博文
Semantic Mathematics Reconstruction towards Subjectivity
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Semantic Mathematics Reconstruction from Subjectivity to Objectivity (Existence Computation and Reasoning, Essence Computing and Reasoning, Purpose Computing and Reasoning)
January 2024 DOI: 10.13140/RG.2.2.32469.81120
Traditional Invention and Innovation Theory 1946-TRIZ Does Not Adapt to the Digital Era
-Innovative problem-solving methods combining DIKWP model and classic TRIZ
Purpose driven Integration of data, information, knowledge, and wisdom Invention and creation methods: DIKWP-TRIZ
(Chinese people's own original invention and creation methods:DIKWP - TRIZ)
Semantic Mathematics Reconstruction from Subjectivity to Objectivity
(Existence Computation and Reasoning, Essence Computing and Reasoning, Purpose Computing and Reasoning)
Prof. Yucong Duan
Benefactor: 
Shiming Gong
DIKWP-AC Artificial Consciousness Laboratory
AGI-AIGC-GPT Evaluation DIKWP (Global) Laboratory
(Email：duanyucong@hotmail.com)
Catalogue
2.1 Subjectivity of mathematical axioms
2.2 Subjective source and objective misunderstanding of mathematics
2.3 Axiom of semantic consistency
2.4 Existence Computation and Reasoning（EXCR）
2.5 Essence Computation and Reasoning（ESCR）
2.6 Practical application and misunderstanding of axioms
2.7 Role of existence computing and reasoning
2.8 Importance of essence computing and reasoning
3 Analysis and solution of content expression and content mixing in artificial intelligence research
3.1 Dstinction between content and expression
3.2 Relationship between content and expression
3.3 Sources of misunderstandings and differences
3.4 Solution of Semantic Computing
3.4.3 Purpose Computation and Reasoning
3.5.3 Purpose Computation and Reasoning
4.1 Subjectivity of mathematical axioms
4.3 Axiom of Semantic Consistency
4.3.1 Existence Computation and Reasoning（EXCR）
4.3.2 Essence Computation and Reasoning（ESCR）
4.4 Subjective source and objective misunderstanding of mathematical knowledge
4.5 Euclidean space observation theorem (EOBS)
4.6 Points, Lines and Faces in Semantic Space
5.1 Semantic Interpretation of Four-color Theorem
5.2 Now let's consider the semantic analysis of existence in different situations.
5.2.1 Examples of areas with no essential line definition
5.2.2 An example of an area defined by an essentially existing line
5.2.3 Examples of areas defined by two essentially existing lines
5.2.4 Examples of areas defined by three essentially existing lines
5.2.5 Examples of areas defined by four or more lines
6.1 Semantic interpretation of Goldbach's Conjecture
6.1.1 Semantic perspective of examples of semantic interpretation types of Goldbach's Conjecture.
6.1.2 Semantic angle of overall type
This paper discusses the dialectical relationship between objectivity and subjectivity of mathematics, and reveals the subjectivity of mathematical axioms and misunderstandings in its practical application from the perspective of DIKWP (data, information, knowledge, wisdom and purpose) theory and semantic mathematics. This paper points out the important roles of Existence Computing and Reasoning (EXCR) and Essence Computing and Reasoning (ESCR) in understanding mathematical axioms, solving the confusion between content and expression, and building artificial intelligence. By introducing the concept of semantic space, combining with EXCR and ESCR, a new semantic explanation is given to points, lines and surfaces in European geometry, and the relative relationship between geometric elements is re-expounded from the semantic point of view. Taking the four-color theorem as an example, this paper makes an innovative semantic space analysis of it by using the methods of Existence Computation and semantic computation, and makes an in-depth analysis of different number of examples of line definition areas. This paper also applies this method to Goldbach's Conjecture, puts forward a semantic interpretation framework of Goldbach's Conjecture based on Existence Computation and semantic computing, and tries to construct the corresponding semantic space model, thus providing a novel and profound way of thinking for the study of mathematical problems.
In the course of the Millennium development of mathematics, the interaction between objectivity and subjectivity has always been a core topic for scholars. Traditionally, mathematics is regarded as an absolutely objective science, and its truth is not influenced by observers. However, when discussing the axiomatic system of mathematics, logical reasoning and the process of knowledge construction, we find that the subjective factors of mathematics can not be ignored. The purpose of this paper is to reveal the inherent subjective characteristics of mathematical axioms through the theoretical framework of DIKWP and a new perspective of semantic mathematics, and to analyze the root causes of its objective misunderstanding.
Firstly, this paper expounds the subjective source of mathematical axioms, and emphasizes the key roles of existence calculation and reasoning (excr) and essence calculation and reasoning (escr) in understanding this problem. At the same time, in view of the confusion of content expression, which is a common problem in artificial intelligence research, a strategy of using semantic computing to distinguish and analyze is proposed.
Further, this paper attempts to build a mathematical semantic space model, and re-examine the basic geometric elements such as points, lines and surfaces in a new semantic dimension to explore how they shape the structure of mathematical knowledge in the interweaving of subjective choice and objective existence. Through concrete examples, such as four-color theorem and Goldbach's Conjunction, this paper shows how the method based on Existence Computation and Semantic Computing can provide a novel and profound interpretation of classical mathematical problems, thus opening up a new path for mathematical research to integrate subjective and objective, promoting the consistency and integrity of internal logic of mathematical disciplines, and providing strong support for the communication and application in interdisciplinary fields.
2 The end of objective mathematics and the subjective regression of mathematics-from the perspective of DIKWP and semantic mathematics
Mathematics has long been regarded as a symbol of objectivity and accuracy, and its cornerstone-axiom (AM) is regarded as an indisputable truth. However, modern research reveals the subjectivity and hypothesis-based nature of mathematical knowledge. With the help of DIKWP model and the framework of semantic mathematics, this report deeply analyzes the process of mathematical knowledge construction and its role and limitations in solving practical problems.
