博文
Semantic Mathematics in Teaching of Algorithms
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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)
World Artificial Consciousness Conference Popular Science Series -
Semantic Mathematics in Teaching of Algorithms: Exploring the Deeper Meanings and Practical Applications of Mathematical Concepts
Yucong Duan
Benefactor: Shiming Gong
DIKWP-AC Artificial Consciousness Laboratory
AGI-AIGC-GPT Evaluation DIKWP (Global) Laboratory
World Association of Artificial Consciousness
(Email：duanyucong@hotmail.com)
The Inaugural World Conference on Artificial Consciousness
(AC2023), August 2023, hosted by DIKWP-AC Research
Catalog
2.1 The core elements of semantic mathematics
2.2 Application of Semantic Mathematics
2.3 Definition of Semantic Mathematics
3 Semantic mathematics applied to Master Theorem
3.1 Overview of Master Theorem
3.2 Interpretation from the Perspective of Semantic Mathematics
4 Using semantic mathematics to derive mathematical formulas
4.1 Closed form formula of Fibonacci sequence (Binet formula)
4.2 Derivation of binomial theorem
4.3 Derivation of Euler's Formula
5 Understanding mathematical concepts through semantic mathematics
5.1 Understanding the essence of Logarithm
5.2 Understanding the essence of Fourier series
5.3 Understanding the essence of Gaussian Transform
5.4 Understanding the Essence of Wavelet Transform
6 The deepening and expansion of semantic mathematics
6.1 The "ecosystem" of mathematical concepts
6.2 Multidimensional nature of mathematics
6.3 Richness of mathematical language
6.4 Cultivation of Mathematical Thinking
With the rapid development of science and technology, the teaching methods of mathematics as a basic subject need to be reformed to meet the needs of the new era. Based on the semantic mathematics method proposed by Professor Yucong Duan, this report discusses how to apply this idea to the teaching of divide-and-conquer algorithm in order to improve students' understanding of the deep meaning of mathematical concepts and their practical application ability. By interpreting the semantic mathematics of the Master Theorem as a case, we analyze the intuitive understanding of the complexity analysis of the divide-and-conquer algorithm and the deep meaning of the mathematical concept.
The report first outlines the core concepts of semantic mathematics, including deepening understanding, intuitive and image thinking, semantic connection and application orientation, and discusses the importance of integrating these concepts into mathematics and computer science teaching. Then, by interpreting the Master Theorem, the report shows how to use semantic mathematics methods to reveal the deep logic and intuitive significance behind complexity analysis, and how to understand the complexity of the algorithm by comparing the relationship between the number of recursive operations and the speed of problem scale reduction.
In addition, the report also discusses the advantages of semantic mathematics in promoting students to understand, remember and apply mathematical concepts, especially how to use mathematical tools more flexibly and creatively in solving practical problems. Finally, the report summarizes the potential of semantic mathematics in mathematics teaching reform, puts forward the possible direction of applying it to broader mathematics concept teaching, and the long-term influence of this method on improving students' innovative thinking and problem-solving ability.
Through the analysis and discussion of this report, we hope to provide new perspectives and strategies for the teaching of mathematics and related fields, and promote the innovation of educational methods, so as to better prepare students for the challenges of the 21st century.
The Master Theorem provides a method to quickly determine the time complexity of the divide-and-conquer recursive algorithm. This theorem is especially applicable to the recursive relation in the form of T(n)=aT(n/b)+f(n), where:
T(n) is the time to solve the problem.
n is the size of the problem.
a≥1 indicates the number of recursive branches at a time.
b>1 indicates the division factor of the problem size in each recursion.
f(n) is the time required in each layer of recursion except recursive calls, which usually represents the cost of segmentation and merging results.
Master Theorem illustrates three main situations:
Situation 1
If it is true for a constant > 0, then
Situation 2
If holds for a constant k≥0, then
Situation 3
If ϵ> 0 holds, and there is af(n/b)≤cf(n) for a constant c<1 and a sufficiently large n, then
These three cases cover most recursive algorithms using divide-and-conquer strategy, which can help to quickly determine the progressive time complexity of the algorithm. By analyzing the division factor, the number of recursions and the computational complexity outside each recursion, Master Theorem provides a powerful tool for algorithm designers to evaluate the performance of the algorithm.
The semantic mathematics proposed by Professor Yucong Duan is an exploratory mathematics teaching and research method, aiming at mastering and applying mathematical knowledge by deeply understanding the essence and significance of mathematical concepts, rather than just relying on the mechanical memory of formulas and algorithms. This method emphasizes the semantic understanding of mathematical thinking, that is, by exploring the meaning and connotation behind mathematical objects, concepts, theorems and methods, and their relationships, we can deepen our understanding and application ability of mathematics.
2.1 The core elements of semantic mathematics
Deepen understanding: not only satisfied with knowing what mathematical formulas or theorems are, but more importantly, understanding why they are like that and how they are derived from basic principles and concepts.
Intuitive and visual thinking: use intuitive and visual ways to understand mathematical concepts and processes, such as revealing the essence of mathematical concepts through graphics, examples or physical meanings.
Semantic connection: Explore the internal connection and logical structure among mathematical concepts, theorems and methods, and understand their position and function in the mathematical system.
Application orientation: emphasize the application value of mathematical knowledge, and understand the practicality and effectiveness of mathematical concepts and methods by solving practical problems.
