A Review of Semantic Mathematics Research Based on the DIKWP

通用人工智能AGI测评DIKWP实验室

2025-11-06

A Review of Semantic Mathematics Research Based on the DIKWP Model

Yucong Duan

Benefactor: Shiming Gong

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

The study of semantic mathematics aims to describe the generation and transmission process of "meaning" by mathematical means. Based on the five-layer cognitive network model of DIKWP (data-information-knowledge-wisdom-purpose), we can classify and analyze semantic mathematics from different research paradigms, and explore the semantic associations, future trends, and structural reorganizations between paradigms. This review will discuss the main directions of semantic mathematics by research paradigm, analyze the emphasis and knowledge generation path of each paradigm at each layer of DIKWP , construct corresponding tables, summarize the semantic network associations of different paradigms, and finally give a trend outlook for the next 10 years.

1.Research paradigm classification dimensions

Semantic mathematics involves multiple research paradigms. The main research directions are divided into paradigms below, including but not limited to formal logic, model-theoretic semantics, category-theoretic semantics, cognitive semantics, computational semantics, intuitive semantics, multimodal semantics, and metaphor/embodied semantics.

2Formal Logic Paradigm

The formal logic paradigm is centered on symbolic deduction and uses axioms and rules for reasoning. Traditional formal logic focuses on symbolic operations and formal proofs, and pays less attention to the semantics behind the symbols. ( PDF ) DIKWP 语义数学的理论、结构与应用简析 For example, first-order logic derives conclusions through grammatical rules, emphasizing the truth validity of propositions, but does not delve into the cognitive meaning of propositions. Formal logic provides a precise symbol system and reasoning mechanism for semantic research, allowing us to deduce knowledge from data and information. However, its limitation is that it lacks the characterization of the wisdom level (such as pragmatics and purpose), and deduction alone may produce problems such as self-referential paradoxes. ( PDF ) DIKWP 语义数学的理论、结构与应用简析 . Representative figures include Aristotle (founder of logic), Frege (father of mathematical logic), Russell, Tarski (founder of formal semantics), etc. Formal logic is currently very mature and is the foundation of mathematics and computer science, but it is out of touch with cognition in terms of semantic understanding. In recent years, formal logic is being combined with probability and uncertainty reasoning, and is being re-emphasized as an explainable reasoning tool in artificial intelligence. In the next 10 years, this paradigm is expected to develop in the direction of integration: combining with machine learning to form a neural symbolic system, improving the adaptability of formal logic in large-scale knowledge processing, and providing stronger explainability for AI.

3Model-based semantic paradigm

The model-theoretic semantics paradigm originates from the truth-value semantics proposed by Tarski, which focuses on how to define “truth” by mapping formal languages onto mathematical structures (models). ( PDF ) DIKWP 语义数学的理论、结构与应用简析 . Model theory achieves the conversion from symbols to information/knowledge by giving symbols interpretations in set theoretic models. For example, a first-order logic formula is true in a certain model, which means that the model provides an information interpretation that makes the formula valid. Model-theoretic semantics is the semantic pillar of formal logic, emphasizing the mapping relationship of data->information->knowledge. Typical representatives include Tarski, Godel, and van Benthem. Its research focuses on meta-logical properties such as truth definition, logical completeness, and model existence. Currently, model theory has developed rich model theories for various logical systems (such as modal logic and first-order logic), and has been applied to database theory, knowledge representation and other fields. Model-theoretic semantics also provides a theoretical basis for knowledge graphs, etc., and regards facts as element relationships in models. With the development of category theory, the views of model theory have been further abstracted and unified: standard set-theoretic semantics have been promoted to the category framework to form a more powerful semantic tool. ( CategoryT h eory ( StanfordEncyclopediaofP h ilosop h y ) ) In the next 10 years, model-theoretic semantics may be further integrated with categorical semantics and probabilistic semantics to cope with complex systems and uncertain information and meet the needs of knowledge representation in artificial intelligence.