2.1 Subjectivity of mathematical axioms
Mathematics is based on axioms (AM), which are regarded as reasonable hypotheses (HP), based on the association of a particular premise S with the result T. The rationality of this connection is subjective (FSUB). In other words, the axioms of mathematics are actually based on our understanding and interpretation of the world.
The definition of axiom: AM := min(HP) means that axiom is the smallest part that can't be simplified, and it is the essence of typed knowledge.
Axiom and Reasoning: Reasoning based on axiom (EXP) is to provide an abstract explanation of the external world. For example, EXP(<S, T>, AM) := <FSUB(S), FSUB(T) >, which means that the relationship between input s and output t is explained through a specific axiom system.
2.2 Subjective source and objective misunderstanding of mathematics
Although axioms are widely regarded as universal facts, their subjective sources are often misunderstood as absolute objectivity. As a formal system based on axioms, mathematics is constantly used to explain examples and determine the instantiation process.
Limitations of mathematical instantiation: The effective action space (EFP) of each concrete mathematics (IM) is actually the combination (ASS) of establishing consistency (CS) between types and examples. This consistency is defined as concrete consistency essential semantics (ES) from the cognitive perspective.
2.3 Axiom of semantic consistency
The axiom of semantic consistency (CS) shows that a specific connection is reasonable only if it belongs to the assumed connection. This axiom emphasizes the dependence on the basic axiom in the process of building mathematical knowledge and the decisive role in instantiation.
Description form: CS (type/type, ins/ins): = {Ass (type/type, ins/ins), Ass (type/type, ins/ins)}, indicating that the specific description should follow the preset type-level contact.
2.4 Existence Computation and Reasoning（EXCR）
EXCR focuses on the expression determination of semantic space from the perspective of cognitive intuition. Based on the conservation axiom of existence (CEX), it emphasizes that in the process of conforming to the consistency operation, the semantic set of existence can only be combined but cannot be denied.
Basic assumption: EX({ex}, CS(TYPE/type, INS/ins)), which emphasizes that the essence of existence will not change when calculating and reasoning.
2.5 Essence Computation and Reasoning（ESCR）
Traceability, expression balance and transformation of instantiation of ESCR processing types at semantic level. The axiom of basic hypothesis (CES) points out that in the process of consistency operation, the specific whole of semantic set has multiple expression forms, and these forms are essentially equivalent.
The realization of axiom: ism (cs (ex)):: = ism (complex (cs (ex)) means that different expression forms can be reduced to the same essential semantic set under the condition of consistent operation.
2.6 Practical application and misunderstanding of axioms
In practical application, it is often futile to try to explain subjective problems (SP) based on concrete mathematics (IM). This is because the objective hypothesis (HP(IM)) of specific mathematics and the hypothesis of subjective problems (HP(SP)) often do not intersect, which leads to the inability to form an effective explanation (EEXP).
Meta-analysis: All conclusions from specific mathematics must be consistent with their axioms. If the assumption of subjective questions does not conform to mathematical axioms, it will be ineffective to seek answers from this mathematical system.
2.7 Role of existence computing and reasoning
The focus of existence computing and reasoning (EXCR) is to form the expression determination of semantic space from the perspective of cognitive intuition and intuitive transfer. This method helps us to understand the process of mathematical knowledge construction and its role and limitations in explaining the real world.
Application of Existence Computing: In a specific mathematical system, a deeper and more comprehensive understanding of the external world can be established through Existence Computing and Reasoning.
2.8 Importance of essence computing and reasoning
Essence Computing and Reasoning (ESCR) pays more attention to the semantic consistency between types and instances. This method enables us to make more in-depth instantiation analysis and explanation on the basis of typed knowledge.
Practice of ESCR: Through ESCR, we can identify and explain practical problems more accurately, while ensuring that these explanations are consistent with basic axioms.
The subjective "regression" of mathematics is not the degradation of mathematical knowledge, but a deeper understanding of the essence of mathematical knowledge. Through the application of DIKWP model and semantic mathematics, we can fully understand the construction process of mathematical axioms and their functions and limitations in practical application. By revealing the subjectivity of mathematical knowledge, we can better understand how to effectively apply mathematical tools to solve practical problems, and keep a clear understanding of the nature of knowledge in the process. This understanding helps us to make more informed decisions in an increasingly complex world and gain deeper insights in the fields of science, technology and philosophy.
3 Analysis and solution of content expression and content mixing in artificial intelligence research
In the current research field of artificial intelligence (AI), a key problem is the confusion between research Content and its Expression. Most studies regard the expression of content as the main body of the study, ignoring the difference between the content itself and its expression. This confusion leads to the deviation between the research objectives, methods and evaluation mechanism and the actual content. The purpose of this report is to discuss this problem in depth and propose a solution based on semantic computing.
3.1 Dstinction between content and expression
In AI research, the confusion between content and expression is common in the interpretation and analysis of data. The expression of research content often does not correspond to the research content completely, correctly, accurately and effectively, which leads to the deviation in understanding and practical application.
3.2 Relationship between content and expression
The relationship between the content and its expression is the mapping between the content itself and the expression carrier of the content. This mapping links the existence of content with the existence of expression carrier, but this association is often misunderstood as semantic equivalence or equivalence.
3.3 Sources of misunderstandings and differences
It is the root of misunderstanding and differences in completeness, correctness, accuracy and effectiveness in many research processes to regard the existence of content as the same as the content expression carrier. This assumption leads to the neglect of the content background and the introduction of the content carrier background by mistake.
3.4 Solution of Semantic Computing
In order to solve this problem, we propose a multi-modal processing solution based on DIKWP model, including Existence Computation, Essence Computation and Purpose Computation and Reasoning.
Concept: Existence Computation focuses on the correlation analysis between the existence of content and the expression carrier, emphasizing the difference between the existence semantics of content and its expression.