2.2 Application of Semantic Mathematics
In teaching, semantic mathematics encourages students to establish a deep understanding of mathematical concepts through exploration, discussion and practice. This method helps students to establish an intuitive feeling of mathematical knowledge, so that they can apply mathematical tools more flexibly and creatively when encountering new problems.
In the research, the perspective of semantic mathematics promotes the exploration of the deep structure of mathematical theory, helps to reveal the relations and laws that are not fully understood in the mathematical knowledge system, and promotes the innovation and development of mathematical theory.
2.3 Definition of Semantic Mathematics
Semantic mathematics is a mathematical thinking method, which focuses on mastering mathematical knowledge and solving practical problems by understanding the deep meaning and relationship of mathematical concepts, theorems and methods. It not only pursues formal correctness, but also emphasizes the connotation understanding, intuitive feeling and practical application of mathematical knowledge.
In a word, semantic mathematics provides a more comprehensive and in-depth path of mathematics learning and research, which emphasizes the deep meaning and application of mathematical knowledge and encourages the establishment of a deep understanding of mathematics through exploration and practice.
3 Semantic mathematics applied to Master Theorem
The semantic mathematics method put forward by Professor Yucong Duan emphasizes mastering mathematical concepts through intuitive and semantic understanding. We apply this method to the interpretation of the universal Master Theorem, so as to intuitively understand the complexity analysis of the divide-and-conquer algorithm.
3.1 Overview of Master Theorem
Master Theorem is suitable for recursive algorithm in the form of T(n)=aT(n/b)+f(n), where:
T(n) represents the time needed to solve the problem with scale n.
a≥1 is the number of subproblems generated by each recursion.
b>1 is the factor for each recursive reduction of the problem scale.
f(n) is the time complexity of the operation performed in each recursion except the recursive call.
3.2 Interpretation from the Perspective of Semantic Mathematics
The division between the number of recursive operations and the scale of the problem: the relationship between a and b intuitively represents the relationship between the speed of each recursive operation and the speed of the problem scale reduction. If the influence of a is greater than that of b, it means that the number of sub-problems that need to be solved increases rapidly as the scale of the problem shrinks, which leads to the complexity of the recursive part becoming the dominant factor of the total complexity.
Case 1: a>bd, recursively dominant
Intuitive understanding: when the number of recursive operations is significantly higher than the speed of problem scale reduction, the time complexity of the whole algorithm is mainly determined by the recursive part. In this case, f(n) is relatively small and can be ignored.
Complexity:
Case 2: a=bd, equilibrium state
Intuitive understanding: when the recursive operation increases at the same speed as the scale of the problem decreases, the recursive part and the non-recursive part contribute equally to the total complexity. This means that both recursive and non-recursive operations have a significant impact on the complexity of the algorithm.
Complexity:
Case 3: a<bd, non-recursive dominant
Intuitive understanding: If the shrinking speed of the problem scale exceeds the increasing speed of recursive operation, it means that with the increase of recursive depth, the non-recursive operation f(n) of each layer has a greater impact on the total complexity.
Complexity:
From the perspective of semantic mathematics, we can understand the time complexity of divide-and-conquer algorithm in an intuitive way. This method emphasizes understanding the relationship between the growth rate of recursive operation and the reduction rate of problem scale, and their influence on the total complexity of the algorithm, thus providing a more intuitive and profound algorithm complexity analysis method.
4 Using semantic mathematics to derive mathematical formulas
Using Professor Yucong Duan's semantic mathematics method to derive mathematical formulas efficiently means that we will deduce them by deeply understanding the essential meaning, intuitive feeling and logical relationship behind the formulas. The following are some examples of derivation of mathematical formulas, which show the application of this method:
4.1 Closed form formula of Fibonacci sequence (Binet formula)
Fibonacci series is defined as F(n) = F(n-1)+F(n-2), which starts with F(0) = 0 and F(1) = 1. Using the method of semantic mathematics, we pay attention to the generation process of Fibonacci sequence and its growth mode.
Intuitive understanding and deduction:
Fibonacci series reflects a natural growth pattern, and each term is the sum of the first two terms, which implies its connection with the golden ratio.
Using the concepts of linear algebra and eigenvalue, we can get the closed form formula of Fibonacci sequence, namely Binet formula.
This deduction emphasizes the intuitive understanding of the growth pattern of series and the deep connection between series and golden ratio, not just mathematical operation.
4.2 Derivation of binomial theorem
The binomial theorem describes the expansion of the expression in the form of (a+b)n. Its standard form is.
Intuitive understanding and deduction:
From the perspective of combinatorics, binomial theorem reflects different ways of choosing objects. When we expand (a+b)n, the coefficient of each term represents the number of methods for selecting k elements from n different elements.
By recursively decomposing (a+b)n into (a+b)n-1·(a+b) and applying the combination rule, we can intuitively understand why the binomial theorem is in this form.
This derivation emphasizes the combinatorial logic behind binomial theorem, rather than simple algebraic operation.
4.3 Derivation of Euler's Formula
Euler's formula eiθ=cos(θ)+isin(θ) is a fundamental formula in mathematics, which occupies a core position in the analysis of complex numbers and transformations.
Intuitive understanding and deduction:
Euler's formula can be derived intuitively by considering the power series expansion of complex numbers and Taylor series expansion of trigonometric functions. The power series expansion of eiθ matches the Taylor series forms of cos(θ) and sin(θ).
This derivation reflects the deep connection between complex exponential function and trigonometric function, and the universality and power of Taylor series in mathematical analysis.
