Number rules the universe.

    数支配着宇宙。
              ——毕达哥拉斯(Pythagoras)

 

 

 

 

    Reason is immortal, all else mortal.

    理性是不朽的,其余一切都会消亡。
              ——毕达哥拉斯(Pythagoras)

 

 

 

 

    The highest form of pure thought is in mathematics”.

    数学是纯粹思维的最高形式。

                                  ——柏拉图(Plato)

 

 

 

 

     

    Nature’s great book is written in mathematical symbol.
    自然这本大书是用数学符号写的。
                             ——伽里略(Galileo)

 

 

 

 

 

 

    Wir müssen wissen, wir werden wissen.

    We must know, we will know.

    我们必须知道,我们必将知道。

 

             ——希尔伯特(Hilbert)

 

 

 

 

    Pure mathematics is,in its way,the poetry of logical ideas.
    纯粹数学,就其本质而言,是逻辑思维的诗篇。
                      ——爱因斯坦(Einstein)

 

 

 



 

   A true mathematician who is not also something of a poet will never be a perfect mathematician.
    一个没有几分诗人气质的数学家永远不可能成为十全十美的数学家。
             ——魏尔斯特拉斯(Weierstrass)





    Mathematics, rightly viewed, 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.
   < 拙译> 数学,当你正确地看待它时,不仅拥有真,而且拥有非凡的美 —— 一种犹如雕塑般冷峻而素朴的美,一种不引诱任何我们的较软弱天性的美,一种没有绘画和音乐那样富丽花俏的装饰的纯净之至的美,同时又能达到一种唯有最伟大的艺术才能表达的严格的完美。
               ——罗素(Bertrand Russell






   The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.
   < 拙译> 正如在诗歌中一样,在数学中同样能找到真正的欢乐的精神,升华的快感,那种超越凡俗接近神祗的美妙感觉——它是最高级的卓越成就的试金石。
               ——罗素(Bertrand Russell







    Besides language and music, it [mathematics] is one of the primary manifestations of the free creative power of the human mind, and it is the universal organ for world-understanding through theoretical construction. Mathematics must therefore remain an essential element of the knowledge and abilities which we have to teach, of the culture we have to transmit, to the next generation.
    < 拙译> 数学是除了语言与音乐之外,人类心灵自由创造力的主要表达方式之一,而且它是通过理论的构建理解宇宙万物的普适工具。因此,数学必须始终是我们得传授给下一代的知识和技能的要素和我们要传承给下一代的文化的要素。
                   ——外尔(Hermann Weyl)

     

 



    What we do may be small , but it has a certain character of permanence ; and to have produced anything of the slightest permanent interest , whether it be a copy of verses or a geometrical theorem , is to have done something utterly beyond the powers of the vast majority of men .
    < 拙译>我们所做的或许微不足道,但它具有某种永恒的特征;而创造了具有一丝一毫永久性兴趣的任何东西,不管是几行诗句也好,或者是一个几何定理也罢,已经是做了某件完全超出芸芸众生绝大多数人能力之上的事情。

    ——哈代G.H.Hardy

         ——摘自《一个数学家的辩白》

            (A Mathematician's Apology)







    Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

    < 拙译> 当埃斯库罗斯被人遗忘之际阿基米德仍会被人铭记,因为语言会死亡而数学思想不会。“长生不死”是一个愚蠢的词,但不管它实际上何所指,数学家当属首选。
   

                            ——哈代G.H.Hardy

         ——摘自《一个数学家的辩白》

            (A Mathematician's Apology)


 

 


研究者们破解了拉马努金在临终病床上写下的神秘公式

  谢国芳(Roy Xie)译

  roixie@163.com

躺在临终病床上,天才的印度数学家拉马努金写下了谜一样离奇难解的函数,他说它们是在他的梦境中出现的,他对这些函数的性态有直觉的预感。一百年后的今天,研究人员说他们证明了拉马努金是对的。“我们解决了他的最后几封神秘的信件中的问题,对于研究这个数学领域的人来说,该问题已经悬而未决九十年了,” 美国艾莫利大学 (Emory University) 的日裔数学家小野(Ken Ono)说。

