Instead of P as Purpose).

This framework allows for a structured progression from raw observations to profound insights, grounding each step in concrete experiences and logical reasoning. In this narrative, I will detail how I used the DIKWP Semantic Framework to explore and develop several mathematical hypotheses. Each chapter will be presented in full length, providing a comprehensive account of my journey from initial observations to the formulation of new mathematical ideas.

• Concept: Raw, unprocessed observations and measurements from the environment.

• Semantics: Data represents the fundamental building blocks upon which information is constructed.

• Concept: Organized or structured data that conveys meaning.

• Semantics: Information emerges when data is categorized, compared, or connected to reveal patterns.

• Concept: The synthesis of information to form coherent concepts, principles, or models.

• Semantics: Knowledge involves understanding the relationships between pieces of information.

• Concept: The application of knowledge to make judicious decisions or to solve complex problems.

• Semantics: Wisdom reflects the ability to use knowledge effectively in various contexts.

• Concept: The overarching principles and values that guide inquiry and understanding.

• Semantics: Philosophy provides the foundational worldview and ethical considerations for exploration.

• Goal: Use the DIKWP framework to explore mathematical hypotheses by progressing from raw observations to deep insights.

• Approach: Ground each mathematical concept in experiential data, build information through pattern recognition, develop knowledge by formulating principles, apply wisdom to solve problems, and reflect philosophically on the implications.

• Data: Listing even numbers greater than 2 (e.g., 4, 6, 8, 10, 12, 14, 16, ...).

• Data: Listing prime numbers (e.g., 2, 3, 5, 7, 11, 13, 17, ...).

• Process: For each even number, find pairs of prime numbers that sum to it.

• Examples:

• 4 = 2 + 2

• 6 = 3 + 3

• 8 = 3 + 5

• 10 = 3 + 7 or 5 + 5

• Observation: Every even number checked can be expressed as the sum of two prime numbers.

• Goldbach Conjecture: Every even integer greater than 2 can be expressed as the sum of two prime numbers.

• Understanding Primes: Primes are the building blocks of natural numbers.

• Implication: The conjecture suggests a deep connection between even numbers and prime numbers.

• Method: Systematically test larger even numbers for compliance.

• Results: Up to a certain limit, the conjecture holds true.

• Limitations: Cannot test all even numbers due to infinity.

• Approach: Seek patterns or proofs that can generalize the observation.

• Significance: If true, the conjecture reveals fundamental properties of numbers.

• Inquiry: Encourages the search for a proof or counterexample.

• Reflection: Recognizes the boundaries between empirical verification and mathematical proof.

• Data: Values of the Riemann zeta function $\zeta (s)$ for various complex numbers $s$.

• Observation: Non-trivial zeros of $\zeta (s)$ appear to lie on the critical line Re(s)=12\text{Re}(s) = \frac{1}{2}Re(s)=21​.

• Observation: All computed non-trivial zeros have a real part of 12\frac{1}{2}21​.

• Visualization: Plotting zeros on the complex plane reveals alignment along the critical line.

• Riemann Hypothesis: All non-trivial zeros of the Riemann zeta function have a real part equal to 12\frac{1}{2}21​.

• Connection to Prime Numbers: The distribution of zeros is linked to the distribution of prime numbers.

• Mathematical Significance: Proving the hypothesis would have profound implications for number theory.

• Analytical Approaches: Exploring functional equations and symmetries of $\zeta (s)$.

• Numerical Methods: Using computational techniques to verify zeros up to high magnitudes.

• Complexity: The hypothesis involves advanced complex analysis.

• Approach: Collaborate with existing mathematical frameworks and seek innovative methods.

• Inquiry: What constitutes proof versus empirical evidence?

• Reflection: The quest to prove the hypothesis embodies the pursuit of absolute certainty in mathematics.

• Observation: Mathematics is a web of interrelated ideas, with the Riemann Hypothesis at a central node.

• Data: Measuring interior angles, side lengths, and areas of various polygons.

• Observation: Regular polygons exhibit symmetry and predictable properties.

• Pattern: The sum of interior angles increases with the number of sides.

• Formula: Sum of interior angles S=(n−2)×180∘S = (n - 2) \times 180^\circS=(n−2)×180∘, where $n$ is the number of sides.

• Experiment: Considering polygons on curved surfaces (spheres, hyperbolic planes).

• Observation: Angle sum formulas differ from Euclidean cases.

• Hypothesis: The sum of the interior angles of a polygon is determined by both the number of sides and the curvature of the space in which it resides.

• Spherical Geometry: S=(n−2)×180∘+(n×curvature constant)S = (n - 2) \times 180^\circ + (n \times \text{curvature constant})S=(n−2)×180∘+(n×curvature constant)

• Hyperbolic Geometry: S=(n−2)×180∘−(n×curvature constant)S = (n - 2) \times 180^\circ - (n \times \text{curvature constant})S=(n−2)×180∘−(n×curvature constant)

• Approach: Develop a unified formula that incorporates curvature.