4Category Theory Semantic Paradigm

The category theory semantic paradigm uses the highly abstract mathematical tools of category theory to unify different semantic structures. Category theory focuses on the relationship between objects and morphisms, which can reveal the commonalities of various structures at a higher level. ( PDF ) DIKWP 语义数学的理论、结构与应用简析 In semantic mathematics, category theory is used to construct functor semantics (proposed by Lawvere), etc., which corresponds syntax and semantics to objects and mappings in categories. ( CategoryT h eory ( StanfordEncyclopediaofP h ilosop h y ) ) For example, category-theoretic semantics can interpret logical systems as topological or categorical objects (Joyal Generalizing Kripke semantics of intuitionistic logic to tomographic semantics ( CategoryT h eory ( StanfordEncyclopediaofP h ilosop h y ) ) ), so that the semantic relationship satisfies the pan-category property. The representatives of this paradigm include Ehrenberg and McLean (founders of category theory), Lowell (Lawvere) (Functor semantics), Lambeck (Category grammar), etc. The research focus is on structural unification: it provides a means to abstractly unify semantics at the knowledge level or even the wisdom level. However, category theory mainly serves the abstract unification of pure mathematical structures and does not provide special mechanisms for cognitive levels or specific semantics. ( PDF ) DIKWP 语义数学的理论、结构与应用简析 . Currently, category semantics occupies an important position in computer science (such as programming language semantics, type theory) and logical foundations, and is regarded as a new framework for "semantics in the 21st century" (Steve Awodey & Erich H. Reck, Completeness and Categoricity. Part I) . In the next 10 years, category semantics is expected to be combined with the DIKWP model to formalize the cognitive semantic hierarchy into category objects and morphisms. ( PDF ) DIKWP 语义数学的理论、结构与应用简析 This will use the powerful abstract tools of category theory to verify the properties of semantic systems and achieve a unified theoretical integration of different semantic paradigms.

5.Cognitive semantic paradigm

The cognitive semantics paradigm studies semantics from the perspective of human cognition, arguing that meaning exists in cognitive conceptual structures rather than just logical truth values. It is a reaction to formal semantics, emphasizing that language meaning is closely related to psychological representation and experience. ( Formalsemantics ( naturallanguage ) - Wikipedia ) . Cognitive semantics (such as Lakoff and Langacker) believe that conceptual categories are not defined by strict axioms, but are determined by prototype effects and experience. For example, the meaning of "bird" is based on the image of a typical bird in the human brain, rather than a set of necessary and sufficient conditions. Representative theories of cognitive semantics include frame semantics (Fillmore), conceptual metaphor (proposed by Lakoff) and mental space theory. George Lakoff and Ronald Langacker is an important figure in this paradigm. They proposed that the meaning of language comes from physical experience and cognitive schema. The focus of cognitive semantics is on the knowledge->wisdom layer: explaining how we elevate sensory data to information, organize it into knowledge structures, and finally gain insights (wisdom) through analogies and metaphors. For example, Lakoff proposed that human abstract reasoning is deeply rooted in sensory-motor experience, and that "the human mind is embodied", that is, even the most abstract thinking is rooted in our senses and bodies. ( Cognitionandt h eembodimentofgeometryinGeorgeLakoff ' smetap h ors - GeometryMatters ) Abstract concepts often use metaphors to gain meaning from concrete experiences, such as using spatial metaphors to understand time. Cognitive semantics is currently being studied in depth in linguistics, psychology, and neuroscience, such as using brain imaging to study the network representation of semantic concepts in the brain. ( W h ereIst h eSemanticSystem ? ACriticalReviewandMeta - Analysisof 120 FunctionalNeuroimagingStudies - PMC ) . With the development of big data and cognitive computing, cognitive semantics is being combined with computational models, and there is a trend of integrating cognitive ontology into AI. In the next 10 years, the cognitive semantic paradigm may deepen its intersection with neuroscience and artificial intelligence: on the one hand, it will simulate human concept classification and metaphorical reasoning through neural networks; on the other hand, it will use cognitive semantic principles to improve the interpretability of AI, so that machines can gradually acquire conceptual understanding and intelligent reasoning capabilities close to those of humans.

