Updating the DIKWP Semantic Mathematics with Philosophy

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)

Abstract

This document presents a detailed recreation of Prof. Yucong Duan's Data-Information-Knowledge-Wisdom-Purpose (DIKWP) Semantic Mathematics framework, integrating philosophical insights from Ludwig Wittgenstein's Logisch-Philosophische Abhandlung and Baruch Spinoza's philosophy. By focusing on the core semantics of Sameness (Data), Difference (Information), and Completeness (Knowledge), we explore how these philosophical perspectives align with and enrich the DIKWP framework. This integration provides a robust mathematical foundation for modeling cognitive processes, enhancing artificial intelligence systems' ability to represent reality, process information, and construct knowledge in a manner aligned with human cognition.

Table of Contents

1. Introduction

• 1.1. Overview

• 1.2. Objectives

2. Overview of the DIKWP Semantic Mathematics Framework

• 2.1. The DIKWP Model

• 2.2. Importance in AI and Cognitive Modeling

3. Mathematical Foundations of DIKWP Semantics

• 3.1. Data: Sameness

• 3.2. Information: Difference

• 3.3. Knowledge: Completeness

• 3.4. Wisdom and Purpose

4. Integration with Philosophical Insights

• 4.2.1. Substance as Data (Sameness)

• 4.2.2. Attributes as Information (Difference)

• 4.2.3. Modes as Knowledge (Completeness)

• 4.2.4. Implications

• 4.1.1. Alignment with DIKWP Semantics

• 4.1.2. Implications

• 4.1. Wittgenstein's Logical Structure

• 4.2. Spinoza's Substance and Modes

5. Detailed Recreation of the DIKWP Semantic Mathematics Framework

• 5.1. Formal Definitions and Theorems

• 5.2. Mathematical Models and Examples

• 5.3. Extended Concepts: Wisdom and Purpose

6. Applications in Artificial Intelligence and Cognitive Modeling

• 6.1. Semantic Representation

• 6.2. Logical Reasoning and Knowledge Construction

• 6.3. Ethical Decision-Making and Purpose Alignment

7. Conclusion

8. References

1. Introduction1.1. Overview

The Data-Information-Knowledge-Wisdom-Purpose (DIKWP) model serves as a foundational framework for understanding cognitive processes and facilitating effective communication between humans and artificial intelligence (AI) systems. Central to this framework are the semantic dimensions of Data, Information, Knowledge, Wisdom, and Purpose, each characterized by distinct mathematical properties. By integrating philosophical insights from Wittgenstein's logical structuring and Spinoza's metaphysical concepts, we aim to enrich the DIKWP framework, providing deeper theoretical underpinnings and practical implications for AI development.

1.2. Objectives

• Recreate the DIKWP Semantic Mathematics framework in detail.

• Integrate philosophical insights from Wittgenstein and Spinoza into the DIKWP model.

• Provide mathematical formalizations of the core semantics: Sameness (Data), Difference (Information), and Completeness (Knowledge).

• Demonstrate applications in AI and cognitive modeling.

• Discuss implications and future directions.

2. Overview of the DIKWP Semantic Mathematics Framework2.1. The DIKWP Model

The DIKWP model outlines a hierarchical structure of cognitive processing:

1. Data (D): Raw, unprocessed facts characterized by sameness.

2. Information (I): Data processed to highlight differences, providing context and meaning.

3. Knowledge (K): Integration of information into a coherent, complete understanding.

4. Wisdom (W): Judicious application of knowledge, reflecting deep understanding and ethical considerations.

5. Purpose (P): Guiding motivations and intentions that direct cognitive processes and actions.

2.2. Importance in AI and Cognitive Modeling

• Semantic Alignment: Ensures AI systems interpret data in ways consistent with human understanding.

• Cognitive Emulation: Models human cognitive processes, enhancing AI's ability to reason and learn.

• Ethical AI: Incorporates wisdom and purpose to align AI actions with ethical principles and human values.

3. Mathematical Foundations of DIKWP Semantics3.1. Data: SamenessDefinition

Data is characterized by the sameness or uniformity among data elements based on shared semantic attributes.