Application: By analyzing the existence characteristics of the content, independent of the expression carrier, the dependence on the expression mode can be reduced and the understanding of the essence of the content can be enhanced.
Concept: Essence Computation focuses on the essential attributes of content and its relationship with expression.
Practice: By mapping the essential attributes of content to an appropriate expression carrier, Essence Computation helps to capture the core characteristics of content more accurately and avoid misleading expression.
3.4.3 Purpose Computation and Reasoning
Concept: Purpose Computation and Reasoning emphasizes the purpose of content and its expression in different situations.
Strategy: By understanding the purpose of the content, we can better choose or design the expression carrier to ensure the effectiveness and accuracy of information transmission.
Solution to the confusion between content and expression in AI research In an artificial intelligence (AI) project, the research team is developing a machine learning model to identify emotional tendencies on social media. The original intention of the project was to accurately identify and classify users' emotional expressions, but the team encountered challenges in processing data and interpreting model output.
Data collection: A large number of social media posts were collected as training data.
Question: The research team found that the same emotion has different expressions under different cultural backgrounds and personal experiences, which leads to the poor performance of the model in specific groups.
Confusion point: the team failed to distinguish between "the existence of emotion" and "the expression carrier of emotion", that is, the specific expression was mistaken for the universal characteristics of emotion itself.
Solution: Applying Semantic Computing
Strategy: Analyze and identify the existence of emotion in each post, independent of its expression carrier.
Practice: Identify the basic features of emotions through text analysis, such as mood and contextual cues, rather than relying solely on obvious emotional expressions.
Strategy: Determine the essential attributes of emotion, such as intensity, duration and influencing factors, and map these attributes to appropriate expression carriers.
Practice: Analyze the differences of emotional expression in different cultural backgrounds to accurately capture the core characteristics of emotions.
3.5.3 Purpose Computation and Reasoning
Strategy: Understand the purpose of emotional expression and choose or design a carrier that can effectively convey emotional intention.
Practice: Adjust the model to identify and explain complex emotional expressions, and consider the user's intention and background.
The improved model can identify and classify diversified and complicated emotional expressions more accurately.
Benefits: The applicability and accuracy of the model in different groups and cultural backgrounds are improved.
By distinguishing the existence of emotion from its expression carrier, and using Existence Computation, Essence Computation and Purpose Computation and Reasoning, AI project successfully improved the accuracy and applicability of the model. This case shows the importance of understanding the difference between content and expression in AI research, and how to solve related problems through semantic calculation.
Confusion between content expression and content itself is a key problem in AI research, which may lead to misunderstanding and inefficiency. Through the DIKWP model and the method of semantic calculation, we can distinguish the content and its expression more accurately, and optimize the research methods and tools. This method not only helps to improve the accuracy and effectiveness of research, but also promotes the innovation and development of AI technology in various application fields. By deeply understanding the relationship between content and expression, we can achieve deeper insight in the fields of science, technology and philosophy.
4 Semantic space in mathematics and existence computing and reasoning (excr) and essence computing and reasoning (escr)
Mathematics has always been regarded as a symbol of objectivity and accuracy, and its foundation is axiom, which is regarded as an indisputable truth. However, recent studies have revealed the subjectivity and hypothesis-based nature of mathematical knowledge. With the help of DIKWP model and the framework of semantic mathematics, this report will deeply analyze the process of mathematical knowledge construction, as well as the roles and limitations of Existence Computing and Reasoning (EXCR) and Essence Computing and Reasoning (ESCR) in explaining geometry.
4.1 Subjectivity of mathematical axioms
Mathematics is based on axioms, which are regarded as reasonable assumptions, based on the relevance of a specific premise to the results. However, the rationality of this connection is actually subjective. The definition of mathematical axioms can be expressed as:
AM := min(HP)
This means that axiom is the smallest part that can no longer be simplified, and it is the essence of typed knowledge. Based on these axioms, we carry out mathematical reasoning to provide an abstract explanation of the external world. For example, mathematical reasoning can be expressed as:
EXP(<S, T>, AM) := <FSUB(S), FSUB(T)>
This means that the relationship between input s and output t is explained through a specific axiom system. Although axioms are widely regarded as universal facts, their subjective sources are often misunderstood as absolute objectivity.
Axiom and reasoning are the key elements in the construction of mathematical knowledge. Reasoning based on axioms is to provide an abstract explanation of the external world. But this explanation actually depends on the choice of axioms, so it is subjective. The selection of axioms and the process of reasoning are interrelated, which together construct the system of mathematical knowledge. However, it also means that the objectivity of mathematical knowledge is limited.
4.3 Axiom of Semantic Consistency
The axiom of semantic consistency emphasizes the dependence on basic axioms and the decisive role in instantiation in the process of mathematical knowledge construction. Its descriptive form can be expressed as:
CS(TYPE/type, INS/ins) := {ASS(TYPE/type, INS/ins), ass(TYPE/type, INS/ins)}
This shows that the specific description should follow the preset type-level connection. The axiom of semantic consistency shows that a specific connection is reasonable only if it belongs to a hypothetical connection. This axiom emphasizes the dependence on the basic axiom in the process of building mathematical knowledge and the decisive role in instantiation.
4.3.1 Existence Computation and Reasoning（EXCR）
Existence computing and reasoning (excr) focuses on the expression determination of semantic space from the perspective of cognitive intuition. Based on the conservation axiom of existence (CEX), it emphasizes that in the process of conforming to the consistency operation, the semantic set of existence can only be combined but cannot be denied. The basic assumptions can be expressed as:
EX({ex}, CS(TYPE/type, INS/ins))
This emphasizes that the essence of existence will not change when calculating and reasoning. EXCR's work focuses on the expression determination of semantic space from the perspective of cognitive intuition and intuitive transfer. This method helps us to understand the process of mathematical knowledge construction and its role and limitations in explaining the real world.
