A Theoretical Report on Semantic Mathematics

A Theoretical Report on Semantic Mathematics: Breaking Free from Formal Shackles

Yucong Duan

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

World Academy for Artificial Consciousness(WAAC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Abstract

Mathematical theorems and conjectures are traditionally based on axiomatic deduction, verifying their truth or falsity within a rigorous logical system. However, this purely formal derivation often overlooks the semantic structure behind the symbols. This article breaks through the traditional deductive framework, based on the DIKWP semantic model proposed by Yucong Duan, to systematically reconstruct the cognitive structure and generation mechanisms in semantic space of classic and modern challenges such as Fermat's Last Theorem, the Riemann Hypothesis, the Twin Prime Conjecture, the Poincaré Conjecture, and the classification of high-dimensional spheres. We introduce the five-layer semantic structure of Data-Information-Knowledge-Wisdom-Purpose (DIKWP), exploring the semantic tension model and self-consistency mechanisms of mathematical discovery and proof: from Data (D) at the symbol layer and Information (I) of pattern differences, through the structural graph and tension closed loop of the Knowledge (K) layer, leaping to the proof form of the Wisdom (W) layer, and finally pointing to the pursuit of universality motivation and structural compression of the Purpose (P) layer. Combining Yucong Duan's Semantic Tension theory and the BUG Consciousness concept, we propose a "conjecture-tension-proof" cognitive chain, characterizing how "BUGs" (incompleteness points) in the cognitive system of mathematical problems trigger high-level semantic regulation and breakthroughs. Through instance analysis of Fermat's Last Theorem, the Riemann Hypothesis, etc., we clarify that: the more concise and universal (highly semantically compressed) a proposition is, the greater its hidden semantic tension, making it more difficult for the traditional W layer (formal proof) to be directly externalized, thus requiring cross-level semantic leaps and knowledge expansion to alleviate the tension and close the semantic structure. Based on this, we construct a "Semantic Mathematics" framework, proposing a measurability of provability parameterized by semantic complexity and tension compression, viewing mathematical structures as stable closed loops in an information-tension system, and regarding deductive reasoning as the externalized path after the K-W-P structure achieves stability. Finally, we suggest constructing a Semantic Theorem Evolution System to simulate the process of conjectures evolving into provable propositions driven by semantic tension, and look forward to the enlightenment of this semantic perspective on artificial intelligence mathematical discovery and the philosophy of mathematics.

Introduction

Mathematical theorems and conjectures are peak products of human cognition. They are often stated in extremely simple forms yet contain profound structures. However, many famous problems (such as Fermat's Last Theorem, the Riemann Hypothesis, etc.), despite their simple and clear statements, have remained difficult to prove for long periods. Traditional mathematical research mainly relies on formal language and logical deduction, proceeding sequentially within an axiomatic system. But relying solely on symbolic reasoning often bogs down the proof process in complex technical details, making it difficult to gain insight into the network of meaning and cognitive driving force behind the theorem. In recent years, the development of artificial intelligence and cognitive science has prompted people to reflect: Is mathematical discovery not just a cold deductive accumulation, but a semantically driven creative process? The DIKWP semantic model proposed by Professor Yucong Duan provides a brand-new perspective, incorporating the five cognitive elements of data, information, knowledge, wisdom, and Purpose into a unified framework, and characterizing the meaning behind symbols in a formal way. This theory of semantic mathematics aims to integrate rigorous logical deduction with real-world semantics, enabling systems like AI to understand the rich meaning of mathematical objects, thereby alleviating the "semantic-loss" problem. At the same time, Yucong Duan's "Consciousness BUG Theory" points out that imperfection and limitations (i.e., "BUGs" in the cognitive process) are, on the contrary, the opportunities that give rise to high-level cognition. Is not every unsolved mystery in the history of mathematics a "BUG" in the knowledge system? From this perspective, conjectures such as Fermat's and Riemann's are precisely the abnormal deviations in the mathematical cognitive network, causing generations of mathematicians to devote huge energy to bridging this semantic gap. Therefore, introducing semantic generation and BUG regulation into the study of mathematical theorems is expected to clarify the internal mechanism of mathematical discovery. Following this introduction, this article will first review the structure and deductive mechanism of the traditional mathematical system, and then introduce the DIKWP semantic model and its role in mathematical construction in detail. On this basis, combined with Yucong Duan's semantic philosophy, we analyze the semantic tension adjustment and self-consistent closed-loop mechanism in the conjecture-proof process. Next, through instances such as Fermat's Last Theorem, the Riemann Hypothesis, the Poincaré Conjecture, and the Twin Prime Conjecture, we reconstruct the semantic structural landscape of these problems, explaining why they are "stubborn" for so long, and how they are finally resolved or still await resolution at the semantic level. Finally, we attempt to construct an overall framework for "Semantic Mathematics," proposing a quantifiable measurability of provability and model concepts, and looking forward to the enlightenment and philosophical reflection of this perspective on future mathematical research and artificial intelligence.

Review of the Traditional Structure and Deductive Mechanism of Mathematical Theorems

In the traditional view, mathematics is regarded as a deductive closed loop built on an axiomatic system. All theorems must start from a set of basic axioms and be derived through formal logic to ensure rigor and consistency. The task of mathematicians is to construct rigorous proof chains on the basis of axioms and known theorems to incorporate new propositions into the system. Under this framework, mathematical structures have the following typical characteristics:

·Axiomatic System and Irreducible Principles: Each mathematical theory starts with a series of axioms. These axioms are regarded as self-evident or agreed-upon truths and are not proven further. Axioms are the atoms of knowledge and should be as independent and irreducible as possible to avoid circular reasoning in the system. For example, Euclid's parallel postulate in geometry, and the Peano axioms for arithmetic, all provide a basic fulcrum for deduction.

·Formal Language and Symbolic Representation: Mathematics uses formal language (such as first-order logic) and symbol systems to describe objects and relationships. Symbol manipulation has clear rules, eliminating the ambiguity of natural language. Through formalization, mathematical statements can be precisely operated, avoiding the pitfalls of intuitive interpretation. This symbolization makes the proof process a process of pure symbol string transformation, seemingly unrelated to its meaning.

·Deductive Reasoning and Logical Closed Loop: A proof is the process of deducing the truth of a proposition step by step according to logical rules (such as modus ponens). Ideally, the entire theory forms a closed logical system: all theorems can be deduced from the axioms, and no external experience is needed in the deduction process. The logical closed loop ensures the necessity and verifiability of the conclusion—every step of reasoning can be checked. Once a proof is established, the theorem is a certain truth within the system, indubitable.

·Inviolable Self-Consistency: A mathematical system requires non-contradiction (compatibility). Through formal axiomatization and logical deduction, the aim is to establish an internally completely self-consistent world. According to the vision of Hilbert's program, mathematical truth is the deductive truth of a formal system. However, the compatibility of the formal system itself sometimes requires higher-level meta-theoretical analysis (such as Gödel's incompleteness theorems on the limitations of the completeness of formal systems). But in specific theories, people assume their axiomatic systems are compatible, thereby ensuring that self-contradictory propositions will not be deduced.

In short, under the traditional mathematical architecture, proof is the core: a mathematical proposition is upgraded to a theorem only after being rigorously proven, otherwise it is at best a conjecture or a pseudo-proposition. This deductivist perspective emphasizes the mechanical reliability of the reasoning process and the certainty of the conclusion. However, this mechanism also brings some limitations: on the one hand, the proof process is often lengthy and complex, filled with technical details, making it difficult for people to grasp the core idea; on the other hand, purely formal deduction conceals the role of intuition and semantics in mathematical discovery. Many major breakthroughs in history often stem from mathematicians' insights into the problem's background and reinterpretation of the meaning of concepts, not just from symbolic calculation. As mathematical propositions become more abstract and complex, relying solely on formal deduction is often stretched thin. Against this background, introducing semantic-level analysis to explore the conceptual network and intentional motivation behind mathematical theorems is expected to supplement the insufficiency of purely deductive methods.

The Role of DIKWP Semantic Structure in Mathematical Construction

The DIKWP model proposed by Yucong Duan divides the cognitive process into five levels: Data (D), Information (I), Knowledge (K), Wisdom (W), and Purpose (P). By introducing this model into the field of mathematics, we can re-examine the construction process of mathematical theorems and conjectures, and understand the roles played by semantics at different levels:

·D Layer (Data Layer): Raw Data of Symbols and Objects. This is the "alphabet" and raw material of mathematics, including basic elements such as numbers, symbols, geometric figures, formulas, etc. In traditional mathematics, the D layer corresponds to the symbolic representation of basic concepts such as definitions, axioms, and postulates, as well as a large number of specific examples (such as known number tables, function graphs, calculation results). The D layer provides the lowest-level factual material and objects of operation. For example, the D-layer elements of Fermat's Last Theorem are the symbols expressing the equation  and some known arithmetic instances; the D-layer of the Riemann Hypothesis contains the analytical expression of the Riemann  function and some calculated non-trivial zero data. At the D layer, semantics are mainly embodied as the direct meaning of symbol denotation (e.g., numbers represent quantity, set symbols represent element belonging, etc.). The information at this layer is "raw material" and has not yet been organized by high-level structures.

·I Layer (Information Layer): Structural Differences and Conjecture Space. The I layer focuses on pattern recognition and difference comparison of the raw data from the D layer. By comparing different instances, mathematicians will find certain discontinuities in patterns, that is, anomalies or novel laws that existing knowledge cannot explain. This is often where conjectures are born. When facing a large amount of data, finding the commonalities and differences within it is precisely the process of refining information. For example, in the study of prime numbers, a large number of calculation examples (D layer) show that the distribution of prime numbers is irregular, but Riemann discovered a frequency fluctuation pattern where the density of prime numbers is related to the zeros of —this is the refinement of I-layer information: discovering the correlation between frequency oscillation and prime number distribution, thus proposing the Riemann Hypothesis to explain this pattern. In the context of Fermat's Last Theorem, the I layer is embodied as the cognitive difference between the Pythagorean theorem case () and the no-solution case for : the sum of two squares can be another square (e.g., ), but no integer solutions were found after trying many exponents . This pattern difference triggered the conjecture: perhaps there are indeed no integer solutions for . The I layer thus opens up the conjecture space: based on the intuitive grasp of existing data patterns, propositions to be verified are proposed. In other words, the I layer captures "where the difference is" and "what it might be," providing direction for the systematization of the K layer.