From the perspective of semantic mathematics, we can not only see the formulas themselves, but also understand the deep meanings and interrelations behind them, which helps us to master mathematical knowledge more deeply and discover new mathematical relations and theorems.
5 Understanding mathematical concepts through semantic mathematics
Through Professor Yucong Duan's semantic mathematics method, we can have a deeper understanding of mathematical concepts and tools in essence. Applying this way of thinking to concepts such as logarithm, Fourier series, Gaussian transform and wavelet transform can not only reveal their mathematical essence, but also understand their application value in solving practical problems. The following is the essential understanding and deduction of these concepts:
5.1 Understanding the essence of Logarithm
Semantic interpretation: Logarithm is actually a labeling and anti-labeling mechanism in power space. It establishes a new expression space, which makes multiplication operations converted into addition operations in this space, thus simplifying the handling of many mathematical problems. This transformation not only makes a lot of operations easier to handle, but also helps us to understand and analyze mathematical problems from different angles.
5.2 Understanding the essence of Fourier series
Semantic interpretation: Fourier series shows the profound meaning that any periodic function can be expressed by the superposition of basic waveforms by decomposing complex periodic signals into the sum of a series of simple sine waves and cosine waves. This decomposition not only reveals the internal structure of periodic function, but also provides a powerful tool to analyze and process signals.
5.3 Understanding the essence of Gaussian Transform
Semantic interpretation: Gaussian Transform usually involves transforming data or functions into a form with Gaussian distribution (normal distribution) characteristics. This transformation is very important in statistical analysis, signal processing and other fields, because Gaussian distribution is ubiquitous in natural and social sciences, and its properties and parameters provide an effective method to describe the characteristics and laws of data.
5.4 Understanding the Essence of Wavelet Transform
Semantic interpretation: Wavelet transform analyzes signals through a series of basis functions (wavelets) with different scales (frequencies) and positions, which can provide both time domain and frequency domain information. The idea of this transformation is based on multi-scale observation of signals, that is, signals can show different characteristics at different scales, thus providing a flexible and powerful tool for signal analysis and processing.
From the perspective of semantic mathematics, we can not only understand the operation mode and calculation rules of these mathematical tools, but also deeply grasp their internal significance and application background, so as to use these tools to solve practical problems more effectively. This method emphasizes the intuitive understanding and practical application behind mathematical concepts, and promotes the deepening and innovation of mathematical thinking.
6 The deepening and expansion of semantic mathematics
Mathematics is not only a science, but also an art. Under this framework, the semantic mathematics proposed by Professor Yucong Duan is not only an innovative teaching and research method, but also the pursuit of deep understanding of mathematics. On this basis, I want to further complete and show semantic mathematics, so as to reveal its far-reaching influence in the field of mathematics.
6.1 The "ecosystem" of mathematical concepts
Semantic mathematics emphasizes that mathematical concepts do not exist in isolation, but form an interdependent and interactive "ecosystem". Every mathematical concept can be regarded as a living body in this ecosystem, and they are connected with each other through logic and application. This perspective urges us not only to pay attention to a single concept or formula, but also to understand their position and function in the whole mathematical system.
6.2 Multidimensional nature of mathematics
Mathematics is a multi-dimensional subject, involving symbols, logic, geometry, application and other aspects. Semantic mathematics advocates crossing these levels in teaching and research and exploring the relationship between them. For example, a geometric problem can be solved by algebraic method, and a logical reasoning may come from geometric intuition. Understanding the multidimensional nature of mathematics helps us to find new ways to solve problems.
6.3 Richness of mathematical language
Mathematical language is not only formulas and symbols, but also graphics, images, stories and metaphors. Semantic mathematics encourages the use of multiple languages to express and understand mathematical concepts, and deepens the understanding of the essence of mathematical concepts through this diversity. For example, we can intuitively understand the function properties by drawing function images, or describe complex mathematical theories with stories.
6.4 Cultivation of Mathematical Thinking
Semantic mathematics not only pays attention to the teaching of mathematical knowledge, but also pays more attention to the cultivation of mathematical thinking. This kind of thinking includes logical reasoning, spatial imagination and abstract thinking. By exploring the meaning and application background of mathematical concepts, semantic mathematics helps students and researchers to establish critical and creative mathematical thinking.
Semantic mathematics is a brand-new perspective, which not only challenges the traditional mathematics teaching and research methods, but also provides a new way to deeply understand and apply mathematics. By exploring the deep meaning of mathematical concepts, understanding their internal relations, and using the multidimensional and rich expressions of mathematics, we can master mathematical knowledge more comprehensively and stimulate the innovation and development of mathematical thinking. As a member of mathematics, we have the responsibility to continue to explore and develop semantic mathematics, making it a powerful tool to promote mathematical progress and educational innovation.
