2024-陆慧-莱布尼茨连续律的哲学研究
博论题目：莱布尼茨连续律的哲学研究
答辩人：陆慧
指导老师：刘大椿
答辩时间：2024年5月25日
目 录
导 论
0.1 莱布尼茨其人
0.1.1 喜忧参半的人生轨道
0.1.2 文理并蓄的全能巨星
0.1.3 生时花未绽，身后名声扬
0.2 聚焦“莱布尼茨连续律”研究
0.2.1 连续律并非莱布尼茨的首创
0.2.2 莱布尼茨连续律的独特性
0.2.3 亟待澄清的争议难题
0.2.4 研究意义
0.3 国内外研究综述
0.4 研究内容、思路、方法
第1章 莱布尼茨连续律的缘源
1.1 “连续律”的词源考释
1.1.1 “Lex Continuitatis”拉丁语释义
1.1.2 “Lex”与“Legem”区分
1.1.3 “Law”与“Principle”辨析
1.2 “连续律”的相关概念
1.2.1 连续性
1.2.2 离散性
1.2.3 连续统
1.3 “连续律”的历史演进
1.3.1 早期朴素直观的连续思维
1.3.2 古希腊自然与哲学连续性
1.3.3 开普勒严格量化的连续性
1.3.4 近代无限小与函数连续律
1.4 莱布尼茨连续律的形成背景
1.4.1 经典线性科学框架
1.4.2 数学虚构主义传统
1.4.3 连续体迷宫的争议
1.4.4 神学最佳合理设计
1.5 本章小结
第2章 莱布尼茨连续律的发展脉络
2.1 莱布尼茨连续律的演化轨迹
2.1.1 早期从“无拐点的曲线”出发：触及数学连续律
2.1.2 中期从“微差异的逼近”出发：界定数学连续律
2.1.3 后期从“自然不做飞跃”出发：升华至哲学连续律
2.2 函数的平滑可导：数学连续律的节点
2.2.1 源于“无限”的和谐
2.2.2 无间断点的函数
2.2.3 检验平滑性的自然律
2.2.4 简明逻辑符号表述
2.3 共存的推演秩序：哲学连续律的节点
2.3.1 时间之流：现在承载过去，孕育未来
2.3.2 粘性之褶：弹性与内聚力
2.3.3 位置之跃：“超创造”予以补救
2.3.4 缺口之美：“断”是“连”的必要
2.4 莱布尼茨连续律的特征
2.4.1 理想性
2.4.2 无界性
2.4.3 自相似性
2.4.4 动态生成性
2.5 本章小结
第3章 数学连续律与微积分的创立和发展
3.1 莱布尼茨微积分的孕育
3.1.1 自然科学计算的需求
3.1.2 曲线切线问题的推进
3.1.3 莱氏的数学逐梦之志
3.1.4 牛顿的运动“流数术”
3.2 连续律与微积分的成长
3.2.1 连续律与无穷小量的结合
3.2.2 连续律为导数提供依据
3.2.3 连续律判断积分的存在值
3.2.4 连续律确保互逆运算
3.3 连续律在数学中的拓展应用
3.3.1 推导有限量规则至无限
3.3.2 保证无穷数列的极限值
3.3.3 消除不规则图形的边界
3.4 本章小结
第4章 从数学连续律到哲学连续律的升华
4.1 有关数学与哲学关系之争议
4.1.1 争议的厘清与对立
4.1.2 争议的根源与解决
4.2 莱布尼茨形而上学的数学化趋势
4.2.1 数学启迪哲学概念的定义
4.2.2 哲学借鉴数学的论证方式
4.2.3 数学符号简化哲学命题
4.2.4 数学与形而上学的共通性
4.3 数学连续律通向哲学连续律的路径
4.3.1 哲学连续律弥补数学的限度
4.3.2 数学连续律提供哲学以方法
4.3.3 哲学连续律借鉴数学的精确推理
4.3.4 上帝对连续律双重性的预定安排
4.4 本章小结
第5章 “粘合剂”：哲学连续律与认知秩序
5.1 用以解释单子与世界的联结
5.1.1 什么是单子
5.1.2 单子的等级·知觉·记忆
5.1.3 单子与世界的连续函数结构
5.2 予以保障前定和谐与因果的协调
5.2.1 什么是前定和谐
5.2.2 因果间的必然与偶然
5.2.3 连续律是前定和谐论的保障
5.3 赋以彰显认知与真理的统一
5.3.1 知觉与心理意识的同一
5.3.2 真理知识与法律的推演
5.3.3 字符与命题主谓的运算
5.3.4 时空与现象的共存秩序
5.4 本章小结
第6章 “脚手架”：哲学连续律与实证科学
6.1 地质科学：化石演化稳定与突变的共生
6.1.1 地球演进的“局部动荡与整体稳定”
6.1.2 海陆冲击的“恒久周期与矢量进步”
6.1.3 矿物金属分解自相似的微小结构
6.2 生物科学：物种进化差异与趋同张力
6.2.1 显微镜下的微观揭秘
6.2.2 预成论与差异中的同一
6.2.3 物种分类的非确定边界
6.3 物理科学：对笛卡尔碰撞定理的批判
6.3.1 死力在撞击中转化为活力
6.3.2 连续律弥补碰撞律的缺陷
6.3.3 静止作为运动消逝的极限
6.4 本章小结
第7章 莱布尼茨连续律的当代审度
7.1 连续律对当代数学的积极影响
7.1.1 康托尔“连续统假设”的启示
7.1.2 拓扑空间连通性的继承
7.1.3 函数论与迭代法的促进
7.2 连续律在科学中的限度
7.2.1 非线性科学的分形质疑
7.2.2 复杂性与突变论的挑战
7.2.3 量子力学波粒能量的间断
7.3 连续律的定位：伟大但非万能
7.3.1 连续律与过程哲学互补
7.3.2 连续律提供可能性思考
7.3.3 辩证审度：非“万能剂”
7.4 本章小结
结 语
参考文献
致 谢
摘 要
莱布尼茨的“连续律”是关于不间断或连贯的规律，强调自然界的变化呈现出平滑的特性，主张特定领域内的运算或转换在极限情况下依然有效，渐进变化的物体在趋近临界点时仍保留某些属性。他试图用定律或原则搭建起沟通数学、哲学和科学的桥梁，连续性定律便是其中的典范，它是统一这三个领域的基本思维原则，属于“元哲学”层面的“基质”。
