Mathematical Definitions of Four Spaces by Networked DIKWP  

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)

Table of Contents

1. Introduction

2. Mathematical Foundations

• 2.1. Sets and Functions

• 2.2. Graph Theory

• 2.3. Transformations and Mappings

3. Defining the Four Spaces

• 3.1. Conceptual Space (ConC)

• 3.2. Cognitive Space (ConN)

• 3.3. Semantic Space (SemA)

• 3.4. Conscious Space (ConsciousS)

4. Integration with Networked DIKWP Transformations

• 4.1. Mapping Transformations to Spaces

• 4.2. Interplay Among Spaces

5. Mathematical Representation of Transformation Modes

• 5.1. Minimal Impact Transformations

• 5.2. Direct Transformations

• 5.3. Indirect and Complex Transformations

• 5.4. Transformation Dynamics

6. Conclusion

7. References

1. Introduction

The Data-Information-Knowledge-Wisdom-Purpose (DIKWP) model serves as a comprehensive framework to understand the progression and transformation of cognitive elements. The Four Spaces Framework—comprising Conceptual Space (ConC), Cognitive Space (ConN), Semantic Space (SemA), and Conscious Space—provides a multidimensional perspective to analyze these transformations. This document offers a mathematical formalization of the Four Spaces within the context of networked DIKWP transformations, leveraging foundational concepts from set theory, graph theory, and functional mappings.

2. Mathematical Foundations

To rigorously define the Four Spaces and their interactions with the DIKWP transformations, we establish the following mathematical foundations.

2.1. Sets and Functions

• Sets: Fundamental collections of distinct objects.

• Notation: Capital letters (e.g., $S$, $C$, $D$) denote sets; lowercase letters (e.g., $s$, $c$, $d$) denote elements.

• Functions: Mappings from one set to another.

• Notation: $f:A\to B$ denotes a function $f$ mapping elements from set $A$ to set $B$.

• Relations: Subsets of Cartesian products of sets, representing connections between elements.

• Notation: $R\subseteq A\times B$ denotes a relation between sets $A$ and $B$.

2.2. Graph Theory

• Graphs: Structures composed of nodes (vertices) and edges (connections).

• Notation: $G=(V,E)$ where $V$ is the set of vertices and $E$ is the set of edges.

• Directed Graphs: Graphs where edges have a direction.

• Notation: $G=(V,E)$ with $E\subseteq V\times V$.

• Undirected Graphs: Graphs where edges do not have a direction.

• Notation: $G=(V,E)$ with $E\subseteq \{\{u,v\}\mid u,v\in V\}$.

2.3. Transformations and Mappings

• Transformation Functions: Functions that convert one cognitive element to another.

• Notation: TXY:X→YT_{XY}: X \rightarrow YTXY​:X→Y, where $X,Y\in \{D,I,K,W,P\}$.

• Composite Functions: Sequential application of multiple functions.

• Notation: $f\circ g$ denotes function $g$ applied first, then function $f$.

3. Defining the Four Spaces

Each of the Four Spaces is mathematically defined to encapsulate specific aspects of the DIKWP model's cognitive processes.

3.1. Conceptual Space (ConC)

Definition:The Conceptual Space (ConC) is a directed graph representing the cognitive representations of concepts, their attributes, and inter-concept relationships.

Mathematical Representation:

GraphConC=(VConC,EConC)\text{GraphConC} = (V_{\text{ConC}}, E_{\text{ConC}})GraphConC=(VConC​,EConC​)

• Vertices (VConCV_{\text{ConC}}VConC​):Each vertex v∈VConCv \in V_{\text{ConC}}v∈VConC​ represents a distinct concept.

• Edges (EConCE_{\text{ConC}}EConC​):Each edge e=(vi,vj)∈EConCe = (v_i, v_j) \in E_{\text{ConC}}e=(vi​,vj​)∈EConC​ represents a relationship from concept viv_ivi​ to concept vjv_jvj​.

• Attributes and Relationships:

• Attributes: Each concept $v$ has an attribute set A(v)={a1(v),a2(v),…,an(v)}A(v) = \{a_1(v), a_2(v), \dots, a_n(v)\}A(v)={a1​(v),a2​(v),…,an​(v)}.

• Relationships: R(vi,vj)R(v_i, v_j)R(vi​,vj​) denotes the relationship type between viv_ivi​ and vjv_jvj​.

Operations:

• Query:QConC(VConC,EConC,q)→{v1,v2,…,vm}Q_{\text{ConC}}(V_{\text{ConC}}, E_{\text{ConC}}, q) \rightarrow \{v_1, v_2, \dots, v_m\}QConC​(VConC​,EConC​,q)→{v1​,v2​,…,vm​}Returns concepts satisfying query $q$ based on attributes or relationships.

• Add Concept:AddConC(VConC,v)\text{Add}_{\text{ConC}}(V_{\text{ConC}}, v)AddConC​(VConC​,v)Adds a new concept $v$ to VConCV_{\text{ConC}}VConC​.

• Update Attributes:UpdateConC(VConC,v,A(v))\text{Update}_{\text{ConC}}(V_{\text{ConC}}, v, A(v))UpdateConC​(VConC​,v,A(v))Updates the attribute set of concept $v$.

3.2. Cognitive Space (ConN)

Definition:The Cognitive Space (ConN) is a functional space where cognitive processing occurs, transforming inputs from one DIKWP component to another through a series of cognitive functions.

Mathematical Representation:

$$\text{ConN}=(R,F)$$

• Relations ($R$):Represents the flow of information and transformations among DIKWP components.

• Function Set ($F$):F={fConN1,fConN2,…,fConNn}F = \{f_{\text{ConN}_1}, f_{\text{ConN}_2}, \dots, f_{\text{ConN}_n}\}F={fConN1​​,fConN2​​,…,fConNn​​}Each function fConNi:Inputi→Outputif_{\text{ConN}_i}: \text{Input}_i \rightarrow \text{Output}_ifConNi​​:Inputi​→Outputi​ represents a specific cognitive processing step.

Input and Output Spaces:

• Input Space (Inputi\text{Input}_iInputi​):Inputi⊆{D,I}\text{Input}_i \subseteq \{ D, I \}Inputi​⊆{D,I}Data or Information inputs received by the cognitive system.

• Output Space (Outputi\text{Output}_iOutputi​):Outputi⊆{I,K,W,P}\text{Output}_i \subseteq \{ I, K, W, P \}Outputi​⊆{I,K,W,P}Processed outputs such as Information classification, Knowledge formation, Wisdom synthesis, or Purpose determination.

Function Decomposition:

fConNi=fConNi(5)∘fConNi(4)∘⋯∘fConNi(1)f_{\text{ConN}_i} = f_{\text{ConN}_i}(5) \circ f_{\text{ConN}_i}(4) \circ \dots \circ f_{\text{ConN}_i}(1)fConNi​​=fConNi​​(5)∘fConNi​​(4)∘⋯∘fConNi​​(1)

Each fConNi(j)f_{\text{ConN}_i}(j)fConNi​​(j) represents a sub-step (e.g., data preprocessing, feature extraction).

3.3. Semantic Space (SemA)

Definition:The Semantic Space (SemA) is a directed graph representing semantic units and their associations, facilitating the communication and interpretation of meaning within the DIKWP model.

Mathematical Representation:

GraphSemA=(VSemA,ESemA)\text{GraphSemA} = (V_{\text{SemA}}, E_{\text{SemA}})GraphSemA=(VSemA​,ESemA​)

• Vertices (VSemAV_{\text{SemA}}VSemA​):Each vertex s∈VSemAs \in V_{\text{SemA}}s∈VSemA​ represents a semantic unit (e.g., word, phrase).

• Edges (ESemAE_{\text{SemA}}ESemA​):Each edge e=(si,sj)∈ESemAe = (s_i, s_j) \in E_{\text{SemA}}e=(si​,sj​)∈ESemA​ represents a semantic relationship between sis_isi​ and sjs_jsj​.

Operations:

• Query:QSemA(VSemA,ESemA,q)→{s1,s2,…,sm}Q_{\text{SemA}}(V_{\text{SemA}}, E_{\text{SemA}}, q) \rightarrow \{s_1, s_2, \dots, s_m\}QSemA​(VSemA​,ESemA​,q)→{s1​,s2​,…,sm​}Returns semantic units satisfying query $q$.

