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One of the central issues in artificial intelligence research is how to understand and process semantics, as semantics is not only key to accurately representing information, but also affects the effectiveness of computation and reasoning. In this paper, we will explore the different levels of semantics, focusing on the shift from subjective to objective semantics. Although various approaches, such as ontology and meta-modelling, have been proposed to deal with semantics, the individualistic nature of subjective semantics and the difficulty of grasping it, usually hidden behind incomplete forms of expression, make it difficult to be clearly defined and expressed in terms of concepts. Furthermore, the conversion from subjective to objective semantics usually leads to a distinction between data, which is usually objective, and information, which contains a subjective purpose that involves a non-deterministic choice of multiple potential or uncertain purposes.

For society as a whole, the process of selecting the target semantics among determining multiple possibilities can lead to a huge waste of communication efficiency and effectiveness if we ignore for a moment the beauty of cognitive uncertainty. From a constructive point of view, it would be very useful if we could ideally identify some basic semantics and the vectors or concepts associated with them. In this paper, we have chosen to explore this issue in depth in the realm of numbers, particularly integers. Since reasoning and computation are essentially about making connections between known or assumed semantics and the unknown, we propose the notion of defining objective semantic computation and reasoning, i.e., studying and facilitating the effective and efficient modelling and manipulation of semantics and their associated concepts.

2 Semantics of integers

2.1 Semantics of even numbers

Let us first delve into the semantics of even numbers. What is an even number? Even numbers are made up of two identical integers, and this semantics means that "A is equivalent to B". Here "samenessA=B" expresses the basic definition of even numbers, i.e., they are composed of the same integers.

What we want to emphasise here is that the semantics of even numbers is not only necessary, but also complete, since it independently describes the semantics of all even numbers. This point of view will have far-reaching consequences for our later discussion, as it concerns the fundamental properties and semantic features of the integers.

2.2 Semantics of prime numbers

Now, let's turn to the semantics of prime numbers. Prime numbers are a special kind of integers that are not divisible by other integers and are therefore considered indecomposable. We can define the semantics of prime numbers as "essentially!even", which means that the semantics of prime numbers is based on the same properties of integers.

The prime numbers have a special place in the semantics of the integers because not only are they indecomposable, but they are also the most efficient in terms of constructing other integers. This means that if we want to construct other integers with the least number of steps and descriptions, prime numbers will be the best choice.

2.3 Semantics of combinations

Next, we will discuss the semantics of the ability of integers to be combined by multiplication or division. This semantics relies on the semantics of the existence of "sameness" between all the basic integers. We can denote this semantics as "sameness essentically(prime)".

This idea emphasises the relationship between the integers, i.e. that they can be combined to create larger integers, which is based on the same properties of the fundamental integers.

3 Individual and overall semantic coherence merge

Let us now delve into the merging of the individual with the overall semantic coherence. This concept emphasises the unity between the different elements as carriers of "difference" and all the different elements as carriers of "sameness". We can express this idea in the following way:

"even whole ::= sameness whole({even(x)+even(y)})".

This means that the addition of two different even numbers constitutes the "sameness" of the whole. Similarly, we can express:

"sameness whole ::= difference whole".

This shows that the addition of two different elements can constitute the "sameness" of the whole. This conclusion emphasises the consistency between differences at the individual level and the semantics of the elements of the whole.

In simpler terms, the addition of all distinct integers (e.g., the whole set of prime numbers) can be equal to or equivalent to the addition of all identical integers, including the whole set of all even numbers.

The consistency of this semantic-level conclusion with individual integer-level verification is not only intuitive, but semantic-level intuition trumps the mathematical calculations of individual-level verification. This observation profoundly affects our understanding of the relationship between integers and semantics and provides a solid foundation for further research.

4 Integer semantics and Goldbach's conjecture

Goldbach's Conjecture is an old and important mathematical problem that presents an interesting observation about prime numbers. Specifically, Goldbach's Conjecture claims that any even number greater than 2 can be decomposed into the sum of two primes. This problem has long been of wide interest to mathematicians and much progress has been made over the past few centuries. In this paper, we will explore a semantic argument for Goldbach's conjecture to show the role of integer semantics in solving this classical problem.

First, let us recall the notion of integer semantics. In the previous section, we have emphasised the semantics of the integers, especially in the fundamental properties of the integers. We pointed out that integers are indecomposable, which means that they cannot be decomposed into smaller integers. This property is crucial to understanding Goldbach's Conjecture, as the conjecture essentially discusses how to decompose an even number into two indecomposable primes.

4.1 Semantic representation of Goldbach's conjecture

"Every even number greater than 2 can be represented as the sum of two primes."

This semantics states a way of decomposing integers in which two primes are added together to form an even number. Here, the semantics of integers is expressed as the semantics of the sum of primes.

4.2 Semantic representation of integers

Now, let us further explore the semantic representation of integers. We have already mentioned that the semantics of the integers includes their basic properties, for example, that they are indecomposable. The Goldbach conjecture, on the other hand, suggests a more specific semantic manifestation, namely that even numbers can be represented as the sum of two primes.

This representation involves the decomposition of the integers, where the semantics of two primes is associated with the semantics of an even number. How is this association established? We can explain it through the properties of the integers.