随着科学技术的快速发展，数学作为基础学科的教学方法亟需改革以满足新时代的需求。本报告基于段玉聪教授提出的语义数学方法，探讨了如何将这一理念应用于分治算法的教学中，以提高学生对数学概念深层含义和实际应用能力的理解。通过对通用分治递推定理（Master Theorem）的语义数学解读为案例，我们分析了分治算法复杂度分析的直观理解和数学概念的深层意义。
报告首先概述了语义数学的核心理念，包括深化理解、直观和形象思维、语义联系及应用导向，并讨论了将这些理念融入数学和计算机科学教学的重要性。接着，通过解读通用分治递推定理，报告展示了如何使用语义数学方法来揭示复杂度分析背后的深层逻辑和直观意义，以及如何通过比较递归操作数量与问题规模缩小速度之间的关系来理解算法复杂度。
此外，报告还探讨了语义数学方法在促进学生理解、记忆和应用数学概念方面的优势，特别是在解决实际问题时如何更加灵活和创造性地运用数学工具。最后，报告总结了语义数学在数学教学改革中的潜力，提出了将其应用于更广泛数学概念教学的可能方向，以及这种方法对提高学生创新思维和解决问题能力的长远影响。
通过本报告的分析和讨论，我们希望能够为数学及相关领域的教学提供新的视角和策略，促进教育方法的创新，从而更好地准备学生面对21世纪的挑战。
通用分治递推定理（Master Theorem）提供了一种快速确定分治递归算法时间复杂度的方法。这个定理特别适用于形式为 T(n)=aT(n/b)+f(n) 的递归关系，其中：
T(n) 是解决问题的时间。
n 是问题的大小。
a≥1 表示每次递归分支的数量。
b>1 表示每次递归中问题大小的分割因子。
f(n) 是在每层递归中除了递归调用外需要的时间，通常代表分割问题和合并结果的代价。
Master Theorem 说明了三种主要情况：
情况 1
如果 对某个常数 ϵ>0 成立，则
情况 2
如果 对某个常数 k≥0 成立，则
情况 3
如果 ϵ>0 成立，并且对于某个常数 c<1 和足够大的 n，有 af(n/b)≤cf(n)，则
这三种情况涵盖了大多数使用分治策略的递归算法，能够帮助快速确定算法的渐进时间复杂度。通过分析递归算法的分割因子、递归次数以及在每次递归外的计算复杂度，Master Theorem 为算法设计者提供了一个强大的工具，以此来评估算法的性能。
段玉聪教授提出的语义数学是一个探索性的数学教学和研究方法，旨在通过深入理解数学概念的本质和意义来掌握和应用数学知识，而不仅仅是依赖公式和算法的机械记忆。这种方法强调数学思维的语义理解，即通过探索数学对象、概念、定理和方法背后的意义和内涵，以及它们之间的关系，来深化对数学的认识和应用能力。
深化理解：不仅仅满足于知道数学公式或定理是什么，而更重要的是理解它们为什么会是那样，以及它们是如何从基本原理和概念中推导出来的。
直观和形象思维：使用直观和形象的方式来理解数学概念和过程，例如通过图形、实例或物理意义来揭示数学概念的本质。
语义联系：探索数学概念、定理和方法之间的内在联系和逻辑结构，理解它们在数学体系中的位置和作用。
应用导向：强调数学知识的应用价值，通过解决实际问题来理解数学概念和方法的实用性和有效性。
在教学中，语义数学鼓励学生通过探索、讨论和实践来建立对数学概念的深入理解。这种方法帮助学生建立起对数学知识的直观感受，从而在遇到新问题时能够更加灵活和创造性地应用数学工具。
在研究中，语义数学的视角促进了对数学理论深层次结构的探究，有助于揭示数学知识体系中未被充分认识的联系和规律，推动数学理论的创新和发展。
语义数学是一种数学思维方法，它侧重于通过理解数学概念、定理和方法的深层含义和相互关系，来掌握数学知识并解决实际问题。它不仅追求形式上的正确，更强调数学知识的内涵理解、直观感受和实际应用。
总之，语义数学提供了一种更加全面和深入的数学学习和研究路径，它强调数学知识的深层意义和应用，鼓励通过探索和实践来建立对数学的深刻理解。
段玉聪教授提出的语义数学方法强调了通过直观和语义层面的理解来掌握数学概念。我们将这种方法应用于通用分治递推定理（Master Theorem）的解读，以直观地理解分治算法的复杂度分析。
Master Theorem 适用于形式为 T(n)=aT(n/b)+f(n) 的递归算法，其中：
T(n) 表示解决规模为 n 的问题所需的时间。
a≥1 是每次递归产生的子问题数量。
b>1 是问题规模每次递归减少的因子。
f(n) 是除了递归调用外在每次递归中执行的操作的时间复杂度。
递归操作数量与问题规模的分割: a 和 b 之间的关系直观地代表了每次递归操作增加的速度与问题规模缩小的速度之间的关系。如果 a 相对于 b 的影响更大，意味着随着问题规模的缩小，需要解决的子问题数量迅速增加，这导致递归部分的复杂度成为总复杂度的主导因素。
情况 1: a>bd ，递归主导
直观理解: 当递归操作的数量显著高于问题规模缩小的速度时，整个算法的时间复杂度主要由递归部分决定。在这种情况下，f(n) 相对较小，可以被忽略。
复杂度: 。
情况 2: a=bd ，平衡状态
直观理解: 当递归操作增加的速度与问题规模缩小的速度相当时，递归部分和非递归部分对总复杂度的贡献相当。这表示递归和非递归操作均对算法的复杂度有显著影响。
复杂度: 。
情况 3: a<bd ，非递归主导
直观理解: 如果问题规模的缩小速度超过了递归操作增加的速度，这意味着随着递归深度的增加，每层的非递归操作 f(n) 对总复杂度的影响更大。
复杂度: 。
通过语义数学的视角，我们能够以直观的方式理解分治算法的时间复杂度。这种方法强调了理解递归操作增长速度与问题规模减少速度之间的关系，以及它们对算法总复杂度的影响，从而提供了一种更加直观和深刻的算法复杂度分析方法。
使用段玉聪教授的语义数学方法进行数学公式的高效推导，意味着我们将通过深入理解公式背后的本质意义、直观感受和逻辑关系，来进行推导。下面是几个数学公式的推导示例，体现了这种方法的应用：