作为莱布尼茨形而上学的数学化趋势下的典型产物，“连续律”在早期是一个数学本体论的动态概念，核心思想在于对定义域内通过各点的无穷小变化对处处可导曲线进行界定。它是莱布尼茨独立于牛顿在微积分方面的“亮点”贡献。后来又升华为对自然法则统一性和协调性的承诺，被巧妙地融入单子论、前定和谐系统、知识论和自然科学理论中。连续律在多个学科中的成功显现，表明现象在一定条件下能够以平稳的方式过渡和转变，以避免某些行为的突变或断裂。连续律概念创建了一个旨在解释自然界运行机制的理想框架。
导论介绍莱布尼茨其人，作为兼具数学家与哲学家身份的学术先驱，他的数学成就为形而上学奠定了根基。喜忧参半的一生铸就了他在思想与行动上远远超前于其所处时代的前瞻性洞察。随着国际莱布尼茨全集的陆续问世，对这位全能巨匠的挖掘之路愈发广阔。本文将聚焦于莱布尼茨连续律的研究，探讨其超越于传统连续律的自然科学价值、动态轨迹、数学起源与争议疑题。
第一章厘清连续性定律的缘源。对“连续律”进行词源考释与辨析，审视其区别于连续性、离散性与连续统等相近概念的独特之处。连续律并非是莱布尼茨的首创，它经历了从早期朴素直观的连续思维到古希腊自然与哲学连续性，到开普勒严格量化的连续性，再到近代无限小与函数连续律的历史演进。此定律诞生于自然哲学与神学的“温床”，受到经典线性科学框架、数学虚构主义传统、连续体迷宫争议以及神学最佳设计的影响。
第二章挖掘莱布尼茨连续律的发展脉络与节点。莱布尼茨对连续律的思考并非“一锤定音”，而是历经从孕育到成熟的发展阶段，从早期对“无拐点曲线”到中期“微差异逼近”的界定，后期升华至“自然不做飞跃”的守恒原则。数学连续律可定义为：函数所遵从的一条处处可导的规律，源于无限，是检验曲线平滑与否的原则；哲学连续律则蕴含着一种共存的推演秩序，涵盖“时间之流、粘性之褶、位置之跃、缺口之美”四个维度，呈现出理想性、无界性、自相似性与动态生成性。
第三章探究数学连续律如何参与并促进了微积分的创立和发展——它承载着数学量的无限推导，充当着弥合有限与无限的桥梁，规定适用于有限数的数学规则同样适用于无限。连续律与微积分相辅相成，体现在连续律与无穷小量的微妙结合、为导数提供理论依据、判断积分存在值的问题、确保微分与积分互逆过程的逻辑一致性。
第四章承上启下，深入论证数学连续律向哲学连续律的升华路径以及连续律是莱布尼茨形而上学的数学化趋势下的典型产物。从“数学启迪哲学概念的直观定义、哲学借鉴数学的几何论证方式、数学符号简化命题、数学与形而上学的共通性”四个方面诠释莱布尼茨思想中形而上学的数学化倾向。从“哲学原则弥补数学限度、数学连续律提供方法、哲学连续律借鉴数学的精确推理、上帝对连续律双重属性的预定安排”维度揭秘连续律从数学至哲学的升华过程。
第五章分析哲学连续律在认知中的“粘合剂”功效——确保哲学范畴的等级与和谐秩序。连续性原则为莱布尼茨拒绝物质原子主义提供了关键依据，它蕴含着事物的不同状态与属性之间的整体联系，构建了单子自身运动与记忆的连续之链，单子与世界之间的函数结构折射出小-大世界间的映射关系。连续律蕴含的因果确定性为前定和谐律的合理性提供了保障，赋予哲学认知与事物间同一性与共存性秩序。
第六章剖析连续律在实证科学中扮演的“脚手架”角色——坚实地支撑物质的溯源与转化，异质事物在连续性思维中被统一化与同质化，解决了临界与跨界问题。莱布尼茨将连续律应用于地质科学，从化石演化的稳定与突变中揭示生命的延续，在生物科学领域探讨物种进化中差异与趋同的张力，在物理科学中借助连续律对看似矛盾的物理状态给予完美的中和，例如活力与死力、静止与运动，进而驳斥笛卡尔的碰撞定律。
第七章审度莱布尼茨连续律在当代数学、科学与哲学中的“功过是非”。尽管它具有诸多优势，但这无法支撑其“万能法则”地位。当代数学的集合论连续性、拓扑连续性与函数论迭代法在一定程度上延续了连续律的核心理念。但由于连续律根植于经典牛顿力学的线性科学框架，携带着固有的时代局限性与滞后性。当下非线性科学、复杂性理论及量子力学等科学现象已然站在连续律的对立面，世界呈现出“离散、分形、不确定、混沌”等样态。因此需要与时俱进地运用更前沿的数学工具和物理框架描述复杂的现象。简言之，连续律伟大但非万能。
关键词：莱布尼茨；数学连续律；微积分；哲学连续律；实证科学
Abstract
Leibniz's “law of continuity” is about uninterrupted or coherent laws, emphasizing that changes in nature show the characteristics of smoothness. It advocates that operations or transformation in specific fields is still valid under extreme cases, and objects undergoing gradual changes will retain some attributes when it approaches the critical points. Leibniz attempts to build a bridge between the fields of mathematics, philosophy, and science using laws or principles. The law of continuity is a typical representative, which is the basic thinking principle of unifying these three fields and belongs to the “substrate” of the “metaphysics” level.