·K Layer (Knowledge Layer): Knowledge Structure Graph and Self-Consistent Closed Loop. The K layer is the level where information is fully organized and deduced to form systematized knowledge. In mathematics, the K layer corresponds to the network of established theories and theorems, i.e., the accumulated mathematical knowledge graph of mankind. The K layer pursues a self-consistent closed loop, meaning that the knowledge structure is logically consistent internally and semantically complete, with no unexplained anomalies (equivalent to semantic "completeness"). If a conjecture is to be elevated to knowledge, it must be integrated into the existing system of the K layer, or promote new theoretical branches. The K layer has compressibility: it compresses a large number of specific facts (D-layer data and I-layer information) into general laws through concepts and axioms. For example, the "Prime Number Theorem" compresses the statistical law of prime number distribution into a concise formula , elevating scattered data to knowledge; classification theorems in topology summarize infinitely many situations into finite types. The compression of the knowledge layer means a high degree of refinement and structurization of information, thereby eliminating semantic redundancy. On the other hand, the K layer strives to form a semantic closed loop, explaining and containing all known facts in the relevant field. If a phenomenon that cannot be explained appears (new anomaly in the I layer), it means that the K layer is not yet complete, and there is a "BUG" in the system. For example, before Fermat's Last Theorem became a theorem, the assertion " has no integer solution ()" existed outside the number theory knowledge system for a long time (a gap in the K layer), becoming a huge knowledge loophole. Only when Wiles proved it and connected it to the theory of modular forms and elliptic curves did the K layer close this semantic gap. At this time, the status of FLT in the K layer changed from an isolated conjecture to a special result of modular form theory, and the knowledge system achieved closure. It is worth noting that the Taniyama-Shimura conjecture itself involves deep knowledge of modern algebraic geometry, which is no longer in the knowledge category of Fermat's time. In order to absorb FLT, the knowledge layer had to undergo huge expansion and integration. Therefore, the K layer represents the self-consistency and completeness of mathematical knowledge, and is also the target state that a proof must achieve.

·W Layer (Wisdom Layer): Formalized Expression and Proof Behavior. The W layer corresponds to the reasoning processes, proof strategies, and methods in mathematical activities, as well as the ability to apply knowledge to solve problems. "Wisdom" here refers not only to general intelligence, but more specifically to the creative thinking and formalized proof skills of mathematicians. At the W layer, we focus on how to prove/solve a conjecture, i.e., finding appropriate proof routes, constructing intermediate lemmas, using existing theorems, etc. Formal deduction, calculation, and logical reasoning all belong to the behavioral manifestations of the W layer. The W layer is the bridge that transforms the conjectures of the I layer into the knowledge of the K layer: through a series of proof steps (logical deduction, analogy, induction, etc.), a path from axiomatic knowledge to the proposition to be proved is built within the formal system. For example, the W-layer exploration of the Riemann Hypothesis includes various strategies such as analytic continuation, functional equations, attempts at proof by contradiction, and constructing the Hilbert-Pólya conjecture, but so far none has successfully reached the destination. This shows that the challenge of the W layer is extremely daunting and requires more sparks of wisdom. It is worth noting that the product of the W layer—the formal proof text—is actually the externalized form of the finally stable knowledge. When a conjecture is truly proven, the reasoning of the W layer will be organized into a standard paper or written proof for others to verify. The existence of this proof itself means that the proposition has been successfully integrated into the K layer knowledge. However, before the proof is invented, the W layer activity is more embodied as exploratory attempts and semantic leaps driven by intuition, rather than mechanical symbol manipulation. The W layer is therefore also the place where semantics and form intersect: the mathematician's intuition, analogy, and insight (semantic components) are trained and finally condensed into a rigorous reasoning roadmap.

·P Layer (Purpose Layer): Mathematical Purpose and the Pursuit of Universality. The P layer is the highest layer of the DIKWP model, representing purposefulness and value-drivenness. In mathematics, the P layer can be understood as the original intention of a mathematician to conduct a certain research, the problem to be solved, the overall goal hoped to be achieved, and even a certain aesthetic or unified pursuit. Purpose guides the choice of research topics and the preference for proof directions; it is a "meaning" at the macro level. For example, the mathematical community's obsession with the twin prime conjecture, at the Purpose layer, lies in the ultimate understanding of the laws of prime number distribution and the persistent belief in "local patterns in an infinite structure." The P-layer motivation behind Fermat's Last Theorem can be said to be a complete understanding of the properties of power sums in the integer domain, as well as the curiosity and challenge of verifying whether Fermat's "marvelous proof" of the time existed. The Purpose layer of the Poincaré conjecture is the ambition to classify the topological structure of three-dimensional space and understand the shape of the universe. The P layer also embodies mathematics's pursuit of conciseness and universality—that is, to use the most general laws possible to describe as many phenomena as possible. For example, if the Riemann Hypothesis is true, it will unify the randomness of prime number distribution with deep analytical structures, which has great universal explanatory power. In addition, the P layer has the goal of compressing structures: just as discovering a theorem can often compress countless special cases. A simple formula of Fermat's Last Theorem contains the compressed conclusion of infinitely many unsolvable equations; the twin prime conjecture, in one sentence, states the grand assertion of the existence of infinitely many prime pairs. It is precisely because of this high degree of compression that if the P-layer Purpose is ahead of current knowledge, it often leads to semantic tension: the Purpose is desired, but the knowledge is temporarily unavailable. In short, the P layer provides the directional gravity for mathematical exploration and is the source of power driving the D-I-K-W chain to continuously cycle upwards.

It needs to be emphasized that the five DIKWP layers are not linear and unidirectional, but form a highly interconnected network. In the process of mathematical creation, there are frequent interactions and feedbacks between the various layers: the upper-level Purpose (P) will affect the collection of data and pattern recognition at the lower levels (P→I, e.g., researchers look for evidence with a conjecture in mind); the knowledge structure (K) also guides which data (D) is worth paying attention to and which information (I) is noise or meaningful; conversely, new and different information at the I layer will shake the existing knowledge K layer, prompting the Wisdom W layer to adjust proof strategies to satisfy the Purpose P, and so on. This multi-level circular interaction forms a semantic closed loop: every complete exploration from data to Purpose, if it can feed back to correct data selection and knowledge updates, completes a cognitive closed loop. Mathematical research often goes through multiple rounds of such closed-loop iterations—the proposal of a conjecture, attempts, failures, revisions, and then attempts again, until the proof is completed and the system is stable. The DIKWP model provides us with a language to understand these iterations: it reveals that mathematical discovery is a process of symbol-meaning integration, and not just the operation of deductive rules.

Integrated Analysis of Yucong Duan's Semantic Theory

Having grasped the meaning of each DIKWP layer, we further introduce the core concepts in Yucong Duan's semantic philosophy to conduct a holistic perspective on the semantic generation and proof of mathematical theorems. The focus here is on: the role of the Semantic Tension Model in the evolution of mathematical structures, how "BUG Consciousness" promotes the conjecture-proof chain, and the mechanism of knowledge layer compression and semantic closure.

Semantic Tension and DIKWP Leaps

So-called Semantic Tension refers to the tension generated between different semantic layers or concepts due to inconsistency or absence, like a stretched rubber band in a knowledge network. When there is a gap between the Purpose of the P layer and the current status of the K layer, or when the K layer temporarily cannot explain the abnormal patterns found by the I layer, semantic tension will be formed. This tension is manifested in mathematics as the cognitive dissonance brought about by unsolved problems: we "feel" that a certain proposition should be true (or need a theory to explain a certain phenomenon), but currently lack proof or theoretical support. This is precisely the reason for the generation and continued existence of mathematical conjectures. The semantic tension model holds that the cognitive system will actively seek to alleviate this tension, thereby driving cross-level efforts (such as guiding the collection of more data from the P layer downwards, or thinking of new proof strategies from the W layer). In the DIKWP framework, semantic tension triggers inter-layer leaps to readjust the structure:

·P→K Compression Leap: When there is a strong Purpose or belief (P layer) but the knowledge base has no corresponding theorem (K layer is blank), it will prompt researchers to try to directly compress the Purpose into knowledge. This is the moment when a conjecture is proposed, and it is also the initial leap: jumping from a vague Purpose to a specific proposition, hoping that it will be established and incorporated into the knowledge system. This step has a bold conjectural component. For example, when Fermat was reading Diophantus's Arithmetica, he suddenly had an idea and wrote down the assertion and the hint of a "marvelous proof" of Fermat's Last Theorem. He actually completed a P→K compression leap: directly condensing the intuition about arithmetic structures (P) into a knowledge proposition (K). This leap produces a very concise assertion, but it will also generate huge semantic tension—because the K layer lacks support, and the W layer proof is absent. In other words, the more extreme the P→K leap compression, the greater the tension. Fermat's Last Theorem, in a short sentence, compresses the situation of infinite integers. This extreme compression leads to extremely high proof difficulty (the W layer cannot be externalized for a long time), and the proposition becomes an unresolved "foreign object" in the knowledge system. Similarly, the Riemann Hypothesis, in one sentence, locks the real part of all non-trivial zeros to . This is also an unusually bold P→K leap, compressing the complexity of prime number distribution into a simple statement, so mathematicians have racked their brains for it for one and a half centuries.

