博文

从弗雷格困境到道函数零点: 融智学序位逻辑对数学基础的彻底重构

已有 70 次阅读 2026-9-18 19:35 |个人分类:学术研究|系统分类:教学心得

Method

File available

From Frege's Predicament to the Dao Function Zero Point

-A Radical Reconstruction of the Foundations of Mathematics via Intelligics Order-Position Logic

In the late nineteenth century, G. Frege attempted to define 0 and 1 from logic, aiming to ground arithmetic and the entire mathematical edifice on logic. The discovery of Rus-sell's Paradox completely destroyed this logicist program. This paper argues that the root of Frege's failure lies in his attempt to artificially define 0 and 1 without finding the true generative source of mathematics. We propose a radical alternative grounded in Intelligics (Smart System Studies): the Order-Position Logic and the Dao Function Zero Point (O D). The Dao is the zero point. From O D , along any direction of the Cartesian 1D number line, 2D coordinates, 3D space, or even Hilbert's N-dimensional and infinite-dimensional coordinate systems, adding one unit yields 1; 0 and 1 constitute the binary system, and its direction is positive. Thus, 0 and 1 are no longer "defined" but "generated." The Unique Conservation of Order-Position Relations ensures the stability of the entire mathematical edifice during its generation. We prove that the Dao Function Zero Point not only resolves Frege's predicament but also completes the fundamental transformation of mathematical foundations from "logical definition" to "order-position generation." Furthermore, we show that the aleph numbers ℵ 0 , ℵ 1 , ℵ 2 ,. .. are merely projections of O D along one particular path-the natural number arithmetic path-and that the Continuum Hypothesis is independent precisely because it conflates different generative paths from O D. The entire mathematical structure, from 0 and 1 through P-ary systems to the eight great forms (char-acter, formula, diagram, table, sound, image, three-dimensional, dynamic), grows out of the Dao Function Zero Point through the Twin Turing Machine and the Linkage Function.

MSC 2020: 03A05 (Philosophical and critical), 03E10 (Ordinal and cardinal numbers), 03E50 (Continuum hypothesis), 03B80 (Other applications of logic), 68Q05 (Models of computation)

从弗雷格困境到道函数零点

融智学序位逻辑对数学基础的彻底重构

中文摘要： 十九世纪末，弗雷格（G. Frege）试图从逻辑定义 0 和 1，旨在将算术乃至整个数学大厦建立在逻辑之上。罗素悖论的发现彻底摧毁了这一逻辑主义纲领。本文认为，弗雷格失败的根源在于：他试图人为定义 0 和 1，而没有找到数学的真正生成源头。我们提出一个植根于融智学（智能系统研究）的彻底替代方案：序位逻辑与道函数零点（OD ）。道即是零点。从 OD 出发，沿笛卡尔一维数轴、二维坐标、三维空间，乃至希尔伯特 N 维与无穷维坐标系的任意一个方向，加上一个单位，就得到 1 ；0 和 1 构成二进制，其方向即为正。因此， 0 和 1 不再是 “被定义的 ” ，而是 “被生成的 ” 。序位关系唯一守恒保证了整个数学大厦在生成过程中的稳定。我们证明：道函数零点不仅解决了弗雷格的困境，更完成了数学基础从 “逻辑定义 ” 到 “序位生成 ” 的根本转变。此外，我们表明：阿列夫数 ℵ0 , ℵ 1 , ℵ2 , ... 仅仅是 OD 沿一条特定路径——自然数算术路径——的投影；连续统假设之所以独立，正是因为它混淆了OD 的不同生成路径。整个数学结构，从 0 和 1 经 P 进制系统到八大形式（字、式、图、表、音、像、立、活），皆通过孪生图灵机与联动函数从道函数零点生长出来。