As a typical product of the metaphysical trend in Leibniz's mathematics, the law of continuity first appears as a dynamic concept in mathematical ontology. Its core idea is to define the everywhere differentiable curves by infinitesimal changes in each point within the definition domain. This is Leibniz's outstanding “highlight” contribution to calculus that is independent of Newton. Later, it evolves into a commitment to the unity and harmony of the laws of nature, and skillfully integrated into theories such as monadology, pre-established harmony, epistemology, and natural science. The successful application of the law of continuity in many disciplines demonstrates that phenomena can transition and transform smoothly under certain conditions, avoiding the sudden mutations or breaks of certain behaviors. The basic concepts of the law of continuity establishes an ideal framework for explaining the operating mechanisms of nature.
The introduction introduces Leibniz, a scholarly pioneer who straddled the roles of mathematician and philosopher. His mathematical achievements laid the groundwork for metaphysics. His tumultuous life shapes remarkably forward-thinking insights in both thought and action, far ahead of his time. With the gradual publication of the international Leibniz complete works, the exploration of this polymath’s legacy becomes increasingly expansive. This paper will focus on the study of Leibniz’s law of continuity, exploring its natural scientific value beyond traditional law, its dynamic trajectory, mathematical origins, and controversial questions.
Chapter 1 clarifies the origins of the law of continuity. I deeply study the etymological explanation of this concept, and expound its unique features that distinguishes it from similar concepts such as continuity, discreteness and continuum. The law of continuity is not invented by Leibniz. It has gone through a historical evolution from simple and intuitive continuous thinking in the early days to the continuity of nature and philosophy in ancient Greece, to the continuity of Kepler's strict quantification, and to the development of modern infinitesimal calculus and the continuity of functions. This law was born in a “nourishing soil” of natural philosophy and theology, including the fusion of classical linear scientific frameworks, the tradition of mathematical fictionalism, the controversy of continuum labyrinth, and the best theological designs.