• Application: Use differential geometry to formalize the relationship.

• Method: Apply the formula to known geometries and compare results.

• Validation: Consistency with established geometric principles confirms the hypothesis.

• Inquiry: How does the nature of space influence geometric properties?

• Reflection: Recognizes that mathematics can describe not just abstract concepts but also the fabric of reality.

• Data: Observing how quantities change over time in physical systems (e.g., falling objects, population growth).

• Observation: Changes can be instantaneous or cumulative over intervals.

• Differentiation: Measures instantaneous rates of change.

• Integration: Accumulates quantities over intervals.

• Fundamental Theorem of Calculus: Differentiation and integration are inverse processes.

• Observation: The area under a curve can be found using antiderivatives.

• Hypothesis: For functions that are continuous and differentiable, the process of integration can be generalized to non-integer orders, leading to fractional calculus.

• Fractional Derivatives: Derivatives of arbitrary (non-integer) order.

• Potential Formula:

  Dqf(x)=1Γ(−q)∫0x(x−t)−q−1f(t)dtD^q f(x) = \frac{1}{\Gamma(-q)} \int_0^x (x - t)^{-q - 1} f(t) dtDqf(x)=Γ(−q)1​∫0x​(x−t)−q−1f(t)dt

• DqD^qDq: Fractional derivative of order $q$

• $\Gamma$: Gamma function

• Physics: Modeling anomalous diffusion and viscoelastic materials.

• Engineering: Signal processing and control systems.

• Complexity: Requires redefining foundational calculus concepts.

• Approach: Develop intuition through practical examples and simulations.

• Inquiry: What are the limits of mathematical operations?

• Reflection: Embraces the idea that mathematics is ever-evolving, with new branches emerging from foundational concepts.

• Observation: Despite diverse topics, underlying patterns connect different areas of mathematics.

• Example: The use of complex analysis in number theory (Riemann Hypothesis) and in solving differential equations (fractional calculus).

• Data: Gathering observations from various mathematical fields.

• Information: Identifying patterns and connections.

• Knowledge: Formulating hypotheses and theories.

• Wisdom: Applying insights to solve complex problems and advance understanding.

• Philosophy: Reflecting on the nature of mathematics and its role in interpreting reality.

Using the DIKWP Semantic Framework, I embarked on a journey to explore and develop mathematical hypotheses. By progressing systematically from raw data to philosophical reflection, I was able to deepen my understanding of complex mathematical concepts and contribute to the broader mathematical discourse.

This approach demonstrates that with curiosity, logical reasoning, and a structured framework, profound insights can be achieved without relying on subjective definitions or advanced prior knowledge. The DIKWP framework not only facilitated the exploration of specific hypotheses but also highlighted the interconnectedness of mathematical ideas.

This narrative illustrates how foundational mathematical principles and hypotheses can be explored through a structured, experience-based approach. In the context of artificial intelligence and cognitive development, it emphasizes the value of the DIKWP Semantic Framework in guiding learning and discovery.

By enabling AI systems to:

• Collect Data: Observe and record information from various sources.

• Process Information: Identify patterns and organize data meaningfully.

• Formulate Knowledge: Develop concepts and models based on information.

• Apply Wisdom: Use knowledge to solve problems and make decisions.

• Reflect Philosophically: Consider the broader implications and ethical dimensions.

We can foster the development of intuitive understanding similar to human learning. This approach avoids reliance on predefined definitions and promotes the natural discovery of mathematical relationships and hypotheses.

Note: This detailed narrative presents the exploration of mathematical hypotheses using the DIKWP Semantic Framework, as if I, an infant, independently observed and reasoned them out. Each concept is derived from basic experiences and logical progression, emphasizing the natural evolution from data to profound mathematical insights. This approach demonstrates that with curiosity and structured reasoning, foundational knowledge about mathematics can be accessed and expanded upon without relying on subjective definitions.

References for Further Reading

1. International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation (DIKWP-SC),World Association of Artificial Consciousness(WAC),World Conference on Artificial Consciousness(WCAC). Standardization of DIKWP Semantic Mathematics of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. October 2024 DOI: 10.13140/RG.2.2.26233.89445 .  https://www.researchgate.net/publication/384637381_Standardization_of_DIKWP_Semantic_Mathematics_of_International_Test_and_Evaluation_Standards_for_Artificial_Intelligence_based_on_Networked_Data-Information-Knowledge-Wisdom-Purpose_DIKWP_Model

2. Duan, Y. (2023). The Paradox of Mathematics in AI Semantics. Proposed by Prof. Yucong Duan:" As Prof. Yucong Duan proposed the Paradox of Mathematics as that current mathematics will not reach the goal of supporting real AI development since it goes with the routine of based on abstraction of real semantics but want to reach the reality of semantics. ".