4.3.2 Essence Computation and Reasoning（ESCR）
Essence Computing and Reasoning (ESCR) deals with the traceability, expression balance and transformation of instantiation at the semantic level of types. The axiom of basic hypothesis points out that in the process of consistency operation, the specific whole of semantic set has many expressions, and these forms are essentially equivalent. The realization of axioms can be expressed as:
ISM(CS(EX)) ::= ISM(Complex(CS(EX)))
This means that different expressions can be reduced to the same essential semantic set under the condition of consistent operation. Through ESCR, we can identify and explain practical problems more accurately, while ensuring that these explanations are consistent with basic axioms.
4.4 Subjective source and objective misunderstanding of mathematical knowledge
Although mathematical axioms are widely regarded as universal facts, their subjective sources are often misunderstood as absolute objectivity. As a formal system based on axioms, mathematics is constantly used to explain examples and determine the instantiation process. However, in practical application, it is often futile to try to explain subjective problems based on concrete mathematics. This is because the objective assumptions of specific mathematics and the assumptions of subjective problems often do not intersect, which leads to the inability to form an effective explanation.
4.5 Euclidean space observation theorem (EOBS)
Euclidean space observation theorem (EOBS) emphasizes that the equivalent transformation of coordinates does not change the semantics of the observed object at the type level. This theorem is of great significance in geometry, which shows that coordinate transformation will not change the essential properties of geometric objects. According to EOBS, the equivalent transformation of the concrete type level of observation coordinates does not change the semantics of the type level of the observed object. This is very useful for explaining geometric objects and phenomena in the real world.
4.6 Points, Lines and Faces in Semantic Space
In order to better understand the role of Existence Computing and Reasoning (EXCR) and Essence Computing and Reasoning (ESCR) in explaining geometry, let's reconsider the relative semantic relationship among points, lines and surfaces from the perspective of semantic space.
A concrete point P is a cognitively concrete existence (P, pl) on a concrete plane PL. The "concrete" here means the meaning of being exactly determined. An abstract point or point type P is a cognitively abstract existence (P, PL) on an abstract plane PL. In the semantic space, the existential semantics ex(p, pl) and EX(P, PL) respectively mean that point P or P is sufficiently limited by reasonable concrete semantics iSCR in the variable space corresponding to the plane.
In a plane COD(X, Y), when any straight line L is cognitively determined, the corresponding straight line can also be semantically described as ASS(L, COD(X, Y)). Any definite line, regardless of the surface semantics of the concept, has only one definite existential semantics exL in its essential existential semantic category.
For any plane PL, when the coordinate space COD(X, Y, Z) is cognitively determined, the corresponding plane can also be semantically described as ASS(PL, COD(X, Y, Z)). From abstract reasoning, it can be directly obtained that there is only one definite existential semantic exPL for an arbitrarily determined plane, regardless of the surface semantics of the concept.
4.7 Re-recognize the relative semantic relationship between points, lines and surfaces from the perspective of semantic space.
On the level of existential semantics, following the conservation axiom of existence (CEX), a reasonable ASS(X, Y, Z) semantically relates only a group of variables X, Y and Z that can't influence each other in the sense of existence.
In the Euclidean coordinate space COD(X, Y, Z), the value spaces of variable X, variable Y and variable Z are defined as real numbers R respectively. According to the axiom of combinatorial consistency (CES), the number of essential variables and the independent components in their combinatorial equivalent forms must not be less than the number of essential variables.
Therefore, it is inferred from COD(X, Y, Z) that any semantic expression target in COD(X, Y, Z) contains no more than three free variables numbers corresponding to the real number field. Inferred from ASS(X, Y, Z), any semantic expression target in ASS(X, Y, Z) cannot contain more than three free variables numbers.
For any plane PL, when the coordinate space COD(X, Y, Z) is cognitively determined, the corresponding plane can also be semantically described as ASS(PL, COD(X, Y, Z)). From abstract reasoning, it can be directly obtained that there is only one definite existential semantic exPL for an arbitrarily determined plane, regardless of the surface semantics of the concept.
Three-dimensional space 3D can be intuitively regarded as the set whole {PL} of plane PL along any real coordinate r. In such a three-dimensional space, the existence meaning of exPL is the existence correspondence of a (PL, r) pair of R values.
Because of the equivalence of coordinate transformation, this R is equivalent to any one of variable X, variable Y and variable Z.. So we can get that the semantics of the plane is the semantic space PL(X, Y) of two variables after one variable is determined in the three-dimensional space.
In a plane COD(X, Y), when any straight line L is cognitively determined, the corresponding straight line can also be semantically described as ASS(L, COD(X, Y)). Any definite line, regardless of the surface semantics of the concept, has only one definite existential semantics exL in its essential existential semantic category.
Two-dimensional space 2D can be intuitively regarded as the set whole {L} of straight line L along any real coordinate r. In such a two-dimensional space, the existential significance of exL is the existential correspondence of R value R of a (L, R) pair.
Because of the equivalence of coordinate transformation, this R is equivalent to any one of variable X and variable Y.. So we can get that the semantics of a straight line is the semantic space L(X) of a variable after a variable is determined in a two-dimensional space.
When any point p in a line COD(X) is cognitively determined, the corresponding point can also be semantically described as ASS(P, COD(X)). From abstract reasoning, it can be directly obtained that any certain point, regardless of the surface semantics of the concept, has only one certain existential semantics exP.
One-dimensional space 1D can be intuitively regarded as the set whole {P} of point p along any real coordinate r. In such a one-dimensional space, the existential significance of exP is the existential correspondence of the R value R of a (p, r) pair.
Therefore, we can get that the semantics of a point is the semantic space P corresponding to the unique variable value X in a one-dimensional space.
Through the frameworks of Existence Computing and Reasoning (EXCR) and Essence Computing and Reasoning (ESCR), we re-examine the semantic relations of points, lines and surfaces in mathematics. This framework provides a new perspective, which enables us to understand the relative semantic relationship between these geometric objects more deeply.
