Dark Integers by Greg Egan

Author:Greg Egan [Egan, Greg]
Language: eng
Format: epub
Published: 2013-07-10T16:00:00+00:00

* * *

In the weeks that followed, Campbell worked on refinements to his theory, emailing Alison and me every few days. Alison had taken my unilateral decision to recruit Campbell with much more equanimity than I’d expected. “Better to have him inside the tent,” was all she’d said.

This proved to be an understatement. While the two of us soon caught up with him on all the technicalities, it was clear that his intuition on the subject, hard-won over many years of trial and error, was the key to his spectacular progress now. Merely stealing his notes and his algorithms would never have brought us so far.

Gradually, the dynamic version of the theory took shape. As far as macroscopic objects were concerned—and in this context, “macroscopic” stretched all the way down to the quantum states of subatomic particles—all traces of Platonic mathematics were banished. A “proof” concerning the integers was just a class of physical processes, and the result of that proof was neither read from, nor written to, any universal book of truths. Rather, the agreement between proofs was simply a strong, but imperfect, correlation between the different processes that counted as proofs of the same thing. Those correlations arose from the way that the primordial states of Planck-scale physics were carved up—imperfectly—into subsystems that appeared to be distinct objects.

The truths of mathematics appeared to be enduring and universal because they persisted with great efficiency within the states of matter and space-time. But there was a built-in flaw in the whole idealization of distinct objects, and the point where the concept finally cracked open was the defect Alison and I had found in our volunteers’ data, which appeared to any macroscopic test as the border between contradictory mathematical systems.

We’d derived a crude empirical rule which said that the border shifted when a proposition’s neighbors outvoted it. If you managed to prove that x+1=y+1 and x-1=y-1, then x=y became a sitting duck, even if it hadn’t been true before. The consequences of Campbell’s search had shown that the reality was more complex, and in his new model, the old border rule became an approximation for a more subtle process, anchored in the dynamics of primordial states that knew nothing of the arithmetic of electrons and apples. The near-side arithmetic Campbell had blasted into the far side hadn’t got there by besieging the target with syllogisms; it had got there because he’d gone straight for a far deeper failure in the whole idea of “integers” than Alison and I had ever dreamed of.

Had Sam dreamed of it? I waited for his next contact, but as the weeks passed he remained silent, and the last thing I felt like doing was calling him myself. I had enough people to lie to without adding him to the list.

Kate asked me how work was going, and I waffled about the details of the three uninspiring contracts I’d started recently. When I stopped talking, she looked at me as if I’d just stammered my way through an unconvincing denial of some unspoken crime.

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