立体矩阵论:开放复杂系统的新数学语言

摘要 微积分支撑了爱因斯坦时代,为连续变化提供了语言;复杂系统时代以涌现、自组织、鲁棒性与开放性为标志,需要的不是微积分的改良,而是本体论预设与之相反的另一门语言。本文论证,SM立体矩阵论(Stereo Matrix,简称SM)是这门语言的成熟候选:它不是凭空出现的记号系统,而是历史推动与选择的结果,已形成公理体系、标准工具库与多领域验证成果。本文提出并论证三项贡献。其一,结构一致性公理:结构一致是计算的前提,连续性受分辨率边界与结构一致性边界双重约束,后者更严苛,连续演化不得逃出结构域。其二,分辨率纲领:离散是本体,连续是离散在不同观测分辨率下的有效逼近,且在SM内部分辨率可由σ谱档位量化。其三,三能力证据:SM作为语言具备判定力、表达力与生产力,并能以负结果入册的方式自检。SM不是对既有数学的替代,而是开放复杂系统数学语言的合格标准示范。

Abstract Calculus underpinned the Einstein era, giving language to continuous change; the era of complex systems, marked by emergence, self-organization, robustness and openness, needs not an improvement of calculus but another language whose ontological presuppositions are its opposite. This paper argues that SM stereo matrix theory (Stereo Matrix, SM for short) is a mature candidate for that language: it is not a notational system that appeared from nowhere, but the result of historical impetus and selection, having already formed an axiom system, a standard toolkit and verification results across many domains. Three contributions are proposed and argued. First, the structural-consistency axiom: structural consistency is the precondition of computation; continuity is doubly constrained by a resolution boundary and a structural-consistency boundary, the latter being the stricter, and continuous evolution may not escape the structural domain. Second, the resolution programme: the discrete is the ontology, the continuous is an effective approximation of the discrete at different observational resolutions, and within SM the resolution can be quantified by σ-spectrum grades. Third, evidence of three capabilities: as a language SM possesses deciding power, expressive power and productive power, and can self-audit by registering negative results. SM is not a replacement of existing mathematics but a qualified standard demonstration of a mathematical language for open complex systems.

本记录含中文版与英文版两个文件(Bilingual: Chinese + English DOCX)。