Starting from the semantic space, we re-examine the expressions of points, lines and surfaces in different dimensions and their semantic relations. This re-examination is helpful for us to better understand the construction process of mathematical knowledge, and the roles of Existence Computing and Reasoning (EXCR) and Essence Computing and Reasoning (ESCR) in explaining geometry.
Existence computing and reasoning (excr) focuses on determining the expression of semantic space from the perspective of cognitive intuition. This method is helpful for us to deeply understand the construction process of mathematical knowledge, and provides a cognitive explanation. At the same time, Essence Computing and Reasoning (ESCR) emphasizes the traceability and expression balance of instantiation at the semantic level of types, which enables us to identify and explain practical problems more accurately.
Through this framework, we realize the balance between subjectivity and objectivity of mathematical knowledge. Although mathematical axioms are based on certain assumptions, they construct an objective mathematical system and provide a powerful tool for explaining the real world. At the same time, the methods of Existence Computing and Reasoning and Essence Computation and Reasoning enable us to understand the essence of mathematical knowledge more deeply, as well as their applications and limitations in explaining geometry and other fields.
In a word, the subjectivity of mathematics is not to belittle its knowledge, but to have a deeper understanding of its essence. Through the application of DIKWP model and semantic mathematics, we can fully understand the construction process of mathematical axioms and their functions and limitations in practical application. By revealing the subjectivity of mathematical knowledge, we can better understand how to effectively apply mathematical tools to solve practical problems, and keep a clear understanding of the nature of knowledge in the process. This understanding helps us to make more informed decisions in an increasingly complex world and gain deeper insights in the fields of science, technology and philosophy. Existence computing and reasoning and Essence Computation and Reasoning provide us with more in-depth mathematical thinking tools to better understand and explore the mysteries of mathematics.
5 Semantic space interpretation of four-color theorem based on Existence Computation and semantic computation
In this chapter, a new method is proposed to explain the four-color theorem, which is regarded as a problem about EXCR and ESCR. We regard the plane as an existence space, the region is represented as a disjoint graph, and the color is represented as semantic discrimination. By analyzing the existing semantics and semantic range, we determine the minimum number of colors needed in different situations, and emphasize that semantic differentiation is carried out within a specific semantic range. We also discussed the situation when there are four or more lines, and pointed out how to determine the number of colors needed. The interpretation of this semantic space provides a new perspective for understanding the four-color theorem and an interesting method for solving similar graph theory and combination problems. By transforming the problem into the analysis of existential semantics and semantic scope, we can understand the demand of color quantity more clearly and provide a new direction for future research.
5.1 Semantic Interpretation of Four-color Theorem
The four-color theorem is a famous graph theory problem, and its semantic interpretation can be carried out through the framework of EXCR and ESCR. The goal of the four-color theorem is to find the combined examples of all regions on a plane, so as to determine how many colors are needed to distinguish these regions at least, so that there will be no same colors between adjacent regions. In this problem, we can regard the plane as a mathematical space, with regions represented as disjoint figures and colors represented as distinguishing semantics. In order to explain the semantics of regions on the plane, we can use existential semantic analysis, in which the existential semantics of points, lines and planes can help us understand the relationship between these regions.
First, let's define the meaning of color in existential semantics. Color C can be regarded as a kind of distinguishing semantics (SM), which is used to distinguish the filled area CZ from the unfilled area NZ. We can express it as:
SM(C) := (CZ, NZ)
Here, CZ represents the filled area and NZ represents the unfilled area. For the four-color theorem, our goal is to find the minimum number of colors so that all regions can be correctly distinguished.
5.2 Now let's consider the semantic analysis of existence in different situations.
5.2.1 Examples of areas with no essential line definition
When there are no lines on the plane, that is, there are no divided areas, we only need one color c1 to fill the whole plane. This is because there is no divided area, so only one color is enough. This can be expressed as:
NUM(Z) = 0 NUM(SMD0 (C)) = NUM(SMD0 (c1)) = 1
From the perspective of existential semantics, NUM(SMD0 ({c1})) means that there is no need for at most one kind of basic area marker existential semantics when there are no lines on the plane.
5.2.2 An example of an area defined by an essentially existing line
When there is a line l1 on the plane, we can divide the plane into two regions Z1 and Z2. These two areas need to be marked with different colors. We can use two colors c1 and c2 to mark these two areas. This can be expressed as:
NUM({Z1, Z2}) = 2 NUM(SMD1 (C)) = NUM(SMD1 ({c1, c2})) = 2
From the perspective of existential semantics, NUM(SMD1 ({c1, c2})) means that there are at most two basic existential semantics when there is a line on the plane.
5.2.3 Examples of areas defined by two essentially existing lines
When there are two lines l1 and l2 on the plane, these two lines further divide the region into four parts, namely, Z11, Z12, Z21 and Z22. These four areas need to be marked with different colors. We can use four colors c1, c2, c3 and c4 to mark these four areas. However, it should be noted here that the four regions are distinguished within a certain semantic range, specifically, they are distinguished within the semantic range of ASS(R(rx), R(ry)), where R(rx) and R(ry) respectively represent the values of horizontal and vertical coordinates. This means that the distinction between these four areas is completely within the scope of these two coordinates. Essentially, this is a specific semantics.
NUM(SMD2, SMD1) = NUM(SMD2 ({c1, c2, c3, c4})) = 4
From the perspective of existential semantics, NUM(SMD2 ({c1, c2, c3, c4}) means that there are at most four basic existential semantics when there are two lines on the plane.
5.2.4 Examples of areas defined by three essentially existing lines
When there are three lines l1, l2 and l3 on the plane, these lines further divide the region and produce more regions. In this case, we need to consider the existence of line l3. Line l3 corresponds to the example of ASS(X, Y), which further divides the existing area into more parts. These new areas will need more colors to mark, but we can still determine the number of colors needed through the analysis of existential semantics.
