Prof. Yucong Duan: Centrality of Semantics in Mathematical Abstraction and the Revolution of Mathematics for Human-AI Coevolution

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 investigates Prof. Yucong Duan's proposal that mathematics should not go contrary to the current practice of abstracting from concrete semantics but should acknowledge that semantics is actually both the source and the target of abstraction. Prof. Duan argues for a revolutionary change in mathematical practice, especially in the context of artificial intelligence (AI) increasingly taking over traditional mathematical roles. He emphasizes the urgent need for humans to rediscover the real meaning of mathematics to guide human and human-AI coevolution. This exploration delves into the implications of integrating semantics as both the foundation and objective of abstraction within the Data-Information-Knowledge-Wisdom-Purpose (DIKWP) Semantic Mathematics framework, and how this shift could influence the future of mathematics, AI development, and human cognition.

1. Introduction1.1. Background

Mathematics has traditionally been characterized by its abstraction from concrete semantics, focusing on formal structures and symbols that are detached from specific meanings. This practice has enabled the development of universal theories and models applicable across various domains. However, with the advent of advanced AI, there is growing concern about the role of humans in mathematics and the potential loss of meaningful engagement with mathematical concepts.

Prof. Yucong Duan's Proposal:

• Centrality of Semantics: Mathematics should recognize that semantics is both the source and the target of abstraction, rather than abstracting away from semantics.

• Revolution in Practice: A radical change is needed in mathematical practice to align with this understanding, especially as AI begins to replace traditional human roles in mathematics.

• Human-AI Coevolution: Humans urgently need to find the real meaning in mathematics to guide the coevolution of humans and AI.

1.2. Objective

This document aims to:

• Investigate Prof. Duan's proposal in depth.

• Explore the role of semantics as both the source and target of abstraction in mathematics.

• Examine the implications for mathematical practice, AI development, and human cognition.

• Discuss the revolutionary changes proposed for current mathematical practices.

• Highlight Prof. Duan's perspectives and arguments supporting this paradigm shift.

2. Understanding the Proposal2.1. Current Practice of Abstraction in Mathematics

• Abstraction from Concrete Semantics:

• Traditional Approach: Mathematics abstracts from specific instances and concrete semantics to develop general theories and models.

• Symbolic Representation: Utilizes symbols and formal systems that are independent of particular meanings.

• Advantages: Allows for universal applicability, logical rigor, and the development of advanced mathematical structures.

• Limitations:

• Detachment from Meaning: May lead to mathematical constructs that lack relevance to real-world applications.

• Barrier to Understanding: Can create a disconnect between mathematical theory and practical understanding for learners and practitioners.

2.2. Prof. Duan's Perspective on Semantics in Mathematics

• Semantics as the Source of Abstraction:

• Origin of Concepts: All mathematical concepts originate from concrete semantics—real-world meanings and experiences.

• Foundation of Abstraction: Abstraction should be grounded in these concrete semantics to retain relevance and meaning.

• Semantics as the Target of Abstraction:

• Purpose of Abstraction: The goal of abstracting is to develop generalized concepts that can be re-applied to various semantic contexts.

• Reapplication of Meaning: Abstracted concepts should ultimately reconnect with semantics to provide insights and solutions to real-world problems.

• Acknowledge Semantics in Abstraction:

• Integration: Mathematics should explicitly acknowledge the role of semantics throughout the abstraction process.

• Continuous Cycle: Recognize the continuous cycle where semantics inform abstraction, and abstraction enhances understanding of semantics.

2.3. The Need for a Revolutionary Change

• Current Situation with AI:

• AI Advancements: AI is increasingly capable of performing complex mathematical computations and proofs.

• Human Role Diminishing: There is a risk that humans may become disconnected from mathematical practice as AI takes over traditional roles.

• Urgent Need for Meaning:

• Rediscovering Meaning: Humans need to find the real meaning in mathematics to maintain relevance and guide the development of AI.

• Human-AI Coevolution: A collaborative evolution where humans and AI advance together, guided by a meaningful understanding of mathematics.

• Revolutionizing Mathematical Practice:

• Shift in Approach: Move away from purely formal and abstract practices to ones that integrate semantics at all levels.

• Educational Reform: Update mathematical education to emphasize the importance of semantics in abstraction.

3. Semantics as Both the Source and Target of Abstraction3.1. Semantics as the Source

• Originating Concepts:

• Empirical Observations: Mathematical concepts often arise from observing patterns and relationships in the real world.

• Cognitive Processes: Human cognition interprets these observations through semantic understanding.

• Grounding Abstraction:

• Meaningful Abstraction: By starting with semantics, abstraction remains connected to real-world meanings.

