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By Root 805 0

automata), one could develop a very different type of mathematics. These cellular automata could be used (in principle, at least) as the basic tools for modeling natural phenomena, instead of the differential equations that have dominated science for three centuries. What was it, then, that drove the ancient civilizations toward discovering and inventing our special “brand” of mathematics? I don’t really know, but it may have had much to do with the particulars of the human perceptual system. Humans detect and perceive edges, straight lines, and smooth curves very easily. Notice, for instance, with what precision you can determine (just by eye) whether a line is perfectly straight, or how effortlessly you are able to distinguish between a circle and a shape that is slightly elliptical. These perceptual abilities may have strongly shaped the human experience of the world, and may have therefore led to a mathematics that was based on discrete objects (arithmetic) and on geometrical figures (Euclidean geometry).

The uniformity in symbolic notation is probably a result of what one might call the “Microsoft Windows effect”: The entire world is using Microsoft’s operating system—not because this conformity was inevitable, but because once one operating system started to dominate the computer market, everybody had to adopt it to allow for ease in communication and for availability of products. Similarly, the Western symbolic notation imposed uniformity on the world of mathematics.

Intriguingly, astronomy and astrophysics may still contribute to the “invention and discovery” question in interesting ways. The most recent studies of extrasolar planets indicate that about 5 percent of all stars have at least one giant planet (like Jupiter in our own solar system) revolving around them, and that this fraction remains roughly constant, on the average, all across the Milky Way galaxy. While the precise fraction of terrestrial (Earth-like) planets is not yet known, chances are that the galaxy is teeming with billions of such planets. Even if only a small (but nonnegligible) fraction of these “Earths” are in the habitable zone (the range of orbits that allows for liquid water on a planet’s surface) around their host stars, the probability of life in general, and of intelligent life in particular, developing on the surface of these planets is not zero. If we were to discover another intelligent life form with which we could communicate, we could gain invaluable information about the formalisms developed by this civilization to explain the cosmos. Not only would we make unimaginable progress in the understanding of the origin and evolution of life, but we could even compare our logic to the logical system of potentially more advanced creatures.

On a much more speculative note, some scenarios in cosmology (e.g., one known as eternal inflation) predict the possible existence of multiple universes. Some of these universes may not only be characterized by different values of the constants of nature (e.g., the strengths of the different forces; the mass ratios of subatomic particles), but even by different laws of nature altogether. Astrophysicist Max Tegmark argues that there should even be a universe corresponding to (or that is, in his language) each possible mathematical structure. If this were true, this would be an extreme version of the “universe is mathematics” perspective—there isn’t just one world that can be identified with mathematics, but an entire ensemble of them. Unfortunately, not only is this speculation radical and currently untestable, it also appears (at least in its simplest form) to contradict what has become known as the principle of mediocrity. As I have described in chapter 5, if you pick a person at random on the street, you have a 95 percent chance that his or her height would be within two standard deviations from the mean height. A similar argument should apply to the properties of universes. But the number of possible mathematical structures increases dramatically with increasing complexity. This means that