·K→W→P Adjustment Leap: After the semantic tension is formed, it needs to be resolved through the activities of the Wisdom layer (W). But the attempts at the W layer are not carried out blindly, but are the result of the joint action of the K layer and the P layer: the K layer provides existing tool theorems, and the P layer provides direction and ultimate standards. When researchers look for a proof at the W layer, they actually often need to jump back and forth between knowledge and Purpose: on the one hand, they choose available methods based on existing theories (K), and on the other hand, they adjust strategies based on the goal they want to prove (P). This is manifested as K→W (drawing inspiration and methods from known knowledge, such as citing relevant theorems, developing new lemmas) and P→W (always reviewing whether the proof is moving in the direction of satisfying the Purpose) two kinds of leaps proceeding alternately. It can be said that the W layer is a process of continuous calibration between local knowledge and the overall goal, which itself is a regulation of structural tension. For example, when Wiles was tackling Fermat's Last Theorem, he first knew from the number theory knowledge base (K) that a direct attack was difficult, so he turned to the idea of the Taniyama-Shimura conjecture (changing the goal under the guidance of the P layer), and then used the theory of modular forms and elliptic curves (advanced knowledge at the K layer) to carry out the proof. In this process, every strategic choice he made balanced the tension between existing knowledge and the final Purpose—introducing families of curves, Iwasawa theory, the method of infinite descent, etc., are all K→W mobilization; while boldly assuming that FLT can be implied by Taniyama-Shimura was P→W guidance. After these cross-layer leaps, Wiles finally constructed the proof, integrating Fermat's Last Theorem into the modular form knowledge system, and the semantic tension was declared resolved.

·I→W/P and W→I Feedback: Sometimes the knowledge system is not yet sufficient to support the proof, and new information or new concepts need to be introduced. This is reflected in the interaction between the I layer and the W/P layers: when attempts at the W layer are blocked, researchers may return to the I layer to seek more inspiration (e.g., calculating special cases, looking for patterns, using numerical experiments). I→W is to use new special examples, unusual phenomena to inspire proof conception. W→I means starting from a proposed proof framework, specifically to collect a certain type of data or counterexamples to verify the direction. If this information does not match expectations, it may prompt the adjustment of the Purpose or the abandonment of a certain plan. This cycle repeats until a feasible path is found.

Through the above various leaps, mathematicians, driven by semantic tension, continuously try self-correction and system reorganization, and finally may make the semantic tension tend to zero—that is, the conjecture is proven, and the knowledge closed loop is formed. Once the proof is completed, the original inter-layer rupture is connected, and the entire cognitive network is re-stretched into a harmonious and stable network. At this time, the former conjecture has become a stable node in the K layer, the proof path in the W layer is also well-known and verified by later generations, and the Purpose of the P layer has been realized or partially realized. The semantic tension model thus reveals: mathematical discovery is not just logical accumulation, but the achievement of a dynamic balance. When a problem is unsolved, the system is in a state of tension and imbalance. Through multi-level leaps and adjustments, a new balance is finally reached.

"BUG" Consciousness and the Conjecture-Tension-Proof Chain

Yucong Duan's Consciousness BUG theory provides a unique analogical perspective for us to understand the role of mathematical problems. This theory holds that the finiteness and imperfection (Bugs) of the cognitive system are, on the contrary, the source of stimulating advanced consciousness. Mapped to mathematics, the unsolved conjectures mentioned above are precisely the "Bugs" or blanks in the knowledge system: they are both flaws and potential breakthrough points. The "BUG consciousness" of mathematicians can be understood as the keen awareness and persistent tracking of these unsolved problems. This consciousness contains the following chain:

1.Discovering Conjectures (Awareness of Bugs): First, mathematicians realize a certain anomaly or absence—for example, finding that a certain property seems to always hold but no one has proven it, or a certain theory lacks an important example for a unified explanation. At this moment, a potential conjecture (Bug) emerges in the cognitive field. For example, Riemann noticed that the lack of regularity in the distribution of prime numbers was a big Bug, and the zeros of  seemed to be related to it, so he proposed the Riemann Hypothesis to "fill" this gap. Another example is the twin prime conjecture. Its Bug is: the fundamental theorem of arithmetic tells us that there are infinitely many prime numbers, but whether there are infinitely many prime pairs with a difference of 2 is unknown. Everyone can understand the problem, but no one can solve it, making it a gap in knowledge.

2.Maintaining Tension (Unresolved): Once a conjecture is proposed, if it remains unconquered for a long time, it exists as a Bug in the system for a long time. At this time, the mathematical community maintains a high "consciousness" of this: generations know that it has not been solved, and its importance and difficulty make it a collective focus. At this stage, the Bug plays a positive driving role: it stimulates countless attempts, forcing people to develop new methods. The Consciousness BUG theory points out that small deviations or Bugs are not purely negative disturbances, but rather promote the generation of the subject's consciousness (here, analogous to the overall wisdom of mathematics) by triggering higher-level integration and reflection. Correspondingly, a long-unsolved conjecture often spawns profound progress in the entire field. For example, to crack the Poincaré conjecture, the method of Ricci flow in differential geometry was invented and greatly promoted the development of geometric analysis; the difficulty of the twin prime conjecture promoted the innovation of sieve methods and analytic number theory techniques, and even welcomed Yitang Zhang's breakthrough on the existence of infinitely many prime pairs with finite gaps in 2013. It can be said that in the years when these Bugs existed, the mathematical system did not stagnate, but continuously emerged with new wisdom under the stimulation of tension.

3.Proof (Fixing the Bug): Finally, when the conditions are ripe, a certain (or a group of) mathematician finds a way to crack it, and the Bug is fixed—the conjecture is proven or disproven, and the semantic tension is released. For example, Perelman's proof of the Poincaré conjecture in 2003 finally ended this century-old Bug. The fixing of a Bug often brings a leap in the knowledge system: the originally isolated assertion is incorporated into a broader theory (such as Fermat's Last Theorem becoming one of the corollaries of the Taniyama-Shimura conjecture, which belongs to a large theory at the intersection of number theory and algebraic geometry), and the mathematical knowledge network thus becomes more rigorous and unified. At this time, the former Bug is transformed into a new tool: the methodologies and concepts developed during the solving process in turn enrich mathematics. For example, topology was deepened due to the proof of the Poincaré conjecture, and people have a deeper understanding of 3-manifolds and the structure of cosmic space.

4.New Bugs: Interestingly, the resolution of every major Bug often reveals deeper problems, like peeling an onion. Once accustomed to the prosperity brought by solving Bugs, mathematicians will soon turn their consciousness to the next Bug. This constitutes the upward spiral of mathematical development: there are always new conjectures appearing, bringing new tensions and challenges. For example, after Fermat's Last Theorem was solved, the number theory community immediately turned its attention to larger puzzles such as the Riemann Hypothesis and the BSD Conjecture; although the twin prime conjecture had Zhang's breakthrough, a complete solution is still on the way. It is foreseeable that even if the Riemann Hypothesis is finally proven, "extended Bugs" such as the generalized Riemann hypothesis for -functions will still follow. BUG consciousness ensures the continuous evolution of mathematics: problems drive research, research solves problems, and then new problems are generated.

In comprehensive terms, the "conjecture-tension-proof" chain shows: unsolved problems are precious, not only for the solution itself, but also because they train and enhance the entire mathematical cognitive apparatus through existence as value. This coincides with the insight of the BUG theory that imperfection brings progress. Therefore, mathematicians need to have "BUG consciousness"—to embrace the unknown and loopholes, and use them as opportunities for innovation rather than obstacles to stop.

Knowledge Compressibility and Semantic Closure Mechanism

Mathematics pursues conciseness and beauty, often expressing profound content in a highly compressed form. However, excessive compression may lead to extreme difficulty in proof. Behind this lies the semantic closure mechanism of the knowledge layer. So-called Knowledge Compressibility refers to the ability to represent a large number of specific cases or complex structures with a general proposition. A successful theorem is often a compression of numerous facts. For example, the sentence "Any simply connected, closed 3-manifold is homeomorphic to the 3-sphere" compresses the conclusion of the classification of infinitely many 3-dimensional shapes. This compression is possible because humans have found patterns or invariants that capture commonality at the knowledge layer. Compression is essentially a semantic refinement, reflecting our ability to understand and abstract.

However, when the compression degree of a proposition is extremely high (information is highly concentrated), semantic closure difficulties may occur: that is, the existing knowledge network cannot easily encompass it, and requires expansion or even leaps to close. For example, Fermat's Last Theorem condenses the "unsolvable" pattern into a concise statement. This statement, when proposed, was far beyond the scope that the knowledge system of the time could explain, leading to a long-term non-closure. In order to close it, the knowledge system had to significantly expand itself (develop algebraic geometry, homology theory, etc.) to cover this proposition. Another example, the Riemann Hypothesis reveals the profound connection between the statistical irregularity of prime numbers and the zeros of an analytic function in one sentence. The depth of this semantic compression is such that its proof has required the development of analytic number theory tools for the entire 20th century and is still not completely conquered.

Semantic Closure refers to the knowledge network incorporating a certain proposition and establishing stable connections with other knowledge, so that there is no more suspense. To achieve closure, the system may need to undergo structural reorganization and expansion. The final closure of Fermat's Last Theorem was completed under the grand framework of generalized modular forms and algebraic curves. The originally isolated proposition became a result of a higher-level theory, and thus was no longer isolated. The Poincaré conjecture is similar. It was integrated into the 3-manifold geometrization theory, making "3D = sphere" a part of a more universal truth, rather than just a special case.

It is worth noting that knowledge closure has unpredictability: before the proof, we often cannot know for sure in what posture the conjecture will be integrated into the knowledge system. Sometimes the proof reveals that the proposition is a corollary of a stronger proposition (e.g., Fermat's Last Theorem → Taniyama-Shimura conjecture), and sometimes a new theoretical framework is needed (e.g., the Riemann Hypothesis may require brand new physical or operator methods to prove). This reflects the creativity of semantic closure: it is not just filling in a link in a chain, but may add new nodes and new connections in the knowledge graph, reorganizing the structure to achieve global self-consistency.

