《除以零》收录在《你一生的故事》中，是我看过的最惊心动魄的爱情故事。作为科幻短篇，这样说似乎言过其实。我难以表达我读后的心情。我想特德姜也发现这难以表达了，于是用上了数学定理。不知道该说是理性崩塌像是爱消失，还是爱的消失如同理性崩塌。这两者在特雷姜的笔下以一种奇特又契合的方式纠缠在一起。从1到9层层递进，最后终于来到了a=b。从不等式到等式，我们的爱就像1=2一样。It can be formally correct, but it’s nonsense.

If mathematical thinking is defective, where are we to find truth and certitude? 希尔伯特的这句话换成If love is defective, where are we to find truth and certitude? 同样成立。除以零在数学规则中不被允许，因为这样做会使其他已经被定义的东西就此丧失意义。数学大厦在稀薄的空气中摇摇欲坠，无数人依旧前赴后继试图证明最后被证明出是不可证明的东西。就像爱一样。

人为何总要去做西西弗呢？特德姜在story notes给出了他的理解：There’s a famous equation that looks like this: e^(iπ) + 1 = 0 When I first saw the derivation of this equation, my jaw dropped in amazement. Let me try to explain why. One of the things we admire most in fiction is an ending that is surprising, yet inevitable. This is also what characterizes elegance in design: the invention that’s clever yet seems totally natural. Of course we know that they aren’t really inevitable; it’s human ingenuity that makes them seem that way, temporarily. Now consider the equation mentioned above. It’s definitely surprising; you could work with the numbers e and i for years, each in a dozen different contexts, without realizing they intersected in this particular way. Yet once you’ve seen the derivation, you feel that this equation really is inevitable, that this is the only way things could be. It’s a feeling of awe, as if you’ve come into contact with absolute truth. A proof that mathematics is inconsistent, and that all its wondrous beauty was just an illusion, would, it seemed to me, be one of the worst things you could ever learn.

我翻译一下他意思就是：因为这实在太美了。欧拉公式如此，对一致性的执念如此。这种执念让罗素可以写上362页的证明试图说1+1=2是对的，虽然他从最开始好像就错了。爱也如此，虽然最后Renee证明出他们的爱不存在，是幻觉。可是，我想说的是，即便最后只能证明出：arithmetic as a formal system cannot guarantee that it will not produce results such as “1 = 2"，也不能否认这一切实在太美了。它能唤起一种强烈罕见而宝贵的情感，这种情感无比动人。在某些情况下，我们可以把这种情感叫做敬畏，抑或是信仰。

在《会饮篇》中，苏格拉底说：爱欲永远是在满足和渴求之间的，不会是完满的。完满了，你就不会再去追求，不会再去爱了。曾经有一个人教给我关于爱的秘密，那是一位女祭司。她教会我的第一堂课，就是爱欲永远是在缺乏和满足之间。如果你是完全的完满，那么你不会去爱；如果你完全缺乏，你也不会去爱，因为你连爱的能力都缺乏。所以，爱永远是中间的形象。爱欲最深的奥秘，在苏格拉底那里，是爱的阶梯。“爱的阶梯”是一种爬升。它不是可被证明之物，它本身即是目的。它处在匮乏与摆脱匮乏之间，它是一种行动。

所以，即便我觉得《除以零》写得真得太好了，但要是换成我来写，我会改变这个悲伤的结局。

1 Dividing a number by zero doesn’t produce an infinitely large number as an answer. The reason is that division is defined as the inverse of multiplication; if you divide by zero, and then multiply by zero, you should regain the number you started with. However, multiplying infinity by zero produces only zero, not any other number. There is nothing which can be multiplied by zero to produce a nonzero result; therefore, the result of a division by zero is literally “undefined.”

2 There is a well-known “proof” that demonstrates that one equals two. It begins with some definitions: “Let a = 1; let b =1.” It ends with the conclusion “a = 2a,” that is, one equals two. Hidden inconspicuously in the middle is a division by zero, and at that point the proof has stepped off the brink, making all rules null and void. Permitting division by zero allows one to prove not only that one and two are equal, but that any two numbers at all—real or imaginary, rational or irrational—are equal.

3 In the Principia Mathematica, Bertrand Russell and Alfred Whitehead attempted to give a rigorous foundation to mathematics using formal logic as their basis. They began with what they considered to be axioms, and used those to derive theorems of increasing complexity. By page 362, they had established enough to prove “1 + 1 = 2.”

4 In the early nineteenth century, mathematicians began exploring geometries that differed from Euclidean geometry; these alternate geometries produced results that seemed utterly absurd, but they didn’t produce logical contradictions. It was later shown that these non-Euclidean geometries were consistent relative to Euclidean geometry: they were logically consistent, as long as one assumed that Euclidean geometry was consistent. The proof of Euclidean geometry’s consistency eluded mathematicians. By the end of the nineteenth century, the best that was achieved was a proof that Euclidean geometry was consistent as long as arithmetic was consistent.

5 At the Second International Congress of Mathematics in 1900, David Hilbert listed what he considered to be the twenty-three most important unsolved problems of mathematics. The second item on his list was a request for a proof of the consistency of arithmetic. Such a proof would ensure the consistency of a great deal of higher mathematics. What this proof had to guarantee was, in essence, that one could never prove one equals two. Few mathematicians regarded this as a matter of much import.

6 In 1931, Kurt Godel demonstrated two theorems. The first one shows, in effect, that mathematics contains statements that may be true, but are inherently unprovable. Even a formal system as simple as arithmetic permits statements that are precise, meaningful, and seem certainly true, and yet cannot be proven true by formal means. His second theorem shows that a claim of the consistency of arithmetic is just such a statement; it cannot be proven true by any means using the axioms of arithmetic. That is, arithmetic as a formal system cannot guarantee that it will not produce results such as “1 = 2"; such contradictions may never have been encountered, but it is impossible to prove that they never will be.

7 In 1936, Gerhard Gentzen provided a proof of the consistency of arithmetic, but to do it he needed to use a controversial technique known as transfinite induction. This technique is not among the usual methods of proof, and it hardly seemed appropriate for guaranteeing the consistency of arithmetic. What Gentzen had done was prove the obvious by assuming the doubtful.

8 Hilbert once said, “If mathematical thinking is defective, where are we to find truth and certitude?”

9 Albert Einstein once said, “Insofar as the propositions of mathematics give an account of reality they are not certain; and insofar as they are certain they do not describe reality.”