英国利兹大学的数学教授Kevin Houston写了一本科普书,How to Think Like a Mathematician:(《如何像数学家一样思考》) 。为了给他的书作宣传,他特地制做了一个幻灯片:像数学家一样思考的10种方式(10 Ways To Think Like A Mathematician),很有趣。这里笔者略作介绍。 1.Question everything [For me one of the truly great beauties of mathematics is that it can be checked. You don’t have to take anyone’s word. If someone says something is true, then you can ask them to prove it. Better still, if you want to really think like a mathematician, then you can try to prove it. Don’t let people spoon feed you!] 2. Write in sentences [But if you want to understand mathematics and to think clearly, then the discipline of writing in sentences forces you to think very carefully about your arguments. If you can’t write the sentence properly, then probably you don’t truly understand what you are writing about. That’s a great opportunity to learn more and develop your skills. And writing well in any subject is a useful skill to possess. One easy way to improve your mathematical writing and thinking is to know how to use the implication symbol 3. ‘What about the converse?’ 4. Use the contrapositive 5. Consider extreme examples 我估计TA也许是完全没有底线!(确实如此,班上有学生考了6分,重修的甚至有0分的!)底线是什么,就是极端情形,这里就是左边的平时分的最乐观估计30、右边的及格线60. 6. Create your own examples 这里有一个来自于数学家陈景润的绝佳例子。 7. Where are the assumptions used? 我想举一个例子。我想应该是很有代表性的。我们在做一件事情(比如写文章、追女友)时,常常想要万事俱备了再行动,简直把自己想象为孔明借箭等得的那阵东风。但问题是,我们为什么要等万事俱备?大多数事情都是一边做一边完成的,等等等,黄花菜都凉了。特别是年轻人,我觉得无需任何假设,青春可以赋予你一切。 再如,有些中国古语,我想很多都是很荒唐的。“一日为师,终身为父”,我就不敢苟同。 另一句,我非常想跟学生讲的是,应该狂一点。为什么清华北大的学生就可以狂,我们西农大的学生就不可以了?谁赋予你狂的权利?你自己——“王侯将相宁有种乎?”(我斗胆揣测,这句话给 毛 爷爷极大的震撼和勇气,甚至一度令他担忧。) 8. Start with complicated side 就像生是复杂的,死是简单的,所以我们从生开始。这是哲学。用到数学上,如果你证明一个等式,左边很复杂,右边很简单,比如说,就是零。那么基本上你只能从左边开始,唯有如此,才能活下去(否则一开头就死翘翘)。 9. Ask ‘What happens if. . . ?’ 10. Communicate! 数学家陈省身的墓就在南开校园里,数学所(陈省身数学所)附近,他曾遗言,要在墓边放黑板,以便后人在附近讨论问题!我相信南开的朋友应该比我更清楚黑板的位置了! 很多人大概不用email,但只要上过正儿八经的研究生,就一定会频繁用email,因为他们需要跟人交流!网络当然提供了极为多样的便捷途径(比如善科网)交流,但也许最正式的还是email. 当我很晚很晚领悟到这一点时,我开始与全世界通信,简直就是“从那一刻起,跟世界上每个可能的人通信”。而当我某一刻发现很久没有收到邮件时,我深深地感到被世界遗忘了:这个世界不再需要我! |
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