In the process of semantic closure, compression and expansion complement each other: first, the proposition appears in a compressed form, causing tension; then the proof process is actually an expansion process—placing the proposition in a broader context to examine, importing related concepts, and even introducing auxiliary propositions and lemmas, to re-refine and expand the compressed information, and finally complete the explanation of the proposition through a long argument. The finally obtained theorem, although its expression form has not changed, has richer "neighbors" in the knowledge network. It is connected to many theoretical nodes, and its semantic meaning is also more substantial. This is like decompressing a compressed file and properly placing it in a system folder, so that the system can call it normally. The deductive proof in mathematics is precisely the tool to achieve this decompression and integration.

Deductive logic plays a key role here: the logical reasoning steps are both the proof and the operational path of semantic closure. Through a series of legal deductions, the new proposition is brought into contact with the existing axioms and theorems, and finally proven. This process actually constructs a path from the new proposition to the knowledge network. Therefore, we can say that the deductive logic chain is the specialized externalized path after the K-W-P structure stabilizes: when the Purpose, knowledge, and wisdom reach a dynamic balance, the proof comes naturally, and can be presented in a formal way for inspection. At this time, logic plays the role of "verification and communication," translating the internally achieved semantic consistency into formal language and unfolding it step by step. However, in the exploratory stage of problem solving, what really plays the main role is often semantic intuition and cross-framework association, and logic is just a reliable means to verify the final result. As many mathematicians have said, the process of discovering a proof is very different from the process of presenting it: the former is full of non-linear attempts and semantic understanding, while the latter is a linear deductive chain. Understanding the mechanism of semantic compression and closure can help us realize that the essence of proof is not only deductive correctness, but also a qualitative change of the semantic system from incoordination to coordination.

Semantic Reconfiguration Instances of Theorems and Conjectures

To illustrate the above theory more concretely, we select several classic theorems and conjectures and re-examine their structural characteristics and the sources of their challenges from a semantic perspective. Through these examples, we can see how semantic tension is hidden behind concise propositions, how the various layers of DIKWP play a role in them, and how the knowledge system finally achieves a closed loop through semantic leaps.

Fermat's Last Theorem: Semantic Tension from Extreme Compression

Fermat's Last Theorem (FLT) states: For any integer , the equation  has no positive integer solutions. This proposition reveals a profound fact with astonishing simplicity, yet it troubled the mathematical community for three and a half centuries. From a semantic perspective, Fermat's Last Theorem is a typical example of P→K extreme compression: Fermat, in 1637, wrote down the assertion and the hint of a "marvelous proof" based on intuition, which is equivalent to directly condensing the Purpose regarding the unsolvability of integer power sums into a knowledge proposition. However, this proposition, when proposed, was completely outside the K layer of number theory at that time—in the 17th century, there were neither conceptual tools nor proof methods to support it, leading to extremely high semantic tension. Analyzing its DIKWP semantic structure:

·D Layer: Fermat's Last Theorem involves basic arithmetic symbols and concepts, such as integers, powers, sums, and equality, which are purely elementary language. This is also its charm: even middle school students can understand its statement. The ease of understanding at the D layer conceals the deep difficulty, and also means that the proof may need to jump out of the existing elementary framework.

·I Layer: By comparing with the Pythagorean theorem ( case), people found that there seemed to be no solution for , which is a "discontinuity" in the pattern. Fermat noticed this difference and guessed that it holds generally. At the I layer, FLT captures the qualitative change in properties from  to , which is a typical example of "structural difference." For example, some calculations showing the non-existence of solutions for small exponents support the conjecture, but cannot prove the infinite case elementarily. This information difference became the soil for the conjecture space.

·K Layer: For a considerable period after it was proposed, FLT remained outside the knowledge system. Although number theory in the 18th-19th centuries confirmed many special cases (e.g., Euler proved , the method of infinite descent proved , etc.), there was a lack of a unified theory to prove it completely. FLT became a big Bug in the K layer: the proof of each special case required different techniques, and no self-consistent theory encompassed all . It was not until the second half of the 20th century, when algebraic number theory and the theory of elliptic curves flourished, that the possibility of incorporating FLT into a grander knowledge framework emerged. Eventually, Wiles and others established a connection between FLT and the Taniyama-Shimura Conjecture. The Taniyama-Shimura conjecture is a grand correspondence between elliptic curves and modular forms, which was proven to imply FLT as a corollary. When Wiles proved a sufficiently general case of the Taniyama-Shimura conjecture, FLT was also proven, embedded in a new knowledge closed loop. At this time, the status of FLT in the K layer changed from an isolated conjecture to a special result of modular form theory, and the knowledge system achieved closure. It is worth noting: the Taniyama-Shimura conjecture itself involves deep knowledge of modern algebraic geometry, which is no longer in the knowledge category of Fermat's time. In order to absorb FLT, the knowledge layer had to undergo huge expansion and integration.

·W Layer: The process of proving FLT can be called an epic at the wisdom level. Countless mathematicians in history have made W-layer efforts: from Fermat himself claiming to have a "marvelous proof" but not leaving it behind, to Euler, Sophie Germain, etc. proving some cases, and then to Kummer introducing ideal numbers to try a general solution. These are all iterations of W-layer activities. Finally, Wiles's proof synthesized the crystallization of wisdom from algebraic geometry and number theory: he used advanced techniques such as Lie algebras, homology groups, level-lowering of elliptic curves, and deformations of infinite Galois representations to crack the key special case of the Taniyama-Shimura conjecture. The entire proof is over a hundred pages long and requires extremely high intellectual arrangement, and is hailed as one of the most complex proofs in mathematical history. The difficulty of the W layer lies in: there is no ready-made paradigm to follow, and new theoretical bridges must be creatively constructed. Wiles's proof had an error when first submitted, and was later repaired with the joint efforts of Taylor, which also reflects the twists and turns of W-layer exploration. The final published proof became a standard deduction in the eyes of later generations, but the process of its invention condensed the cumulative leap of number theory wisdom of several generations.

·P Layer: The deep attraction of Fermat's Last Theorem lies to a large extent in its strong challenge and beauty at the Purpose layer. First, as the last unproven Fermat's conjecture, it carried a sense of historical mission, and mathematicians had the obsession to "must prove it." Bell's book called it "the last problem," and even predicted that civilization might end before it was solved, which shows its status at the Purpose layer. Second, the statement of FLT is concise but unexpected. This counter-intuitive characteristic (equations of higher powers are more strictly unsolvable than squares) inspires the pursuit of mathematical beauty. Many people believe that such a beautiful proposition should have a beautiful proof, which itself is an aesthetic driving force at the Purpose layer. Third, Fermat's claim to have a marvelous proof but not revealing it seems to have left an intellectual legacy for later generations to discover, inspiring generations of Purpose to uncover the mystery. It can be said that Fermat's Last Theorem has become a mythical existence, driving the career ideals of countless mathematicians (Wiles's determination to crack this problem since the age of 10 is an example). This powerful Purpose-driven traction kept the conjecture alive and well until the mission was completed.

Semantically, the difficulty of Fermat's Last Theorem is that its extremely simple statement contains an extremely rich structure, requiring the integration of ideas across fields to analyze. It connects arithmetic, algebra, geometry, and analysis (bridging number theory and analysis through the Taniyama-Shimura correspondence). Such a semantic span means that the tension of a single-domain knowledge is difficult to eliminate, and a semantic leap to a broader unified perspective is necessary. This also explains why it was solved only in modern times: mathematics did not mature and intersect in related fields until the end of the 20th century, and the nodes of the semantic network were dense enough to finally weave a path to the solution. The completion of the proof of Fermat's Last Theorem not only solved one problem, but also marked that the mathematical semantic system had reached a new height of integration.

Riemann Hypothesis: The Tension Band in Frequency Fluctuations

The Riemann Hypothesis is known as the "king of conjectures." Its content involves an amazing connection between prime number distribution and the zeros of a complex function: all non-trivial zeros of the Riemann  function should have a real part of . Intuitively, this means that the "frequency" fluctuations of the appearance of prime numbers have a certain perfect symmetry and regularity. Riemann proposed this conjecture in 1859, and it remains unsolved to this day, a jewel in the crown of mathematics. From a semantic perspective, the difficulty of the Riemann Hypothesis is that it attempts to fuse randomness and determinism, compressing the chaotic distribution of prime numbers into the orderly structure of an analytic function, forming a highly taut semantic tension band:

·D Layer: The raw data involved in the Riemann Hypothesis includes the sequence of prime numbers (2, 3, 5, 7, ...) and the definition of the Riemann  function. The distribution of prime numbers seems to have no rules and belongs to basic number theory data; while the properties of  in the complex domain belong to basic objects of analysis. Data from two completely unrelated fields collide here. In the mid-19th century, people had verified that many non-trivial zeros of  fall on the  line, but this was computational data. The statistical error of the prime-counting function  also served as data to support Riemann's idea. Therefore, the D layer provides a series of numerical and functional graph hints of a certain correlation.

·I Layer: Riemann's key insight was to discover that the frequency of prime number appearance is closely related to the distribution of the zeros of . This is a cross-domain pattern correspondence, which is an extraordinary piece of information at the I layer. Before the Prime Number Theorem was proven, Riemann transformed the problem of prime counting into a problem of the zeros of  through analytic continuation and the Euler product, which itself was an I-layer feat of identifying structural differences and establishing connections. The Riemann Hypothesis asserts that all zeros are on the line , which is equivalent to believing that the error term of the prime number distribution exhibits extremely regular oscillation frequencies. This translates the "irregularity" of prime numbers into the "neatness" of zeros, which is a filling of the gap at the pattern discontinuity: the original prime number distribution had no concise pattern, but the conjecture assumes a hidden spectral order. This is clearly a great refinement and assumption of I-layer information.