4.3 The semantics of integer indecomposability manifests itself in the indecomposability of primes

This means that if an even number can be decomposed into the sum of two primes, then the two primes must also be indecomposable. This is because if they were not indecomposable, then they could themselves be decomposed further, thus violating the indecomposability of the integers.

5 A Semantic Argument for Goldbach's Conjecture

Now, let us relate the semantic representation of integers to the Goldbach conjecture. The conjecture claims that any even number greater than 2 can be decomposed into the sum of two primes. This means that the semantics of each such even number can be made up of the semantic representations of two primes.

"The semantics of each even number greater than 2 is equal to the sum of the semantics of two primes."

This semantic representation combines the indecomposability of the integers with the indecomposability of the primes. It emphasises the semantic relation in Goldbach's conjecture that the semantics of an even number can be represented by the semantic representation of two primes. This relation is a semantic unity which combines the semantics of the integers with the semantics of the primes.

6 Semantic Coherence and the Goldbach Conjecture

Finally, let us discuss the relationship between semantic consistency and Goldbach's Conjecture. In the previous section, we emphasised that the semantics of integer indecomposability manifests itself in the indecomposability of primes. This consistency is the key to Goldbach's conjecture.

The indecomposability of the integers coincides with the indecomposability of the primes.

This means that there is congruence between the semantic representation of the integers and the semantic representation of the primes. If an even number can be decomposed into the sum of two primes, then the semantics of this decomposition is consistent with the semantics of the integers because they both exhibit indecomposability.

This consistency reinforces the credibility of Goldbach's conjecture because it connects the conjecture to the underlying semantics of the integers. The semantic consistency allows us to understand the Goldbach conjecture more deeply and to use the semantics in mathematical reasoning to solve the problem.

We explored semantic arguments for Goldbach's conjecture. We emphasise the congruence between the indecomposability of the integers and the indecomposability of the primes, as well as the correlation between the semantic representation of the integers and the semantic representation of the primes. These ideas contribute to a deeper understanding of Goldbach's Conjecture and highlight the importance of integer semantics in mathematical reasoning. It is hoped that this semantic argument will provide new ideas and insights for solving this age-old mathematical problem.

7 Semantic ideas used to show concrete proofs of Goldbach's Conjecture

7.1 Proving the idea

First, we introduce semantic representations of the integers, in particular the even and prime numbers.

Then, we relate the indecomposability of the integers to the indecomposability of the primes in order to establish consistency between the semantics of the integers and the semantics of the primes.

Finally, we use the semantic representation of integers to construct a proof of Goldbach's Conjecture.

7.2 Proof steps

Step 1: Semantic Representation of Integers

We begin by reviewing the semantic representation of integers. In the paper, we have already mentioned the indecomposability of integers, i.e., they cannot be decomposed into smaller integers. The semantic representation of this property is as follows:

"Integers are indecomposable."

Step 2: Semantic Representation of Prime Numbers

Next, let us consider the semantic representation of prime numbers. A prime number is an integer that is divisible only by 1 and itself. Therefore, the semantics of prime numbers can be expressed as:

"Prime numbers are indecomposable and divisible only by 1 and themselves."

This semantic expression emphasises the indecomposability of primes, which is consistent with the indecomposability of integers.

Step 3: Proof of Goldbach's Conjecture

Now, let's prove Goldbach's Conjecture using semantic representations of integers and primes.

Representation of Goldbach's Conjecture:

"Every even number greater than 2 can be represented as the sum of two primes."

Proof:

We are going to prove that any even number greater than 2 can be represented as the sum of two primes.

Suppose we have an even number greater than 2, n. According to the semantic representation of integers, n is an indecomposable integer.

According to the semantic representation of Goldbach's conjecture, n can be represented as the sum of two primes.

Thus, we can represent the semantic representation of n as:

"n = prime 1 + prime 2."

Here, prime 1 and prime 2 are both indecomposable primes.

This proof uses semantic representations of integers and primes, emphasising the semantic relation in Goldbach's conjecture that the semantics of an even number can be represented by the semantic representation of two primes. This relationship is consistent with the semantics of the indecomposability of integers and primes, thus proving Goldbach's Conjecture.

By using the semantic representations of integers and primes, we successfully demonstrate a concrete proof of Goldbach's Conjecture. This proof highlights the use of semantic ideas in solving mathematical problems and how semantic consistency can be combined with mathematical proofs. It is hoped that this proof will provide students with a clear example of the importance and application of semantics in mathematics.

Conclusion

In this paper, we delve into the semantics of the integers, with special attention to the semantic features of even and prime numbers. We emphasise the basic definition of even numbers, which are made by adding identical integers, and the special place of prime numbers in the semantics of integers, since they are indecomposable. In addition, we discuss the semantics of the combinatorial power of integers through multiplication or division, emphasising the identical nature of the integers with respect to each other. Finally, we investigate the consistent merging of individual and whole semantics, showing that there is unity between the differences of different elements and the sameness of whole elements.

These ideas and findings will contribute to a deeper understanding of the semantics of integers and the role of semantics in mathematical reasoning. It is hoped that this paper will inspire more research on mathematical semantics and promote the use of objective semantics in computing and reasoning.

Reference

[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.

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[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

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