Mathematical Formalization

• Equivalence Relation (∼): A binary relation on set $D$ satisfying:

1. Reflexivity: $d\sim d$ for all $d\in D$.

2. Symmetry: If d1∼d2d_1 \sim d_2d1​∼d2​, then d2∼d1d_2 \sim d_1d2​∼d1​.

3. Transitivity: If d1∼d2d_1 \sim d_2d1​∼d2​ and d2∼d3d_2 \sim d_3d2​∼d3​, then d1∼d3d_1 \sim d_3d1​∼d3​.

• Equivalence Classes: Partition $D$ into disjoint subsets where each element is equivalent under $\sim$.

Example

Let D={d1,d2,d3,d4}D = \{ d_1, d_2, d_3, d_4 \}D={d1​,d2​,d3​,d4​} with attributes $\text{Color}$ and $\text{Size}$.

• d1d_1d1​ and d2d_2d2​ share the same attributes: $\text{Red},\text{Large}$.

• Equivalence class [d1]={d1,d2}[d_1] = \{ d_1, d_2 \}[d1​]={d1​,d2​}.

3.2. Information: DifferenceDefinition

Information quantifies the difference or variability between data elements.

Mathematical Formalization

• Distance Metric (δ): A function $\delta :D\times D\to \mathbb{R}$ satisfying:

1. Non-negativity: δ(d1,d2)≥0\delta(d_1, d_2) \geq 0δ(d1​,d2​)≥0.

2. Identity of Indiscernibles: δ(d1,d2)=0  ⟺  d1=d2\delta(d_1, d_2) = 0 \iff d_1 = d_2δ(d1​,d2​)=0⟺d1​=d2​.

3. Symmetry: δ(d1,d2)=δ(d2,d1)\delta(d_1, d_2) = \delta(d_2, d_1)δ(d1​,d2​)=δ(d2​,d1​).

4. Triangle Inequality: δ(d1,d3)≤δ(d1,d2)+δ(d2,d3)\delta(d_1, d_3) \leq \delta(d_1, d_2) + \delta(d_2, d_3)δ(d1​,d3​)≤δ(d1​,d2​)+δ(d2​,d3​).

• Information Set (I): I={δ(di,dj)∣di,dj∈D,di≠dj}I = \{ \delta(d_i, d_j) \mid d_i, d_j \in D, d_i \neq d_j \}I={δ(di​,dj​)∣di​,dj​∈D,di​=dj​}.

Example

Using Euclidean distance for numerical attributes:

δ(di,dj)=∑k(fk(di)−fk(dj))2\delta(d_i, d_j) = \sqrt{\sum_{k} (f_k(d_i) - f_k(d_j))^2}δ(di​,dj​)=∑k​(fk​(di​)−fk​(dj​))2​.

3.3. Knowledge: CompletenessDefinition

Knowledge represents a complete and consistent understanding derived from information.

Mathematical Formalization

• Formal System (K): $K=(S,⊢)$ where:

• $S$: Set of axioms derived from $I$.

• $⊢$: Deduction relation.

• Logical Completeness: For every proposition $\varphi$ in language $\mathbb{L}$, either $\varphi$ or $\neg \varphi$ is derivable in $K$.

• Consistency: No contradictions are derivable; $⊬\varphi \wedge \neg \varphi$.

Example

Given $I$, construct $K$ such that all logical consequences of $I$ are included.

3.4. Wisdom and PurposeWisdom (W)

• Definition: The judicious application of knowledge, incorporating ethical considerations and deep understanding.

• Mathematical Formalization: Wisdom can be modeled using utility functions or decision-theoretic frameworks that evaluate the application of knowledge.

Purpose (P)

• Definition: The guiding motivations and intentions that direct actions and cognitive processes.

• Mathematical Formalization: Purpose can be represented as goal-oriented functions or optimization objectives that influence decision-making.

4. Integration with Philosophical Insights4.1. Wittgenstein's Logical Structure4.1.1. Alignment with DIKWP Semantics

• Logical Structuring: Wittgenstein's propositions align with the hierarchical structuring of DIKWP semantics.

• Data (Sameness): The world as the totality of facts corresponds to data elements characterized by sameness.

• Information (Difference): Facts and states of affairs introduce differences, aligning with the information level.

• Knowledge (Completeness): Logical pictures and propositions represent knowledge derived from information.

4.1.2. Implications

• Emphasizes Logical Form: Highlights the importance of logical structure in representing reality.

• Limits of Language: Wittgenstein's focus on what can be meaningfully expressed aligns with ensuring completeness and consistency in knowledge.