It should be noted here that the distinction between new regions is within a new semantic range, namely, ASS(R(rx), R(ry), ASS(R(rx), R(ry)). This means that the distinction between new areas is completely within the scope of these two coordinates and the new semantic scope. From the perspective of existential semantics, we can continue to apply the same method to determine the number of colors needed.
NUM(SMD3 ({c1, c2, c3, c4})) = 4
From the perspective of existential semantics, NUM(SMD3 ({c1, c2, c3, c4}) means that there are at most four basic existential semantics when there are three lines on the plane.
5.2.5 Examples of areas defined by four or more lines
When there are four or more lines on the plane, we can continue to apply the same method to determine the number of colors needed. Specifically, if the newly introduced line is parallel to the existing line set, no new color requirements will be introduced, otherwise, we can determine the required number of colors according to Theorem (rZCO) or Theorem (rZCI).
To sum up, we use the framework of EXCR and ESCR to explain the semantics of the four-color theorem. Under this framework, we regard the plane as an existence space, the region is represented as disjoint graphics, and the color is represented as semantic discrimination. By analyzing the semantic meaning and semantic range, we can determine the minimum number of colors needed in a given situation.
We also discussed the semantic analysis of existence in different situations, including the situation that there is no line, one line, two lines and three lines. In each case, we have determined the number of colors needed and explained the semantic relationship between these colors. In particular, we emphasize that distinguishing semantics is carried out within a specific semantic range, which is very important for understanding the semantic interpretation of the four-color theorem.
Finally, we also discuss the situation when there are four or more lines, and point out how to determine the number of colors needed in these situations. Generally speaking, by transforming the four-color theorem into the problem of Existence Computation and semantic computation, we can understand this problem more clearly and determine the minimum number of colors needed for the solution.
The interpretation of this semantic space provides a new perspective, which can help us better understand the four-color theorem and provide an interesting method to solve this classic problem. By transforming the problem into the analysis of existential semantics and semantic scope, we can understand the demand of color quantity more clearly and provide a new direction for future research. This method can also be applied to other similar graph theory and combinatorial problems, which provides a powerful tool for us to understand and solve these problems.
6 Semantic Interpretation and Semantic Space Construction of Goldbach's Conjecture Based on Existence Computation and Semantic Computing
This chapter aims to reinterpret the famous number theory problem Goldbach's Conjecture, and explore a new perspective of this problem by constructing semantic space. Goldbach's Conjecture claims that every even number greater than 2 can be expressed as the sum of two prime numbers. However, although this problem has aroused widespread concern in the field of mathematics, a general solution has not yet been found. In this paper, we reinterpret Goldbach's Conjecture as a question about EXCR and ESCR, and analyze the importance of existential semantics and semantic scope. We analyze Goldbach's Conjecture from the semantic perspective of type examples and overall types, and emphasize the key concepts of semantic existence and semantic scope. By establishing semantic associations with natural number types and prime number types, we get the type semantics of Goldbach's conjugate. Finally, we discuss the potential influence of this new interpretation method on mathematical research and its application prospect in solving similar graph theory and combinatorial problems. This research provides a new theoretical framework for Goldbach's Congress and a new direction for future research.
6.1 Semantic interpretation of Goldbach's Conjecture
Goldbach's Conjunction is a famous number theory problem, which claims that every even number greater than 2 can be expressed as the sum of two prime numbers. Although this problem has aroused widespread interest in mathematics, a general solution has not yet been found. In this paper, we reinterpret Goldbach's Conjecture as a question about EXCR and ESCR, and explore this classic question by constructing a semantic space (SCR).
6.1.1 Semantic perspective of examples of semantic interpretation types of Goldbach's Conjecture.
First of all, we explain it from the semantic point of view of examples of types of Goldbach's Conjecture. Suppose we have an even number e, which is an instance of Goldbach's Conjunction. We can use the instantiation relation INS(E) to represent the instance of e, where e represents the specific value of e. According to Goldbach's Conjunction, we can express e as the sum of two prime numbers p, that is, E = P1+P2.
This can be expressed in the following ways:
INS(E) := ASS((INS(P), INS(P)), REL(+))
Here, INS(P) represents an example of prime number p, and p1 and p2 are specific values of p respectively. Through the relation REL(+), we can add two instances of prime numbers, and thus get an instance of e.
6.1.2 Semantic angle of overall type
Next, we will consider the semantic interpretation of Goldbach's Conjunction from the semantic perspective of the whole type. For any instance of even number E, we can equivalently deduce the corresponding semantic association at the type level through the cross-type instance level and according to the basic assumption axiom of Existence Computing and Reasoning Excr, that is, the Conservation of Existence Set (CEX).
In this case, even type E can establish an existential semantic association with natural number type Z through type level semantic association E(x) := R(y)+R(y), because E can be expressed as the sum of two natural numbers. Here, R(y) represents the existential semantics of the natural number y.
This can be expressed in the following ways:
INS(E) := ASS((INS(Z), INS(Z), REL(+)))
Here, INS(Z) represents an example of the natural number z. Through the relation REL(+), we can add two instances of natural numbers to get an instance of e.
Similarly, we can express the semantic association at the type level between prime number type P and natural number type Z, because prime number P cannot be divisible by other natural numbers except 1 and itself, which can be expressed as P = ASS(Z,! ())。 Here! () indicates a semantic relationship that is not equal to (*), that is, it cannot be multiplied by other natural numbers.
On the basis of confirming that there is semantics at the type level of prime number P, we can use the basic hypothesis axiom of Essence Computing and Reasoning Escr, that is, the CES (Consistency of Compounded Essential Set). According to the concrete semantic relationship of the semantic relationship ASS(P, Z) between the instance INS(P) of prime number P and the instance INS(Z) of natural number type Z, the corresponding semantic relationship at the type level is established through the mapping between the type and the instance level. Finally, the type semantics of Goldbach's Conjecture P+P = E is obtained.