• Add Semantic Unit:AddSemA(VSemA,s)\text{Add}_{\text{SemA}}(V_{\text{SemA}}, s)AddSemA​(VSemA​,s)Adds a new semantic unit $s$ to VSemAV_{\text{SemA}}VSemA​.

• Update Relationship:UpdateSemA(ESemA,si,sj,r)\text{Update}_{\text{SemA}}(E_{\text{SemA}}, s_i, s_j, r)UpdateSemA​(ESemA​,si​,sj​,r)Updates the relationship $r$ between semantic units sis_isi​ and sjs_jsj​.

3.4. Conscious Space (ConsciousS)

Definition:The Conscious Space (ConsciousS) encapsulates the ethical, reflective, and value-based dimensions of cognition, integrating Purpose into the cognitive and semantic processes.

Mathematical Representation:

ConsciousS=(VConsciousS,EConsciousS,P)\text{ConsciousS} = (V_{\text{ConsciousS}}, E_{\text{ConsciousS}}, P)ConsciousS=(VConsciousS​,EConsciousS​,P)

• Vertices (VConsciousSV_{\text{ConsciousS}}VConsciousS​):Each vertex c∈VConsciousSc \in V_{\text{ConsciousS}}c∈VConsciousS​ represents an ethical or reflective concept.

• Edges (EConsciousSE_{\text{ConsciousS}}EConsciousS​):Each edge e=(ci,cj)∈EConsciousSe = (c_i, c_j) \in E_{\text{ConsciousS}}e=(ci​,cj​)∈EConsciousS​ represents an ethical or reflective relationship between concepts cic_ici​ and cjc_jcj​.

• Purpose ($P$):$P$ denotes the set of Purpose-driven functions influencing transformations within Conscious Space.

Operations:

• Ethical Evaluation:EvaluateConsciousS(K,W,P)→W\text{Evaluate}_{\text{ConsciousS}}(K, W, P) \rightarrow WEvaluateConsciousS​(K,W,P)→WIntegrates Knowledge and Purpose to generate Wisdom.

• Purpose Definition:DefineConsciousS(P)\text{Define}_{\text{ConsciousS}}(P)DefineConsciousS​(P)Establishes or refines Purpose based on ethical considerations.

4. Integration with Networked DIKWP Transformations

The integration of the Four Spaces with the networked DIKWP transformations involves mapping each transformation mode to specific spaces and defining how these spaces interact during transformations.

4.1. Mapping Transformations to Spaces

Each transformation mode TXY:X→YT_{XY}: X \rightarrow YTXY​:X→Y within the DIKWP model is mapped to one or more spaces based on the nature of the transformation.

TransformationXYMapped Space(s)DescriptionTD→DDDConNData maintenance via cognitive processingTD→IDIConN, ConCData processed into InformationTD→KDKConN, ConCData analyzed into KnowledgeTD→WDWConN, ConsciousSData synthesized into WisdomTD→PDPConC, ConsciousSData-driven Purpose definitionTI→DIDConNInformation deconstructed into DataTI→IIISemA, ConNInformation refinementTI→KIKConN, SemAInformation organized into KnowledgeTI→WIWConN, ConsciousSInformation integrated into WisdomTI→PIPSemA, ConsciousSInformation leveraged to define PurposeTK→DKDConN, SemAKnowledge translated back to DataTK→IKISemA, ConNKnowledge communicated as InformationTK→KKKConC, ConNKnowledge refinement and expansionTK→WKWConN, ConsciousSKnowledge synthesized into WisdomTK→PKPConC, ConsciousSKnowledge used to define PurposeTW→DWDConN, ConCWisdom applied to generate DataTW→IWISemA, ConNWisdom translated into InformationTW→KWKConC, ConNWisdom refines KnowledgeTW→WWWConsciousSWisdom refinement through reflectionTW→PWPConsciousSWisdom shapes PurposeTP→DPDConNPurpose-directed Data generationTP→IPISemA, ConNPurpose-guided Information processingTP→KPKConC, ConNPurpose-driven Knowledge developmentTP→WPWConsciousSPurpose integrates with WisdomTP→PPPConsciousSPurpose refinement and redefinition\begin{array}{|c|c|c|c|c|} \hline \text{Transformation} & X & Y & \text{Mapped Space(s)} & \text{Description} \\ \hline T_{D \rightarrow D} & D & D & \text{ConN} & \text{Data maintenance via cognitive processing} \\ T_{D \rightarrow I} & D & I & \text{ConN, ConC} & \text{Data processed into Information} \\ T_{D \rightarrow K} & D & K & \text{ConN, ConC} & \text{Data analyzed into Knowledge} \\ T_{D \rightarrow W} & D & W & \text{ConN, ConsciousS} & \text{Data synthesized into Wisdom} \\ T_{D \rightarrow P} & D & P & \text{ConC, ConsciousS} & \text{Data-driven Purpose definition} \\ T_{I \rightarrow D} & I & D & \text{ConN} & \text{Information deconstructed into Data} \\ T_{I \rightarrow I} & I & I & \text{SemA, ConN} & \text{Information refinement} \\ T_{I \rightarrow K} & I & K & \text{ConN, SemA} & \text{Information organized into Knowledge} \\ T_{I \rightarrow W} & I & W & \text{ConN, ConsciousS} & \text{Information integrated into Wisdom} \\ T_{I \rightarrow P} & I & P & \text{SemA, ConsciousS} & \text{Information leveraged to define Purpose} \\ T_{K \rightarrow D} & K & D & \text{ConN, SemA} & \text{Knowledge translated back to Data} \\ T_{K \rightarrow I} & K & I & \text{SemA, ConN} & \text{Knowledge communicated as Information} \\ T_{K \rightarrow K} & K & K & \text{ConC, ConN} & \text{Knowledge refinement and expansion} \\ T_{K \rightarrow W} & K & W & \text{ConN, ConsciousS} & \text{Knowledge synthesized into Wisdom} \\ T_{K \rightarrow P} & K & P & \text{ConC, ConsciousS} & \text{Knowledge used to define Purpose} \\ T_{W \rightarrow D} & W & D & \text{ConN, ConC} & \text{Wisdom applied to generate Data} \\ T_{W \rightarrow I} & W & I & \text{SemA, ConN} & \text{Wisdom translated into Information} \\ T_{W \rightarrow K} & W & K & \text{ConC, ConN} & \text{Wisdom refines Knowledge} \\ T_{W \rightarrow W} & W & W & \text{ConsciousS} & \text{Wisdom refinement through reflection} \\ T_{W \rightarrow P} & W & P & \text{ConsciousS} & \text{Wisdom shapes Purpose} \\ T_{P \rightarrow D} & P & D & \text{ConN} & \text{Purpose-directed Data generation} \\ T_{P \rightarrow I} & P & I & \text{SemA, ConN} & \text{Purpose-guided Information processing} \\ T_{P \rightarrow K} & P & K & \text{ConC, ConN} & \text{Purpose-driven Knowledge development} \\ T_{P \rightarrow W} & P & W & \text{ConsciousS} & \text{Purpose integrates with Wisdom} \\ T_{P \rightarrow P} & P & P & \text{ConsciousS} & \text{Purpose refinement and redefinition} \\ \hline \end{array}TransformationTD→D​TD→I​TD→K​TD→W​TD→P​TI→D​TI→I​TI→K​TI→W​TI→P​TK→D​TK→I​TK→K​TK→W​TK→P​TW→D​TW→I​TW→K​TW→W​TW→P​TP→D​TP→I​TP→K​TP→W​TP→P​​XDDDDDIIIIIKKKKKWWWWWPPPPP​YDIKWPDIKWPDIKWPDIKWPDIKWP​Mapped Space(s)ConNConN, ConCConN, ConCConN, ConsciousSConC, ConsciousSConNSemA, ConNConN, SemAConN, ConsciousSSemA, ConsciousSConN, SemASemA, ConNConC, ConNConN, ConsciousSConC, ConsciousSConN, ConCSemA, ConNConC, ConNConsciousSConsciousSConNSemA, ConNConC, ConNConsciousSConsciousS​DescriptionData maintenance via cognitive processingData processed into InformationData analyzed into KnowledgeData synthesized into WisdomData-driven Purpose definitionInformation deconstructed into DataInformation refinementInformation organized into KnowledgeInformation integrated into WisdomInformation leveraged to define PurposeKnowledge translated back to DataKnowledge communicated as InformationKnowledge refinement and expansionKnowledge synthesized into WisdomKnowledge used to define PurposeWisdom applied to generate DataWisdom translated into InformationWisdom refines KnowledgeWisdom refinement through reflectionWisdom shapes PurposePurpose-directed Data generationPurpose-guided Information processingPurpose-driven Knowledge developmentPurpose integrates with WisdomPurpose refinement and redefinition​​

Explanation of Mapped Spaces:

• ConC (Conceptual Space): Involved when transformations require the formulation or refinement of concepts.

