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By Root 2076 0

physics, that spacetime is continuous) a batted ball passes through an infinite number of different points as it travels from home plate to left field.9 Keeping track of a ball through an infinity of locations, and of an infinity of possible speeds at those locations, will always remain beyond reach. At best, computers can perform highly refined but still approximate calculations, tracking a ball every millionth or billionth or trillionth of a centimeter, for instance. That’s fine for many purposes, but it’s still an approximation. Quantum mechanics and quantum field theory, by introducing various forms of discreteness, help in some ways. But both make extensive use of continuously varying numbers (values of probability waves, values of fields, and so on). The same reasoning holds for all the other standard equations of physics. A computer can approximate the math, but it can’t simulate the equations exactly.*

There are other types of mathematical functions, however, for which a computer simulation can be absolutely precise. They’re part of a class called computable functions, which are functions that can be evaluated by a computer running through a finite set of discrete instructions. The computer may need to cycle through the collection of steps repeatedly but sooner or later it will produce the exact answer. No originality or novelty is needed at any step; it’s just a matter of grinding out the result. In practice, then, to simulate the motion of a batted ball, computers are programmed with equations that are computable approximations to the laws of physics that you learned in high school. (Typically, continuous space and time are approximated on a computer by a fine grid.)

By contrast, a computer trying to calculate a noncomputable function will churn away indefinitely without coming to an answer, regardless of its speed or memory capacity. Such would be the case for a computer seeking the exact continuous trajectory of that batted ball. For a more qualitative example, imagine a simulated universe in which a computer is programmed to provide a wonderfully efficient simulated chef who provides meals for all those simulated inhabitants—and only those simulated inhabitants—who don’t cook for themselves. As the chef furiously bakes, fries, and broils, he works up quite an appetite. The question is: Whom does the computer charge with feeding the chef?10 Think about it, and it makes your head hurt. The chef can’t cook for himself as he only cooks for those who don’t cook for themselves, but if the chef doesn’t cook for himself, he is among those for whom he is meant to cook. Rest assured, the computer’s head would hardly fare better than yours. Noncomputable functions are much like this example: they stymie a computer’s ability to complete its calculations, and so the simulation being run by the computer would hang. The successful universes constituting the Simulated Multiverse would therefore be based on computable functions.

The discussion suggests an overlap between the Simulated and Ultimate Multiverses. Consider a scaled-down version of the Ultimate Multiverse that includes only universes arising from computable functions. Then, rather than merely being posited as a resolution to one particular question—Why is this universe real, while other possible universes are not?—the scaled-down version of the Ultimate Multiverse can emerge from a process. An army of future computer users, perhaps not much different in temperament from today’s Second Life enthusiasts, could spawn this multiverse through their insatiable fascination with running simulations based on ever-different equations. These users wouldn’t generate all universes contained in the Mathematical Library of Babel, because the ones based on noncomputable functions wouldn’t get off the ground. But the users would continually work their way through the library’s computable wing.

The computer scientist Jürgen Schmidhuber, extending earlier ideas of Zuse, has come to a similar conclusion from a different angle. Schmidhuber realized that it’s actually easier to program