• Relevance: Ensures that abstract concepts are relevant and can be related back to practical applications.

3.2. Semantics as the Target

• Applying Abstract Concepts:

• Problem Solving: Abstract mathematical concepts are used to solve concrete problems, reconnecting with semantics.

• Interdisciplinary Applications: Mathematics informs other fields (physics, engineering, economics) through the application of abstract concepts to semantic contexts.

• Feedback Loop:

• Refinement of Concepts: The application of mathematics to real-world problems can lead to new insights, refining the abstract concepts.

• Continuous Development: This creates a continuous loop where semantics and abstraction inform and enhance each other.

3.3. The Cycle of Abstraction and Semantics

• Integrative Process:

• Cycle: Semantics → Abstraction → Application to Semantics → Refined Abstraction.

• Dynamic Relationship: Recognizes that mathematics is not a one-way process but a dynamic interplay between meaning and abstraction.

• Implications for Mathematical Practice:

• Holistic Approach: Encourages mathematicians to consider the semantic origins and destinations of their work.

• Enhanced Understanding: Leads to deeper insights and more meaningful mathematical developments.

4. Implications for Mathematics4.1. Revolutionizing Mathematical Practice

• Shift in Focus:

• From Form to Meaning: Transition from a focus on formalism to an emphasis on the meanings behind mathematical constructs.

• Semantics-Driven Development: Prioritize the development of mathematical theories that are grounded in and aimed at enhancing semantic understanding.

• Educational Changes:

• Curriculum Reform: Incorporate the importance of semantics in mathematical education at all levels.

• Emphasizing Context: Teach mathematics within the context of its real-world applications and meanings.

• Research and Collaboration:

• Interdisciplinary Work: Foster collaborations between mathematicians and experts in other fields to ground mathematical research in practical semantics.

• Problem-Oriented Research: Encourage research that addresses real-world problems, integrating semantics throughout the process.

4.2. Impact on Mathematical Abstraction

• Meaningful Abstraction:

• Retention of Semantics: Ensure that abstraction does not strip away the essential meanings of concepts.

• Balanced Approach: Maintain a balance between generalization and specificity, keeping abstraction connected to semantics.

• Innovation:

• New Perspectives: By integrating semantics, mathematicians may discover new approaches and solutions.

• Enhanced Creativity: A semantic focus can stimulate creativity, leading to breakthroughs that pure formalism may not achieve.

5. The Role of AI in Mathematics5.1. AI Replacing Human Roles

• Computational Capabilities:

• Automation of Tasks: AI can perform calculations, proofs, and even generate mathematical conjectures.

• Efficiency: AI can process and analyze data much faster than humans.

• Challenges:

• Loss of Engagement: Risk of humans becoming passive consumers of mathematical results generated by AI.

• Dependence: Over-reliance on AI could lead to a decline in human mathematical skills and understanding.

5.2. Guiding Human-AI Coevolution

• Collaborative Development:

• Complementary Roles: Humans provide semantic understanding and purpose, while AI offers computational power.

• Mutual Enhancement: AI can assist in exploring mathematical concepts, while humans guide AI with meaningful direction.

• Maintaining Human Centrality:

• Purposeful Use of AI: Utilize AI as a tool to enhance human understanding rather than replace it.

• Ethical Considerations: Ensure AI development aligns with human values and the meaningful progression of mathematics.

5.3. AI and Semantics

• Integrating Semantics into AI:

• Semantic AI: Develop AI systems that not only process symbols but also understand and manipulate semantics.

• Enhanced Capabilities: AI that can grasp semantic meanings may be more effective in assisting with complex mathematical problems.

• Human Oversight:

• Guidance: Humans need to oversee AI to ensure that the integration of semantics is meaningful and aligned with human understanding.

• Education: Mathematicians should be trained to work effectively with AI, leveraging its strengths while providing semantic direction.

6. Urgent Need for Real Meaning in Mathematics6.1. Rediscovering Meaning

• Reconnecting with Semantics:

• Deep Understanding: Move beyond mechanical manipulation of symbols to a deeper comprehension of the meanings behind mathematical concepts.

• Personal Engagement: Encourage individuals to engage with mathematics on a meaningful level, fostering a personal connection to the subject.

• Cultural Shift:

• Valuing Meaning: Shift cultural perceptions to value the meaningful aspects of mathematics rather than just technical proficiency.

• Public Engagement: Promote public understanding and appreciation of the meaningful role of mathematics in society.

6.2. Guiding Human and Human-AI Coevolution

• Shared Goals:

• Common Purpose: Establish shared purposes between humans and AI in the advancement of mathematics.