斐波那契数列定义为 F(n)=F(n−1)+F(n−2) 以F(0)=0,F(1)=1 开始。使用语义数学方法，我们关注于斐波那契数列的生成过程和它的增长模式。
直观理解与推导:
斐波那契数列反映了一种自然增长的模式，每一项是前两项之和，这暗示了它与黄金比例 的联系。
使用线性代数和特征值的概念，我们可以得到斐波那契数列的封闭形式公式，即Binet公式： 。
这种推导强调了数列增长模式的直观理解和数列与黄金比例之间的深层联系，而不仅仅是数学操作。
二项式定理描述了形如 (a+b)n 的表达式的展开。它的标准形式是 。
直观理解与推导:
从组合学的角度来看，二项式定理反映了选择对象的不同方式。当我们展开 (a+b)n 时，每一项的系数 代表了从 n 个不同元素中选择 k 个元素的方法数。
通过递归地将 (a+b)n 分解为 (a+b)n−1·(a+b) 并应用组合规则，我们可以直观地理解为什么二项式定理的形式是这样的。
这种推导强调了二项式定理背后的组合逻辑，而非简单的代数操作。
欧拉公式 eiθ=cos(θ)+isin(θ) 是数学中的一个根本公式，它在复数和变换分析中占有核心地位。
直观理解与推导:
欧拉公式可以通过考虑复数的幂级数展开和三角函数的泰勒级数展开来直观地推导。eiθ的幂级数展开与 cos(θ) 和 sin(θ) 的泰勒级数形式相匹配。
这种推导体现了复数指数函数与三角函数之间深层的联系，以及泰勒级数在数学分析中的通用性和强大性。
通过语义数学的视角，我们不仅仅看到了公式本身，还能够理解它们背后的深层含义和相互联系，这有助于我们更深刻地掌握数学知识，发现新的数学关系和定理。
通过段玉聪教授的语义数学方法，我们可以从本质上更深刻地理解数学概念和工具。将这种思考方式应用于对数、傅里叶级数、高斯变换和小波变换等概念，不仅能够揭示它们的数学本质，还能理解它们在解决实际问题中的应用价值。下面是这些概念的本质理解和推导思路：
语义解读：对数实际上是一种幂空间的标记和逆标记机制，它建立了一个新的表达空间，使得乘法运算在这个空间中转换为加法运算，从而简化了许多数学问题的处理。这种转换不仅使得大量运算变得容易处理，还帮助我们从不同的角度理解和分析数学问题。
语义解读：傅里叶级数通过将复杂的周期信号分解为一系列简单的正弦波和余弦波的和，展现了任何周期函数都可以用基本波形的叠加来表示的深刻内涵。这种分解不仅揭示了周期函数的内在结构，也提供了一种强大的工具来分析和处理信号。
语义解读：高斯变换通常涉及到将数据或函数转换为具有高斯分布（正态分布）特征的形式。这种变换在统计分析、信号处理等领域极为重要，因为高斯分布在自然和社会科学中普遍存在，它的性质和参数提供了一种有效描述数据特征和规律的方法。
语义解读：小波变换通过一系列具有不同尺度（频率）和位置的基函数（小波）来分析信号，能够同时提供时域和频域信息。这种变换的思想基于对信号的多尺度观察，即信号可以在不同的尺度上展现不同的特征，从而为信号的分析和处理提供了一种灵活而强大的工具。
通过语义数学的视角，我们不仅可以理解这些数学工具的操作方式和计算规则，还能够深刻把握它们的内在意义和应用背景，从而更有效地利用这些工具来解决实际问题。这种方法强调了数学概念背后的直观理解和实际应用，促进了数学思维的深化和创新。
数学不仅是一门科学，也是一种艺术。在这个框架下，段玉聪教授提出的语义数学不仅是一种创新的教学和研究方法，而且是对数学深层次理解的追求。在此基础上，我想对语义数学进行进一步的补全和展示，从而揭示其在数学领域的深远影响。
语义数学强调数学概念之间不是孤立存在的，而是形成了一个相互依存、相互影响的“生态系统”。每一个数学概念都可以看作是这个生态系统中的一个生命体，它们之间通过逻辑和应用关系相互连接。这种视角促使我们不仅关注单个概念或公式，而是理解它们在整个数学体系中的位置和作用。
数学是一门多维的学科，涉及符号、逻辑、几何、应用等多个层面。语义数学提倡在教学和研究中穿越这些层面，探索它们之间的联系。例如，一个几何问题可以通过代数方法解决，一个逻辑推理可能源自几何直觉。理解数学的多维性有助于我们发现新的解决问题的路径。
数学语言不仅仅是公式和符号，还包括图形、图像、故事和比喻。语义数学鼓励使用多种语言表达和理解数学概念，通过这种多样性加深对数学概念本质的理解。比如，通过绘制函数图像来直观理解函数性质，或者用故事来描述复杂的数学理论。
语义数学不仅关注数学知识的传授，更重视数学思维的培养。这种思维包括逻辑推理、空间想象、抽象思考等。通过深入探讨数学概念的意义和应用背景，语义数学帮助学生和研究者建立起批判性和创造性的数学思维。
语义数学是一种全新的视角，它不仅挑战了传统的数学教学和研究方法，还提供了一种深入理解和应用数学的新路径。通过探索数学概念的深层含义，理解它们之间的内在联系，以及利用数学的多维性和丰富的表达方式，我们可以更全面地掌握数学知识，激发数学思维的创新和发展。作为数学界的一员，我们有责任继续探索和发展语义数学，使其成为推动数学进步和教育创新的强大工具。
[1] 段玉聪(Yucong Duan). (2024). 大语言模型(LLM)偏见测评(种族偏见)(Large Language Model (LLM) Racial Bias Evaluation). DOI: 10.13140/RG.2.2.33162.03521. https://www.researchgate.net/publication/377963440_Large_Language_Model_LLM_Racial_Bias_Evaluation_--DIKWP_Research_Group_International_Standard_Evaluation_Prof_Yucong_Duan.
