The story is written in sections numbering one through nine, which detail some odd, historical aspects of mathematics. Each number has "A" and "B" subsections relating Renee and Carl's perspectives, respectively.
Commentary:The key here is Chiang's own comment on the story:
"A proof that mathematics is inconsistent, and all its wondrous beauty was just an illusion, would, it seemed to me, be one of the worst things you could ever learn."For those in the sciences, the sciences fall down a spectrum of scientific rigor--rather, the degree to which it is predictable and replicatable:
- Psychology (not a science or a "pseudo-science" as some scientists consider it)
- Sociology (see psychology)
You can plot a course for the path of an orbit with a high degree of accuracy. Chemicals combine and repeat the same result. And then things get messy, in terms of science. Of course, this is an oversimplification. The physics of the electron and below get strange, and organic chemistry often produces mixed results at some ratio. Biology is the combination of multiple organic and inorganic chemicals (read: an exponential increase in variables), directed by non-sentient(?) cells, which complicates predictability. Groups of cells can organize into larger and more specialized behavior, which complicates predictability further (read: another exponential increase in variables).
Then we get into consciousness--a scientific map that remains largely uncharted--and how that affects human individuals (read: yet another exponential increase in variables) and groups of individuals divided by genetics and culture interacting on large scales (read: the final exponential increase in variables). It doesn't help that a recent report stated that these human-based sciences have studies with about a fifty-percent chance of "replicability" (Some see this an issue of bias in the field).
The degree to which an individual prefers order may affect the type of career one might choose.
Most scientists see mathematics as a tool for interpreting the science. Apparently, mathematicians can see their study as further purity, stripped bare of scientific study--the encumbrances of any kind of messy consideration. Mathematics could be considered the foundation that all other aspects of the universe are built upon. It, from the view of the pure mathematician, would lie at the top of the heap. This sheds light on Chiang's statement: What if there were no true tool of studying the universe?
Seeing mathematics as a study that strips away the extraneous brings new perspective to Chiang's work. One might call this Chiang's MO--modus operandi--at least of his early work. Characters are stripped to their essence so that the ideas can remain pure, unfiltered, unbiased. Renee is mathematics (at least, from a human perspective--the clear, unfettered idea) while Carl studies or is biology (life, messy). The characters provide perfect foils for one another.
When Carl was a graduate student, he attempted suicide. Therefore, when Renee attempts, Carl experiences deja vu, although this also mirrors the story idea that all numbers are equivalent. Presumably, that same story idea explains the nonlinear story time (if all numbers are equivalent) since it begins at the end (the psych ward after an attempted suicide).
Carl cannot understand Renee's dilemma because he is used to the messiness of his science. Renee, representing order, is supposed order herself. Since Carl is messy, it is unsurprising that he attempts suicide as a young man, but Renee's attempt surprises, coming after she uncovers the idea that even mathematics might have its own mess. This brings us to the title. Division by zero represents a mathematical equation that has no answers.