·K Layer: After being proposed in 1859, the Riemann Hypothesis was quickly regarded as the holy grail of number theory. Some partial results, such as the Prime Number Theorem (proven in 1896), relied on assuming the Riemann Hypothesis was true. It can be said that the Riemann Hypothesis has become an important pillar conjecture in the fields of number theory and analysis: many profound propositions are known to be true under its premise. However, it has never been incorporated into the knowledge base as a theorem. After much effort, mathematicians have established a broad theoretical foundation related to it: such as the development of analytic number theory (the equivalence conditions between the distribution of zeros and number theoretic results), the system of propositions equivalent to the Riemann Hypothesis, random matrix models, etc., all enriching the K-layer content. Despite this, the conjecture itself remains unresolved, and the K layer is still missing a crucial piece of the puzzle. It can be seen that the Riemann Hypothesis has triggered the expansion and deepening of the knowledge layer—whether it is the Prime Number Theorem, the generalized Riemann hypothesis for elliptic curves, or the richly rewarded Clay Millennium Prize, all reflect its core status in the knowledge system. To date, people have verified that the first tens of billions of zeros all conform to the conjecture. The empirical support at the K layer is extremely strong, but the lack of a rigorous proof makes the system not closed. The Riemann Hypothesis is clearly one of the last fortresses of the K layer.

·W Layer: Over the past century and a half, attempts to prove the Riemann Hypothesis have been endless, which can be described as a historical epitome of wisdom. In the early days, there were Riemann's own analysis of the distribution of zeros, and attempts by Burnside, Dedekind, etc. In the 20th century, a series of propositions equivalent to the Riemann Hypothesis were introduced, providing alternative attack routes at the W layer, including: the equivalence of the distribution of zeros and the error term of the prime number distribution, certain inequalities with Fourier transforms, etc. Hilbert and Pólya once envisioned that proving the conjecture would require finding a self-adjoint operator whose spectrum gives the zeros of . This became the Hilbert-Pólya conjecture, providing a possible direction for the W layer (translating the problem into spectral analysis). On the other hand, attempting proof by contradiction and looking for contradictory results of zeros deviating from the  line was also a line of thought, but it was not successful. In modern times, random matrix theory surprisingly revealed that the spacing of high-order zeros conforms to a certain random spectral distribution, which suggests that the zeros of  seem to be related to some unknown quantum system. Tao and others are studying the results of assuming zeros deviate, and whether abnormal zeros exist. These efforts all reflect the multi-track advancement of the W layer: analysis, algebra, geometry, and random methods are all used, but none has yet been effective. This shows the high difficulty of the W layer of the Riemann Hypothesis, which requires the integration of wisdom across fields. Perhaps new mathematical ideas or even physical principles are needed to break through the current bottleneck. It is precisely because of this that the unsolved status of the Riemann Hypothesis itself has spawned many new theories (such as the development of the entire analytic number theory, L-function theory), and W-layer exploration and K-layer new knowledge complement each other.

·P Layer: The profound meaning of the Riemann Hypothesis gives it a special position at the Purpose layer. On the one hand, it is related to the ultimate answer to the mystery of integers: prime numbers are called the atoms of number theory. Grasping the laws of prime number distribution is like understanding the mystery of the structure of natural numbers. This is an incomparable attraction for mathematicians. From Hilbert listing it as the 8th problem to the Clay Millennium Prize offering a million-dollar reward, all reflect the unwavering will of the mathematical community to conquer it. On the other hand, the beauty of the Riemann Hypothesis lies in its elegant connection of seemingly unrelated fields. This pursuit of unity itself conforms to the desire for harmony and order in mathematical Purpose. Just imagine, if random prime numbers really come from a symmetrical spectrum, what deep harmony the mathematical world will present! This ideological beauty has inspired countless people. Furthermore, the truth or falsity of the Riemann Hypothesis also affects the fate of a large number of other conjectures, and its weight is fascinating and awe-inspiring. Many mathematicians believe that the Riemann Hypothesis "should" be true. This belief itself is a P-layer preference and driving force. This is not entirely blind faith, but stems from a belief in the overall beauty and unity of mathematics: if the Riemann Hypothesis were not true, many existing structures would collapse or at least become strange and unacceptable.

Semantically, the Riemann Hypothesis creates a tension band across number theory and analysis: one end is the immeasurable chaos of prime number distribution, and the other end is the precise arrangement of the zeros of an analytic function. To prove it, a bridge mechanism needs to be found in mathematics that can both explain the irregularity of prime numbers and prove that the regularity is hidden in the symmetry of . This challenge is far greater than Fermat's Last Theorem, because the latter "only" involves arithmetic itself, while the Riemann Hypothesis involves a wider range. From a semantic level, it may be necessary to introduce new semantic elements—for example, introducing the concept of time evolution through physical analogy (Hilbert-Pólya's vision), or understanding -functions through a higher, more general algebraic framework. Current mathematical knowledge has not fully covered this cross-domain connection. Therefore, the Riemann Hypothesis continues to exist as a Bug, constantly raising our cognitive level with its existence. If one day a proof appears, it will surely mark a deep fusion of the mathematical semantic network, just like two long-separated continents are finally connected by a bridge. At that time, the former tension band will become a stable avenue.

Poincaré Conjecture: The Uniqueness Hypothesis of Topological Closed Loops

The Poincaré Conjecture asserts: Any simply connected, closed 3-manifold is homeomorphic to the 3-sphere. A simple analogy: if a 3-dimensional space has no holes and is finite, then it is a 3D-sphere, just as the surface of an orange has no holes and can be shrunk to a point. This conjecture has a fundamental meaning in topology. It was proven by Perelman in 2003, becoming the first of the Millennium Prize Problems to be solved. Semantically, the Poincaré conjecture explores the semantic closed loop of topological space: the structural uniqueness of 3D simply connected closed manifolds. Its difficulty and significance can be understood from the DIKWP level as follows:

·D Layer: Involves the basic concepts and objects of topology, including "3-manifold," "simply connected," "closed," "sphere," etc. These definitions sound simple: a 3-manifold is a space that is locally like , simply connected means there are no small looped holes that cannot be shrunk, and closed means finite and without boundary. The sphere  is the classic model. The D layer also has some specific example data, such as known 3D space models: ,  (a generalization of a torus), etc. In his early years, Poincaré himself constructed a "Poincaré homology sphere" as an attempt at a counterexample, but it had a non-trivial fundamental group and was not a true sphere. Instead, it inspired the conjecture itself. So the D-layer information shows: to determine whether a shape is a sphere, deep topological invariants (the fundamental group) need to be considered. The Poincaré conjecture is precisely the problem refined from these D-layer clues.

·I Layer: The Poincaré conjecture focuses on a topological difference: in the 2D case, it is known that every closed surface without holes is a sphere (proven by topological classification), the 4D and higher dimensions were unknown at the time, but it was later found that high dimensions are easy while 3D is stuck. When Poincaré proposed the conjecture, his I-layer motivation came from: in his work of determining a sphere using homology theory, he found that homology conditions were not enough, and the fundamental group condition was needed. So the conjecture was proposed with trivial fundamental group as a necessary and sufficient condition. That is, believing that "no holes" (trivial fundamental group) is sufficient to determine a sphere. The extension of this pattern is natural: true in low dimensions, conjecture it holds in high dimensions. At that time, he could not prove the 3D case, and the conjecture was born. The I layer is also manifested in the discovery of comparing topological classifications in different dimensions: 5D and above were quickly proven by Smale, and 4D was later solved by Freedman, while 3D was left to the last. This shows the peculiarity of 3D. At the I layer, 3D became a "weird point" of pattern discontinuity. This difference aroused great interest in the topology community: why is 3D so special? The difficulty of the conjecture lies here.

·K Layer: The importance of the Poincaré conjecture in the K layer of topology is undoubtedly extremely high. It is regarded as one of the fundamental problems of topology, like "the Yangtze and Yellow Rivers in geometry" (Shing-Tung Yau's words). For more than a century, it has driven the development of topology. Many topological invariants and techniques were born to serve the purpose of cracking this conjecture, such as Dehn surgery, Heegaard splitting, and other 3-manifold topology techniques. The K layer gradually accumulated rich partial results and equivalent forms: for example, Thurston's Geometrization Conjecture included Poincaré as a special case in a broader classification. In the 1960s, it was surprisingly found that the generalized Poincaré conjecture in high dimensions () was easier to prove (Smale's h-cobordism theorem). This made the K-layer situation very special: high dimensions OK, low dimensions in suspense, and 3D became the last fortress. In the 1980s, Thurston proposed the 3-manifold geometrization program, decomposing 3-manifolds into eight types of geometric pieces. The Poincaré conjecture corresponds to one of these pieces, which can only be spherical. This grand conjecture gave the K layer a clear direction. Finally, Perelman published papers in 2002-2003, realizing the geometrization conjecture using the method of Ricci flow. After verification, the Poincaré conjecture was proved. He was awarded the Fields Medal (which he declined) and the Clay Millennium Prize. Since then, the classification of 3-manifold topology has been settled, the K-layer knowledge has reached a closed loop, and topology has since entered a new era. It is worth mentioning: the solution to the Poincaré conjecture introduced new methods of analysis and geometry (differential equation flow), which is an example of the K layer cross-border integration in order to close the 3D sphere problem.

·W Layer: The proof process of the Poincaré conjecture is a twists-and-turns display of creativity. In the early days, "proofs" were announced and then overturned multiple times. The difficulty of 3-manifold topology is the lack of surgical and simplification means of high dimensions. W-layer attempts in the 1930s-50s mainly focused on constructive topological methods, such as the results of Papakyriakopoulos, but still did not get the point. It was not until Richard Hamilton introduced the idea of Ricci flow in the 1980s, gradually smoothing the shape of the manifold, that a brand new W-layer path was provided: to see the shape evolve by solving partial differential equations. If the shape is not a sphere, the Ricci flow may develop singularities, and analyzing the forms of these singularities may lead to a contradiction. Although Hamilton laid the framework, he encountered difficulties in analyzing singularities. Perelman's contribution at the W layer was to overcome these obstacles: he introduced an entropy-increasing function and surgical techniques to ensure the Ricci flow could be continued, and proved that the manifold eventually "rounds" completely into a sphere through superb analysis. This proof used a lot of knowledge such as differential geometry, partial differential equations, and topology, and was extremely complex but beautiful in its ideas. It is worth noting that Perelman's proof was not a one-step traditional topological argument, but a hybrid product of analysis + topology, reflecting cross-layer wisdom: injecting geometric evolution (new I-layer information) into the topological problem, breaking through the proof path. Finally, after Huai-Dong Cao and Xi-Ping Zhu completed the details and verification, the W-layer proof work was successfully completed. Perelman's choice to withdraw from the mathematical community also became a good story.

