Representations of finite groups of Lie type (2nd edn.) by Francois Digne and Jean Michel, pp 172, £37.99 (paper), ISBN 978-1-10872-262-9, Cambridge University Press (2020)
Representations of finite groups of Lie type (2nd edn.) by Francois Digne and Jean Michel, pp 172, £37.99 (paper), ISBN 978-1-10872-262-9, Cambridge University Press (2020)
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DOI：
10.1017/mag.2022.99
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发表时间：
2022-06
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通讯作者：
Mark Hunacek
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作者：
Mark Hunacek
维拉尼的写作令人信服，我不禁想起其他著名数学家的观察，他们试图解释他们对这一学科的美学吸引力的迷恋。伯特兰·罗素（Bertrand Russell）有句名言：“数学……不仅拥有真理，而且拥有至高无上的美——一种冷酷而朴素的美，就像雕塑的美一样，不会吸引我们脆弱本性的任何部分，没有绘画或音乐的华丽装饰，但却极其纯粹，并且能够达到只有最伟大的艺术才能展现的严格完美”。 G.H.哈代也试图解释这种吸引力，他写道：“定义数学美可能非常困难，但这对于任何形式的美来说都是如此——我们可能不知道一首美丽的诗意味着什么，但这并不妨碍我们在读它时认出一首诗”。因此，维拉尼的作品无疑被视为解决这些问题的传统的一部分。然而，他具有罗素和哈代的优势，因为他在将该学科发展到21世纪方面发挥了重要作用。然而，我对数学是“科学的一部分”这一基本断言提出质疑。也许这是一个语言问题——毫无疑问，维拉尼意识到了语言问题，因为他在某一时刻插入了一个脚注，对比法语中的“la mathématique”和“les mathématiques”两种形式——这一问题在英语中不会出现。维拉尼渴望在“数学”和“科学”实践之间建立相似之处，但不可否认的是，也存在根本的差异。例如，假设您希望显示某些几何配置中的三条线是并发的。使用动态几何包很容易，它可以对情况进行建模，并通过拉动图形的各个元素来显示所讨论的线继续穿过同一点。然而，没有人会认为这是并发性证明——这需要独立于特定数字的严格论证，并且实际上能够解决图表依赖性的各种问题。我记得，在这种情况下，已故的克里斯托弗·泽曼（Christopher Zeeman）曾经对一位讲师说，他们正在研究科学而不是数学，因为他们的方法依赖于特定案例以及混乱的观察和证据。数学真理具有普遍性，这将其与自然科学结果的偶然性区分开来。如果我暂时暗示像维拉尼这样杰出的数学家并没有敏锐地意识到这些区别，那我就太无耻了。然而，如果他多花一点时间来强调数学作为一门不依赖实验和观察的学科的特殊性，我会更高兴。但这只是对数学中创造力和审美敏感性的非凡探索的一个小吹毛求疵，我完全推荐给任何想要理解为什么它会继续激励和吸引所有爱好者的人。 10.1017/mag.2022.98 © 作者，2022 GERRY LEVERSHA 由剑桥大学出版社出版 15, Maunder Road，代表数学协会 Hanwell，伦敦 W7 3PN 电子邮件：g.leversha@btinternet.com
Villani writes convincingly, and I could not help but be reminded of observations by other celebrated mathematicians who have attempted to explain their fascination with the aesthetic appeal of the subject. Bertrand Russell famously said that ‘Mathematics ... possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show’. G. H. Hardy also tried to explain this appeal when he wrote that ‘it may be very hard to define mathematical beauty, but that is just as true of beauty of any kind—we may not know what we mean by a beautiful poem, but that does not prevent us from recognising one when we read it’. So Villani's work is surely to be seen as forming part of a tradition which tackles these questions. However, he has the advantage of Russell and Hardy in that he has been instrumental in developing the discipline into the 21st century. I would, however, query the fundamental assertion that mathematics is ‘part of science’. Perhaps this is a linguistic problem—and there is no doubt that Villani is aware of the issues of language, for he inserts, at one point, a footnote contrasting the two forms la mathématique and les mathématiques in French—an issue which does not arise in English. Villani is eager to establish similarities between ‘mathematical’ and ‘scientific’ practice, but it cannot be denied that there are also fundamental differences. Suppose, for example, that you wish to show that three lines in some geometrical configuration are concurrent. It is easy to use a dynamic geometry package which models the situation and show, by pulling around various elements of the figure, that the lines in question continue to pass through the same point. Nobody, however, would consider this to be a proof of concurrency—that would require a rigorous argument independent of a particular figure and, indeed, able to address various issues of diagram-dependency. I can remember, in this context, the late Christopher Zeeman once remarking to a lecturer that they were doing science and not mathematics, since their approach was dependent upon particular cases and muddled observation with proof. There is a universal quality to mathematical truth which distinguishes it from the contingency of results in the natural science. It would be outrageous of me to suggest for one moment that someone of the mathematical eminence of Villani is not acutely aware of these distinctions. I would, however, have been happier if he had devoted a little more time to emphasising the specificity of mathematics as a discipline which is not reliant upon experiment and observation. But that is a minor cavil about this remarkable exploration of creativity and aesthetic sensitivity in mathematics, and I would wholly recommend it to anyone who wants to appreciate why it will continue to inspire and fascinate all of its devotees. 10.1017/mag.2022.98 © The Authors, 2022 GERRY LEVERSHA Published by Cambridge University Press 15, Maunder Road, on behalf of The Mathematical Association Hanwell, London W7 3PN e-mail: g.leversha@btinternet.com