Immortal but Illiterate Monkeys

Is it an infinite number of monkeys, or is it infinitely-lived monkeys?  If what you want is Shakespeare with probability 1 it matters.  Because Hamlet is a fixed finite string of characters.  That means the monkey has to stop typing when the string is complete.  If we model the monkey as a process which every second taps a random key from the keyboard according to a fixed probability distribution, then to produce the Dithering Dane he must eventually repeat the space bar (or equivalently no key at all) until his terminal date.

If that terminal date is infinity, i.e. the monkey is given infinite time, then this event has probability zero.  On the other hand, an infinite number of monkeys who each live long enough, but not infinitely long, will Exuent with probability 1 as desired.

(If your criterion is simply that the text of Hamlet appear somewhere in the output string, then a) you are sorely lacking in ambition and b) it no longer matters which version of infinity you have.)

Mortarboard Missive: Marginal Revolution.

Reductionism and Superman

Via kottke, here is a paper proposing A Unified Theory of Superman’s Powers.  The abstract reads as follows.

Since Time immemorial, man has sought to explain the powers of Kal-El, a.k.a. Superman. Siegel et al. Supposed that His mighty strength stems from His origin on another planet whose density and as a result, gravity, was much higher than our own. Natural selection on the planet of krypton would therefore endow Kal El with more efficient muscles and higher bone density; explaining, to first order, Superman’s extraordinary powers. Though concise, this theory has proved inaccurate. It is now clear that Superman is actually flying rather than just jumping really high; and His freeze-breath, x-ray vision, and heat vision also have no account in Seigel’s theory.

In this paper we propose a new unfied theory for the source of Superman’s powers; that is to say, all of Superman’s extraordinary powers are manifestation of one supernatural ability, rather than a host. It is our opinion that all of Superman’s recognized powers can be unified if His power is the ability to manipulate, from atomic to kilometer length scales, the inertia of His own and any matter with which He is in contact.

The paper goes on to show how the theory can explain Superman’s super strength, ability to fly, super senses, and even his heat vision and freeze breath.  It’s an elegant theory but the analysis has one significant gap.  It is not enough to find a simple principle from which all of Superman’s powers follow.   It is necessary to also show that the principle would not imply powers that Superman does not have.

If we do not insist on the latter, then there is an even simpler theory that does the trick:  Superman can do everything. (Although that comes with its own difficulties.)

The Largest Number Game

Its a standard example of a game that has no Nash equilibrium.  But what exactly are the rules of the game?  How about these:

You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number—not an infinity—on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.

Hmm… maybe it does have a Nash equilbirium.  But after reading the article (highly recommended), I am still not sure.  I think it comes down to whether or not the players are Turing machines.  (Fez flip: The Browser)

Useful Math?

A post at Language Log explores the use of mathematics in linguistics.  It closes with

Anyhow, my conclusion is that anyone interested in the rational investigation of language ought to learn at least a certain minimum amount of mathematics.

Unfortunately, the current mathematical curriculum (at least in American colleges and universities) is not very helpful in accomplishing this — and in this respect everyone else is just as badly served as linguists are — because it mostly teaches thing that people don’t really need to know, like calculus, while leaving out almost all of the things that they will really be able to use. (In this respect, the role of college calculus seems to me rather like the role of Latin and Greek in 19th-century education:  it’s almost entirely useless to most of the students who are forced to learn it, and its main function is as a social and intellectual gatekeeper, passing through just those students who are willing and able to learn to perform a prescribed set of complex and meaningless rituals.)

Before getting into economics and after getting out of physics, I took calculus and found it very useful and interesting for its own sake.  I do see that the way calculus is taught in the US is geared toward engineers and physicists, but I have a hard time thinking of what mathematics would substitute for calculus in the undergraduate curriculum if the goal was to teach students something useful.  It can’t be analysis or topology.  I took abstract algebra as an undergraduate and found it esoteric and boring.  Discrete mathematics?  OK maybe statistics, but don’t you need integration for that?  Help me out here, if you had the choice, what would you replace calculus with? And remember the goal is to teach something useful.

Procedural Animation

“These are relatively simple physical equations, so you program them into the computer and therefore kind of let the computer animate things for you, using those physics,” said May. “So in every frame of the animation, (the computer can) literally compute the forces acting on those balloons, (so) that they’re buoyant, that their strings are attached, that wind is blowing through them. And based on those forces, we can compute how the balloon should move.”

This process is known as procedural animation, and is described by an algorithm or set of equations, and is in stark contrast to what is known as key frame animation, in which the animators explicitly define the movement of an object or objects in every frame.

Why stop there?  Next, we can use models from the behavioral sciences, program a few equations and let the characters, dialog, and action animate themselves by following the solution of the model.  Don’t believe me? Here’s how to procedurally animate Romeo and Juliet.