anthony_aguirre's picture
Professor of Physics, University of California, Santa Cruz; Author, Cosmological Koans
Indexical Information

“Information” is a term with many meanings, but in physics or information theory we can quantify information as the specificity of a subset of many possibilities. Consider an 8-digit binary string like 00101010. There are 256 such strings, but if we specify that the first digit is “0” we define a subset of 128 strings with that property; this reduction corresponds (via a base-2 logarithm in the formula) to one bit of information. The string 00101010, one out of 256, has 8 bits of information. This information, which is created by pointing to particular instances out of many possibilities, can be called indexical information. (A closely related concept is statistical information, in which each possibility is assigned a probability.)

This notion of information has interesting implications. You might imagine that combining many strings like 00101010 would necessarily represent more information, but it doesn’t! Throw two such strings in a bag, and your bag now contains less than 8 bits of information: two of the 256 possibilities are there, which is less specific than either one.        

This paradoxical situation is the basis for Borges’ story of the Library of Babel, which contains all possible 410-page books. In having all possibilities, it is information-free, and completely useless. Yet suppose one had an index, which pointed to all the books composed of actual words, then among those the books in sensible English, and of those, the books of competent philosophy. Suddenly, there is information—and the index has created it! This fable also makes clear that more information is not necessarily better. Pointing to any single book creates the maximal amount of indexical information, but a good index would create a smaller quantity of more useful information.       

This creation of indexical information by pointing to what is important to us underlies many creative endeavors. One could write a computer program to spit out all possible sequences of musical notes, or all possible mathematical theorems provable from some set of axioms. But it would not be writing music, nor doing mathematics—those endeavors select the tiny subset of possibilities that are interesting and beautiful.        

What are most people most interested in? Themselves! Of all the people ever to live, you are you, and this creates indexical information particular to you. For example, indexical information results from “our” position in space-time. Why are there no dinosaurs about? Why is the ocean water rather than methane? Those exist in other times and places—but we are here, and now. Without here and now, there is of course no fact of the matter about whether dinosaurs exist; but we are accustomed to considering here, and especially now, to be objective facts of the world. Modern physics indicates this is unlikely.        

Indeed, some modern physical theories such as the “many worlds” view of quantum mechanics, or that of an infinite universe, this indexical information takes on fundamental importance. In both scenarios there are many or infinitely many copies of “you” in existence that are indistinguishable in their experiences but that are embedded in different larger environments. Through time, as you make observations—say of whether it starts to rain—you narrow down the subset of “yous” compatible with what you have seen, and the information you have gained is indexical.        

In a sufficiently large Universe we can indeed even ask if there is anything but indexical information. The information about no-dinosaurs, yes-water, and what-weather is missing from the World, which may contain little or no information, but it is present in our World, which contains an immense amount. This information is forged by our point of view, and created by us, as our individual view takes a simple, objective world with little or no content, and draws from it a rich, interesting information structure we call reality.