• ConN (Cognitive Space): Central to processing and transforming data and information.

• SemA (Semantic Space): Engaged when meanings and communications are restructured or interpreted.

• ConsciousS (Conscious Space): Integral when ethical, reflective, or purpose-driven considerations influence transformations.

4.2. Interplay Among Spaces

Transformations often involve multiple spaces working in tandem. The interplay among spaces ensures that transformations are coherent, ethically grounded, and contextually relevant.

Example 1: TK→WT_{K \rightarrow W}TK→W​ (Knowledge to Wisdom)

ConN:Synthesize structured Knowledge into higher-order insightsConsciousS:Integrate ethical and contextual considerations into WisdomResult:Wisdom that is both intellectually robust and ethically sound\begin{align*} & \text{ConN}: \text{Synthesize structured Knowledge into higher-order insights} \\ & \text{ConsciousS}: \text{Integrate ethical and contextual considerations into Wisdom} \\ & \text{Result}: \text{Wisdom that is both intellectually robust and ethically sound} \end{align*}​ConN:Synthesize structured Knowledge into higher-order insightsConsciousS:Integrate ethical and contextual considerations into WisdomResult:Wisdom that is both intellectually robust and ethically sound​

Example 2: TI→PT_{I \rightarrow P}TI→P​ (Information to Purpose)

SemA:Shape information semantics to align with strategic goalsConsciousS:Ensure Purpose definition adheres to ethical standardsResult:Purpose-driven objectives informed by ethical and semantic alignment\begin{align*} & \text{SemA}: \text{Shape information semantics to align with strategic goals} \\ & \text{ConsciousS}: \text{Ensure Purpose definition adheres to ethical standards} \\ & \text{Result}: \text{Purpose-driven objectives informed by ethical and semantic alignment} \end{align*}​SemA:Shape information semantics to align with strategic goalsConsciousS:Ensure Purpose definition adheres to ethical standardsResult:Purpose-driven objectives informed by ethical and semantic alignment​

Example 3: TD→WT_{D \rightarrow W}TD→W​ (Data to Wisdom)

ConN:Process raw Data into actionable insightsConsciousS:Apply ethical considerations to synthesize WisdomResult:Ethical Wisdom derived directly from Data\begin{align*} & \text{ConN}: \text{Process raw Data into actionable insights} \\ & \text{ConsciousS}: \text{Apply ethical considerations to synthesize Wisdom} \\ & \text{Result}: \text{Ethical Wisdom derived directly from Data} \end{align*}​ConN:Process raw Data into actionable insightsConsciousS:Apply ethical considerations to synthesize WisdomResult:Ethical Wisdom derived directly from Data​

Example 4: TW→PT_{W \rightarrow P}TW→P​ (Wisdom to Purpose)

ConsciousS:Utilize Wisdom to define or refine PurposeResult:Purpose aligned with ethical Wisdom\begin{align*} & \text{ConsciousS}: \text{Utilize Wisdom to define or refine Purpose} \\ & \text{Result}: \text{Purpose aligned with ethical Wisdom} \end{align*}​ConsciousS:Utilize Wisdom to define or refine PurposeResult:Purpose aligned with ethical Wisdom​

5. Mathematical Representation of Transformation Modes

Each transformation mode TXY:X→YT_{XY}: X \rightarrow YTXY​:X→Y can be represented as a function that maps elements from one DIKWP component to another within the appropriate spaces.

5.1. Minimal Impact Transformations (D→D, I→I, K→K, W→W, P→P)

Definition:Minimal impact transformations involve maintaining the integrity and consistency of existing elements without significant alteration.

Mathematical Representation:

TXX:X→XT_{XX}: X \rightarrow XTXX​:X→X

Where $X\in \{D,I,K,W,P\}$.

Mapped Space:

$$\text{ConN}$$

Since these transformations primarily involve internal reinforcement within the Cognitive Space.

Examples:

• TD→DT_{D \rightarrow D}TD→D​: Data verification and reinforcement.

• TI→IT_{I \rightarrow I}TI→I​: Information refinement and clarification.

• TK→KT_{K \rightarrow K}TK→K​: Knowledge consolidation and updating.

• TW→WT_{W \rightarrow W}TW→W​: Wisdom refinement and ethical review.

• TP→PT_{P \rightarrow P}TP→P​: Purpose reaffirmation and goal adjustment.

5.2. Direct Transformations (D→I, D→K, D→W, D→P)

Definition:Direct transformations involve processing raw data into more refined constructs or aligning data with specific purposes.

Mathematical Representation:

TXY:D→YT_{XY}: D \rightarrow YTXY​:D→Y

Where $Y\in \{I,K,W,P\}$.

Mapped Spaces:

$$\text{ConN, ConC}\text{ (and }\text{ConsciousS}\text{ for }W\text{ and }P)$$

Examples:

• TD→IT_{D \rightarrow I}TD→I​:

  TD→I:SD×CD×ID→SI×CI×IIT_{D \rightarrow I}: \mathcal{S}_D \times \mathcal{C}_D \times \mathcal{I}_D \rightarrow \mathcal{S}_I \times \mathcal{C}_I \times \mathcal{I}_ITD→I​:SD​×CD​×ID​→SI​×CI​×II​

  Data processed into Information by identifying patterns and relationships.

• TD→KT_{D \rightarrow K}TD→K​:

  TD→K:SD×CD×ID→SK×CK×IKT_{D \rightarrow K}: \mathcal{S}_D \times \mathcal{C}_D \times \mathcal{I}_D \rightarrow \mathcal{S}_K \times \mathcal{C}_K \times \mathcal{I}_KTD→K​:SD​×CD​×ID​→SK​×CK​×IK​

  Data analyzed into Knowledge by structuring and organizing information.

• TD→WT_{D \rightarrow W}TD→W​:

  TD→W:SD×CD×ID→SW×CW×IWT_{D \rightarrow W}: \mathcal{S}_D \times \mathcal{C}_D \times \mathcal{I}_D \rightarrow \mathcal{S}_W \times \mathcal{C}_W \times \mathcal{I}_WTD→W​:SD​×CD​×ID​→SW​×CW​×IW​

  Data synthesized into Wisdom by integrating ethical and contextual insights.

• TD→PT_{D \rightarrow P}TD→P​:

  TD→P:SD×CD×ID→SP×CP×IPT_{D \rightarrow P}: \mathcal{S}_D \times \mathcal{C}_D \times \mathcal{I}_D \rightarrow \mathcal{S}_P \times \mathcal{C}_P \times \mathcal{I}_PTD→P​:SD​×CD​×ID​→SP​×CP​×IP​

  Data-driven Purpose definition aligning with organizational or personal goals.

5.3. Indirect and Complex Transformations (I→D, I→I, I→K, I→W, I→P; K→D, K→I, K→K, K→W, K→P; W→D, W→I, W→K, W→W, W→P; P→D, P→I, P→K, P→W, P→P)

Definition:Indirect and complex transformations facilitate the evolution of elements through multiple interconnected processes, often involving multiple spaces.

Mathematical Representation:

TXY:X→YT_{XY}: X \rightarrow YTXY​:X→Y

Where $X,Y\in \{I,K,W,P\}$ and $X=Y$.

Mapped Spaces:

$$\text{Multiple Spaces}(\text{ConC, ConN, SemA, ConsciousS})$$

Examples:

• TI→KT_{I \rightarrow K}TI→K​:

  TI→K:SI×CI×II→SK×CK×IKT_{I \rightarrow K}: \mathcal{S}_I \times \mathcal{C}_I \times \mathcal{I}_I \rightarrow \mathcal{S}_K \times \mathcal{C}_K \times \mathcal{I}_KTI→K​:SI​×CI​×II​→SK​×CK​×IK​

  Information organized into Knowledge frameworks by establishing logical and semantic connections.

