Sunday, November 23, 2008

Something of which there is a googol

A future Facebook friend of mine posed this puzzle: "Name any material, physical, thing, of which there exist a googol." The "material" restriction excludes things like, e.g., 1/googol of a meter, or the number of different combinations of hydrogen molecules. He asserted there was nothing that existed in the quantity of a googol. Here's a possible (more interesting than satisfactory) answer.

Let f(t) be the highest number someone could reasonably give at time t to answer the question "What is the largest number of any object in the universe?"

The function f I guess has been increasing over the last couple centuries as scientists have probed deeper into the microstructure and macrostructure of the universe. If you assume it will continue with the same trajectory in the future, you can plot when it will hit a googol. So there are a googol of whatever particle scientists will have discovered at that time. You could challenge this as not being a "material, physical" object as it hasn't been discovered yet, but it is nevertheless an actual existing entity in the world now. We only need better futurology to predict its discovery with greater certainty. If you have a futurology that confidently predicts a discovery, is not the discovery as good as made? How about a futurology that is only good enough to discover the discoveries of future futurologists? If one continued this reasoning too far one might eventually get in a futurological panic analogous to the current financial panic (science research grinds to a halt as scientists re-examine all the particles and chemicals and biological processes they've been studying, disentangling the real things from discredited future projections; embarrassed science teachers announce that classes have been cancelled indefinitely) but surely any good futurology would have already taken this issue into account, the way that a good statistician makes sure to take multiple comparisons into account.

I guess my brain's been warped by the books I've recently read: Stanislaw Lem's "The Futurological Congress" (science fiction, enthusiastically recommended) and Ray Kurzweil's "The Singularity is Near" (putative science non-fiction, more cautiously recommended).

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