6Computational Semantic Paradigm

The computational semantic paradigm aims to allow machines to process and understand semantics, involving the semantic web, knowledge graphs, natural language understanding and other directions. It uses formal semantics and statistical methods to transform semantics into computer-operable data->knowledge structures. A typical example is the semantic web technology : adding semantic annotations to web page data through ontology to achieve automatic extraction of information and knowledge from massive data. ( PDF ) DIKWP 语义数学的理论、结构与应用简析 . Tim Berners-Lee's vision of the semantic web and knowledge graphs (such as Google's knowledge graph) are representative applications of computational semantics, which use a combination of logical reasoning (OWL ontology, rules) and statistical learning. This paradigm focuses on the information layer and the knowledge layer: extracting information from unstructured data, integrating it into a knowledge base, and acquiring new knowledge through reasoning. For example, natural language sentences are converted into logical triple data through semantic parsing, stored in a knowledge graph, and then the implicit relationship is inferred. Representative figures/ideas include John Sowa 's concept graph, Guha etc. on ontology languages, as well as recent deep semantic models. Computational semantics is developing rapidly in AI . On the one hand, there are knowledge bases and reasoning systems based on logic, and on the other hand, there are distributed semantics based on deep learning (such as word embeddings and pre-trained language models). However, these two paths have their own shortcomings: symbolic methods are interpretable but lack robustness, and statistical methods are supported by massive data but are difficult to explain reasoning at the intelligent level. For this reason, the academic community has begun to explore semantic models of neural-symbolic fusion, combining symbolic knowledge with neural networks. ( MultiplayerKnowledgeGrap h Creation , Expert - LedHuman ... ) In the next 10 years, computational semantic paradigms are expected to converge and evolve: that is, to merge logical knowledge and multimodal perception in semantic representation, and to combine deduction and deep learning in reasoning mechanisms. Such development will enable AI systems to use large-scale data training to obtain "intuitive" semantics (similar to the wisdom accumulated by human experience), while also following semantic consistency constraints to avoid becoming an opaque "black box". ( PDF ) DIKWP 语义数学的理论、结构与应用简析 In short, the computational semantics paradigm will move towards building human-like semantic intelligence.

7Intuitive semantic paradigm

The intuitive semantics paradigm refers to a semantics view based on intuitionistic logic and constructivism. Unlike classical logic that emphasizes truth value, intuitionistic semantics believes that the meaning of a proposition lies in our constructive method of proving its truth, rather than the objective binary value of true/false. ( 直觉主义逻辑 - 知乎专栏 ) This paradigm originated from Brouwer's intuitionistic mathematical philosophy and was formalized by Heyting and Kolmogorov as the BHK interpretation: semantics is not judged by truth tables, but explained by construction processes. ( 直觉主义逻辑 - 知乎专栏 ) . Simply put, under intuitionistic semantics, a statement such as "there exists an object that satisfies property P" must have a constructible instance to be considered true. Arend Heyting, AA Markloff and others laid the semantic foundation for intuitionistic logic, and Saul Kripke further gave the Kripke model semantics of intuitionistic logic. The focus of intuitionistic semantics research is on the knowledge layer, emphasizing the knowledge generation process itself: only constructed knowledge is real knowledge. This semantic form is close to the intuitive characteristic of human cognition that "only personal proof can understand". At present, intuitionistic semantics has important applications in computer science, such as type theory (such as Persistence). Martin-Löf type theory) and proof verification (such as Coq, Agda proof assistant) are all built on the semantics of intuitionistic logic. Through these systems, mathematical proofs are regarded as computational objects, and the semantics is equivalent to the existence of the proof. This further integrates the intuitionistic semantics paradigm with category theory, such as connecting intuitionistic logic and category theory through topology and layer semantics. ( CategoryT h eory ( StanfordEncyclopediaofP h ilosop h y ) ) In the next 10 years, intuitive semantics is expected to play a greater role in formal knowledge and trustworthy AI. On the one hand, constructive semantics can improve the credibility of AI reasoning results (each conclusion is supported by "proof"); on the other hand, with the development of homology type theory, intuitive semantics will be combined with higher algebraic topology, which may give rise to a new generation of mathematical foundations and affect a wide range of knowledge representation and reasoning theories.

