|The ``cold war'' between this universe with our mathematical laws and a bordering universe with different ones (which began in "Luminous") heats up when the numerical experiments of a mathematical physicist on our side are perceived as an attack on the other side. The title refers to the name the mathematical physicist has come up with for the anomalous numbers, ``dark integers'', just as cosmologists have come to use names like ``dark matter'' and ``dark energy'' for those things they cannot explain using conventionally accepted theories.
As with the original story, I am not sure I completely understand this idea of mathematical theorems being `proved' by different physical processes resulting in different laws of mathematics in different locations. This time, due to the presence of the mathematical physicist character, things are even more `physical', with the real papers of Gerard t'Hooft on deterministic quantum mechanics (see, for example, here) playing a seemingly important role.
IMHO, this is a really well written story (despite the fact that it doesn't quite make sense when you think about it too hard). The different tensions (academic, diplomatic and domestic) interact in a very interesting way .
A sequel to "Luminous". Ten years later, Bruno, Alison, and Yuen have maintained a secret peace treaty with the "other side", the world that exists with other axioms and laws of arithmetic; they can now communicate with an entity there they call "Sam", and neither side attempts incursions on the other. But while Bruno, Alison, and Yuen have been forthcoming about our world, the other side is mum about theirs. Then, one day, Bruno gets a message from Sam: someone managed to get a region of the otherspace, far from the "border", begin to obey our axioms; the other side is worried, and wants to know what is going on. Bruno tracks the incursion to Tim Campbell, a mathematician in Wellington. They bring him into the cabal, but they also keep the details from Sam; Campbell's intuition into the problem allows the cabal to start mapping the other side, and learn more about them. And, then, suddenly, the financial systems of our side begin to crash. Is this a pre-emptive attack by the other side?
Published originally as Asimov's Science Fiction, October/November 2007, pp. 20-51 and now available in the collection Dark Integers and Other Stories.
It seemed to me that the math in this book falls into the category in which the appearance to a layperson is more important than what the words and form would imply to a mathematician. Given the size of the set of integers, it is hard to imagine that the described boundary boundary between the worlds could be continuous, which makes some of the later story elements rather confusing vis-a-vis the smoothing that takes place. Still, it is undeniable that math plays a central role in the story, and it is always nice to see terms like equivalence classes appearing outside of scholarly publications. The overall story was good and I would recommend it to anyone with an interest in math.