[2] 段玉聪(Yucong Duan). (2024). 人为什么不愿意被别人改变：DIKWP和语义数学的深入探讨(Why People Don't Want to Be Changed by Others: Insight from DIKWP and Semantic Mathematics). DOI: 10.13140/RG.2.2.17961.77927. https://www.researchgate.net/publication/377726002_Why_People_Don't_Want_to_Be_Changed_by_Others_Insight_from_DIKWP_and_Semantic_Mathematics.
[3] 段玉聪(Yucong Duan). (2024). 语义新质生产力：原理与技术(Semantic New Quality Productivity: Principles and Techniques). DOI: 10.13140/RG.2.2.14606.33607. https://www.researchgate.net/publication/377726380_Semantic_New_Quality_Productivity_Principles_and_Techniques.
[4] 段玉聪(Yucong Duan). (2024). DIKWP与语义心理学(Semantic Psychology and DIKWP). DOI: 10.13140/RG.2.2.12928.61449. https://www.researchgate.net/publication/377726404_Semantic_Psychology_and_DIKWP.
[5] 段玉聪(Yucong Duan). (2024). 基于"主观客观化"的语义不确定性处理(Semantic Uncertainty Handling Based on "Subjective Objectivisation"). DOI: 10.13140/RG.2.2.31383.55206. https://www.researchgate.net/publication/377726442_Semantic_Uncertainty_Handling_Based_on_Subjective_Objectivisation.
[6] 段玉聪(Yucong Duan). (2024). DIKWP与语义数学：创造新质生产力的融合(Semantic Mathematics and DIKWP : Creating New Qualities of Productivity). DOI: 10.13140/RG.2.2.19639.50085. https://www.researchgate.net/publication/377726532_Semantic_Mathematics_and_DIKWP_Creating_New_Qualities_of_Productivity.
[7] 段玉聪(Yucong Duan). (2024). 语义法学与DIKWP：以英美法系与大陆法系分析为例(Semantic Jurisprudence and DIKWP: Common Law vs. Continental Law). DOI: 10.13140/RG.2.2.28028.10889. https://www.researchgate.net/publication/377726622_Semantic_Jurisprudence_and_DIKWP_Common_Law_vs_Continental_Law.
[8] 段玉聪(Yucong Duan). (2024). DIKWP新质生产力与传统生产力的对比分析(DIKWP New Quality Productivity vs. Traditional Productivity Analysis). DOI: 10.13140/RG.2.2.21317.22242. https://www.researchgate.net/publication/377726626_DIKWP_New_Quality_Productivity_vs_Traditional_Productivity_Analysis.
[9] 段玉聪(Yucong Duan). (2024). 语义物理化学(Semantic Physical Chemistry). DOI: 10.13140/RG.2.2.21261.51684. https://www.researchgate.net/publication/377439785_Semantic_Physical_Chemistry.
[10] 段玉聪(Yucong Duan). (2024). DIKWP与语义认知学(DIKWP and Semantic Cognition). DOI: 10.13140/RG.2.2.14052.55680. https://www.researchgate.net/publication/377415901_DIKWP_and_Semantic_Cognition.