It should be noted that for the semantics of composite number c, we can't simply establish semantic association at the type level as the number of pixels. Because the type of composite number is not the essential semantic existence, the semantic definition of composite number itself involves the semantic existence of higher level types. The existential dependence of semantics cannot be established in reverse, which is a supplement to the axiom of conservation of existence (CEX) and the axiom of combinatorial consistency of global integrity of essential sets (CES).
Through the methods based on EXCR and ESCR, we reinterpret Goldbach's society and construct a semantic space (SCR) to study this number theory problem deeply. We analyze Goldbach's Conjecture from the semantic point of view of type examples and overall types, and emphasize the importance of semantic existence and semantic scope. This new explanation provides us with a different perspective, enables us to understand Goldbach's Congress more clearly, and provides a new direction for future research.
By transforming the problem into the analysis of existential semantics and semantic scope, we can discuss mathematical problems more deeply, and at the same time provide an interesting method for solving similar graph theory and combinatorial problems. As a classical problem of number theory, Goldbach's Conjecture has always attracted the interest of mathematicians. We hope that this new interpretation method can provide more inspiration for solving this problem, and also provide a new tool and perspective for mathematical research.
Focusing on exploring the subjective and objective essence of mathematics, this paper reveals the inherent subjectivity of mathematical axioms and their objective misunderstanding in practical application by deeply analyzing the conceptual framework of DIKWP theory and semantic mathematics. The research emphasizes the importance of Existence Computing and Reasoning (EXCR) and Essence Computing and Reasoning (ESCR) in understanding and handling this contradiction, and successfully applies them to solve the problem of confusing content and expression in the field of artificial intelligence.
This paper innovatively constructs a mathematical semantic space model, gives a new semantic explanation to the basic elements of European geometry-points, lines and surfaces, and discusses the relative semantic relationship between them in a deep level. Through a new interpretation of four-color theorem and Goldbach's Conjecture based on Existence Computation and semantic computation, this paper shows how this novel method provides a deeper and more intuitive understanding way for complex mathematical problems.
This paper not only deepens our understanding of the interaction between subjectivity and objectivity in mathematics, but also demonstrates the great potential of semantic computing in expanding the theoretical boundary of mathematics, promoting interdisciplinary communication and solving practical problems through examples, which provides valuable theoretical tools and thinking perspectives for the future development of mathematics research and related fields.
[1] Duan Y. Which characteristic does GPT-4 belong to? An analysis through DIKWP model. DOI: 10.13140/RG.2.2.25042.53447. https://www.researchgate.net/publication/375597900_Which_characteristic_does_GPT-4_belong_to_An_analysis_through_DIKWP_model_GPT-4_shishenmexinggeDIKWP_moxingfenxibaogao. 2023.
[2] Duan Y. DIKWP Processing Report on Five Personality Traits. DOI: 10.13140/RG.2.2.35738.00965. https://www.researchgate.net/publication/375597092_wudaxinggetezhide_DIKWP_chulibaogao_duanyucongYucong_Duan. 2023.
[3] Duan Y. Research on the Application of DIKWP Model in Automatic Classification of Five Personality Traits. DOI: 10.13140/RG.2.2.15605.35047. https://www.researchgate.net/publication/375597087_DIKWP_moxingzaiwudaxinggetezhizidongfenleizhongdeyingyongyanjiu_duanyucongYucong_Duan. 2023.
[4] Duan Y, Gong S. DIKWP-TRIZ method: an innovative problem-solving method that combines the DIKWP model and classic TRIZ. DOI: 10.13140/RG.2.2.12020.53120. https://www.researchgate.net/publication/375380084_DIKWP-TRIZfangfazongheDIKWPmoxinghejingdianTRIZdechuangxinwentijiejuefangfa. 2023.
[5] Duan Y. The Technological Prospects of Natural Language Programming in Large-scale AI Models: Implementation Based on DIKWP. DOI: 10.13140/RG.2.2.19207.57762. https://www.researchgate.net/publication/374585374_The_Technological_Prospects_of_Natural_Language_Programming_in_Large-scale_AI_Models_Implementation_Based_on_DIKWP_duanyucongYucong_Duan. 2023.
[6] Duan Y. The Technological Prospects of Natural Language Programming in Large-scale AI Models: Implementation Based on DIKWP. DOI: 10.13140/RG.2.2.19207.57762. https://www.researchgate.net/publication/374585374_The_Technological_Prospects_of_Natural_Language_Programming_in_Large-scale_AI_Models_Implementation_Based_on_DIKWP_duanyucongYucong_Duan. 2023.
[7] Duan Y. Exploring GPT-4, Bias, and its Association with the DIKWP Model. DOI: 10.13140/RG.2.2.11687.32161. https://www.researchgate.net/publication/374420003_tantaoGPT-4pianjianjiqiyuDIKWPmoxingdeguanlian_Exploring_GPT-4_Bias_and_its_Association_with_the_DIKWP_Model. 2023.
[8] Duan Y. DIKWP language: a semantic bridge connecting humans and AI. DOI: 10.13140/RG.2.2.16464.89602. https://www.researchgate.net/publication/374385889_DIKWP_yuyanlianjierenleiyu_AI_deyuyiqiaoliang. 2023.
[9] Duan Y. The DIKWP artificial consciousness of the DIKWP automaton method displays the corresponding processing process at the level of word and word granularity. DOI: 10.13140/RG.2.2.13773.00483. https://www.researchgate.net/publication/374267176_DIKWP_rengongyishide_DIKWP_zidongjifangshiyiziciliducengjizhanxianduiyingdechuliguocheng. 2023.
[10] Duan Y. Implementation and Application of Artificial wisdom in DIKWP Model: Exploring a Deep Framework from Data to Decision Making. DOI: 10.13140/RG.2.2.33276.51847. https://www.researchgate.net/publication/374266065_rengongzhinengzai_DIKWP_moxingzhongdeshixianyuyingyongtansuocongshujudaojuecedeshendukuangjia_duanyucongYucong_Duan. 2023.