• TK→WT_{K \rightarrow W}TK→W​:

  TK→W:SK×CK×IK→SW×CW×IWT_{K \rightarrow W}: \mathcal{S}_K \times \mathcal{C}_K \times \mathcal{I}_K \rightarrow \mathcal{S}_W \times \mathcal{C}_W \times \mathcal{I}_WTK→W​:SK​×CK​×IK​→SW​×CW​×IW​

  Knowledge synthesized into Wisdom by integrating ethical and contextual insights.

• TW→PT_{W \rightarrow P}TW→P​:

  TW→P:SW×CW×IW→SP×CP×IPT_{W \rightarrow P}: \mathcal{S}_W \times \mathcal{C}_W \times \mathcal{I}_W \rightarrow \mathcal{S}_P \times \mathcal{C}_P \times \mathcal{I}_PTW→P​:SW​×CW​×IW​→SP​×CP​×IP​

  Wisdom shapes Purpose by aligning goals with ethical standards.

General Representation:

TXY:SX×CX×IX→SY×CY×IYT_{XY}: \mathcal{S}_X \times \mathcal{C}_X \times \mathcal{I}_X \rightarrow \mathcal{S}_Y \times \mathcal{C}_Y \times \mathcal{I}_YTXY​:SX​×CX​×IX​→SY​×CY​×IY​

Where SX,CX,IX\mathcal{S}_X, \mathcal{C}_X, \mathcal{I}_XSX​,CX​,IX​ and SY,CY,IY\mathcal{S}_Y, \mathcal{C}_Y, \mathcal{I}_YSY​,CY​,IY​ represent semantic attributes, concepts, and instances in respective spaces.

5.4. Transformation Dynamics

Synergistic Interactions:Many transformations involve synergistic interactions between spaces, enhancing the depth and applicability of cognitive and societal processes.

Ethical Integration:Conscious Space (ConsciousS) consistently plays a crucial role in transformations involving wisdom, ensuring that ethical considerations are integral to the process.

Mathematical Representation of Synergy:

Synergy(X,Y)=fConN(X)+fConsciousS(Y)\text{Synergy}(X, Y) = f_{\text{ConN}}(X) + f_{\text{ConsciousS}}(Y)Synergy(X,Y)=fConN​(X)+fConsciousS​(Y)

Where $X$ and $Y$ are transformation modes that require cognitive and ethical processing.

Example: Innovation Cycle

Innovation Cycle=ConC→TConC→ConNConN→TConN→SemASemA→TSemA→ConsciousSConsciousS→TConsciousS→ConCConC\text{Innovation Cycle} = \text{ConC} \xrightarrow{T_{ConC \rightarrow ConN}} \text{ConN} \xrightarrow{T_{ConN \rightarrow SemA}} \text{SemA} \xrightarrow{T_{SemA \rightarrow ConsciousS}} \text{ConsciousS} \xrightarrow{T_{ConsciousS \rightarrow ConC}} \text{ConC}Innovation Cycle=ConCTConC→ConN​​ConNTConN→SemA​​SemATSemA→ConsciousS​​ConsciousSTConsciousS→ConC​​ConC

This cycle represents a continuous loop where ideas are generated, processed, communicated, ethically evaluated, and refined.

5. Mathematical Representation of Transformation Modes

Each transformation mode TXY:X→YT_{XY}: X \rightarrow YTXY​:X→Y can be represented as a function that maps elements from one DIKWP component to another within the appropriate spaces.

General Representation:

TXY:SX×CX×IX→SY×CY×IYT_{XY}: \mathcal{S}_X \times \mathcal{C}_X \times \mathcal{I}_X \rightarrow \mathcal{S}_Y \times \mathcal{C}_Y \times \mathcal{I}_YTXY​:SX​×CX​×IX​→SY​×CY​×IY​

Where:

• SX\mathcal{S}_XSX​ and SY\mathcal{S}_YSY​ are semantic attribute sets for components $X$ and $Y$.

• CX\mathcal{C}_XCX​ and CY\mathcal{C}_YCY​ are concept sets within ConC for $X$ and $Y$.

• IX\mathcal{I}_XIX​ and IY\mathcal{I}_YIY​ are instance sets within SemA for $X$ and $Y$.

Specific Examples:

1. TD→IT_{D \rightarrow I}TD→I​ (Data to Information)

   TD→I:SD×CD×ID→SI×CI×IIT_{D \rightarrow I}: \mathcal{S}_D \times \mathcal{C}_D \times \mathcal{I}_D \rightarrow \mathcal{S}_I \times \mathcal{C}_I \times \mathcal{I}_ITD→I​:SD​×CD​×ID​→SI​×CI​×II​

• Within ConN and ConC:Data is processed and conceptualized into meaningful Information by identifying patterns and relationships.

2. TI→KT_{I \rightarrow K}TI→K​ (Information to Knowledge)

   TI→K:SI×CI×II→SK×CK×IKT_{I \rightarrow K}: \mathcal{S}_I \times \mathcal{C}_I \times \mathcal{I}_I \rightarrow \mathcal{S}_K \times \mathcal{C}_K \times \mathcal{I}_KTI→K​:SI​×CI​×II​→SK​×CK​×IK​

• Within ConN and SemA:Information is organized and structured into Knowledge frameworks by establishing logical and semantic connections.

3. TK→WT_{K \rightarrow W}TK→W​ (Knowledge to Wisdom)

   TK→W:SK×CK×IK→SW×CW×IWT_{K \rightarrow W}: \mathcal{S}_K \times \mathcal{C}_K \times \mathcal{I}_K \rightarrow \mathcal{S}_W \times \mathcal{C}_W \times \mathcal{I}_WTK→W​:SK​×CK​×IK​→SW​×CW​×IW​

• Within ConN and ConsciousS:Knowledge is synthesized into Wisdom by integrating ethical and contextual insights.

4. TP→WT_{P \rightarrow W}TP→W​ (Purpose to Wisdom)

   TP→W:SP×CP×IP→SW×CW×IWT_{P \rightarrow W}: \mathcal{S}_P \times \mathcal{C}_P \times \mathcal{I}_P \rightarrow \mathcal{S}_W \times \mathcal{C}_W \times \mathcal{I}_WTP→W​:SP​×CP​×IP​→SW​×CW​×IW​

• Within ConsciousS:Purpose-driven considerations shape Wisdom by aligning goals with ethical standards.

5. Composite Transformation Example: TD→WT_{D \rightarrow W}TD→W​ (Data to Wisdom)

TD→W=TD→I∘TI→K∘TK→WT_{D \rightarrow W} = T_{D \rightarrow I} \circ T_{I \rightarrow K} \circ T_{K \rightarrow W}TD→W​=TD→I​∘TI→K​∘TK→W​

• Mathematical Breakdown:TD→W:SD×CD×ID→SW×CW×IWT_{D \rightarrow W}: \mathcal{S}_D \times \mathcal{C}_D \times \mathcal{I}_D \rightarrow \mathcal{S}_W \times \mathcal{C}_W \times \mathcal{I}_WTD→W​:SD​×CD​×ID​→SW​×CW​×IW​

• Steps:

1. Data to Information: TD→IT_{D \rightarrow I}TD→I​

2. Information to Knowledge: TI→KT_{I \rightarrow K}TI→K​

3. Knowledge to Wisdom: TK→WT_{K \rightarrow W}TK→W​

6. Conclusion

By mathematically defining the Four Spaces and their integration with the networked DIKWP transformations, we establish a robust framework for analyzing and understanding the complex cognitive processes involved in transforming Data, Information, Knowledge, Wisdom, and Purpose. This formalization facilitates precise mapping, enhances clarity in transformation pathways, and supports the development of computational models and AI systems aligned with human cognitive structures.

7. References

• Arnheim, R. (1969). Visual Thinking. University of California Press.

• Chalmers, D. J. (1996). The Conscious Mind: In Search of a Fundamental Theory. Oxford University Press.

• Danto, A. (1997). After the End of Art. Princeton University Press.

• Duan, Y. (2022). The End of Art - The Subjective Objectification of DIKWP Philosophy. ResearchGate.

• Dennett, D. C. (1991). Consciousness Explained. Little, Brown and Company.

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

• Gombrich, E. H. (1950). The Story of Art. Phaidon Press.

• Heidegger, M. (1971). Poetry, Language, Thought. Harper & Row.

• Kurzweil, R. (2005). The Singularity Is Near: When Humans Transcend Biology. Viking Press.

• Manovich, L. (2001). The Language of New Media. MIT Press.

• Moravec, H. (1988). Mind Children: The Future of Robot and Human Intelligence. Harvard University Press.