8Multimodal semantic paradigm

The multimodal semantics paradigm focuses on forming semantic representations across multiple senses or data modalities. Unlike traditional semantics that only studies textual language, multimodal semantics incorporates visual, auditory, tactile and other information into the scope of semantic analysis. For example, in image semantic understanding, the image (visual data) and text description need to be aligned for the semantics to be complete; another example is the study of the joint meaning of gestures and language. Multimodal semantics emphasizes the fusion of data/information layers: extracting information from data of different modalities and fusing it into a unified knowledge representation. Cognitively, the human brain stores semantics in a multimodal manner: the semantic network in the brain involves the posterior multimodal association cortex, the prefrontal heterogeneous association area, and the limbic system, etc. ( W h ereIst h eSemanticSystem ? ACriticalReviewandMeta - Analysisof 120 FunctionalNeuroimagingStudies - PMC ) ——In other words, the brain integrates information from various channels such as vision and hearing to form conceptual understanding. Representative studies include conceptual fusion. Blending) theory, and multimodal machine learning in artificial intelligence in recent years. In AI applications, multimodal semantics is used for tasks such as visual question answering (VQA), image and text retrieval, and video understanding. By embedding image features and text semantics into the same vector space, semantic alignment is achieved. At present, multimodal semantics is a hot topic at the forefront of AI. Large-scale pre-trained models (such as CLIP, GPT-4, etc.) have demonstrated the powerful semantic association ability of cross-modal images and texts, which confirms the importance of the multimodal paradigm. In the next 10 years, multimodal semantics is expected to make breakthroughs: first, cross-modal knowledge graphs may emerge, integrating visual, language and other sensor data to build a more comprehensive knowledge network; second, cognitive science will conduct in-depth research on how the brain integrates multimodal information to guide the development of artificial systems. The multimodal semantic paradigm will further narrow the distance between machine understanding and human cognition, and achieve more comprehensive and intuitive semantic modeling at the intelligence level.

9Metaphor and the Embodied Semantic Paradigm

The metaphorical semantics and embodied semantics paradigms originate from cognitive linguistics and provide a unique perspective on the formation mechanism of semantics. The conceptual metaphor theory (proposed by Lakoff and Johnson) believes that humans understand abstract concepts through existing experience in specific fields - metaphor is not only a rhetoric, but also a cognitive tool. ( Cognitionandt h eembodimentofgeometryinGeorgeLakoff ' smetap h ors - GeometryMatters ) The book "Metaphors We Live By" proposes that metaphors shape our thinking by allowing reasoning from the sensorimotor domain to be applied to the abstract domain . ( Cognitionandt h eembodimentofgeometryinGeorgeLakoff ' smetap h ors - GeometryMatters ) For example, "high" is used to describe status, and "bright" is used to describe intelligence. These metaphors map spatial and visual experiences to social and intellectual concepts. Embodied semantics emphasizes that semantics originate from physical experience and perceptual actions. Lakoff et al. proposed the "embodied nature of the mind", arguing that the mind is not a computer separated from the body, and human cognition is deeply rooted in the sensory and motor systems. ( Cognitionandt h eembodimentofgeometryinGeorgeLakoff ' smetap h ors - GeometryMatters ) . The brain's processing of semantics involves the activation of sensory and motor areas, indicating that meaning and physical experience are difficult to separate. Embodied semantics also includes the discovery of mirror neurons: the semantics of understanding other people's actions is similar to the neural activity when we perform our own actions, which reflects embodied understanding. In addition to Lakoff, typical representatives include Mark Johnson (who jointly proposed the idea of embodied metaphor), Lakoff and Nunes (who applied the concept of embodiment to mathematical concepts), etc. The metaphor/embodied semantics paradigm focuses on the wisdom layer and the purpose layer: through metaphor mapping, humans sublimate low-level concrete knowledge into high-level wisdom, and often use metaphors to inspire thinking for the purpose of communication or problem solving. At present, this paradigm has penetrated into artificial intelligence and human-computer interaction research, such as robots understanding spatial concepts through embodied learning, or NLP systems trying to parse metaphorical meanings. Embodied semantics is also supported by neuroscience. For example, fMRI studies have shown that metaphors that process abstract concepts activate brain areas corresponding to concrete sensations. In the next 10 years, metaphor/embodied semantics is expected to make progress in interdisciplinary fields: on the one hand, cognitive neuroscience will further reveal the brain mechanism of embodied meaning; on the other hand, artificial intelligence will try to achieve embodied cognition, such as allowing virtual agents to learn semantics through the experience of virtual environments. From a broader perspective, the embodied semantics paradigm may promote a revolution in knowledge representation - moving away from a purely symbolic logic framework and turning to a new system that integrates sensor data, robot actions, and semantic reasoning, giving machines human-like "understanding" and adaptability.