4.2. Spinoza's Substance and Modes4.2.1. Substance as Data (Sameness)

• Single Substance: Spinoza's concept of a single, self-sufficient substance maps to the data level characterized by sameness.

• Uniformity: All entities are manifestations of the same substance, reflecting uniformity among data elements.

4.2.2. Attributes as Information (Difference)

• Attributes: Different ways the intellect perceives substance correspond to differences in data, forming information.

• Variability: Attributes introduce distinctions, aligning with the measurement of differences.

4.2.3. Modes as Knowledge (Completeness)

• Modes: Particular expressions or states derived from substance and attributes represent knowledge.

• Complete Understanding: Knowledge encompasses all modes, ensuring completeness.

4.2.4. Implications

• Unified Reality: Emphasizes the interconnectedness of all entities.

• Pursuit of Understanding: Aligns with the DIKWP's aim to construct complete and consistent knowledge.

5. Detailed Recreation of the DIKWP Semantic Mathematics Framework5.1. Formal Definitions and TheoremsData Semantics: Sameness

• Equivalence Relation ($\sim$):

• Definition: For d1,d2∈Dd_1, d_2 \in Dd1​,d2​∈D, d1∼d2d_1 \sim d_2d1​∼d2​ if they share identical attributes.

• Mathematical Representation:

  d1∼d2  ⟺  ∀f∈S, f(d1)=f(d2)d_1 \sim d_2 \iff \forall f \in S, \ f(d_1) = f(d_2)d1​∼d2​⟺∀f∈S, f(d1​)=f(d2​).

• Equivalence Classes ($[d]$):

• Definition: [d]={d′∈D∣d′∼d}[d] = \{ d' \in D \mid d' \sim d \}[d]={d′∈D∣d′∼d}.

• Properties:

• Partitioning: D=⋃[d][d]D = \bigcup_{[d]} [d]D=⋃[d]​[d], with [di]∩[dj]=∅[d_i] \cap [d_j] = \emptyset[di​]∩[dj​]=∅ for $i=j$.

Information Semantics: Difference

• Distance Metric ($\delta$):

• Euclidean Distance:

  δ(d1,d2)=∑f∈S(f(d1)−f(d2))2\delta(d_1, d_2) = \sqrt{\sum_{f \in S} (f(d_1) - f(d_2))^2}δ(d1​,d2​)=∑f∈S​(f(d1​)−f(d2​))2​.

• Hamming Distance:

  δ(d1,d2)=∑f∈SI(f(d1)≠f(d2))\delta(d_1, d_2) = \sum_{f \in S} \mathbb{I}(f(d_1) \neq f(d_2))δ(d1​,d2​)=∑f∈S​I(f(d1​)=f(d2​)).

• Definition: $\delta :D\times D\to \mathbb{R}$, measuring dissimilarity based on attributes.

• Examples:

• Information Set ($I$):

• Definition: I={δ(di,dj)∣di,dj∈D,di≠dj}I = \{ \delta(d_i, d_j) \mid d_i, d_j \in D, d_i \neq d_j \}I={δ(di​,dj​)∣di​,dj​∈D,di​=dj​}.

Knowledge Semantics: Completeness

• Formal System ($K=(S,⊢)$):

• Axioms ($S$): Derived from information $I$.

• Deduction Relation ($⊢$): Rules for deriving propositions.

• Logical Completeness:

• Criterion: $\forall \varphi \in \mathbb{L},\varphi \in K\vee \neg \varphi \in K$.

• Consistency:

• Criterion: $⊬\varphi \wedge \neg \varphi$, avoiding contradictions.

5.2. Mathematical Models and ExamplesExample Dataset

• Data Elements:

  D={d1,d2,d3,d4}D = \{ d_1, d_2, d_3, d_4 \}D={d1​,d2​,d3​,d4​}.

• Attributes:

  $S=\{\text{Color},\text{Shape}\}$.

• Attribute Functions:

• fColor(d)f_{\text{Color}}(d)fColor​(d): Maps $d$ to its color value.

• fShape(d)f_{\text{Shape}}(d)fShape​(d): Maps $d$ to its shape value.

Equivalence Classes

• Based on Sameness:

• [d1]={d1,d2}[d_1] = \{ d_1, d_2 \}[d1​]={d1​,d2​} if d1∼d2d_1 \sim d_2d1​∼d2​.