• Paul, C. (2015). Digital Art. Thames & Hudson.

• Russell, S., & Norvig, P. (2020). Artificial Intelligence: A Modern Approach (4th ed.). Pearson.

• Searle, J. R. (1980). "Minds, Brains, and Programs." Behavioral and Brain Sciences, 3(3), 417-457.

• Zeki, S. (1999). Inner Vision: An Exploration of Art and the Brain. Oxford University Press.

• Additional Works by Duan, Y. Various publications on the DIKWP model and its applications in artificial intelligence, philosophy, and societal analysis.

Appendix: Detailed Mathematical Formulations and Examples

To further elaborate on the mathematical representations and their applications, the following sections provide comprehensive formulations and illustrative examples.

A. Conceptual Space (ConC) in Detail

Graph Representation:

GraphConC=(VConC,EConC)\text{GraphConC} = (V_{\text{ConC}}, E_{\text{ConC}})GraphConC=(VConC​,EConC​)

• Vertices (VConCV_{\text{ConC}}VConC​):Each vertex v∈VConCv \in V_{\text{ConC}}v∈VConC​ is defined as:

  $$v=\{\text{Concept ID},A(v)\}$$

  Where:

• Concept ID: A unique identifier for the concept.

• Attributes ($A(v)$):A(v)={a1(v),a2(v),…,an(v)}A(v) = \{a_1(v), a_2(v), \dots, a_n(v)\}A(v)={a1​(v),a2​(v),…,an​(v)}Each ai(v)a_i(v)ai​(v) is an attribute describing the concept $v$.

• Edges (EConCE_{\text{ConC}}EConC​):Each edge e=(vi,vj)∈EConCe = (v_i, v_j) \in E_{\text{ConC}}e=(vi​,vj​)∈EConC​ is defined as:

  e={Concept IDi,Concept IDj,R(vi,vj)}e = \{ \text{Concept ID}_i, \text{Concept ID}_j, R(v_i, v_j) \}e={Concept IDi​,Concept IDj​,R(vi​,vj​)}

  Where:

• R(vi,vj)R(v_i, v_j)R(vi​,vj​): The type of relationship between concepts viv_ivi​ and vjv_jvj​.

Graph Operations:

1. Query Operation:

   QConC(VConC,EConC,q)={v∈VConC∣Q(v,q)}Q_{\text{ConC}}(V_{\text{ConC}}, E_{\text{ConC}}, q) = \{ v \in V_{\text{ConC}} \mid Q(v, q) \}QConC​(VConC​,EConC​,q)={v∈VConC​∣Q(v,q)}

• Where:$Q(v,q)$ is a predicate function that returns true if concept $v$ satisfies the query $q$.

2. Add Operation:

   AddConC(VConC,v)=VConC∪{v}\text{Add}_{\text{ConC}}(V_{\text{ConC}}, v) = V_{\text{ConC}} \cup \{ v \}AddConC​(VConC​,v)=VConC​∪{v}

3. Update Operation:

   UpdateConC(VConC,v,A(v))={v′∈VConC∣v′=v}∪{v′∣v′ has updated attributes A(v)}\text{Update}_{\text{ConC}}(V_{\text{ConC}}, v, A(v)) = \{ v' \in V_{\text{ConC}} \mid v' = v \} \cup \{ v' \mid v' \text{ has updated attributes } A(v) \}UpdateConC​(VConC​,v,A(v))={v′∈VConC​∣v′=v}∪{v′∣v′ has updated attributes A(v)}

Example:

• Concept Addition:

• Adding the concept "Car" with attributes:$$A(\text{"Car"})=\{\text{wheels}=4,\text{purpose}=\text{"transportation"},\text{capacity}=\text{"passengers or goods"}\}$$

• Adding relationships:$$R(\text{"Car"},\text{"Transportation"})=\text{"is a type of"}$$

B. Cognitive Space (ConN) in Detail

Functional Representation:

$$\text{ConN}=(R,F)$$

• Relations ($R$):Represent the flow and dependencies among cognitive processes.

• Function Set ($F$):Each function fConNif_{\text{ConN}_i}fConNi​​ can be represented as:

  fConNi:Inputi→Outputif_{\text{ConN}_i}: \text{Input}_i \rightarrow \text{Output}_ifConNi​​:Inputi​→Outputi​

  Where:

• Input (Inputi\text{Input}_iInputi​):A subset of $\{D,I\}$ representing Data and Information inputs.

• Output (Outputi\text{Output}_iOutputi​):A subset of $\{I,K,W,P\}$ representing the transformed outputs.

Function Decomposition:

fConNi=fConNi(5)∘fConNi(4)∘⋯∘fConNi(1)f_{\text{ConN}_i} = f_{\text{ConN}_i}(5) \circ f_{\text{ConN}_i}(4) \circ \dots \circ f_{\text{ConN}_i}(1)fConNi​​=fConNi​​(5)∘fConNi​​(4)∘⋯∘fConNi​​(1)

Where:

• fConNi(j)f_{\text{ConN}_i}(j)fConNi​​(j) represents the $j$-th sub-step in the cognitive process (e.g., data preprocessing, feature extraction).

Example:

• Function fConN1f_{\text{ConN}_1}fConN1​​ for Data to Information Transformation:fConN1=fConN1(3)∘fConN1(2)∘fConN1(1)f_{\text{ConN}_1} = f_{\text{ConN}_1}(3) \circ f_{\text{ConN}_1}(2) \circ f_{\text{ConN}_1}(1)fConN1​​=fConN1​​(3)∘fConN1​​(2)∘fConN1​​(1)

• Sub-Steps:

1. Data Preprocessing: Cleaning and normalizing raw data.

2. Feature Extraction: Identifying relevant features from data.

3. Pattern Recognition: Detecting patterns and trends in data.

Function Composition Example:

fConN1(3)∘fConN1(2)∘fConN1(1)(D)=If_{\text{ConN}_1}(3) \circ f_{\text{ConN}_1}(2) \circ f_{\text{ConN}_1}(1) (D) = IfConN1​​(3)∘fConN1​​(2)∘fConN1​​(1)(D)=I

Where:

• $D$ is raw Data.

• $I$ is processed Information.

C. Semantic Space (SemA) in Detail

Graph Representation:

GraphSemA=(VSemA,ESemA)\text{GraphSemA} = (V_{\text{SemA}}, E_{\text{SemA}})GraphSemA=(VSemA​,ESemA​)

• Vertices (VSemAV_{\text{SemA}}VSemA​):Each vertex s∈VSemAs \in V_{\text{SemA}}s∈VSemA​ represents a semantic unit (e.g., word, phrase).

• Edges (ESemAE_{\text{SemA}}ESemA​):Each edge e=(si,sj)∈ESemAe = (s_i, s_j) \in E_{\text{SemA}}e=(si​,sj​)∈ESemA​ represents a semantic relationship between sis_isi​ and sjs_jsj​.

Graph Operations:

1. Query Operation:

   QSemA(VSemA,ESemA,q)={s∈VSemA∣Q(s,q)}Q_{\text{SemA}}(V_{\text{SemA}}, E_{\text{SemA}}, q) = \{ s \in V_{\text{SemA}} \mid Q(s, q) \}QSemA​(VSemA​,ESemA​,q)={s∈VSemA​∣Q(s,q)}

• Where:$Q(s,q)$ is a predicate function that returns true if semantic unit $s$ satisfies the query $q$.

2. Add Operation:

   AddSemA(VSemA,s)=VSemA∪{s}\text{Add}_{\text{SemA}}(V_{\text{SemA}}, s) = V_{\text{SemA}} \cup \{ s \}AddSemA​(VSemA​,s)=VSemA​∪{s}

3. Update Operation:

   UpdateSemA(ESemA,si,sj,r)={e′∈ESemA∣e′=(si,sj,r)}\text{Update}_{\text{SemA}}(E_{\text{SemA}}, s_i, s_j, r) = \{ e' \in E_{\text{SemA}} \mid e' = (s_i, s_j, r) \}UpdateSemA​(ESemA​,si​,sj​,r)={e′∈ESemA​∣e′=(si​,sj​,r)}

Example:

• Semantic Unit Addition:

• Adding semantic units "Driving" and "Fuel Consumption" and establishing relationships:$$R(\text{"Driving"},\text{"Fuel Consumption"})=\text{"causality"}$$

3.4. Conscious Space (ConsciousS) in Detail

Definition:The Conscious Space (ConsciousS) encapsulates the ethical, reflective, and value-based dimensions of cognition, integrating Purpose into the cognitive and semantic processes.