10DIKWP semantic hierarchy analysis table

The above research paradigms DIKWP (Data-Information, Information-Information, Knowledge-Knowledge, Wisdom-Wisdom, Purpose) has different focuses on the cognitive level. The following table summarizes the main DIKWP level distribution, representative figures' thoughts, research focus, current status and evolution forecast for the next 10 years for each paradigm:

DIKWP level emphasis, representative figures' ideas, research focus, current status and future evolution prediction of each research paradigm of semantic mathematics

Research Paradigm	DIKWP level main distribution	Typical representative ideas/characters	Research Focus	Current Development Status	Evolution prediction for the next 10 years
Formal Logic	Data → Information → Knowledge Layer	Aristotle; Frege; Russell; Tarski, etc. ( PDF ) DIKWP 语义数学的理论、结构与应用简析	Symbolic reasoning, axiomatic systems, deductive consistency	Mature and stable; widely used in mathematics and computer science foundations, but limited in semantic expression	Combined with probability and learning, it enhances uncertain reasoning and explainable AI, maintaining the core position of logical reasoning
Model-theoretic semantics	Information → Knowledge Layer	Tarski (truth semantics); van Benthem; Montag (formal semantics), etc.	Model interpretation, truth value definition, and knowledge representation of logical languages	Mature and complete; supports applications such as databases and knowledge graphs; expands model theories such as modality and temporality	Integrate new methods such as category theory to extend to a richer system and combine statistical methods to handle large-scale uncertain information
Category-theoretic semantics	Knowledge → Wisdom Layer	Ehrenberg & McLean (category theory); Lawvere (functor semantics); Lambek (category grammar), etc. ( PDF ) DIKWP 语义数学的理论、结构与应用简析	Unified representation of mathematical structures and functor mapping of syntax and semantics	Frontier and in-depth; plays a role in logic and computer theory, but has not yet directly touched the cognitive level semantics ( PDF ) DIKWP 语义数学的理论、结构与应用简析	Combined with cognitive model, formalizing DIKWP hierarchy ( PDF ) DIKWP 语义数学的理论、结构与应用简析 ; Achieved breakthroughs in areas such as homology type theory, becoming a new basic framework for semantics
Cognitive semantics	Knowledge → Wisdom → Purpose Layer	Lakoff ( Cognitionandt h eembodimentofgeometryinGeorgeLakoff ' smetap h ors - GeometryMatters ) ; Langacker; Fillmore; Talmy wait	Concept categorization, prototype theory, metaphor and frame semantics	Flourishing; cross-integration with psychology and neuroscience, verifying hypotheses such as embodied cognition; but less connected with formal methods	Deepen interdisciplinary integration and combine with AI to improve machine semantic understanding; theoretically integrate with formal semantics to bridge the cognitive-symbolic gap ( Formalsemantics ( naturallanguage ) - Wikipedia )
Computational semantics	Data → Information → Knowledge Layer	Berners-Lee (Semantic Web); Google Knowledge Graph; Sowa (Concept Graph), etc.	Semantic representation standardization, ontology construction, automatic reasoning and question answering	Rapid development; knowledge graph and deep learning are advancing in parallel, with a trend of neural-symbolic hybrid; knowledge fragmentation and black box problems exist	A semantic system that realizes neural-symbolic fusion ( PDF ) DIKWP 语义数学的理论、结构与应用简析 ; Multimodal knowledge fusion; Support more intelligent question-answering and decision-making systems, and more complete semantic technology standards
Intuitive semantics	Information → Knowledge Layer	Brouwer, Heyting, Kripke, etc. ( 直觉主义逻辑 - 知乎专栏 )	BHK construct semantics, proof as semantics, constructive knowledge acquisition	The profession has made steady progress; it has become the cornerstone of computer verification and type theory; it has little involvement in mainstream AI	Incorporate new mathematical foundations (such as homology type theory); provide a reliable reasoning kernel for AI (with proof at every step); further integrate and expand with category theory
Multimodal semantics	Data → Information → Knowledge Layer	CLIP multimodal model; Barsalou (perceptual symbol system) etc. ( W h ereIst h eSemanticSystem ? ACriticalReviewandMeta - Analysisof 120 FunctionalNeuroimagingStudies - PMC )	Perceptual symbol fusion, cross-modal alignment, and unified representation space	Emerging hotspot; Multimodal pre-training models have significant effects, but the unified understanding mechanism is still under exploration	Construct cross-modal knowledge graphs to achieve complex scene understanding; clarify the brain's multimodal semantic integration mechanism at the cognitive level and feed back to artificial models
Metaphorical/embodied semantics	Knowledge → Wisdom → Purpose Layer	Lakoff & Johnson (Conceptual Metaphor) ( Cognitionandt h eembodimentofgeometryinGeorgeLakoff ' smetap h ors - GeometryMatters ) ;Gibbs;Glenberg wait	The influence of metaphor mapping mechanism and physical experience on concept formation	Theory matures; embodied cognition is supported by experimental evidence, influencing philosophy and AI; metaphor processing is a difficult point in NLP	Apply the principle of embodiment to robots and interactive AI to realize intelligent agents that truly "understand" the environment; improve NLP metaphor recognition and generation to enable machines to master flexible human-like use of meaning