[11] 段玉聪(Yucong Duan). (2024). DIKWP与语义生物学：拓展跨学科的知识领域(DIKWP and Semantic Biology: Expanding Interdisciplinary Knowledge Areas). DOI: 10.13140/RG.2.2.27474.32962. https://www.researchgate.net/publication/377416091_DIKWP_and_Semantic_Biology_Expanding_Interdisciplinary_Knowledge_Areas
[12] 段玉聪(Yucong Duan). (2024). DIKWP体系与语义数学结合构建传染病防治指标体系(DIKWP System Combined with Semantic Mathematics to Construct an Indicator System for Infectious Disease Prevention and Control). DOI: 10.13140/RG.2.2.12374.83521. https://www.researchgate.net/publication/377416103_DIKWP_System_Combined_with_Semantic_Mathematics_to_Construct_an_Indicator_System_for_Infectious_Disease_Prevention_and_Control
[13] 段玉聪(Yucong Duan). (2024). DIKWP与语义哲学(DIKWP and Semantic Philosophy). DOI: 10.13140/RG.2.2.34185.21606. https://www.researchgate.net/publication/377416120_DIKWP_and_Semantic_Philosophy
[14] 段玉聪(Yucong Duan). (2024). 语义物理与创新发展(Semantic Physics and Innovation Development). DOI: 10.13140/RG.2.2.19085.72167. https://www.researchgate.net/publication/377416222_Semantic_Physics_and_Innovation_Development
[15] 段玉聪(Yucong Duan). (2024). 语义认知学：连接人类思维与计算机智能的未来(Semantic Cognition: Connecting the Human Mind to the Future of Computer Intelligence). DOI: 10.13140/RG.2.2.29152.05129. https://www.researchgate.net/publication/377416321_Semantic_Cognition_Connecting_the_Human_Mind_to_the_Future_of_Computer_Intelligence
[16] 段玉聪(Yucong Duan). (2024). 语义物理：理论与应用(Semantic Physics: Theory and Applications). DOI: 10.13140/RG.2.2.11653.93927. https://www.researchgate.net/publication/377401736_Semantic_Physics_Theory_and_Applications
[17] 段玉聪(Yucong Duan). (2024). 基于语义数学的美国和中国经济增长分析(Semantic Mathematics based Analysis of Economic Growth in the United States and China). DOI: 10.13140/RG.2.2.35980.90246. https://www.researchgate.net/publication/377401731_Semantic_Mathematics_based_Analysis_of_Economic_Growth_in_the_United_States_and_China
[18] 段玉聪(Yucong Duan). (2024). Collatz Conjecture的语义数学探索(Collatz Conjecture's Semantic Mathematics Exploration). DOI: 10.13140/RG.2.2.28517.99041. https://www.researchgate.net/publication/377239567_Collatz_Conjecture's_Semantic_Mathematics_Exploration
[19] 段玉聪(Yucong Duan). (2024). 语义数学与 DIKWP 模型（本质计算与推理、存在计算与推理以及意图计算与推理）(Semantic Mathematics and DIKWP Model (Essence Computation and Reasoning, Existence Computation and Reasoning, and Purpose Computation and Reasoning)). DOI: 10.13140/RG.2.2.24323.68648. 377239628_Semantic_Mathematics_and_DIKWP_Model_Essence_Computation_and_Reasoning_Existence_Computation_and_Reasoning_and_Purpose_Computation_and_Reasoning
[20] 段玉聪(Yucong Duan). (2024). 从主观到客观的语义数学重构（存在计算与推理、本质计算与推理、意图计算与推理）(Semantic Mathematics Reconstruction from Subjectivity to Objectivity (Existence Computation and Reasoning, Essence Computing and Reasoning, Purpose Computing and Reasoning)). DOI: 10.13140/RG.2.2.32469.81120. https://www.researchgate.net/publication/377158883_Semantic_Mathematics_Reconstruction_from_Subjectivity_to_Objectivity_Existence_Computation_and_Reasoning_Essence_Computing_and_Reasoning_Purpose_Computing_and_Reasoning
[21] 段玉聪(Yucong Duan). (2024). DIKWP与语义数学在车票订购案例中的应用(DIKWP and Semantic Mathematics in the Case of Ticket Ordering). DOI: 10.13140/RG.2.2.35422.20800. https://www.researchgate.net/publication/377085570_DIKWP_and_Semantic_Mathematics_in_the_Case_of_Ticket_Ordering
[22] 段玉聪(Yucong Duan). (2024). DIKWP与语义数学分析《论语》“君子和而不同，小人同而不和”(DIKWP and Semantic Mathematical Analysis The Confluent Analects Gentleman is harmonious but different, while petty people are the same but not harmonious). DOI: 10.13140/RG.2.2.28711.32165. https://www.researchgate.net/publication/377085455_DIKWP_and_Semantic_Mathematical_Analysis_The_Confluent_Analects_Gentleman_is_harmonious_but_different_while_petty_people_are_the_same_but_not_harmonious
[23] 段玉聪(Yucong Duan). (2023). DIKWP 人工意识芯片的设计与应用(DIKWP Artificial Consciousness Chip Design and Application). DOI: 10.13140/RG.2.2.14306.50881. https://www.researchgate.net/publication/376982029_DIKWP_Artificial_Consciousness_Chip_Design_and_Application
[24] 段玉聪(Yucong Duan). (2024). 直觉的本质与意识理论的交互关系(The Essence of Intuition and Its Interaction with theory of Consciousness). DOI: 10.13140/RG.2.2.16556.85127. https://www.researchgate.net/publication/378315211_The_Essence_of_Intuition_and_Its_Interaction_with_theory_of_Consciousness
[25] 段玉聪(Yucong Duan). (2024). 意识中的“BUG”：探索抽象语义的本质(Understanding the Essence of "BUG" in Consciousness: A Journey into the Abstraction of Semantic Wholeness). DOI: 10.13140/RG.2.2.29978.62409. https://www.researchgate.net/publication/378315372_Understanding_the_Essence_of_BUG_in_Consciousness_A_Journey_into_the_Abstraction_of_Semantic_Wholeness
[26] 段玉聪(Yucong Duan). (2024). 个人和集体的人造意识(Individual and Collective Artificial Consciousness). DOI: 10.13140/RG.2.2.20274.38082. https://www.researchgate.net/publication/378302882_Individual_and_Collective_Artificial_Consciousness
[27] 段玉聪(Yucong Duan). (2024). 人工意识系统的存在性探究：从个体到群体层面的视角(The Existence of Artificial Consciousness Systems: A Perspective from Group Consciousness). DOI: 10.13140/RG.2.2.28662.98889. https://www.researchgate.net/publication/378302893_The_Existence_of_Artificial_Consciousness_Systems_A_Perspective_from_Collective_Consciousness
[28] 段玉聪(Yucong Duan). (2024). 意识与潜意识：处理能力的有限性与BUG的错觉(Consciousness and Subconsciousness: from Limitation of Processing to the Illusion of BUG). DOI: 10.13140/RG.2.2.13563.49447. https://www.researchgate.net/publication/378303461_Consciousness_and_Subconsciousness_from_Limitation_of_Processing_to_the_Illusion_of_BUG
[29] 段玉聪(Yucong Duan). (2024). 如果人是一个文字接龙机器，意识不过是BUG(If Human is a Word Solitaire Machine, Consciousness is Just a Bug). DOI: 10.13140/RG.2.2.13563.49447. https://www.researchgate.net/publication/378303461_Consciousness_and_Subconsciousness_from_Limitation_of_Processing_to_the_Illusion_of_BUG
[30] 段玉聪(Yucong Duan). (2024). 超越达尔文：技术、社会与意识进化中的新适应性(Beyond Darwin: New Adaptations in the Evolution of Technology, Society, and Consciousness). DOI: 10.13140/RG.2.2.29265.92001. https://www.researchgate.net/publication/378290072_Beyond_Darwin_New_Adaptations_in_the_Evolution_of_Technology_Society_and_Consciousness
[31] 段玉聪(Yucong Duan). (2024). 【人物】段玉聪：未来人工意识的发展：消除“bug”之路. 应用观察. https://mp.weixin.qq.com/s/q0eA97OPW0f30D9rXEKuPQ
[32] 段玉聪(Yucong Duan). (2024). 【视角】段玉聪：直觉的本质与意识理论的交互关系. 应用观察. https://mp.weixin.qq.com/s/8nZJZobAFpqIdriahe-wMQ
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.
