Visible Structures in Number Theory
Visible Structures in Number Theory
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数论中的可见结构
DOI：
10.1080/00029890.2001.11919824
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发表时间：
2001
期刊：
影响因子：
--
通讯作者：
Loki Jörgenson
中科院分区：
文献类型：
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作者：
P. Borwein;Loki Jörgenson
这是这位伟大的法国数学家在证明存在大于 11 的素数时用来描述他最初想法的词语 [11, p. 11]。 76]。他将他最终的心理形象描述为“……混乱的质量和第一点之间的某个地方”。在他引人入胜但古怪的专着中评论这一点时，他问道：“如此奇怪和阴暗的图像有什么用处？”。哈达玛认为数学思想是视觉的，文字只会产生干扰。当他询问他最杰出的中世纪同事的思维过程时，他发现他们中的大多数人在某种程度上都同意（一个值得注意的例外是乔治·波利亚）。对于非专业人士来说，数学家“看到”他们的想法的想法可能会令人惊讶。然而，数学史上有许多基于视觉的显着发展。例如，笛卡尔引入的“笛卡尔”坐标可以说是上个千年数学中最重要的进步。它从根本上重塑了数学家思考数学的方式，正是因为它让他们能够更好地“看到”数学。事实上，数学家很早就意识到可视化的重要性，并付出了巨大的努力来开发它。卡尔·弗里德里希·高斯在给海因里希·克里斯蒂安·舒马赫的一封信中感叹，绘制做出准确猜想所需的图片是多么困难。高斯被许多人认为是有史以来最伟大的数学家，他在引用他对代数基本定理的第一个证明所附的图表时写道：
These are the words the great French mathematician used to describe his initial thoughts when he proved that there is a prime number greater than 11 [11, p. 76]. His final mental image he described as "... a place somehere between the confused mass and the first point". In commenting on this in his fascinating but quirky monograph, he asks "What may be the use of such a strange and cloudy imagery?". Hadamard was of the opinion that mathematical thought is visual and that words only interfered. And when he inquired into the thought processes of his most distinguished mid-century colleaugues, he discovered that most of them, in some measure, agreed (a notable exception being George Polya). For the non-professional, the idea that mathematicians "see" their ideas may be surprising. However the history of mathematics is marked by many notable developments grounded in the visual. Descartes' introduction of "cartesian" co-ordinates, for example, is arguably the most important advance in mathematics in the last millenium. It fundamentally reshaped the way mathematicians thought about mathematics, precisely because it allowed them to "see" better mathematically. Indeed, mathematicians have long been aware of the significance of visualization and made great effort to exploit it. Carl Friedrich Gauss lamented, in a letter to Heinrich Christian Schumacher, how hard it was to draw the pictures required for making accurate conjectures. Gauss, whom many consider the greatest mathematician of all time, in reference to a diagram that accompanies his first proof of the fundamental theorem of algebra, wrote