11Semantic Network Associations of Different Paradigms

Although each semantic paradigm has different focuses, they are not developed in isolation, but are interwoven into a semantic research network through the flow of concepts and methods:

Formal logic → Model theory semantics: Model theory provides a semantic interpretation for formal logic, making symbolic deduction correspond to the real world. ( PDF ) DIKWP 语义数学的理论、结构与应用简析 The model-theoretic semantics of classical logic is the starting point of semantic mathematics, and a series of subsequent semantic paradigms are more or less based on the extension or reflection of model theory.

Formal logic → cognitive semantics: Cognitive semantics is a backlash against the limitations of traditional formal semantics ( Formalsemantics ( naturallanguage ) - Wikipedia ) Formal logic emphasizes truth conditions, while cognitive semantics turns its attention to the human brain's intuitive grasp of meaning. The two were initially opposed, but in recent years there have been signs of integration, such as frame semantics attempting to express cognitive conceptual structures in a formal way, and cognitive semantics also using logical tools to express conceptual relationships. ( Formalsemantics ( naturallanguage ) - Wikipedia ) .

Model-theoretic semantics ↔ Category-theoretic semantics: Category theory can be seen as a further abstraction and unification of model theory. ( CategoryT h eory ( StanfordEncyclopediaofP h ilosop h y ) ) In the second half of the 20th century, logical semantics showed a trend of "categorization": Lawvere's functor semantics categorized the model theory viewpoint, and the models of different logical systems can be regarded as objects in the category, which can be uniformly described by functors at a higher level. ( CategoryT h eory ( StanfordEncyclopediaofP h ilosop h y ) ) At the same time, the development of category theory also depends on the verification of the results of model theory. The two promote each other and make semantics have a more general mathematical form.

Model-theoretic semantics → Intuitionistic semantics: Intuitionistic semantics was first proposed by BHK, and then Kripke gave a formal model. However, in the 1970s and 1980s, people found that the semantics of intuitionistic logic can be understood by categories and topology. For example, Joyal and others generalized Kripke semantics to sheaf semantics. ( CategoryT h eory ( StanfordEncyclopediaofP h ilosop h y ) ) This is actually an extended application of category-theoretic semantics to model-theoretic semantics, further enriching the model of intuitive semantics. Therefore, intuitive semantics is closely related to category semantics, and together they constitute a new perspective on classical semantics.

Cognitive semantics → metaphor/embodied semantics: Metaphorical semantics and embodied semantics can be seen as the deepening of the branches within cognitive semantics. Cognitive semantics generally emphasizes experience and concepts, while metaphor theory specifically reveals that the concepts at the wisdom level come from the mapping of experience at the knowledge level. ( Cognitionandt h eembodimentofgeometryinGeorgeLakoff ' smetap h ors - GeometryMatters ) ; Embodied semantics highlights the shaping of meaning by purpose/action ( Cognitionandt h eembodimentofgeometryinGeorgeLakoff ' smetap h ors - GeometryMatters ) These two branches have further enriched the cognitive semantics paradigm and also impacted the traditional philosophical semantics. For example, embodied semantics opposes the mind/body dichotomy and promotes semantic research to consider biological factors.