Mathematical Representation:

ConsciousS=(VConsciousS,EConsciousS,P)\text{ConsciousS} = (V_{\text{ConsciousS}}, E_{\text{ConsciousS}}, P)ConsciousS=(VConsciousS​,EConsciousS​,P)

• Vertices (VConsciousSV_{\text{ConsciousS}}VConsciousS​):Each vertex c∈VConsciousSc \in V_{\text{ConsciousS}}c∈VConsciousS​ represents an ethical or reflective concept.

• Edges (EConsciousSE_{\text{ConsciousS}}EConsciousS​):Each edge e=(ci,cj)∈EConsciousSe = (c_i, c_j) \in E_{\text{ConsciousS}}e=(ci​,cj​)∈EConsciousS​ represents an ethical or reflective relationship between concepts cic_ici​ and cjc_jcj​.

• Purpose ($P$):$P$ denotes the set of Purpose-driven functions influencing transformations within Conscious Space.

Mathematical Operations:

1. Ethical Evaluation Function:

   EvaluateConsciousS:(K×P)→W\text{Evaluate}_{\text{ConsciousS}}: (K \times P) \rightarrow WEvaluateConsciousS​:(K×P)→W

• Where:$K$ represents Knowledge inputs, and $P$ represents Purpose inputs. The function integrates these to produce Wisdom $W$.

2. Purpose Definition Function:

   DefineConsciousS:P→P′\text{Define}_{\text{ConsciousS}}: P \rightarrow P'DefineConsciousS​:P→P′

• Where:P′P'P′ is the refined or newly defined Purpose based on ethical considerations.

Example:

• Ethical Evaluation:

  EvaluateConsciousS(K,P)=W\text{Evaluate}_{\text{ConsciousS}}(K, P) = WEvaluateConsciousS​(K,P)=W

  Where Knowledge $K$ is integrated with Purpose $P$ to generate Wisdom $W$.

• Purpose Definition:

  DefineConsciousS(P)=P′\text{Define}_{\text{ConsciousS}}(P) = P'DefineConsciousS​(P)=P′

  Purpose $P$ is refined to P′P'P′ based on ethical deliberations.

4. Integration with Networked DIKWP Transformations

The integration of the Four Spaces with the networked DIKWP transformations involves mapping each transformation mode to specific spaces and defining how these spaces interact during transformations.

4.1. Mapping Transformations to Spaces

Each transformation mode TXY:X→YT_{XY}: X \rightarrow YTXY​:X→Y within the DIKWP model is mapped to one or more spaces based on the nature of the transformation.

TransformationXYMapped Space(s)DescriptionTD→DDDConNData maintenance via cognitive processingTD→IDIConN, ConCData processed into InformationTD→KDKConN, ConCData analyzed into KnowledgeTD→WDWConN, ConsciousSData synthesized into WisdomTD→PDPConC, ConsciousSData-driven Purpose definitionTI→DIDConNInformation deconstructed into DataTI→IIISemA, ConNInformation refinementTI→KIKConN, SemAInformation organized into KnowledgeTI→WIWConN, ConsciousSInformation integrated into WisdomTI→PIPSemA, ConsciousSInformation leveraged to define PurposeTK→DKDConN, SemAKnowledge translated back to DataTK→IKISemA, ConNKnowledge communicated as InformationTK→KKKConC, ConNKnowledge refinement and expansionTK→WKWConN, ConsciousSKnowledge synthesized into WisdomTK→PKPConC, ConsciousSKnowledge used to define PurposeTW→DWDConN, ConCWisdom applied to generate DataTW→IWISemA, ConNWisdom translated into InformationTW→KWKConC, ConNWisdom refines KnowledgeTW→WWWConsciousSWisdom refinement through reflectionTW→PWPConsciousSWisdom shapes PurposeTP→DPDConNPurpose-directed Data generationTP→IPISemA, ConNPurpose-guided Information processingTP→KPKConC, ConNPurpose-driven Knowledge developmentTP→WPWConsciousSPurpose integrates with WisdomTP→PPPConsciousSPurpose refinement and redefinition\begin{array}{|c|c|c|c|c|} \hline \text{Transformation} & X & Y & \text{Mapped Space(s)} & \text{Description} \\ \hline T_{D \rightarrow D} & D & D & \text{ConN} & \text{Data maintenance via cognitive processing} \\ T_{D \rightarrow I} & D & I & \text{ConN, ConC} & \text{Data processed into Information} \\ T_{D \rightarrow K} & D & K & \text{ConN, ConC} & \text{Data analyzed into Knowledge} \\ T_{D \rightarrow W} & D & W & \text{ConN, ConsciousS} & \text{Data synthesized into Wisdom} \\ T_{D \rightarrow P} & D & P & \text{ConC, ConsciousS} & \text{Data-driven Purpose definition} \\ T_{I \rightarrow D} & I & D & \text{ConN} & \text{Information deconstructed into Data} \\ T_{I \rightarrow I} & I & I & \text{SemA, ConN} & \text{Information refinement} \\ T_{I \rightarrow K} & I & K & \text{ConN, SemA} & \text{Information organized into Knowledge} \\ T_{I \rightarrow W} & I & W & \text{ConN, ConsciousS} & \text{Information integrated into Wisdom} \\ T_{I \rightarrow P} & I & P & \text{SemA, ConsciousS} & \text{Information leveraged to define Purpose} \\ T_{K \rightarrow D} & K & D & \text{ConN, SemA} & \text{Knowledge translated back to Data} \\ T_{K \rightarrow I} & K & I & \text{SemA, ConN} & \text{Knowledge communicated as Information} \\ T_{K \rightarrow K} & K & K & \text{ConC, ConN} & \text{Knowledge refinement and expansion} \\ T_{K \rightarrow W} & K & W & \text{ConN, ConsciousS} & \text{Knowledge synthesized into Wisdom} \\ T_{K \rightarrow P} & K & P & \text{ConC, ConsciousS} & \text{Knowledge used to define Purpose} \\ T_{W \rightarrow D} & W & D & \text{ConN, ConC} & \text{Wisdom applied to generate Data} \\ T_{W \rightarrow I} & W & I & \text{SemA, ConN} & \text{Wisdom translated into Information} \\ T_{W \rightarrow K} & W & K & \text{ConC, ConN} & \text{Wisdom refines Knowledge} \\ T_{W \rightarrow W} & W & W & \text{ConsciousS} & \text{Wisdom refinement through reflection} \\ T_{W \rightarrow P} & W & P & \text{ConsciousS} & \text{Wisdom shapes Purpose} \\ T_{P \rightarrow D} & P & D & \text{ConN} & \text{Purpose-directed Data generation} \\ T_{P \rightarrow I} & P & I & \text{SemA, ConN} & \text{Purpose-guided Information processing} \\ T_{P \rightarrow K} & P & K & \text{ConC, ConN} & \text{Purpose-driven Knowledge development} \\ T_{P \rightarrow W} & P & W & \text{ConsciousS} & \text{Purpose integrates with Wisdom} \\ T_{P \rightarrow P} & P & P & \text{ConsciousS} & \text{Purpose refinement and redefinition} \\ \hline \end{array}TransformationTD→D​TD→I​TD→K​TD→W​TD→P​TI→D​TI→I​TI→K​TI→W​TI→P​TK→D​TK→I​TK→K​TK→W​TK→P​TW→D​TW→I​TW→K​TW→W​TW→P​TP→D​TP→I​TP→K​TP→W​TP→P​​XDDDDDIIIIIKKKKKWWWWWPPPPP​YDIKWPDIKWPDIKWPDIKWPDIKWP​Mapped Space(s)ConNConN, ConCConN, ConCConN, ConsciousSConC, ConsciousSConNSemA, ConNConN, SemAConN, ConsciousSSemA, ConsciousSConN, SemASemA, ConNConC, ConNConN, ConsciousSConC, ConsciousSConN, ConCSemA, ConNConC, ConNConsciousSConsciousSConNSemA, ConNConC, ConNConsciousSConsciousS​DescriptionData maintenance via cognitive processingData processed into InformationData analyzed into KnowledgeData synthesized into WisdomData-driven Purpose definitionInformation deconstructed into DataInformation refinementInformation organized into KnowledgeInformation integrated into WisdomInformation leveraged to define PurposeKnowledge translated back to DataKnowledge communicated as InformationKnowledge refinement and expansionKnowledge synthesized into WisdomKnowledge used to define PurposeWisdom applied to generate DataWisdom translated into InformationWisdom refines KnowledgeWisdom refinement through reflectionWisdom shapes PurposePurpose-directed Data generationPurpose-guided Information processingPurpose-driven Knowledge developmentPurpose integrates with WisdomPurpose refinement and redefinition​​

Explanation of Mapped Spaces:

• ConC (Conceptual Space): Involved when transformations require the formulation or refinement of concepts.