• Collaborative Progress: Work towards coevolution where both humans and AI contribute to and benefit from mathematical developments.

• Ethical and Philosophical Considerations:

• Human Values: Ensure that the coevolution aligns with human values, ethics, and well-being.

• Responsibility: Recognize the responsibility humans have in guiding AI development and integration into mathematics.

7. Prof. Yucong Duan's Perspectives7.1. Emphasizing Semantics in Mathematics

• Semantics as Central:

• Opinion: Prof. Duan asserts that semantics is both the source and the target of abstraction, and mathematics should acknowledge this.

• Argument: Ignoring semantics leads to a disconnection between mathematics and reality, limiting its applicability and understanding.

• Revolutionizing Practice:

• Call to Action: He calls for a revolution in mathematical practice, moving away from abstraction that disregards semantics.

• Alignment with Reality: Mathematics should reflect the reality of how humans think, understand, and apply concepts.

7.2. Human-AI Coevolution

• Urgency of Rediscovering Meaning:

• Concern: With AI taking over traditional mathematical roles, there is an urgent need for humans to find real meaning in mathematics.

• Guidance: This meaning will guide both human development and the coevolution with AI.

• AI Integration:

• Human Role: Humans must remain central in guiding AI, ensuring it develops in ways that enhance human understanding and align with human values.

• Semantic AI: Advocates for AI that can understand and process semantics, enhancing its utility in mathematics.

8. Challenges and Considerations8.1. Implementing the Revolution

• Resistance to Change:

• Tradition: The mathematical community may resist changing long-established practices.

• Solution: Promote awareness of the benefits and necessity of integrating semantics.

• Educational Reform:

• Curriculum Development: Updating educational programs is a significant undertaking.

• Solution: Start with incremental changes, integrating semantics into existing courses.

8.2. Balancing Abstraction and Semantics

• Maintaining Rigor:

• Scientific Standards: Ensure that the inclusion of semantics does not compromise mathematical rigor.

• Solution: Develop methodologies that integrate semantics without sacrificing formal precision.

• Complexity Management:

• Increased Complexity: Integrating semantics may add complexity to mathematical models.

• Solution: Use abstraction levels strategically to manage complexity while retaining semantic connections.

8.3. Ethical Implications with AI

• Dependence on AI:

• Over-Reliance: Risk of humans becoming overly dependent on AI for mathematical understanding.

• Solution: Emphasize human oversight and the importance of maintaining human mathematical skills.

• Control and Responsibility:

• Guiding AI Development: Ensure that AI development is guided by ethical considerations and human values.

• Solution: Establish clear guidelines and frameworks for AI integration into mathematics.

9. Conclusion

Prof. Yucong Duan's proposal advocates for a fundamental shift in mathematical practice, emphasizing that semantics is both the source and the target of abstraction. He calls for a revolution in how mathematics is conducted, especially in the context of AI increasingly taking over traditional mathematical roles. By acknowledging the centrality of semantics, mathematics can become more meaningful, relevant, and aligned with human cognition and the realities of the world.

This shift has significant implications for mathematical education, research, and the development of AI. It encourages a collaborative evolution between humans and AI, guided by a deep understanding of mathematics that integrates semantics at every level. Embracing this approach can lead to more innovative, applicable, and ethically aligned advancements in mathematics and technology.

10. Future Directions10.1. Research and Development

• Methodological Frameworks: Develop new methodologies for integrating semantics into mathematical abstraction.

• Interdisciplinary Collaboration: Work with experts in cognitive science, philosophy, and AI to enrich mathematical practices.

10.2. Educational Reform

• Curriculum Changes: Introduce courses that emphasize the role of semantics in mathematics.

• Teacher Training: Prepare educators to teach mathematics with a focus on meaning and real-world applications.

10.3. AI Development

• Semantic AI Systems: Invest in developing AI that can understand and process semantics.

• Ethical Guidelines: Establish ethical frameworks to guide the integration of AI into mathematical practice.

11. 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. Lakoff, G., & Núñez, R. E. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books.

4. Hersh, R. (1997). What Is Mathematics, Really? Oxford University Press.

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

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

12. Author Information

For further discussion on Prof. Yucong Duan's proposal regarding the centrality of semantics in mathematical abstraction and the revolution of mathematical practice in the age of AI within the DIKWP Semantic Mathematics framework, please contact [Author's Name] at [Contact Information].

Keywords: DIKWP Semantic Mathematics, Semantics in Mathematics, Prof. Yucong Duan, Mathematical Abstraction, Human-AI Coevolution, Mathematical Revolution, Semantics Integration, Artificial Intelligence, Human-Centered Mathematics, Meaningful Mathematics.