Cognitive semantics ↔ computational semantics: The theories of cognitive semantics are gradually being applied to computational semantics, and the two are learning from each other. For example, conceptual frameworks are used to construct ontological word networks, and the prototype effect of cognition inspires corpus-based distributed representations. In the field of AI, large language models have captured the implicit structure of human semantic space to a certain extent, but lack clear conceptual boundaries; this has prompted researchers to introduce knowledge of cognitive semantics, such as using human common sense concept networks to calibrate model outputs. In turn, the development of computational semantics has also provided an experimental platform for cognitive semantics, which deepens the understanding of human semantics by simulating and testing the feasibility of cognitive semantic theories (such as metaphor understanding) on machines.

Computational semantics ↔ multimodal semantics: The rise of multimodal semantics depends largely on the development of computational semantic technology. With the maturity of computer vision and speech processing, computational semantics has expanded to process data forms such as images and audio, thus forming multimodal research. Today's multimodal pre-training models combine a large amount of annotated data (computational semantic assets) for cross-modal alignment. The two complement each other: computational semantics provides algorithms and frameworks, and multimodal research broadens the data foundation of semantics. At the same time, the results of multimodal cognitive research (such as how the brain associates images and sounds) are fed back to the design of artificial systems, inspiring fusion mechanisms that are closer to humans.

Computational semantics → Category theory semantics: As the scale of knowledge representation grows, computational semantics encounters challenges of complexity and consistency, and begins to seek more abstract mathematical tools. The combinatorial paradigm provided by category theory can improve the design of large-scale semantic networks and make the mapping relationship between different knowledge subnets clearer. Recently, some scholars have tried to use the category paradigm to describe knowledge graphs and ontology mappings, treating knowledge resources as category objects to unify reasoning rules. This intersection improves the mathematical rigor of the knowledge base and reduces the ambiguity of semantic integration.

Intuitive semantics ↔ computational semantics: Although intuitionistic semantics is currently mainly in the academic field, its concept has a far-reaching potential impact on computational semantics. Modern AI has increasing requirements for reliability and verifiability, and the "witnessed truth" advocated by intuitive semantics meets this demand. Some reasoning systems have adopted intuitive logic to avoid non-constructive reasoning, thereby generating traceable conclusion proofs. This trend indicates that intuitive semantics will be deeply combined with computational semantics in the field of AI security and explainability, giving AI systems the ability to prove their rationality.

In summary, different paradigms together constitute a network spectrum of semantic mathematics research: the formal/model theory paradigm lays the formal foundation of semantics, the cognitive/embodied paradigm injects a humanistic perspective, the category/intuition paradigm provides new mathematical tools, and the computational/multimodal paradigm applies theory to real AI systems. ( Cognitionandt h eembodimentofgeometryinGeorgeLakoff ' smetap h ors - GeometryMatters ) These paradigms penetrate each other and evolve continuously, pushing the semantic mathematics system towards a more comprehensive and powerful direction.

12Future Trends, Core Challenges and Prospects

In the next 10 years, semantic mathematics is expected to see a situation where fusion innovation and structural reorganization coexist, with challenges and opportunities coexisting at all levels:

Paradigm fusion and the formation of new frameworks: The boundaries between different semantic paradigms will become more blurred, and a comprehensive framework will replace them. For example, the fusion of cognitive semantics and formal semantics may give birth to "cognitive formal semantics", which has both rigorous symbols and human conceptual intuition. Category theory may serve as a meta-framework to formalize the semantic processes of each layer of DIKWP, thereby uniformly describing the transformation chain from data to wisdom. ( PDF ) DIKWP 语义数学的理论、结构与应用简析 This fusion will restructure the structure of semantic mathematics, making it a network system covering symbols, algorithms, and cognition, truly running through all levels of DIKWP.