• ConN (Cognitive Space): Central to processing and transforming data and information.

• SemA (Semantic Space): Engaged when meanings and communications are restructured or interpreted.

• ConsciousS (Conscious Space): Integral when ethical, reflective, or purpose-driven considerations influence transformations.

4.2. Interplay Among Spaces

Transformations often involve multiple spaces working in tandem. The interplay among spaces ensures that transformations are coherent, ethically grounded, and contextually relevant.

Example 1: TK→WT_{K \rightarrow W}TK→W​ (Knowledge to Wisdom)

ConN:Synthesize structured Knowledge into higher-order insightsConsciousS:Integrate ethical and contextual considerations into WisdomResult:Wisdom that is both intellectually robust and ethically sound\begin{align*} & \text{ConN}: \text{Synthesize structured Knowledge into higher-order insights} \\ & \text{ConsciousS}: \text{Integrate ethical and contextual considerations into Wisdom} \\ & \text{Result}: \text{Wisdom that is both intellectually robust and ethically sound} \end{align*}​ConN:Synthesize structured Knowledge into higher-order insightsConsciousS:Integrate ethical and contextual considerations into WisdomResult:Wisdom that is both intellectually robust and ethically sound​

Example 2: TI→PT_{I \rightarrow P}TI→P​ (Information to Purpose)

SemA:Shape information semantics to align with strategic goalsConsciousS:Ensure Purpose definition adheres to ethical standardsResult:Purpose-driven objectives informed by ethical and semantic alignment\begin{align*} & \text{SemA}: \text{Shape information semantics to align with strategic goals} \\ & \text{ConsciousS}: \text{Ensure Purpose definition adheres to ethical standards} \\ & \text{Result}: \text{Purpose-driven objectives informed by ethical and semantic alignment} \end{align*}​SemA:Shape information semantics to align with strategic goalsConsciousS:Ensure Purpose definition adheres to ethical standardsResult:Purpose-driven objectives informed by ethical and semantic alignment​

Example 3: TD→WT_{D \rightarrow W}TD→W​ (Data to Wisdom)

ConN:Process raw Data into actionable insightsConsciousS:Apply ethical considerations to synthesize WisdomResult:Ethical Wisdom derived directly from Data\begin{align*} & \text{ConN}: \text{Process raw Data into actionable insights} \\ & \text{ConsciousS}: \text{Apply ethical considerations to synthesize Wisdom} \\ & \text{Result}: \text{Ethical Wisdom derived directly from Data} \end{align*}​ConN:Process raw Data into actionable insightsConsciousS:Apply ethical considerations to synthesize WisdomResult:Ethical Wisdom derived directly from Data​

Example 4: TW→PT_{W \rightarrow P}TW→P​ (Wisdom to Purpose)

ConsciousS:Utilize Wisdom to define or refine PurposeResult:Purpose aligned with ethical Wisdom\begin{align*} & \text{ConsciousS}: \text{Utilize Wisdom to define or refine Purpose} \\ & \text{Result}: \text{Purpose aligned with ethical Wisdom} \end{align*}​ConsciousS:Utilize Wisdom to define or refine PurposeResult:Purpose aligned with ethical Wisdom​

5. Mathematical Representation of Transformation Modes

Each transformation mode TXY:X→YT_{XY}: X \rightarrow YTXY​:X→Y within the DIKWP model is represented as a function that maps elements from one component to another within the appropriate spaces.

5.1. Minimal Impact Transformations (D→D, I→I, K→K, W→W, P→P)

Definition:Minimal impact transformations involve maintaining the integrity and consistency of existing elements without significant alteration.

Mathematical Representation:

TXX:X→XT_{XX}: X \rightarrow XTXX​:X→X

Where $X\in \{D,I,K,W,P\}$.

Mapped Space:

$$\text{ConN}$$

Since these transformations primarily involve internal reinforcement within the Cognitive Space.

Examples:

• TD→DT_{D \rightarrow D}TD→D​: Data verification and reinforcement.

• TI→IT_{I \rightarrow I}TI→I​: Information refinement and clarification.

• TK→KT_{K \rightarrow K}TK→K​: Knowledge consolidation and updating.

• TW→WT_{W \rightarrow W}TW→W​: Wisdom refinement and ethical review.

• TP→PT_{P \rightarrow P}TP→P​: Purpose reaffirmation and goal adjustment.

5.2. Direct Transformations (D→I, D→K, D→W, D→P)

Definition:Direct transformations involve processing raw data into more refined constructs or aligning data with specific purposes.

Mathematical Representation:

TXY:D→YT_{XY}: D \rightarrow YTXY​:D→Y

Where $Y\in \{I,K,W,P\}$.

Mapped Spaces:

$$\text{ConN, ConC}(\text{and }\text{ConsciousS}\text{ for }W\text{ and }P)$$

Examples:

• TD→IT_{D \rightarrow I}TD→I​:

  TD→I:SD×CD×ID→SI×CI×IIT_{D \rightarrow I}: \mathcal{S}_D \times \mathcal{C}_D \times \mathcal{I}_D \rightarrow \mathcal{S}_I \times \mathcal{C}_I \times \mathcal{I}_ITD→I​:SD​×CD​×ID​→SI​×CI​×II​

  Data processed into Information by identifying patterns and relationships.

• TD→KT_{D \rightarrow K}TD→K​:

  TD→K:SD×CD×ID→SK×CK×IKT_{D \rightarrow K}: \mathcal{S}_D \times \mathcal{C}_D \times \mathcal{I}_D \rightarrow \mathcal{S}_K \times \mathcal{C}_K \times \mathcal{I}_KTD→K​:SD​×CD​×ID​→SK​×CK​×IK​

  Data analyzed into Knowledge by structuring and organizing information.

• TD→WT_{D \rightarrow W}TD→W​:

  TD→W:SD×CD×ID→SW×CW×IWT_{D \rightarrow W}: \mathcal{S}_D \times \mathcal{C}_D \times \mathcal{I}_D \rightarrow \mathcal{S}_W \times \mathcal{C}_W \times \mathcal{I}_WTD→W​:SD​×CD​×ID​→SW​×CW​×IW​

  Data synthesized into Wisdom by integrating ethical and contextual insights.

• TD→PT_{D \rightarrow P}TD→P​:

  TD→P:SD×CD×ID→SP×CP×IPT_{D \rightarrow P}: \mathcal{S}_D \times \mathcal{C}_D \times \mathcal{I}_D \rightarrow \mathcal{S}_P \times \mathcal{C}_P \times \mathcal{I}_PTD→P​:SD​×CD​×ID​→SP​×CP​×IP​

  Data-driven Purpose definition aligning with organizational or personal goals.

5.3. Indirect and Complex Transformations (I→D, I→I, I→K, I→W, I→P; K→D, K→I, K→K, K→W, K→P; W→D, W→I, W→K, W→W, W→P; P→D, P→I, P→K, P→W, P→P)

Definition:Indirect and complex transformations facilitate the evolution of elements through multiple interconnected processes, often involving multiple spaces.

Mathematical Representation:

TXY:X→YT_{XY}: X \rightarrow YTXY​:X→Y

Where $X,Y\in \{I,K,W,P\}$ and $X=Y$.

Mapped Spaces:

$$\text{Multiple Spaces}(\text{ConC, ConN, SemA, ConsciousS})$$

Examples:

• TI→KT_{I \rightarrow K}TI→K​:

  TI→K:SI×CI×II→SK×CK×IKT_{I \rightarrow K}: \mathcal{S}_I \times \mathcal{C}_I \times \mathcal{I}_I \rightarrow \mathcal{S}_K \times \mathcal{C}_K \times \mathcal{I}_KTI→K​:SI​×CI​×II​→SK​×CK​×IK​

  Information organized into Knowledge frameworks by establishing logical and semantic connections.

