The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [165]
Simulators employing emergent strategies would have to iron out mismatches arising from the disparate methods, and they’d need to ensure that the meshing was smooth. This would require fiddles and tweaks which, to an inhabitant, might appear as sudden, baffling changes to the environment with no apparent cause or explanation. And the meshing might fail to be fully effective; the resulting inconsistencies could build over time, perhaps becoming so severe that the world became incoherent, and the simulation crashed.
A possible way to obviate such challenges would be to use a different approach—call it the “ultra-reductionist strategy”—in which the simulation would proceed by a single set of fundamental equations, much as physicists imagine is the case for the real universe. Such simulations would take as input a mathematical theory of matter and the fundamental forces and a choice of “initial conditions” (how things were at the starting point of the simulation); the computer would then evolve everything forward in time, thereby avoiding the meshing issues of the emergent approach. But simulations of this kind would encounter their own computational problems, even beyond the staggering computational burden of simulating “everything,” right down to the behavior of individual particles. If the equations our descendants have in their possession are similar to those we work with today—involving numbers that can vary continuously—then the simulations would necessarily invoke approximations. To exactly follow a number as it varies continuously, we would need to track its value to an infinite number of decimal places (for instance, as such a quantity varies, say, from .9 to 1, it would pass through numbers like .9, .95, .958, .9583, .95831, .958317, and on and on, with an arbitrarily large number of digits required for full accuracy). That’s something a computer with finite resources can’t manage: it will run out of time and memory. So, even if the deepest equations were used, it’s still possible that computer-based calculations would inevitably be approximate, allowing errors to build up over time.*
Of course, by “error” I mean a deviation between what occurs in the simulation and the description inherent in the most refined physical theories the simulator has at his or her disposal. But to those like you who are within the simulation, the mathematical rules driving the computer would be your laws of nature. The issue, then, is not how closely the mathematical laws used by the computer model the external world; we’re imagining that you don’t observe the external world from within the simulation. Rather, the problem for a simulated universe is that when a computer’s necessary approximations permeate otherwise exact mathematical equations, calculations easily lose their stability. Round-off errors, when accumulated over a great many computations, can yield inconsistencies. You and other simulated scientists might witness anomalous results from experiments; cherished laws might start yielding inaccurate predictions; measurements that had long since converged on a single widely confirmed result might start producing different answers. For long stretches, you and your simulated colleagues would think that you’d encountered evidence, much as your forebears had throughout the previous centuries and millennia, that your final theory wasn’t so final after all. Collectively, you’d closely reexamine the theory, perhaps coming up with new ideas, equations, and principles that better described the data. But, assuming the inaccuracies didn’t result in contradictions that crashed the program, at some point you’d hit a wall.
After an exhaustive search through possible explanations, none of which was