You Know More Things Than Laplace’s Demon

So, it turns out the average human knows more facts than some varieties of omniscient being.

Picture the least omniscient thing that still deserves that adjective. I’m thinking of something a couple notches up from that.
The thing I’m thinking of is a hypothetical something that knows everything there is to know about the universe and all the objects in it, from quarks to galaxies and beyond. And it knows nothing else. Obviously this thing can fairly be called omniscient. But it doesn’t necessarily have to be intelligent. It might be, or it might have no concept at all of logic or reasoning.

The thing I’m talking about is basically identical to (the weakest possible version of) Laplace’s Demon. So I’ll call it that. It knows everything knowable about the state of the universe and the laws of physics. It can extrapolate backward to see exactly how Socrates died, so you bet it knows he was mortal. And it knows exactly how many humans die. But it might not understand the “therefore” in “all men are mortal, Socrates is a man, therefore Socrates is mortal.”

It can check throughout all of time and space and make sure that every time there are two of something, and two more of that thing, there are four of it. But it doesn’t know that 2+2=4, because it doesn’t understand “equals.” It doesn’t need to. (If it can think, that won’t hurt it. The one in the Trivia Contest could. But it doesn’t need to think to be a functioning Laplace’s Demon.)

I claim that this version of Laplace’s Demon knows at most a countable infinite number of things.
In our universe, it doesn’t make much sense to divide distances into smaller and smaller pieces forever. It’s conceivable that some universe might work that way, but ours doesn’t. In ours, you hit a limit, somewhere around one decillionth of a centimeter. Anything smaller than that limit is literally unmeasurable, as Laplace’s demon knows better than anyone. Two locations separated by less distance than that might as well be the same place, and that is an actual fact about the universe not about measurement technology. (Current tech is nowhere near that good anyway.) And in any finite distance, there is only a finite number of Planck lengths.

Same thing with time. The universe started a long time ago, and the shortest measurable length of time is very small. There have been an awful lot of Planck time intervals since the universe started, and there will be an awful lot more before it ends. But it’s still a finite number.

I, like most currently existing humans, don’t know whether or not the universe is infinite in size. (Laplace’s Demon does.) If so, that’s a countable infinity. It’ll be infinite in the sense of “go one Planck length at a time, keep going forever. For any given point, you’ll be at that point eventually, but you’ll never run out of space.” This is the smallest possible infinity.
It is a property of infinity that infinity times itself is the same size of infinity*, so if the universe goes on forever in all three dimensions and we have to cube that, then we’re still talking about the smallest possible infinity.

Laplace’s Demon knows everything about every point in the universe at every time. It can look at any space-time coordinates and tell you exactly what particle, if any, was/will be there. (Can there be multiple particles within the Planck distance of each other? I should probably know that, but I don’t. If so, the Demon can tell you exactly which ones were where. Anyway, I’d be surprised if there can be an infinite number of particles in the same Planck volume and shocked if it can be an uncountable infinity.)

We’re multiplying a countable infinity (from the size of the universe, measured in Planck units) by large finite numbers (from the duration of the universe, and the number and type of particles in each space). So the result is still a countable infinity. And if the universe turns out to not go on forever, then Laplace’s Demon only knows a finite number of things. That’s almost impressively small.

By comparison, you (yes, you) know an uncountably infinite number of true things. Watch this: “For any real number greater than three, that number is greater than two.” That single obvious sentence generates an infinite number of things you know. You know that 4>2, that 3.001>2, that π>2, and so on. There are an infinite number of things you know, and that infinity, the uncountable number of real numbers above three, is bigger than the countable infinity of things that the Demon knew. Since humans have the awesome and underrated power of abstract thought, we don’t need to limit our knowledge to concrete facts.

In fact (and you can skip this part if you want to avoid math vocabulary), I’m pretty sure that the set of true statements that I know has cardinality Beth Omega. That’s big, even for an infinity.
Let Xn be the set of maps from Xn-1 to Xn-1. X0 is the natural numbers. The number of maps is the cardinality of Xn.  Since each Xn is the powerset of the one before, the cardinality of Xn is  \bethn.

\beth is the Hebrew letter Beth; \beth0 is the cardinality of the natural numbers. n can be any finite natural number.  \bethn means 2בn-1. (That’s two to the power of the previous infinity.) The smallest infinity larger than all \bethn is called \beth_\omega. Lower case omegas don’t look nearly as cool as the capital ones. For any n,  \bethn+1 is bigger than  \bethn. So the cardinality of the union of Xn over all n must be greater than any \bethn. It is therefore at least \beth_\omega

I happen to know that every map in that infinite union is the image of another one. (Trivially true: For an arbitrary map in Xn, there is a map in Xn+1 that just turns all maps in Xn into that map. So any given map in Xn (for any n) is the image of something.) Therefore, there are at least Beth Omega true statements I can claim to know.**

Even without the confusing part, I think it’s safe to say that a normal human knows an uncountably infinitely larger amount of true things than that omniscient Demon did. (No comment on who gets a better selection of things.)
So; next time you want to win an Internet argument by appealing to credibility, make this claim. Tell people your knowledge exceeds that of some omniscient beings, and that it would be impossible to list all the things you know even if you had infinite time to do it inYou’re welcome.

*Sort of. The cardinality of an infinite set crossed with itself is equal to the cardinality of that set, but the usual kind of multiplication doesn’t really make sense with infinities. But if you promise not to repeat this in front of any mathematicians and definitely promise not to tell people that said it, then sure. Infinity times itself is the same size of infinity.

**The previous version of this claimed that the number of continuous functions from Xn to Xn is \bethn, which is a smaller infinity than \bethn+1. It feels like that’s probably true for some form of continuity, but I didn’t think it through. If the old version actually is right, I got lucky. Would this be a good time to mention that the number of things I don’t know is also hugely infinite?

Advertisements

4 thoughts on “You Know More Things Than Laplace’s Demon

    1. Nate Gabriel Post author

      I…wasn’t. I guess I was assuming that since the continuity thing works for the first couple infinities it’ll probably continue to work. Of all the possible places to make a mistake, this is one of them. I’ll fix it.

      Reply

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s