·P Layer: The reason why the Poincaré conjecture has attracted top mathematicians for a long time is that its meaning at the Purpose layer is significant. First, it is related to the understanding of the shape of the universe: 3-manifolds can be regarded as models of the 3D universe, and the conjecture says that if the universe has no holes, it must be a deformation of the 3-sphere. This inspires physical and philosophical curiosity. Second, it is equivalent to a fundamental problem in the field of topology: just like classifying the periodic table of elements, classifying all 3D shapes is the crown achievement of topology. Mathematicians are naturally unwilling to leave this matter unresolved. Furthermore, in the context that high dimensions have been resolved, completing the last piece of the puzzle for 3D has a strong Purpose of completing the entire topological puzzle. Every topologist regards cracking the Poincaré conjecture as the pinnacle of their career. Indeed, the solver Perelman is regarded as a hero for this. His refusal of the award further highlights his pure pursuit of truth itself. In general, the P-layer Purpose of the Poincaré conjecture includes the confirmation of uniqueness, the desire for comprehensive classification knowledge, and a determination to not give up until the goal is achieved. At the national level, it even became a symbol of honor, such as the contributions made by the Chinese mathematician Zhu/Cao team in perfecting the proof, which was also rendered as a major victory by the domestic media.

The entire process of the Poincaré conjecture perfectly interprets the semantic mathematics framework: a simple topological statement leads to a series of profound concepts (fundamental group, Ricci flow, etc.), and is finally proven through the cross-domain integration of the knowledge network. The semantic tension in it is manifested as the confusing peculiarity of 3D, and the solving process, by introducing the new continuous parameter (time flow) thinking, achieved the continuous cracking of the original discrete topological problem. This implies that solving high-tension problems often requires introducing additional semantic dimensions (such as the concept of time evolution here), so as to transform the nature of the problem. The proof of the Poincaré conjecture marks the completion of a major closed loop in the topological semantic system, and also shows that humanity has taken a big step forward on the road to understanding the structure of space.

Twin Prime Conjecture: The Minimal Tension Path of Prime Pairs

The Twin Prime Conjecture asserts: There are infinitely many pairs of primes  such that . That is, apart from (2,3), there are infinitely many prime "twins" such as (3,5), (11,13), (17,19)... This conjecture is

easy to understand and is part of Hilbert's 8th problem, and it remains unproven to this day. However, in 2013, Yitang Zhang proved that there are infinitely many prime pairs with a gap less than 70 million. The Polymath project reduced this upper bound to 246, which is much closer to the twin prime conjecture but has not yet fully reached a gap=2. Semantically, the twin prime conjecture studies the local proximity relationship in the prime number sequence, and can be regarded as a hypothesis of low-tension connection in the distribution of prime numbers. Analyzing its semantic structure:

·D Layer: The D-layer elements of the twin prime conjecture are specific prime numbers and prime pairs, as well as the property of having a difference of 2. A large amount of computational data shows that twin primes appear frequently in the range of small numbers up to hundreds of millions, but gradually become rarer. Brun has proven that the harmonic series of twin primes converges (i.e., the problem of whether the series diverges or not due to a finite number of twin primes, the result shows that twin primes should be infinite, otherwise the series convergence and divergence would be contradictory). These numerical facts and shallow results form the D-layer basis.

·I Layer: At the information layer, prime pairs  are striking as a special pattern. Euler and others conjectured very early that this type of pairing is infinite. The core of the I layer is the local fluctuation of prime number distribution: the overall distribution of prime numbers is sparse but irregular, yet it still allows two prime numbers to appear close together at times. Why is it impossible that after a certain point, there are no more twins? Intuitive information tends to believe that prime numbers "can still be close to each other occasionally." Furthermore, the Hardy-Littlewood second conjecture (H-L second conjecture) on the distribution of prime pairs even gives an asymptotic formula for the frequency of twin primes. This shows that the information layer believes: twin primes are not only infinite, but their asymptotic number can also be quantitatively described. Although the H-L conjecture is not proven, a large amount of numerical verification supports it. Therefore, the I layer provides a strong hint: prime numbers should locally show the pattern of a difference of 2 infinitely many times.

·K Layer: The twin prime conjecture is located in one of the core areas of the number theory knowledge network. It, along with the Riemann Hypothesis, Goldbach's Conjecture, etc., is regarded as the last fortress of the prime number puzzle. The main tools mastered by the K layer are sieve methods, analytic number theory methods, etc. Although it is not yet possible to prove the twin prime conjecture, the K layer has made partial progress: for example, Jingrun Chen proved in 1973 that there are infinitely many prime pairs with a difference of at most a certain even number (e.g., 2, 3, 5, 7, 11), which initiated the study of the "weak twin prime conjecture"; and Yitang Zhang's breakthrough in 2013, which reduced the "finite gap" to a specific number, 70 million, for the first time. These achievements have enriched the K-layer knowledge, gradually "closing in" on the conjecture. Polymath further worked together to reduce the gap value. Currently, the K layer can prove: there are infinitely many prime gaps less than or equal to 246. This means that the knowledge layer has almost proven the twin prime conjecture, only lacking the reduction of 246 to 2. This gap still has huge difficulty, but the K layer can be said to be on the verge of victory, and the semantic closed loop is almost complete. The twin prime conjecture is also related to the weak form of the Riemann Hypothesis (such as the Bombieri-Vinogradov theorem, etc.). Some results assume that the GRH can deduce approximate results of the twin prime conjecture being true. In general, the K layer's understanding of the twin prime conjecture is much deeper than that of the Riemann Hypothesis, and what is lacking may just be the final overcoming of technical difficulties. It is no longer like a completely isolated conjecture, but more like a rough piece of knowledge waiting to be polished.

·W Layer: Attempts to prove the twin prime conjecture are mainly carried out within the framework of sieve methods in analytic number theory. Classic large sieve and small sieve techniques have been gradually improved over the years: the Erdős–Rankin result in the mid-20th century showed that there are infinitely many prime gaps that are much smaller than the average gap; subsequently, double linear sieve, triple sieve, etc. were developed. Chen's method introduced the "almost prime" concept, proving that there are infinitely many solutions when  has at most two prime factors (i.e., infinitely many  such that  or  is prime)—this is actually a proof of the "weak twin prime conjecture." Yitang Zhang's breakthrough came from adopting the basic sieve architecture of the Goldston–Pintz–Yıldırım (GPY) trio and introducing new even elements, and multi-complex variable analysis techniques, to press down the calculation of the correlation value that needed to be verified. The important concept of his work is to use "smooth numbers" to handle, which can handle large prime factors in the sieve method. The Polymath project, on the other hand, pooled wisdom to improve various links. These advances in the W layer reflect progressive cumulative wisdom: each generation improves the sieve method, introduces new technologies, and compresses the difference step by step. It should also be mentioned that some people have tried non-sieve methods, such as considering expanding the framework of the Riemann Hypothesis or arithmetic geometry methods, but at present, it still mainly relies on sieve methods. The W layer has not completely eliminated the gap, which means there are still difficulties. But from the historical trend, this is a steady convergence process, as if we are climbing up the slope of the semantic tension gradient, constantly approaching the target peak.

·P Layer: The attraction of the twin prime conjecture at the human Purpose layer lies in its approachable statement and profound meaning. As one of the most simple prime number conjectures, it has great attraction for both professionals and non-professionals: anyone can understand the problem of two prime numbers differing by 2, but solving it is so difficult, creating the charm of an intellectual challenge. For mathematicians, proving the twin prime conjecture means thoroughly grasping the local limit behavior of prime number distribution—this is the holy grail in number theory. Overall, prime numbers are irregular, but if the twin prime conjecture is confirmed, then an eternal thin line is found in the irregularity: at least the proximity of a difference of 2 appears infinitely many times. This undoubtedly satisfies the mathematical Purpose's desire for discovery of order. More practically, the solution of the twin prime conjecture will prove Hardy-Littlewood's distribution formula, verify the limit capabilities of tools such as sieve methods, and provide a reference for other parts of number theory. To some extent, it is the ultimate test of the energy of analytic number theory. Therefore, the number theory community is determined to win it. The story of Yitang Zhang's rise has cast a legendary color on this conjecture, inspiring many successors to devote themselves to it. The P-layer driving force of the twin prime conjecture is also that it is closely related to the Riemann Hypothesis, etc. Solving one often leads to a string of results, so its significance is not limited to the conjecture itself, but to the whole of number theory.

Semantically, the twin prime conjecture explores the "minimal tension channel" in the set of prime numbers: the closest distance between two prime numbers is 2, which can be seen as a relaxation allowed by the internal structure of prime numbers (much smaller than the general average gap). The conjecture claims that no matter how the prime number gap increases with the whole, there are always some prime pairs that maintain this minimum distance of 2 and continue to infinity. If prime numbers are seen as random points, the infinite number of pairs with a difference of 2 indicates that this "coupling" structure appears repeatedly, and there is no complete dissipation caused by entropy increase. This is like finding infinitely many low-entropy islands in an entropy-increasing system, which is full of metaphorical meaning. Solving the twin prime conjecture requires a thorough understanding of the source of the correlation between prime numbers, which is equivalent to understanding the properties of  near the critical line. This actually has a certain internal connection with the Riemann Hypothesis. The semantic picture may need to see prime numbers as a network rather than independent points to explain the infinite continuation of the twin phenomenon. As sieve methods get closer and closer, perhaps the final proof is not far away. Once the proof is completed, the cognition of prime number distribution will step up to a new level. We will confirm that in the prime number world with extremely large information entropy, there still exists a micro-line of order, which is undoubtedly a great consolation for mathematics's understanding of nature.