拉马努金是一个出身在印度南部一个农村的自学成材的数学家,据小野说,他因为花太多时间思考数学,曾两度被勒令从印度的大学退学。可他不折不饶地给数学家们写信描述他的工作,他们当中最杰出的之一——英国数学家哈代辨识出了这个印度男孩的天才,并邀请他到英国剑桥大学学习。在剑桥,拉马努金发表了三十多篇论文,并被选为皇家学会的正式会员。“在短短的五年时间里,他像一团火焰点燃了整个数学王国,”小野对《科学前沿》(LiveScience)说。可是英国的严寒天气最终损害了拉马努金的健康,当他生命垂危时,他回到了他的祖国印度。

1920年,躺在病床上奄奄一息的拉马努金在给哈代的一封信中描述了一种酷似theta函数或者说模形式的神秘的函数,像正弦余弦等三角函数一样,theta函数也有一种重复的范式,但这种范式比简单的正弦函数要复杂和微妙得多。theta函数还是“超对称的”,意思是如果一种叫麦比乌斯变换的特别函数作用于其上,它们会变成自身。因为其高度对称性,这些函数在很多数学和物理领域中都很有用,包括弦论。

拉马努金相信他发现的十七个新的函数是伪模形式,当写成无穷乘积时它们看上去很像theta函数(其系数以同样的方式增大),但却不是超对称的。拉马努金是一个虔诚的印度教教徒,他认为是女神娜玛吉丽向他揭示了这些规律。

拉马努金没能证明他的直觉性的预感就离开了人世,但九十多年以后,小野和他的团队证实了这些函数的确酷似模形式,但却不具有其本质特征如超对称性。伪模形式的展开式可以帮助物理学家计算黑洞的熵或者说无序程度。

小野说,拉马努金发展的伪模形式大大超越了他的时代。直到2002年,数学家们才搞清楚这些方程属于哪个数学分支,现在人们意识到,拉马努金留下的遗产比任何人在他去世时所能想象的都要重要得多。

拉马努金(Ramanujan, 1887-1920)

英文原文(original English text):

Researchers Unlock Mysterious Formula Written
By Ramanujan On His Death Bed

While on his death bed, the brilliant Indian mathematician Srinivasa Ramanujan cryptically wrote down functions he said came to him in dreams, with a hunch about how they behaved. Now 100 years later, researchers say they've proved he was right."We've solved the problems from his last mysterious letters. For people who work in this area of math, the problem has been open for 90 years," Emory University mathematician Ken Ono said.

Ramanujan, a self-taught mathematician born in a rural village in South India, spent so much time thinking about math that he flunked out of college in India twice, Ono said. But he sent mathematicians letters describing his work, and one of the most preeminent ones, English mathematician G. H. Hardy, recognized the Indian boy's genius and invited him to Cambridge University in England to study. While there, Ramanujan published more than 30 papers and was inducted into the Royal Society. "For a brief window of time, five years, he lit the world of math on fire," Ono told LiveScience. But the cold weather eventually weakened Ramanujan's health, and when he was dying, he went home to India.

It was on his deathbed in 1920 that he described mysterious functions that mimicked theta functions, or modular forms, in a letter to Hardy. Like trigonometric functions such as sine and cosine, theta functions have a repeating pattern, but the pattern is much more complex and subtle than a simple sine curve. Theta functions are also "super-symmetric," meaning that if a specific type of mathematical function called a Moebius transformation is applied to the functions, they turn into themselves. Because they are so symmetric these theta functions are useful in many types of mathematics and physics, including string theory.

Ramanujan believed that 17 new functions he discovered were "mock modular forms" that looked like theta functions when written out as an infinte sum (their coefficients get large in the same way), but weren't super-symmetric. Ramanujan, a devout Hindu, thought these patterns were revealed to him by the goddess Namagiri.

Ramanujan died before he could prove his hunch. But more than 90 years later, Ono and his team proved that these functions indeed mimicked modular forms, but don't share their defining characteristics, such as super-symmetry. The expansion of mock modular forms helps physicists compute the entropy, or level of disorder, of black holes.

In developing mock modular forms, Ramanujan was decades ahead of his time, Ono said; mathematicians only figured out which branch of math these equations belonged to in 2002. "Ramanujan's legacy, it turns out, is much more important than anything anyone would have guessed when Ramanujan died," Ono said.