Development driven by artificial intelligence: The development of AI poses two major demands and challenges to semantic mathematics: one is large-scale semantic acquisition, how to automatically obtain reliable knowledge and wisdom from massive data (this involves the challenges of knowledge extraction and verification); the other is explainable intelligence, how to give AI's reasoning process a semantic scaffolding so that its decision-making is based on evidence and has a clear purpose ( PDF ) DIKWP 语义数学的理论、结构与应用简析 . To this end, semantic mathematics needs to develop a neural-symbolic hybrid method to combine the "data intuition" of deep learning with the "knowledge rationality" of symbolic semantics. This is both a technical challenge and a theoretical challenge - it requires rethinking the representation and acquisition of knowledge and wisdom. In addition, multimodal fusion and situational semantic understanding (the impact of context and purpose on meaning) are also difficult problems facing AI semantics, which will force semantic theory to respond in the future.

Closure and self-feedback at the semantic level: DIKWP The model emphasizes closed-loop feedback from data to purpose ( PDF ) DIKWP 语义数学的理论、结构与应用简析 . Future semantic systems will pay more attention to feedback mechanisms: the application of knowledge in turn generates new data, and the conclusions of wisdom in turn correct the knowledge base and achieve self-evolution. For example, a large model can continuously update its knowledge representation by interacting with the environment (data layer) and adjust the internal semantic structure according to the achievement of the goal (purpose layer). This adaptive semantic evolution requires a mathematically rigorous framework to ensure consistency, otherwise semantic drift or self-contradiction may occur. Therefore, the development of dynamic semantic theory and evolutionary ontology will be key to ensure the stability and effectiveness of semantic closed-loop operation. ( PDF ) DIKWP 语义数学的理论、结构与应用简析 .

Core challenges: Despite its broad prospects, semantic mathematics still faces many challenges in the future. One of them is the barrier to interdisciplinary communication: the understanding of "semantics" varies greatly in different fields. How to form a common language? This requires scholars to have a composite background and integrate the concepts of philosophy, linguistics, mathematics, and computers. The second is the challenge of complexity: the semantic model that fully covers the entire DIKWP link may be extremely complex. How to ensure the computability and verifiability of the model? Perhaps new abstraction and simplification strategies need to be introduced. The third is the data and knowledge gap: in the era of big data, the amount of data far exceeds the ability of human knowledge to summarize, and how to automatically sublimate wisdom from data has not yet been determined. This involves new algorithms and theoretical breakthroughs, such as whether reinforcement learning can play a role at a higher level of intelligence and whether metaphorical analogies can be algorithmic. Finally, thinking at the ethical and purpose levels is also indispensable-when machines begin to form their own "purpose" semantics, how do we ensure that it is consistent with human values? The DIKWP model puts "purpose" at the top level, suggesting that future semantic systems must have built-in considerations of purpose and value. ( PDF ) DIKWP 语义数学的理论、结构与应用简析 .

Prediction of structural reorganization: In combination with the above trends, it can be foreseen that the organizational structure of semantic mathematics research itself will be adjusted. For example, universities and research institutions may establish cross-departmental "semantic intelligence" research centers to bring together experts in computer science, cognitive science, and philosophy of logic to jointly develop a unified semantic framework. In academic publications, single-paradigm papers will decrease, replaced by cross-paradigm results. Standardization work will also be put on the agenda, such as the formulation of unified semantic representation standards, covering formal logic, ontology, and metaphor annotation, so that semantic knowledge of different systems can be interoperable. Equally important is the adjustment at the educational level. In the future, students will be trained to dabble in logic, AI, and cognitive psychology at the same time to meet the needs of the new semantic discipline.

13Conclusion

In summary, as a bridge connecting human intelligence and formal symbols, semantic mathematics will move towards deeper integration and transformation in the next decade. Each paradigm will jointly shape a closed-loop system of integrated data-information-knowledge-wisdom-purpose through network connections. In this process, we must not only face the challenges brought by big data and AI, but also persist in philosophical reflection on the essence of semantics. It can be foreseen that a new semantic mathematics paradigm that is both "down to earth" (data, information) and "looking up to the stars" (wisdom, purpose) will eventually take shape, laying a more solid and comprehensive foundation for human understanding of intelligence. ( Cognitionandt h eembodimentofgeometryinGeorgeLakoff ' smetap h ors - GeometryMatters ) Through this evolution, we are expected to crack the complex network of semantics, usher in a leap forward in cognitive computing and artificial intelligence, and achieve a deep integration of the human-machine semantic world.