Classification of High-Dimensional Spheres: Dimensionality Increase and Tension Resolution

In addition to the specific conjectures mentioned above, it is worth mentioning the classification problem of high-dimensional spheres. This problem can be regarded as the generalization of the Poincaré conjecture to higher dimensions: for an -dimensional (  ) simply connected closed manifold, is it homeomorphic to the -dimensional sphere ? Historically, this was called the Generalized Poincaré Conjecture, and its solution surprisingly varied by dimension: 5 dimensions and above were proven by Smale in 1961, 4 dimensions were solved by Freedman in 1981, and 3 dimensions were not completed until 2003 by Perelman. This reveals an abnormal phenomenon: the higher-dimensional problems were solved first, and the low dimensions were more difficult. What does this mean semantically?

From the perspective of semantic tension, some difficulties in high-dimensional topology become easier to handle in high dimensions, because high dimensions provide more space for deformation and generalization methods, thus reducing the tension. Smale used the handlebody theory of high dimensions () to avoid the obstacles of 3 and 4 dimensions, achieved topological surgery, and proved the high-dimensional Poincaré conjecture. This shows that in high dimensions, the knowledge K layer may be more complete, and the tools (such as the h-cobordism theorem) are more powerful, resulting in less semantic tension. The 4-dimensional case required the invention of special techniques such as the Casson handle. Freedman introduced a simple homology structure, and finally conquered it. However, 4 dimensions still have special complexities in concept and technology compared to 3 dimensions (e.g., smooth structures have singularities in 4 dimensions). Therefore, 3D became the last fortress.

This phenomenon can be explained as: Semantic complexity is not completely monotonic with dimension. Although objects in high dimensions are more complex, methods such as induction, analogy, and dimension reduction can be used, making it easier to form a closed loop; while in low dimensions, because there is no higher degree of freedom, many means fail, and brand-new ideas are needed. Semantically, the high-dimensional results coming out first made people believe that 3D should also be true (consistent with the expectations of the Purpose layer), but the long-term inability to prove 3D formed a huge tension. This tension was only released by introducing new semantic carriers from across fields (such as Ricci flow analysis). This again confirms: When the existing semantic network cannot be closed, it may be necessary to jump to a higher-dimensional semantic space to solve it. In the Poincaré problem, time was added as the "4+1st dimension," giving the 3D problem a continuous flow dimension, and it was solved.

In addition, high-dimensional spheres also lead to the classification problem of differential structures: Milnor discovered in 1956 that  has different smooth structures, i.e., "exotic spheres." This result shocked the topology community, because topologically equivalent spaces can have different differential structures. Kervaire and Milnor further classified the number of smooth structures on high-dimensional spheres, expressing it with algebraic K-theory. This phenomenon tells us: when the semantic requirements are stricter (such as considering differential information instead of just topology), the knowledge closed loop is broken again, and new tension appears. The discovery of exotic spheres spawned a new field of differential topology, and also shows that our understanding of space is constantly being refined and deepened. Semantically, the existence of exotic spheres means that objects that were thought to be simple actually hide complex semantic distinctions, requiring a more precise classification system (this is equivalent to the I layer discovering new differences in the DIKWP model, and the K layer having to expand its classification knowledge). Currently, the differential structure classification of 4-dimensions still has unsolved mysteries (e.g., whether the 4-sphere has an exotic structure is not yet determined), which is one of the major current BUGs in topology.

In short, the study of the classification of high-dimensional spheres shows the different behaviors of the mathematical semantic network as the dimension and structural requirements change: sometimes higher dimensions reduce tension and make it easier to prove, and sometimes additional structures introduce new tensions to challenge cognition. This reminds us that semantic tension is not a simple function, but depends on the complexity and maturity of the semantic network in which the problem is located. Mathematicians continuously adjust models at different semantic levels (topological/smooth) and different dimensions, looking for ways to balance and close the loop. This process has greatly enriched the knowledge K layer of topology, and has also deepened the understanding of the meaning of "what is a sphere." From the perspective of semantic mathematics, this is an attempt at multiple semantic closed loops for the same concept in different contexts, and each challenge promotes the overall cognition.

Constructing a "Semantic Mathematics" Theoretical Framework

Combining the above analysis, we preliminarily draw the theoretical framework prototype of Semantic Mathematics. This framework aims to quantitatively and qualitatively characterize the provability of mathematical theorems, structural stability, etc., in semantic space, so as to transcend the traditional practice of measuring difficulty only by logical complexity. The core ideas include:

·Provability Measure Function: Introduce a function based on semantic complexity and tension compression to measure the "provability" of a conjecture. Semantic complexity can be measured by indicators such as the conceptual hierarchy involved in the proposition, the cross-domain scope, and the number of new concepts required; tension compression refers to the degree to which the proposition compresses information (e.g., the breadth of covering infinite cases) and its deviation from existing knowledge. These factors can be combined into an indicator to roughly estimate the difficulty of proving the theorem and the new resources needed. For example, we might give a proposition like Fermat's Last Theorem a high compression ratio (simple statement but covers infinite integer conditions) and a high cross-domain degree (the proof introduces modular forms), so the provability measure is low (corresponding to high difficulty); while for a technical lemma, both compression and cross-domain degrees are low, so the provability is high (easy to prove). Although this measure is difficult to be precise quantitatively, it can be used as a conceptual tool to help understand why some conjectures are more difficult than others, and can also be used for AI systems to select suitable goals among many unsolved problems (e.g., prioritizing solving problems with slightly lower semantic tension).

·Semantic Closed-Loop Perspective of Mathematical Structures: We propose to regard mathematical structures (such as a theoretical system or a theorem network) as stable closed entities in an "information-tension-semantic system." That is to say, a mature mathematical theory is a self-consistent collection of several basic semantic elements connected by axioms. All internal tensions have been resolved or limited to a controllable range, so that no contradictions or undecided propositions will be produced. Conversely, a theory with huge internal tension is unstable, indicating that it needs to be expanded or reformed. For example, the shaking of the foundations of arithmetic at the end of the 19th century (tension) led to the reconstruction of set theory and axiomatic systems in the 20th century, which was to re-close the semantic structure. The resolution of every unsolved conjecture also makes the relevant theory more stable and complete. Therefore, stability can be used as a measure of theoretical maturity, and stability depends on whether the internal tension has been resolved. This view can also explain the phased nature of mathematical development: when the main tension points in a field are resolved (such as the approximate solution of the three-body problem in celestial mechanics in classical mechanics), the field enters a relatively stable period; new breakthroughs often come from discovering new sources of tension (such as the introduction of quantum views to challenge classical theories). In the history of mathematics, various disciplines have continuously created-resolved tension, promoting theoretical replacement and integration.

·Semantic Interpretation of Deductive Logic: Traditionally, logic is regarded as a formal process unrelated to semantics, but in the semantic mathematics framework, we see that logical reasoning is actually a representation of semantic stability. When a certain conjecture is integrated into the knowledge system through proof, its proof process can be seen as the trajectory of semantic tension gradually decreasing to zero. The logical steps strictly ensure that the tension does not rebound at each step (i.e., no new contradictions are introduced), and finally a closed loop is reached. Therefore, the deductive reasoning chain can be understood as the projection of the K-W-P structure after it stabilizes: when the proof is completed, the knowledge K layer and the Purpose P layer are coordinated and consistent, and the exploration of the wisdom W layer has converged to fixed steps. At this time, the reasoning can be displayed in a purely formal way. Conversely, when it has not yet stabilized, logical attempts are often interrupted (the proof cannot proceed) or diverge (run into contradictions), which corresponds to the semantic structure not being closed. Therefore, the success of deductive logic itself is a signal of semantic consistency. With this cognition, we can also better understand the difference between "machine proof" and "human proof": machines can verify the correctness of a given reasoning, but discovering a new proof requires perceiving semantic tension and finding ways to resolve it, which is difficult for purely formal methods to complete automatically. The semantic mathematics framework strives to bring this human intuition into model-based discussion, so that the discovery process of proof is no longer a black box.

·Semantic Space Graph Model: Formally, one can imagine taking mathematical concepts and propositions as nodes, and the reasoning relationships and semantic associations between them as directed edges, forming a huge directed graph. In this graph, the proof of a theorem corresponds to a directed path from the axiom node set to the target proposition node. A closed loop of a theory means that there are multiple loops covering the relevant node set (axioms returning to themselves through deduction). A conjecture is manifested as a node that has not yet been connected to the axiom network. Semantic tension can be represented by some disconnected or weakly connected situations in the graph: for example, the conjecture node only has a few long-distance connections with the axiom subgraph, or is reached through many intermediate nodes, indicating that the proof requires a long chain. We can even define a measure of Tension = Distance: the longer the "shortest path length" between the conjecture node and the known network, the greater the tension. The proof process is the process of shortening the distance on the graph, by introducing new nodes (lemmas) or new edges (establishing new connections) to achieve a finite and reachable distance. The final success of the proof means that the distance has become a finite and specific path, and the conjecture node is merged into the main graph. This graph model is different from traditional mathematical knowledge graphs in that it emphasizes the unconnected parts and the difficulty of connection. Through this model, the distribution of "tight clusters" and "loose ends" on the map of mathematical knowledge can be intuitively visualized, thereby identifying the weak links in the overall research, and also providing support for predicting which conjectures may be solved first (those nodes that are already quite related to the main network may be easier to conquer).

In summary, the semantic mathematics framework aims to bridge the rigor of symbols and the intuition of semantics, providing new tools for understanding mathematical discovery and proof. It does not deny the role of traditional deductive logic, but gives it a background meaning, examining mathematical activities in the broader vision of cognitive science and information science. From this panoramic perspective, mathematics is reduced to a special "semantic compression" game for humans to understand the world: we continuously find patterns, compress descriptions, and then use logic to expand and verify, ensure the compression is correct, and then repeat this process. Semantic mathematics is precisely the abstraction and generalization of this process.

