Renee is a brilliant mathematician. At the age of seven, she found beauty and comfort in the mathematical patterns associated with perfect squares, and felt this same sense of "rightness" throughout her career as she discovered new theorems that got her acclaim (and job offers). However, this is precisely why she is so devastated when her own research leads to a contradiction that demonstrates that mathematics itself is inconsistent. (In particular, she is able to correctly prove that any two numbers are, in fact, equal to each other.) Her biologist husband feels that he should understand her situation as he picks her up from the hospital after her suicide attempt, since he also attempted to kill himself once many years earlier, but he is not able to grasp her dispair over this mathematical result. There is literary irony in the comparison of Renee's feelings about mathematics and her husband's discovery that marriage is not exactly what he thought it was either.

Gödel's research is often misquoted or misused in popular press and fiction. Here, however, it is not only described accurately but used correctly in a very fascinating piece of speculative fiction. One major consequence of Gödel's work is the fact that we cannot prove the consistency of our axiomatic systems (such as arithmetic itself)...at least not from within. Logicians may argue that reality itself as a model demonstrates the consistency of mathematics, but this is clearly a metamathematical statement that relies on the validity of our understanding of reality, which is not completely certain itself. Renee's husband tries such an argument, pointing out that mathematics has been very useful in predicting and understanding reality, but Renee is not comforted by this when she knows now that the numbers 2, pi and 0 are all equal.

The story starts with a nice description of why division by zero is not possible, as well as a famous example of what sorts of contraditions you can get if you forget to exclude it. Russell, Whitehead, Hilbert and Godel are mentioned, as is a 1936 paper of Gerhard Gentzen which uses transfinite induction to prove the consistency of mathematics.

In the author's notes at the end of SOYLAO (see below), Chiang says: "Now consider [Euler's formula]. It's definitely surprising; youcould work with the numbers e, Pi 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 ineveitable, that this is the only way things could be. It's a feeling of awe, as if you've come into contact with the 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."

Originally published in the 1991 anthology Full Spectrum 3, this wonderful story has just been republished in the Ted Chiang collection: Stories of Your Life and Others. (Thanks to Steven H Silver for mentioning SOYLAO to me...)