数学家Jacob T. Schwartz（纽约大学Courant数学研究所数学与计算机教授）写过一篇很刺眼的文章：“The Pernicious Influence of Mathematics on Science”（文章重印见Discrete Thoughts: Essays on Mathematics, Science, and Philosophy, edited by Mark Kac, Gian-Carlo Rota and Jacob T. Schwartz, 1992），大概说数学只能玩儿简单游戏，复杂的就不行了：

Mathematics is able to deal successfully only with the simplest of situations, more precisely, with a complex situation only to the extent that rare good fortune makes this complex situation hinge upon a few dominant simple factors. Beyond the well-traversed path, mathematics loses its bearings in a jungle of unnamed special functions and impenetrable combinatorial particularities. Thus, the mathematical technique can only reach far if it starts from a point close to the simple essentials of a problem which has simple essentials. That form of wisdom which is the opposite of single-mindedness, the ability to keep many threads in hand, to draw for an argument from many disparate sources, is quite foreign to mathematics. 

那天说的Gregory Chaitin“证明达尔文”在正文前引用过这段话。我过了一眼Jacob的文章，现在向大家传达精神：

1）计算机死心眼儿，没想象力，头脑简单（single-mindedness, literal-mindedness, simple-mindedness），这些毛病全是数学感染的。

2）即使最精确的科学也多少带着病态的近似（如Schrodiner方程）

3）数学只能处理明确界定（well-defined）的情形，数学家把科学家的方便假定转化为公理，然后遵公理做事，把科学家也带坏了

4）因为数学无聊，没想象力（literal-mindedness，只会照本本做事，没创意），假如要适当用于科学，那么它需要证明的假定还得从更大的视野去寻找，而它自己是看不见的。（The literal-mindedness of mathematics thus makes it essential, if mathematics is to be appropriately used in science, that the assumptions upon which mathematics is to elaborate be correctly chosen from a larger point of view, invisible to mathematics itself.）（接下来就是前面引的那段话。）

5）数学没想象力，自然头脑简单（simple-mindedness），想证明一切——不管是辉煌还是荒谬，都给披上一套夺目的公式和定理的外衣（to dress scientific brilliancies and scientific absurdities alike in the impressive uniform of formulae and theorems）。穿了衣服的荒谬比赤裸裸的荒谬更有诱惑。

6）数学是骗人的戏法儿。如在统计力学中起着基本作用的Birkhoff各态历经定理，恰好证明了它只不过是各态历经假定上的一个无关紧要的超结构（a questionably relevant superstructure upon this hypothesis）。

7）数学论证巧妙隐藏了一些我们无知的东西，让我们看不到它省略的东西——这就是所谓数学的“一贯道”（single-mindedness）。

如果Hardy见了Jacob对数学的批判，大概会说，看见了吧，数学本来是美好的，可惜被用坏了，用丑了。