Modelable Suggestions and Philosophical Reflections

Simulation of Theorem Evolution Based on Semantic Models

The semantic mathematics framework notG only provides descriptive tools, but also inspires us to build computational models to simulate the process of mathematical discovery. A possible bold idea is a "Semantic Theorem Evolution System": to realize the mechanism of conjecture generation and proof evolution in the DIKWP semantic space in the form of a computer program. Its key points include:

1.Knowledge Graph Initialization: Using existing mathematical knowledge as the initial K layer, construct semantic network nodes and connections (can start from existing databases of theorems, definitions, and relationships, such as the Mizar mathematical knowledge base). This graph marks which nodes have been proven (stable nodes) and which are conjectures (hanging nodes). The semantic attributes of the nodes are also marked (field they belong to, related concepts, etc.).

2.Tension Detection Module: The program finds high-semantic-tension areas in the knowledge graph, i.e., unsolved conjectures or weak chains, according to certain heuristics (such as the "Tension = Distance" or semantic complexity indicators mentioned above). Select the target conjecture node, analyze its shortest path to the knowledge network, associated nodes, etc., to understand the difficulties that need to be solved.

3.Semantic Leap Simulation: Design algorithms to simulate the exploratory steps of human mathematicians. For example, introducing new intermediate nodes (i.e., introducing lemmas or new concepts) to try to connect the conjecture node with the main network; adjusting the target, i.e., if it is difficult to prove directly, one can try to prove a more general/weaker proposition (P layer Purpose change), to see if it is easier to connect; or introducing parallel knowledge (transferring tools from other fields, which is equivalent to temporarily networking external knowledge from across domains). These correspond to the cross-layer leap behaviors of DIKWP. The program needs to have certain rules to guide these operations, such as: selecting known nodes that share some attributes with the conjecture node, trying to construct connections; or inferring possible lemma forms based on existing proof pattern templates.

4.Verification and Feedback: Once the program "proposes" a new connection (equivalent to guessing a possible reasoning step or lemma), it needs to call a theorem prover or computation to check its truth or provability. If it fails, then backtrack and try other leaps. If it succeeds, add it to the knowledge graph, shortening the distance of the conjecture node. If the distance is not yet 0, continue to iterate this process. This loop is similar to the trial and error and correction process in human proof.

5.Iteration and Evolution: The system continuously self-improves. On the one hand, every time a sub-lemma is solved, the knowledge base is expanded, and the tension of the original conjecture is reduced; on the other hand, the system also records which strategies are effective, to strengthen the application of similar scenarios. This machine learning component can improve exploration efficiency.

Such a simulation system may be difficult to achieve full functionality at present, but it can start from simple fields (such as small problems in combinatorics or algebra) and continuously improve. Its significance is: it allows us to experimentally test the effectiveness of the semantic model for mathematical discovery. If the machine can re-discover some known theorems driven by semantics, it will be a huge victory. Even if it cannot be fully automated, it can at least serve as an assistant to mathematicians, providing new association hints. For example, the system might propose: "Conjecture A is similar in position in the knowledge graph to the previous conjecture B, and B was proven by introducing concept C. It is recommended to try a similar C or its generalization." This type of hint may decompose complex proofs into reasonable small steps under human-machine collaboration.

Artificial Intelligence and Semantic Mathematics

Current automatic theorem proving, multi-agent collaborative proof systems, etc., are mostly based on logical search and pattern matching, lacking an understanding of semantic Purpose. The semantic mathematics framework can provide a new direction for the next generation of AI, enabling it to have the prototype of mathematical intuition. In this framework, AI not only retrieves theorems and performs formal deduction, but also evaluates the meaning and difficulty of propositions, and actively adjusts goals. For example, for a target theorem, AI can judge: "This proposition has a large amount of information and may be difficult to prove directly. It is better to consider proving a special case of it to obtain a pattern." This ability requires AI to have a certain semantic concept graph and tension assessment module.

Some existing explorations, such as the application of OpenAI's Codex or DeepMind's AlphaZero in mathematical conjecture verification, are limited to search or neural network prediction, and have not explicitly introduced semantic layer knowledge. Semantic mathematics suggests that AI can adopt layered cognition: dividing computation into DIKWP steps. For example: D layer collects data (tries small examples to test the conjecture), I layer looks for patterns (perhaps uses neural networks to induce laws), K layer calls knowledge (associates with existing theorems), W layer tries to prove (calls a prover or search), and P layer adjusts strategies based on the results. In this way, the AI's workflow will be closer to the human thinking process rather than pure brute-force search. In the long run, this may enhance AI's ability to solve difficult problems, allowing AI to no longer only solve problems from a question bank or short reasoning, but to make a difference on the open frontiers of mathematics.

Philosophical Reflections on Mathematics

The changes brought by semantic mathematics are not only at the technical level, but also at the level of the philosophy of mathematics. Traditionally, the Platonist view holds that mathematical truth exists objectively, and humans just discover it; formalism simply equates mathematics with a symbol game, without talking about meaning; while the humanist view emphasizes the human brain factor in mathematical creation. Semantic mathematics synthesizes these views: it admits that mathematics is presented as a symbol game, but behind it is driven by cognitive semantics; it also believes that mathematical discovery has subjective factors, but the structures revealed (such as Fermat's Last Theorem) have objective necessity once proven. Here, the semantic space can be seen as a modern interpretation of Plato's "world of ideas"—except that this world of ideas does not exist abstractly and perfectly, but is presented through the human cognitive framework, and its structure is affected by the limitations of human thinking and paradigms. In other words, mathematical truth has both an objective core and a cognitive projection, and we approach the truth in the cognitive space. In different historical stages, the cognitive abilities and conceptual semantic networks are different, and the corresponding mathematical systems are also different (e.g., ancient Greece did not have analytical concepts, and could not raise questions like the Riemann Hypothesis). Therefore, the philosophy of semantic mathematics holds: Mathematics is a cognitive construction represented by symbols, and a successful construction corresponds to a realistic characterization of an objective structure. This explains why mathematics is so effectively applied to reality—the patterns we perceive often reflect natural structures, and also explains why mathematics sometimes gets lost in formalism—because symbol manipulation detached from semantics may temporarily deviate from a meaningful direction ("meaningless true theorems" exist but are of little value).

Another reflection is on mathematical aesthetics. Many mathematicians emphasize the role of beauty in guiding discovery. However, beauty is a subjective semantic experience. In the semantic mathematics framework, beauty can be understood as the satisfaction brought by the just-right resolution of semantic tension. A beautiful theorem is usually a high degree of fit between the Purpose of the problem and the proof method: the tension is greatly released, giving people a sense of sudden enlightenment. For example, the proof of the Poincaré conjecture using Ricci flow is hailed as "seamless," because the idea of continuous deformation perfectly fits the needs of topological classification. Beauty is also reflected in the balance between the conciseness and profundity of semantic compression: the formula is short but the meaning is broad, which is amazing (e.g.,  is called "God's formula"). Semantic mathematics provides a language for this aesthetic discussion—we can say that this formula has an extremely high semantic compression rate, yet is completely closed-loop, and is therefore aesthetically awe-inspiring. In the future, one might even try to quantitatively analyze the aesthetic feeling of theorems: the degree of compression, high-level semantic association, and the degree of mapping between the proof Purpose and the method, as indicators of aesthetic feeling. This may partially explain why some proofs, although correct, are regarded as ugly (perhaps lengthy, tedious, and lacking a unified Purpose), while some solutions are additionally respected (the path maps out deep connections, killing two birds with one stone).

Outlook and Conclusion

Semantic mathematics is still a nascent concept, but it provides us with a mirror to think about the nature and future of mathematics. In the age of artificial intelligence, computers are expected to assist or even independently prove more and more theorems. We must think: when proof is automated, what are the roles and values of human mathematicians? The answer of semantic mathematics may be: the ability to create new meaning and ask profound questions. This is difficult for artificial formal systems to replace. In other words, humans are good at discovering semantic tension and beauty, while machines are good at executing established logic. The combination of the two will create a new model of mathematical discovery. The "mathematician" of the future may be a human-machine hybrid team: humans give machines the overall Purpose, semantic goals, and machines explore the reasoning detail paths, collaborating with each other to complete challenges that were previously unattainable. At that time, the paradigm of mathematical research will be innovated again.

Philosophically, semantic mathematics prompts us to re-examine "what does a proof mean." Perhaps a proof is not just to ensure that a proposition is true, but also a process of eliminating doubts in our hearts and making our knowledge feel complete. This has a subjective color, but it is also precisely what makes mathematics charming: the proof of a theorem often makes people feel pleased and at ease, because a certain sense of order has been restored. Just as Yucong Duan's theory of consciousness relativity says, for different intelligent agents, the judgment of consciousness depends on their respective understanding. Similarly, the value of a proof also depends on whether the reader understands its semantic meaning. For those who cannot understand a proof, it has no aesthetic feeling; for those who understand its ideas, it is like listening to a symphony. Semantic mathematics emphasizes the status of understanding in mathematics—a proof must not only exist, but also be comprehensively understood to be truly "complete." From this perspective, what we pursue is not just proof, but understandable proof, that is, the realization of a semantic closed loop in the collective mind.

In summary, this article uses semantics as a clue to connect mathematical knowledge and cognition, and interprets classic difficult problems in a new framework. Many of the discussions are still exploratory, but we believe this represents a useful perspective. Mathematics is not only a collection of theorems, but also the weaving of meaning; the work of mathematicians is not only deductive reasoning, but also navigating in semantic space, discovering new continents, and filling in blanks. With the development of artificial intelligence and the deepening of human understanding of their own cognition, new forms of mathematics will surely appear. Semantic mathematics may be a cornerstone towards future mathematics, helping us find a balance between form and meaning, and letting mathematics, the bright pearl of human wisdom, shine more brightly in the new era.

References:

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