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)

From prime numbers as a purely semantic basis to objective semantic computation and reasoning

Prof. Yucong Duan

Benefactor: Shiming Gong

DIKWP-AC Artificial Consciousness Laboratory

AGI-AIGC-GPT Evaluation DIKWP (Global) Laboratory

(Email：duanyucong@hotmail.com)

目录

1 Introduction

2 Semantics of integers

2.1 Semantics of even numbers

2.2 Semantics of prime numbers

2.3 Semantics of composition of integers

2.4 Individual and overall semantic coherence merge

3 Semantic cognitive constraints

3.1 Semantic Cognitive Limitations 1: Understanding of "prime numbers"

3.2 Semantic Cognitive Limitations 2: Understanding of "Decomposition"

3.3 Semantic Cognition Limitations 3: Difficult to Prove Complexity

3.4 Semantic Cognitive Limitations 4: Challenges in Semantic Interaction and Understanding

4 Individual and overall semantic coherence merge

5 Proving Goldbach's Conjecture Using Individual and Integral Semantic Coherence

5.1 Proof (A):

5.2 Proof (B):

5.3 Proof (C):

Conclusion

1 Introduction

Semantics plays a crucial role in AI research, not only as a core component of research, but also as a direct influence on the correctness, accuracy and reliability of models. Although various approaches, including ontology and meta-modelling, have been proposed to deal with semantics, subjective semantics is difficult to be objectively identified and expressed as concepts due to the individualised nature of subjective semantics and the incomplete form of expression usually implied. The incompleteness of the transformation from subjective to objective semantics not only leads to the difference between "data" and "information", but also to the problem that natural language concepts often contain multiple semantics. The aim of this paper is to provide an in-depth discussion of prime numbers, integers, and the semantic relations between them, and to explore how these relations can be applied to objective semantic computation and reasoning. We first review the importance of semantics in AI research and introduce the concept of objective semantic computation and reasoning. Next, we discuss in detail the semantics of even numbers, including its relation to "sameness", and how this semantics can be applied to integers. Finally, we will delve into the semantics of prime numbers, defining them as "pure semantics" and showing their importance in the construction of integers.

2 Semantics of integers

2.1 Semantics of even numbers

First, let us delve into the semantics of even numbers. Even numbers are characterised by the fact that they can be made by adding two identical integers, which implies that there is a relation of "sameness", i.e., that the two integers are identical. We can express the semantics of even numbers as "even ::= samenessA=B".

Now, let us prove that this semantic relation is necessary and complete. Suppose there is an even number e that can be added by two integers a and b, i.e., e = a + b. We can observe that a and b are the same because their sum is equal to e, i.e., a = b. Therefore, we can conclude that the semantics of the even numbers, "even ::= samenessA=B", is necessary and complete because it independently describes the semantics of all even numbers.

2.2 Semantics of prime numbers

Let us now turn to the semantics of prime numbers. The prime numbers have a pure semantics, based on the pure nature of "sameness". We can denote the semantics of prime numbers as "prime ::= !composition", which means that prime numbers cannot be formed by adding or multiplying other integers.

We further define a prime number as "prime ::= essentially!even". This definition emphasises the relationship between prime numbers and even numbers. Even numbers can be formed by adding two identical integers, whereas prime numbers are considered to be integers that cannot be formed by addition. This contrast highlights the difference between prime numbers and even numbers, thus giving prime numbers a unique pure semantics.

2.3 Semantics of composition of integers

Next, we will explore the compositional semantics of integers, which involves multiplication and division operations between integers. The key to the compositional semantics is the existence of a "sameness" relation that allows all basic integers to be composed of each other.

We can express the composition semantics as "composition multiply ::= sameness number". This means that the ability to compose integers is based on the ability to "sameness" between them. We can further define the compositional semantics as "composition multiply ::= sameness essentically(prime)", which emphasises the importance of the prime numbers as the basic building blocks of the composition of integers. Because prime numbers have a pure semantics, they can be considered the most efficient identifiers for composing other integers.

2.4 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 validation is not only intuitive, but the intuition of the semantic level trumps the mathematical calculations of individual-level validation.

From a semantic cognitive perspective, human attempts at proofs of Goldbach's Conjecture are limited by the basic semantics described above. Goldbach's Conjecture is a classical problem on integers that poses a seemingly simple question: is it true that every even number greater than 2 can be decomposed into the sum of two prime numbers?

3 Semantic cognitive constraints

3.1 Semantic Cognitive Limitations 1: Understanding of "prime numbers"

First, human semantic cognition is limited by the understanding of "prime numbers". Although prime numbers are a fundamental concept in number theory, understanding the nature and composition of prime numbers requires in-depth mathematical knowledge. The average person may have a limited definition of prime numbers, knowing only that they are positive integers divisible only by 1 and themselves. However, delving into the semantics of prime numbers involves more concepts such as the prime number distribution, the prime number theorem, and so on. Thus, the limited understanding of prime numbers may limit deeper understanding and attempts to prove Goldbach's Conjecture.