Introduction of Prof. Yucong Duan
l Founder of the DIKWP-AC Artificial Consciousness (Global) Team
l Founder of the AGI-AIGC-GPT Evaluation DIKWP (Global) Laboratory
l Initiator of the World Artificial Consciousness Conference (Artificial Consciousness 2023, AC2023, AC2024)
l Initiator of the International Data, Information, Knowledge, Wisdom Conference (IEEE DIKW 2021, 2022, 2023)
l The only one selected for the "Lifetime Scientific Impact Leaderboard" of top global scientists in Hainan Information Technology by Stanford
l The sole recipient of the national award in the field of AI technology invention in Hainan (Wu Wenjun Artificial Intelligence Award)
l Holder of the best record for the China Innovation Method Contest Finals (representing Hainan)
l The individual with the highest number of granted invention patents in the field of information technology in Hainan Province
l Holder of the best achievement for Hainan in the National Enterprise Innovation Efficiency Contest
l Holder of the best performance for Hainan in the National Finals of the AI Application Scenario Innovation Challenge
l Hainan Province's Most Outstanding Science and Technology Worker (also selected as a national candidate)
Professor at Hainan University, doctoral supervisor, selected as part of the first batch for the Hainan Province South China Sea Eminent Scholars Plan and Hainan Province Leading Talents. Graduated from the Institute of Software, Chinese Academy of Sciences in 2006, he has worked and studied at Tsinghua University, Capital Medical University, POSTECH in South Korea, French National Centre for Scientific Research, Charles University in Prague, University of Milan-Bicocca, and Missouri State University in the USA. He currently serves as a member of the Academic Committee of the College of Computer Science and Technology at Hainan University, leader of the DIKWP Innovation Team at Hainan University, senior advisor to the Beijing Credit Association, distinguished researcher at Chongqing Police College, leader of the Hainan Province Double Hundred Talents Team, vice president of the Hainan Inventors Association, vice president of the Hainan Intellectual Property Association, vice president of the Hainan Low-Carbon Economic Development Promotion Association, vice president of the Hainan Agricultural Products Processing Enterprise Association, director of the Hainan Cyber Security and Informatization Association, director of the Hainan Artificial Intelligence Society, member of the Medical and Engineering Integration Branch of the China Health Care Association, visiting researcher at Central Michigan University, and member of the PhD advisory committee at the University of Modena in Italy. Since being introduced to Hainan University as a Class D talent in 2012, he has published over 260 papers, with more than 120 indexed by SCI, 11 highly cited by ESI, and over 4500 citations. He has designed 241 Chinese national and international invention patents for various industries and fields, including 15 PCT patents, and has been granted 85 patents as the first inventor. In 2020, he received the Third Prize of the Wu Wenjun Artificial Intelligence Technology Invention Award; in 2021, he independently initiated the first IEEE DIKW 2021 as the chair of the program committee; in 2022, he served as the chair of the steering committee for IEEE DIKW 2022; in 2023, he served as the chair of IEEE DIKW 2023. In 2022, he was named the most beautiful science and technology worker in Hainan Province (and recommended for national recognition); in 2022 and 2023, he was consecutively listed in the "Lifetime Scientific Impact Leaderboard" of the world's top 2% scientists published by Stanford University. He has participated in the development of 2 international standards for the IEEE Financial Knowledge Graph and 4 industry standards for knowledge graphs. In 2023, he initiated and co-organized the first World Artificial Consciousness Conference (Artificial Consciousness 2023, AC2023).
Prof. Yucong Duan
DIKWP-AC Artificial Consciousness Laboratory
AGI-AIGC-GPT Evaluation DIKWP (Global) Laboratory
World Association of Artificial Consciousness
duanyucong@hotmail.com
The 2nd World Congress of Artificial Consciousness (AC2024) looks forward to your participation
http://yucongduan.org/DIKWP-AC/2024/#/
https://blog.sciencenet.cn/blog-3429562-1424382.html
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