Data can be regarded as a concrete manifestation of the same semantics in our cognition. Often, Data represents the semantic confirmation of the existence of a specific fact or observation, and is recognised as the same object or concept by corresponding to some of the same semantic correspondences contained in the existential nature of the cognitive subject's pre-existing cognitive objects. When dealing with data, we often seek and extract the particular identical semantics that labels that data, and then unify them as an identical concept based on the corresponding identical semantics. For example, when we see a flock of sheep, although each sheep may be slightly different in terms of size, colour, gender, etc., we will classify them into the concept of "sheep" because they share our semantic understanding of the concept of "sheep". The same semantics can be specific, for example, when identifying an arm, we can confirm that a silicone arm is an arm based on the same semantics as a human arm, such as the same number of fingers, the same colour, the same arm shape, etc., or we can determine that the silicone arm is not an arm because it doesn't have the same semantics as a real arm, which is defined by the definition of "can be rotated". It is also possible to determine that the silicone arm is not an arm because it does not have the same semantics as a real arm, such as "rotatable".
Information, on the other hand, corresponds to the expression of different semantics in cognition. Typically, Information refers to the creation of new semantic associations by linking cognitive DIKWP objects with data, information, knowledge, wisdom, or purposes already cognised by the cognising subject through a specific purpose. When processing information, we identify the differences in the DIKWP objects they are cognised with, corresponding to different semantics, and classify the information according to the input data, information, knowledge, wisdom or purpose. For example, in a car park, although all cars can be classified under the notion of 'car', each car's parking location, time of parking, wear and tear, owner, functionality, payment history and experience all represent different semantics in the information. The different semantics of the information are often present in the cognition of the cognitive subject and are often not explicitly expressed. For example, a depressed person may use the term "depressed" to express the decline of his current mood relative to his previous mood, but this "depressed" is not the same as the corresponding information because its contrasting state is not the same as the corresponding information. However, the corresponding information cannot be objectively perceived by the listener because the contrasting state is not known to the listener, and thus becomes the patient's own subjective cognitive information.
Knowledge corresponds to the complete semantics in cognition. Knowledge is the understanding and explanation of the world acquired through observation and learning. In processing knowledge, we abstract at least one concept or schema that corresponds to a complete semantics through observation and learning. For example, we learn that all swans are white through observation, which is a complete knowledge of the concept "all swans are white" that we have gathered through a large amount of information.
Wisdom corresponds to information in the perspective of ethics, social morality, human nature, etc., a kind of extreme values from the culture, human social groups relative to the current era fixed or individual cognitive values. When dealing with Wisdom, we integrate this data, information, knowledge, and wisdom and use them to guide decision-making. For example, when faced with a decision-making problem, we integrate various perspectives such as ethics, morality, and feasibility, not just technology or efficiency.
Purpose can be viewed as a dichotomy (input, output), where both input and output are elements of data, information, knowledge, wisdom, or purpose. Purpose represents our understanding of a phenomenon or problem (input) and the goal we wish to achieve by processing and solving that phenomenon or problem (output). When processing purposes, the AI system processes the inputs according to its predefined goals (outputs), and gradually brings the outputs closer to the predefined goals by learning and adapting.
Yucong Duan, male, currently serves as a member of the Academic Committee of the School of Computer Science and Technology at Hainan University. He is a professor and doctoral supervisor and is one of the first batch of talents selected into the South China Sea Masters Program of Hainan Province and the leading talents in Hainan Province. He graduated from the Software Research Institute of the Chinese Academy of Sciences in 2006, and has successively worked and visited Tsinghua University, Capital Medical University, POSCO University of Technology in South Korea, National Academy of Sciences of France, Charles University in Prague, Czech Republic, Milan Bicka University in Italy, Missouri State University in the United States, etc. He is currently a member of the Academic Committee of the School of Computer Science and Technology at Hainan University and he is the leader of the DIKWP (Data, Information, Knowledge, Wisdom, Purpose) Innovation Team at Hainan University, Distinguished Researcher at Chongqing Police College, Leader of Hainan Provincial Committee's "Double Hundred Talent" Team, Vice President of Hainan Invention Association, Vice President of Hainan Intellectual Property Association, Vice President of Hainan Low Carbon Economy Development Promotion Association, Vice President of Hainan Agricultural Products Processing Enterprises Association, Visiting Fellow, Central Michigan University, Member of the Doctoral Steering Committee of the University of Modena. Since being introduced to Hainan University as a D-class talent in 2012, He has published over 260 papers, included more than 120 SCI citations, and 11 ESI citations, with a citation count of over 4300. He has designed 241 serialized Chinese national and international invention patents (including 15 PCT invention patents) for multiple industries and fields and has been granted 85 Chinese national and international invention patents as the first inventor. Received the third prize for Wu Wenjun's artificial intelligence technology invention in 2020; In 2021, as the Chairman of the Program Committee, independently initiated the first International Conference on Data, Information, Knowledge and Wisdom - IEEE DIKW 2021; Served as the Chairman of the IEEE DIKW 2022 Conference Steering Committee in 2022; Served as the Chairman of the IEEE DIKW 2023 Conference in 2023. He was named the most beautiful technology worker in Hainan Province in 2022 (and was promoted nationwide); In 2022 and 2023, he was consecutively selected for the "Lifetime Scientific Influence Ranking" of the top 2% of global scientists released by Stanford University in the United States. Participated in the development of 2 international standards for IEEE financial knowledge graph and 4 industry knowledge graph standards. Initiated and co hosted the first International Congress on Artificial Consciousness (AC2023) in 2023.
Prof. Yucong Duan
DIKWP-AC Artificial Consciousness Laboratory
AGI-AIGC-GPT Evaluation DIKWP (Global) Laboratory
DIKWP research group, Hainan University
duanyucong@hotmail.com
https://blog.sciencenet.cn/blog-3429562-1416777.html
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