Chapter 2 explores the development process and nodes of Leibniz’s law of continuity. Leibniz's thinking on the law of continuity is not achieved overnight, but has gone through several stages from conception to maturity. It evolves from early considerations of “curve without turning points” to mid-term definition of “infinitesimal approximation” and finally sublimate to the principle of “nature does not make leaps” in later stages. Mathematically, the law of continuity can be defined as a rule followed by functions that is differentiated everywhere, originating from infinity, as a criterion for testing the smoothness of curves. Philosophically speaking, the law of continuity implies a coexisting deductive order, which includes four dimensions: “the flow of time, the folding of adhesive, the jumping of position, and the beauty of gaps”, and shows these characteristics of ideality, unboundedness, self-similarity, and dynamic generation.
Chapter 3 delves into how the mathematical law of continuity participates in and promotes the creation of calculus - it carries the infinite derivation of mathematical quantities, acting as a bridge between the finite and infinite, and stipulates that the mathematical rules of finite numbers are also applicable to the infinite. The law of continuity and calculus complement each other, which is embodied in the subtle integration of continuity and infinitesimal, providing theoretical basis for derivatives, judging the existence value of integrals, and ensuring the logical consistency in the inverse processes of differentiation and integration.
Chapter 4 builds on the previous contents and demonstrates in depth the sublimation path from mathematical law of continuity to philosophical law of continuity in detail, and elaborates that law of continuity is a typical product of the metaphysical trend of Leibniz’s mathematics. I interpret the metaphysical tendency of mathematics in Leibniz's thought from four aspects: “the intuitive definition of philosophical concepts inspired by mathematics, the geometric argumentation method that philosophy draws lessons from mathematics, proposition of simplification of mathematical symbols, and the commonality between mathematics and metaphysics”. The sublimation process of the law of continuity from mathematics to philosophy is revealed from the dimension of “the philosophical principles that supplements the limitations of mathematics, mathematical law of continuity providing methods, philosophical law of continuity drawing lessons from mathematical precise reasoning, and God's predetermined arrangements of the dual attributes of the law of continuity.”
Chapter 5 analyzes the “adhesive” function of the law of continuity in philosophical cognition - ensuring the hierarchy and harmonious order of philosophical categories. The principle of continuity provided a vital basis for Leibniz to reject the atomism of matter, which embodies the holistic connection between different states and attributes of things. It has built a continuous chain of motion and memory for monads. The functional structure of the monads and the world reflects the mapping relationship between the microcosm and the macrocosm. The certainty of cause and effect adhered to by the law of continuity provides a guarantee for the rationality of the predetermined harmony, endowing philosophical cognition with the order of identity and coexistence for matter.
Chapter 6 interprets the role of the law of continuity as “scaffolding” in empirical science - solidly supporting the traceability and transformation of matter, heterogeneous entities are unified and homogenized in continuous thinking, resolving issues of thresholds and boundaries. Leibniz applies the law of continuity to geological science, revealing the continuation of life from the stability and mutation of fossil evolution. In the field of biological science, he explored the tension between differences and convergences in species evolution. In physical sciences, he uses the law of continuity to reconcile seemingly contradictory physical states, such as living force and dead force, stillness and motion, thus refuting Descartes' law of collision.
Chapter 7 evaluates the advantages and disadvantages of Leibniz’s law of continuity in contemporary mathematics, science and philosophy. Although the law of continuity, as a core principle running through Leibniz's thought, possesses numerous advantages. But these cannot support its “omnipotent” position. Contemporary mathematical theories such as set-theoretic continuity, topological continuity, and iterative methods in functional analysis to some extent extend and develop the core ideas of the law of continuity. However, because the law of continuity is rooted in the linear scientific framework of classical Newtonian mechanics, it has its inherent limitations and lags of the era. Current scientific phenomena, such as nonlinear science, complexity theory, and wave-particle duality of quantum mechanics are already on the opposite side of the law of continuity, and the world appears to be “discrete, fractal, uncertain, and chaotic.” Therefore, it is necessary to keep pace with the times and use cutting-edge mathematical tools as well as physical frameworks to describe complex phenomena. In short, its position can be summarized in one sentence: the law of continuity is great but not omnipotent.
Keywords: Leibniz; Mathematical law of continuity; Calculus; Philosophical law of continuity; Empirical Science
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