论牛顿代数研究的数学实验思想
On the Mathematical Experimental Ideas in the Study of Newtonian Algebra
DOI: 10.12677/pm.2024.147278, PDF,   
作者: 杨欣童：清华大学人文学院科学史系，北京
关键词: 牛顿；代数学；数学实验；普遍算术；Newton； Algebra； Mathematical Experiments； Universal Arithmetic
摘要: 牛顿在自己的代数学研究过程中曾运用过多种方法，但数学实验的思想方法是其应用最多的。在《普遍算术》一书中，牛顿给出的若干重要结论几乎都运用了数学实验的思想方法。牛顿对于数学实验思想方法的运用经常以“代入一些有特点的整数或整数列”为“探针”进行探索。牛顿的这一做法与数学历史发展早期人们用来解决代数问题的方法在思想上是相同的。也许正因如此，牛顿才比当时的许多数学家在研究代数学时走得更远了一些。
Abstract: Newton has used various methods in the study of algebra, but the ideas and methods of mathematical experiments are the most widely used. Almost all of Newton’s important conclusions in the book Universal Arithmetic were based on the ideas and methods of mathematical experiments. Newton often explored the application of mathematical experimental ideas and methods by using “introducing some characteristic integers or integer columns” as “probes”. Newton’s approach is ideologically the same as the method used by people to solve algebraic problems in the early development of mathematics history. Perhaps because of this, Newton went further than many mathematicians at that time in studying algebra.
文章引用：杨欣童. 论牛顿代数研究的数学实验思想[J]. 理论数学, 2024, 14(7): 113-122. https://doi.org/10.12677/pm.2024.147278

参考文献

[1] Jonathan, B. and David, B. (2008) Mathematics by Experiment Plausible Reasoning in the 21st Century. A K Peters/CRC Press, 1-5.
[2] 刘邦奇. 数学实验方法及其哲学思考[J]. 科学技术与辩证法, 1994(3): 5-11.
[3] 邵光华, 卞忠运. 数学实验的理论研究与实践[J]. 课程.教材.教法, 2007(3): 39-43.
[4] 劳汉生, 张爱国, 朱熙湖. 数学实验方法的历史脉络[J]. 科学学研究, 2006(S1): 21-24.
[5] 黄毅蓉, 李有慧. 从数学方法论看“数学实验” [J]. 成都航空职业技术学院学报, 2005(4): 12-14.
[6] Beutelspacher, A. (2018) Mathematical Experiments—An Ideal First Step into Mathematics. In: Kaiser, G., Forgasz, H., Graven, M., Kuzniak, A., Simmt, E. and Xu, B., Eds., Invited Lectures from the 13th International Congress on Mathematical Education. ICME-13 Monographs, Springer, 19-29. [Google Scholar] [CrossRef] 
[7] Merzbach, U. and Boyer, B. (2010) A History of Mathematics. John Wiley & Sons, Inc., 40-126.
[8] 李文林. 数学史概论[M]. 北京: 高等教育出版社, 2002: 20, 32-38, 52-57, 196-206.
[9] Dunham, W. (1990) Journey through Genius: The Great Theorems of Mathematics. John Wiley & Sons, Inc., 223-244, 155-183.
[10] 贝尔. 数学精英[M]. 徐源, 译. 北京: 商务印书馆, 1997: 102-133.
[11] 克莱因. 古今数学思想∙第一册[M]. 张理京, 张锦炎, 江泽涵, 等, 译. 上海: 上海科学技术出版社, 2014: 284-313.
[12] Richards, W. (1980) Never at Rest. Cambridge University Press, 176-237.
[13] Niccolò, G. (2011) Isaac Newton on Mathematical Certainty and Method. The MIT Press, 61-63.
[14] Newton, I. (1972) The Mathematical Papers of Isaac Newton: Volume V, 1683-1684. Cambridge University Press, 1-21.
[15] Stedall, J. (2011) From Cardano’s Great Art to Lagrange’s Reflections: Filling a Gap in the History of Algebra. European Mathematical Society Publishing House, 71-75. [Google Scholar] [CrossRef] 
[16] Ball, W.W.R. (2009) A History of the Study of Mathematics at Cambridge. Cambridge University Press, 272-274. [Google Scholar] [CrossRef] 
[17] Cajori, F. (1894) A History of Mathematics. Macmilian & Co., 216-217.
[18] 杨欣童. 牛顿关于初等代数学的研究[J]. 中学数学教学, 2024(2): 38-44.
[19] Newton, I. (1707) Arithmetica universalis: Sive de compositione et resolutione arithmetica liber. Impersis Benj. Tooke, 42-55, 242-248, 255-258.
[20] Katz, V.J. (1988) A History of Mathematics. 2nd Edition, Addison Wesley Longman, 12-15.
[21] 梁宗巨, 王青建. 世界数学通史[M]. 沈阳: 辽宁教育出版社, 2001: 18.
[22] 伊夫斯. 数学史概论[M]. 太原: 山西人民出版社, 1986: 23.
[23] 卡茨. 数学史通论[M]. 北京: 高等教育出版社, 2004: 16.
[24] 张苍. 九章算术[M]//任继愈. 中国科学技术典籍通汇(数学卷一). 郑州: 河南教育出版社, 1994: 98-214.
[25] Viete, F. (2006) Two Treatises on the Understanding and Amendment of Equations. In: Viète, F., Ed., The Analytic Art, Dover Publications, 159-310.
[26] Descartes, R. (1925) The Geometry of Rene Descartes. Dover Publications, 379-382.
[27] 笛卡儿. 几何[M]. 武汉: 武汉出版社, 1992: 68-79.
[28] 杨欣童. 关于牛顿著作《广义算术》形成及其内容与影响的研究[J]. 交叉科学快报, 2023, 7(1): 10-16.
[29] 钱宝琮. 盈不足術的发展史[J]. 数学教学, 1955(1): 1-3.
[30] 邹大海. 从《算数书》盈不足问题看上古时代的盈不足方法[J]. 自然科学史研究, 2007(3): 312-323.
[31] Chabert, J.-L. (2012) Methods of False Position. In: Chabert, J.-L., Ed., A History of Algorithms: From the Pebble to the Microchip, Springer Science & Business Media, 82-91. [Google Scholar] [CrossRef] 
[32] Schwartz, R.K. (2004) Issues in the Origin and Development of Hisāb al-Khatā’ayn (Calculation by Double False Position). Eighth North African Meeting on the History of Arab Mathematics, Radès, 18-20 December 2004.
[33] Smith, D.E. (1958) History of Mathematics. Dover Publications, 437-441.

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