• TK→WT_{K \rightarrow W}TK→W​:

  TK→W:SK×CK×IK→SW×CW×IWT_{K \rightarrow W}: \mathcal{S}_K \times \mathcal{C}_K \times \mathcal{I}_K \rightarrow \mathcal{S}_W \times \mathcal{C}_W \times \mathcal{I}_WTK→W​:SK​×CK​×IK​→SW​×CW​×IW​

  Knowledge synthesized into Wisdom by integrating ethical and contextual insights.

• TW→PT_{W \rightarrow P}TW→P​:

  TW→P:SW×CW×IW→SP×CP×IPT_{W \rightarrow P}: \mathcal{S}_W \times \mathcal{C}_W \times \mathcal{I}_W \rightarrow \mathcal{S}_P \times \mathcal{C}_P \times \mathcal{I}_PTW→P​:SW​×CW​×IW​→SP​×CP​×IP​

  Wisdom shapes Purpose by aligning goals with ethical standards.

General Representation:

TXY:SX×CX×IX→SY×CY×IYT_{XY}: \mathcal{S}_X \times \mathcal{C}_X \times \mathcal{I}_X \rightarrow \mathcal{S}_Y \times \mathcal{C}_Y \times \mathcal{I}_YTXY​:SX​×CX​×IX​→SY​×CY​×IY​

Where SX,CX,IX\mathcal{S}_X, \mathcal{C}_X, \mathcal{I}_XSX​,CX​,IX​ and SY,CY,IY\mathcal{S}_Y, \mathcal{C}_Y, \mathcal{I}_YSY​,CY​,IY​ represent semantic attributes, concepts, and instances in respective spaces.

5.4. Transformation Dynamics

Synergistic Interactions:Many transformations involve synergistic interactions between spaces, enhancing the depth and applicability of cognitive and societal processes.

Ethical Integration:Conscious Space (ConsciousS) consistently plays a crucial role in transformations involving wisdom, ensuring that ethical considerations are integral to the process.

Mathematical Representation of Synergy:

Synergy(X,Y)=fConN(X)+fConsciousS(Y)\text{Synergy}(X, Y) = f_{\text{ConN}}(X) + f_{\text{ConsciousS}}(Y)Synergy(X,Y)=fConN​(X)+fConsciousS​(Y)

Where $X$ and $Y$ are transformation modes that require cognitive and ethical processing.

Example: Innovation Cycle

Innovation Cycle=ConC→TConC→ConNConN→TConN→SemASemA→TSemA→ConsciousSConsciousS→TConsciousS→ConCConC\text{Innovation Cycle} = \text{ConC} \xrightarrow{T_{ConC \rightarrow ConN}} \text{ConN} \xrightarrow{T_{ConN \rightarrow SemA}} \text{SemA} \xrightarrow{T_{SemA \rightarrow ConsciousS}} \text{ConsciousS} \xrightarrow{T_{ConsciousS \rightarrow ConC}} \text{ConC}Innovation Cycle=ConCTConC→ConN​​ConNTConN→SemA​​SemATSemA→ConsciousS​​ConsciousSTConsciousS→ConC​​ConC

This cycle represents a continuous loop where ideas are generated, processed, communicated, ethically evaluated, and refined.

5.5. Mathematical Formalization of the DIKWP Graphing System

The DIKWP graphing system maps elements of the digital world and the cognitive world to five main components: Data Graph (DG), Information Graph (IG), Knowledge Graph (KG), Wisdom Graph (WG), and Purpose Graph (PG). Each graph is further subdivided into three levels of mapping: the semantic level, the conceptual level, and the instance level. Thus, each graph $g\in G$ is a triplet mapping:

$$g:S\times C\times I\to g$$

Where:

• $G$: Set of all graphs ($G=\{DG,IG,KG,WG,PG\}$).

• $S$: Set of semantic levels.

• $C$: Set of concepts.

• $I$: Set of instances.

Function $f$:

$$f:G\times G\to G$$

Represents the transformation function that maps the interaction between two graphs into another graph.

Example Transformation:

f(DG,KG)=TDG→KGf(DG, KG) = T_{DG \rightarrow KG}f(DG,KG)=TDG→KG​

This function represents the transformation from Data Graph to Knowledge Graph.

6. Conclusion

By mathematically defining the Four Spaces and their integration with the networked DIKWP transformations, we establish a robust framework for analyzing and understanding the complex cognitive processes involved in transforming Data, Information, Knowledge, Wisdom, and Purpose. This formalization facilitates precise mapping, enhances clarity in transformation pathways, and supports the development of computational models and AI systems aligned with human cognitive structures.

Key Insights:

• Comprehensive Mapping:All 25 transformation modes within the networked DIKWP model are effectively accommodated within the Four Spaces Framework, ensuring no gaps in coverage.

• Interconnectedness:The synergistic interactions among the four spaces enhance the depth and applicability of transformations, fostering a holistic understanding of cognitive and societal processes.

• Ethical Integration:Conscious Space (ConsciousS) plays a crucial role in embedding ethical and contextual insights into transformations, ensuring responsible and meaningful outcomes.

• Mathematical Rigor:Formal mathematical representations provide precision and clarity, enabling the application of the framework in computational models and AI systems.

Implications:

• Theoretical Robustness:The framework bridges abstract cognitive concepts with precise mathematical structures, enhancing theoretical understanding.

• Practical Applicability:Enables the design of AI systems that mimic human cognitive transformations, ensuring alignment with ethical and purpose-driven objectives.

• Future Research:Offers a foundation for further exploration into complex cognitive processes and their mathematical modeling, facilitating advancements in cognitive science and AI.

7. References

• Arnheim, R. (1969). Visual Thinking. University of California Press.

• Chalmers, D. J. (1996). The Conscious Mind: In Search of a Fundamental Theory. Oxford University Press.

• Danto, A. (1997). After the End of Art. Princeton University Press.

• Duan, Y. (2022). The End of Art - The Subjective Objectification of DIKWP Philosophy. ResearchGate.

• Dennett, D. C. (1991). Consciousness Explained. Little, Brown and Company.

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

• Gombrich, E. H. (1950). The Story of Art. Phaidon Press.

• Heidegger, M. (1971). Poetry, Language, Thought. Harper & Row.

• Kurzweil, R. (2005). The Singularity Is Near: When Humans Transcend Biology. Viking Press.

• Manovich, L. (2001). The Language of New Media. MIT Press.

• Moravec, H. (1988). Mind Children: The Future of Robot and Human Intelligence. Harvard University Press.

• Paul, C. (2015). Digital Art. Thames & Hudson.

• Russell, S., & Norvig, P. (2020). Artificial Intelligence: A Modern Approach (4th ed.). Pearson.

• Searle, J. R. (1980). "Minds, Brains, and Programs." Behavioral and Brain Sciences, 3(3), 417-457.

• Zeki, S. (1999). Inner Vision: An Exploration of Art and the Brain. Oxford University Press.

• Additional Works by Duan, Y. Various publications on the DIKWP model and its applications in artificial intelligence, philosophy, and societal analysis.

• Yucong Duan, etc. (2024). DIKWP Conceptualization Semantics Standards of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. 10.13140/RG.2.2.32289.42088.  

• Yucong Duan, etc.  (2024). Standardization of DIKWP Semantic Mathematics of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. 10.13140/RG.2.2.26233.89445.  

• Yucong Duan, etc.  (2024). Standardization for Constructing DIKWP -Based Artificial Consciousness Systems ----- International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. 10.13140/RG.2.2.18799.65443.  

• Yucong Duan, etc.  (2024). Standardization for Evaluation and Testing of DIKWP Based Artificial Consciousness Systems - International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. 10.13140/RG.2.2.11702.10563. 

Final Remarks:

The Four Spaces Framework initiated by Prof. Yucong Duan, when integrated with the networked DIKWP model, provides a mathematically rigorous and conceptually comprehensive tool for dissecting and understanding the intricate transformations between Data, Information, Knowledge, Wisdom, and Purpose. This formalization not only enhances theoretical comprehension but also paves the way for practical applications in AI, cognitive science, and knowledge management, fostering advancements that are both intelligent and ethically grounded.

转载本文请联系原作者获取授权，同时请注明本文来自段玉聪科学网博客。
链接地址：https://blog.sciencenet.cn/blog-3429562-1460296.html

上一篇：The Sufficiency of the Four Spaces by Networked DIKWP Model
下一篇：Pathways Among the Four Spaces in Networked DIKWP Model