分形的成因及影响探析——纪念芒德勃罗诞辰100周年  

Analysis on the Cause and Influence of Fractal:In Memory of the 100th Anniversary of Mandelbrot’s Birthday

作　　者：江南[1,2] 王昌[2] 杜永峰[3] JIANG Nan;WANG Chang;DU Yong-feng(School of Science,Xi’an Shiyou University,Xi’an 710065;Institute for Advanced Study in History of Science,Northwest University,Xi’an 710127;School of Marxism,Xi’an Shiyou University,Xi’an 710065,China)

机构地区：[1]西安石油大学理学院,西安710065 [2]西北大学科学史高等研究院,西安710127 [3]西安石油大学马克思主义学院,西安710065

出　　处：《自然辩证法研究》2024年第9期115-122,共8页Studies in Dialectics of Nature

基　　金：陕西省社会科学基金年度项目“分形理论的形成及其非线性思想研究”(2023C004);中国博士后科学基金资助项目“分形几何的历史演变探究”(2020M683693XB)。

摘　　要：分析严格化后期,魏尔斯特拉斯函数、康托尔集、科赫曲线等许多病态数学客体不断涌现,与之相关的数学问题层出不穷。在它们的驱使下,豪斯多夫等数学家创建和完善了分数维理论,切萨罗等数学家建立和发展了自相似理论。与此同时,海岸线长度悖论等实际问题也层见迭出。内外因的交互作用促使芒德勃罗从个性中提炼共性,从无序中寻找有序,从外在表象嵌入内在本质,深入破解问题,抽象升华理论,最终创立了分形几何。严格的分形概念历经演变,至今仍在求索,建立在无穷小理论和非线性思想之上的分数维和自相似特征揭示了它的内涵和外延。作为新生事物的分形虽在创立之初招致强烈反对,但与生俱来的优良属性一定会使它未来可期。In the later period of analysis rigor,many ill conditioned mathematical objects such as Weierstrass function,Cantor set,Koch curve and so on continue to emerge,and mathematical problems related to them also emerge in endlessly.Driven by these problems,Hausdorff and other mathematicians created and improved the theory of fractional dimension.Cesàro and other mathematicians established and developed the self-similarity theory.At the same time,practical issues such as the paradox of coastline length also emerge one after another.The interaction between internal and external factors led Mandelbrot to extract commonalities from individuality,search for order from disorder,embed internal essence from external appearances,deeply solve problems,abstract and sublimate theory,and ultimately establish fractal geometry.The strict concept of fractal has undergone evolution and is still being explored to this day.The fractal dimension and self similarity features based on infinitesimal theory and nonlinear thinking reveal its connotation and extension.Although fractal,as a new thing,was opposed at the beginning of its establishment,its own unique excellent attributes will certainly make it perfect in the future.

关 键 词：数学史 分形 分数维 芒德勃罗 

分 类 号：N09[自然科学总论—科学技术哲学]