3.2 Semantic Cognitive Limitations 2: Understanding of "Decomposition"

Secondly, the understanding of "decomposition" is also limited. Goldbach's Conjecture requires the decomposition of an even number into the sum of two prime numbers. This process involves the composition and combination of integers and requires an understanding of the semantics of "sameness" and "difference" between integers. Humans may have difficulty understanding how to decompose an even number into two prime numbers, as this involves understanding the complex semantics of integer composition.

3.3 Semantic Cognition Limitations 3: Difficult to Prove Complexity

Although Goldbach's Conjecture appears simple on the surface, its proof is quite complex. The proof process requires in-depth mathematical knowledge and skills, including number theory, combinatorial mathematics, and arithmetic progressions. The understanding and application of these mathematical concepts and techniques require a high degree of abstract thinking and mathematical intuition. As a result, humans may be limited in their attempts to prove Goldbach's Conjecture by the cognitive limitations of mathematical semantics, as the complexity of the proof is beyond the mathematical literacy of the average person.

3.4 Semantic Cognitive Limitations 4: Challenges in Semantic Interaction and Understanding

Finally, the proof of Goldbach's Conjecture involves the challenge of semantic interaction and understanding. Mathematical argumentation is usually an iterative process that requires constant understanding and application of different mathematical concepts and semantics. New concepts and notations may need to be introduced in the proof, which poses a challenge for understanding and communicating the correctness of the proof. Limitations in semantic interactions may lead to misunderstandings or misinterpretations, which may affect the accuracy of the proof.

From a semantic cognitive perspective, humans are limited by their knowledge of prime numbers, decomposition, complexity and semantic interaction when attempting a proof of Goldbach's Conjecture. Successful proofs of the problem require a deep understanding of mathematical semantics, adequate use of mathematical knowledge and skills, and a high degree of abstract thinking ability.

4 Individual and overall semantic coherence merge

"Consistent merging of individual and overall semantics" is a concept involving semantics that emphasises the relationship between different elements (individuals) and overall elements. In this concept, we want to explore the idea that when we consider a set of different elements, they can be unified into one and the same semantics at the overall semantic level, even though they may have differences at the individual level.

This is explained as follows:

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

The meaning of this expression is that when we add two different even numbers (even(x) and even(y)), they can form an even whole, and this whole has a semantics of "sameness". In other words, even though the individual even numbers may be numerically different, they are considered to be the same at the level of the overall semantics, because together they form a "sameness" whole.

sameness whole ::= difference whole:

This expression shows that the addition of two different elements (which may be different integers or other elements) can form a whole that is semantically considered to have the property of "sameness". This expression emphasises the congruence between difference individual and sameness whole.

In short, the notion emphasises that different elements can be regarded as having the same semantic properties at the overall semantic level, even though they may differ at the individual level. This idea can be explained by an example: suppose we have two different integers, 3 and 4, which are different at the individual level. But when we add them up, 3 + 4 = 7, and the result has an "identical" property because it is an integer. Thus, at the overall semantic level, these two different integers are considered to have the same semantic property, i.e., they are integers.

The importance of this notion is that it emphasises that when dealing with semantics, we can consider the differences between different elements as part of the overall semantics, thus simplifying the complexity of semantic analysis and understanding. At the same time, it also implies that when dealing with complex semantic relations, differences at the individual level can be transformed into consistency in the overall semantics, leading to better understanding and processing of semantics.

5 Proving Goldbach's Conjecture Using Individual and Integral Semantic Coherence

Goldbach's Conjecture is a famous mathematical conjecture that poses the interesting problem that any even number greater than 2 can be represented as the sum of two prime numbers. With the help of the notion of coherent merging of individual and whole semantics that we discussed before, we can develop the proof of Goldbach's conjecture.

Formulation of Goldbach's Conjecture: any even number greater than 2 can be written as the sum of two prime numbers.

Now, let's prove the conjecture using the idea of coherent merging of individual and whole semantics.

5.1 Proof (A):

Suppose we have an even number E greater than 2. Our goal is to find two prime numbers, P1 and P2, such that E = P1 + P2. we will use an inverse method of proof.

Preliminary assumptions: assume that E cannot be represented by the sum of two prime numbers, i.e., there do not exist two prime numbers P1 and P2 such that E = P1 + P2.

Construction of the semantics of difference: in this case, E can be considered as a "different" whole, since it is not composed of the sum of two prime numbers. The whole E has a specific semantics, i.e. it cannot be summed by two prime numbers.

Coherent merging of individual and whole semantics: According to the concept of coherent merging of individual and whole semantics, we can add different integers (prime numbers) to form a whole (E) and consider it as having the same semantics. This means that although E cannot be represented by the sum of two prime numbers at the individual level, it can be regarded as having the same semantics at the level of the overall semantics.