• Partitioning Data:

• D=[d1]∪[d3]∪[d4]D = [d_1] \cup [d_3] \cup [d_4]D=[d1​]∪[d3​]∪[d4​].

Calculating Differences

• Using Euclidean Distance:

• δ(d1,d3)=(fColor(d1)−fColor(d3))2+(fShape(d1)−fShape(d3))2\delta(d_1, d_3) = \sqrt{(f_{\text{Color}}(d_1) - f_{\text{Color}}(d_3))^2 + (f_{\text{Shape}}(d_1) - f_{\text{Shape}}(d_3))^2}δ(d1​,d3​)=(fColor​(d1​)−fColor​(d3​))2+(fShape​(d1​)−fShape​(d3​))2​.

Constructing Knowledge

• Axioms ($S$):

• Derived from $I$, such as "If δ(di,dj)\delta(d_i, d_j)δ(di​,dj​) is small, then did_idi​ and djd_jdj​ share common properties."

• Deriving Propositions:

• Use deduction rules to infer new knowledge.

5.3. Extended Concepts: Wisdom and PurposeWisdom

• Utility Function ($U$):

• Definition: $U:K\to \mathbb{R}$, assigning value to knowledge propositions.

• Decision-Making:

• Choose actions maximizing expected utility based on knowledge.

Purpose

• Goal Function ($G$):

• Definition: $G:D\to \text{Outcomes}$, guiding data processing toward desired outcomes.

• Optimization Objective:

• Formulate as an optimization problem: max⁡aU(a∣K,P)\max_{a} U(a \mid K, P)maxa​U(a∣K,P), where $a$ is an action, and $P$ represents purpose.

6. Applications in Artificial Intelligence and Cognitive Modeling6.1. Semantic Representation

• Data Modeling:

• Represent real-world entities as data elements with attributes.

• Information Processing:

• Quantify differences to identify patterns and relationships.

6.2. Logical Reasoning and Knowledge Construction

• Knowledge Bases:

• Build complete and consistent knowledge bases using formal systems.

• Inference Engines:

• Implement deduction relations to derive new propositions.

6.3. Ethical Decision-Making and Purpose Alignment

• Incorporating Wisdom:

• Use utility functions to evaluate the ethical implications of actions.

• Aligning with Purpose:

• Ensure AI systems act in accordance with predefined goals and values.

7. Conclusion

By recreating the DIKWP Semantic Mathematics framework in detail and integrating philosophical insights from Wittgenstein and Spinoza, we have established a robust mathematical foundation for modeling cognitive processes. This integration enriches the framework, providing deeper theoretical underpinnings and practical applications in artificial intelligence and cognitive modeling. The alignment with philosophical perspectives emphasizes the importance of logical structure, unified reality, and the pursuit of comprehensive understanding, enhancing AI systems' ability to reason, learn, and act ethically.

8. References

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. ".

3. Wittgenstein, L. (1921). Logisch-Philosophische Abhandlung (Tractatus Logico-Philosophicus). (Various translations).

4. Spinoza, B. (1677). Ethics. (Translated editions available).

5. Russell, B. (1903). The Principles of Mathematics. Cambridge University Press.

6. Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379–423.

7. Russell, S., & Norvig, P. (2021). Artificial Intelligence: A Modern Approach (4th ed.). Pearson.

8. Floridi, L. (2011). The Philosophy of Information. Oxford University Press.

9. Gärdenfors, P. (2000). Conceptual Spaces: The Geometry of Thought. MIT Press.

10. Frege, G. (1892). On Sense and Reference.

11. Hilbert, D., & Ackermann, P. (1928). Principles of Mathematical Logic. Chelsea Publishing.

Keywords: DIKWP Semantic Mathematics, Sameness, Difference, Completeness, Data, Information, Knowledge, Wisdom, Purpose, Wittgenstein, Spinoza, Artificial Intelligence, Semantic Representation, Logical Reasoning, Cognitive Modeling.

Note: This document provides a detailed recreation of the DIKWP Semantic Mathematics framework, integrating philosophical insights from Wittgenstein and Spinoza. By focusing on the core semantics of Sameness (Data), Difference (Information), and Completeness (Knowledge), and extending to Wisdom and Purpose, we establish a comprehensive mathematical foundation for modeling cognitive processes in AI systems. The integration with philosophical concepts enriches the framework, highlighting the importance of logical structure and unified understanding in representing reality.