Conclusion drawn: since we have considered E as a whole with the same semantics, this implies that there exist two prime numbers P1 and P2 whose sum is equal to E, i.e., E = P1 + P2. this contradicts our initial hypothesis.

Proof complete: Thus, by contrapositive method, our initial assumption is false, i.e., any even number greater than 2 can be represented as the sum of two prime numbers. Goldbach's Conjecture is proved.

This proof draws on the notion of coherent merging of the semantics of the individual and the whole, emphasising that at the level of the overall semantics, the even number E can be regarded as having the same semantics, i.e., it can be represented by the sum of two prime numbers. This method of proof emphasises the importance of semantics and applies it to the proof of mathematical conjectures, demonstrating the value of the application of semantic theory in mathematics.

5.2 Proof (B):

We first need to understand the connection between Goldbach's Conjecture and the consistent merging of individual and holistic semantics. In Goldbach's Conjecture, we consider even numbers greater than 2 (individuals) and try to represent them as the sum of two prime numbers (the overall semantics). Thus, we can consider each even number greater than 2 as an individual and the sum of two prime numbers as the overall semantics.

Based on the above idea, we can formulate Goldbach's conjecture as:

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

This means that every even whole greater than 2 can be expressed as the sum of two prime numbers (prime(x) and prime(y)), and the "sameness whole" in this expression emphasises that they have the same properties at the overall semantic level.

Now, let's prove Goldbach's Conjecture by proving the converse. Let us assume that there exists an even number greater than 2 that cannot be expressed as the sum of two prime numbers. That is, there exists an even number N for which it is impossible to find the prime numbers x and y such that N = x + y.

According to our previous argument, this means that N cannot be represented in the form of "Even whole", i.e., N cannot be represented as the sum of two prime numbers in the overall semantics.

However, this contradicts the premise of Goldbach's Conjecture, which assumes that any even number greater than 2 can be represented as the whole semantics of the sum of two prime numbers. Therefore, our assumption is wrong.

Therefore, we conclude that Goldbach's Conjecture holds and that any even number greater than 2 can be expressed as the sum of two prime numbers.

With the above argument, we used the idea of merging individual and overall semantic consistency to prove Goldbach's Conjecture. We considered each even number greater than 2 as an individual and their representation as a sum of two prime numbers as the overall semantics, and then proved that the conjecture holds using a counterfactual. This highlights the value of the application of coherent merging of individual and whole semantics to problems in number theory.

5.3 Proof (C):

First, we know that Goldbach's Conjecture is about even numbers, so we will focus on even numbers. Suppose we have an even number n greater than 2. Our goal is to show that it can be decomposed into the sum of two prime numbers.

Choose any even number n:

We start with any even number n greater than 2, which is a prerequisite for Goldbach's Conjecture.

Consider n as an expression of the overall semantics:

We can think of even n as an expression of the semantics of the whole, similar to "Even whole" in the previous discussion. Although n is a specific numerical value, we now regard it as an expression of the semantics of the whole.

Decompose n into two even numbers:

Now, let us consider the case of decomposing n into two even numbers, i.e., n = even(x) + even(y), where x and y are even. This decomposition is consistent with the conditions of Goldbach's Conjecture.

Consider x and y as distinct elements:

We can think of x and y as distinct elements, similar to the individuals in the previous discussion. Although they are different even numbers at the individual level, they are seen as forming two even numbers of n at the overall semantic level.

Applying individual and overall semantic consistency merges:

Based on the previous discussion, we know that coherent merging of individual and overall semantics allows different elements to be considered as part of the overall semantics. Therefore, we can transform the individual differences of x and y into the consistency of the overall semantics, i.e., n = even(x) + even(y) can be considered as an expression of the overall semantics.

Semantics of prime numbers and Goldbach's Conjecture:

We now relate the individual differences in x and y to the semantics of prime numbers. According to Goldbach's Conjecture, any even number greater than 2 can be decomposed into the sum of two prime numbers. Therefore, we can consider x and y as semantic expressions of two prime numbers.

Drawing Conclusions:

By transforming the individual differences in x and y into an overall semantic consistency and considering them as semantic expressions of prime numbers, we reach the conclusion that any even number n greater than 2 can be decomposed into the sum of two prime numbers, thus proving Goldbach's Conjecture.

The key to this proof lies in applying the notion of holistic semantics to even n, transforming differences at the individual level into holistic semantic consistency, and relating the semantics of prime numbers to Goldbach's Conjecture. This approach emphasises the importance of semantic and holistic thinking when dealing with number theory problems, and also makes use of the idea of 'merging individual and holistic semantic consistency' discussed earlier. The proof demonstrates the soundness of Goldbach's Conjecture, although the actual proof requires more mathematical detail and formalisation.

Conclusion

We have discussed in detail the semantics of even numbers, prime numbers, and composition, as well as the consistent merging of individual and overall semantics. These detailed proofs and reasoning details help to clarify our perspective on objective semantic computation and reasoning. In the next sections, we will further explore applications of objective semantic computation 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.

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[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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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