# Built on Facts

## Off the Normal Path

#### July 26th, 2008 · 3 Comments

Let there be pi?

Mark Chu-Carroll at Good Math, Bad Math has an interesting post taking down a guy who thinks messages from God are encoded into pi.

Stare at any number, and set of numbers, or any numeric coding of a text. If you try hard enough and long enough, then you will find some interesting patterns. Looked at probabilistically, the chances of finding a large sequence of numbers or letters where we can’t find any pattern is vanishingly small. So given a pattern, we need to ask, is this pattern just the result of randomness? Just because you found an apparent pattern doesn’t mean that it’s deliberate or meaningful. In fact, it probably isn’t.

Very true, and I don’t think I can improve on his analysis. You should read the whole thing, it’s quite fascinating.

There are two points I’d like to add, one mathematical and one theological. For the mathematical point, let’s take as our jumping-off point this, from Mark’s post

Pi isn’t random. No one who understands what “random” means would say that. In fact, pi is very much not random. It’s a highly compressible number: there’s a simple algorithm for computing it, which means that by definition, it’s not random.

Absolutely correct. However, there’s an interesting property of numbers like pi with regard to the decimal digits of the number. As you read through the trillions of digits of pi that have been computed, you’ll notice that while the numbers certainly aren’t random (they’re exactly the digits of pi!), in a way they behave as though they were random. If you throw a dart at a printout of the digits of pi, the odds that you’ll hit a 3 are about 1/10. The odds that you’ll hit your 7-digit phone number are about 1/107. In other words, though the string of digits is not random, the digits themselves are statistically distributed in the same kinds of ways actually random digits would be distributed. If you go far enough, eventually you’ll find any given sequence of digits. A number whose decimal digits exhibit this property is called a normal number.

Well, those are the properties of normal numbers and pi gives every indication that it is in fact a normal number. But this hasn’t actually been proved yet. Maybe there’s some subtle statistical irregularity that has not yet been detected in pi. For instance, maybe you won’t ever get Shakespeare’s Hamlet in binary no matter how far you go in pi. Nonetheless, mathematicians have very good reasons to believe pi, e, and most other numbers (in a technical sense) are normal. It’s not a certainty, but it’s pretty darn likely.

So that’s kind of neat. A string of digits that looks random might actually be something completely nonrandom, like the digits of pi. And correspondingly, a very orderly string of digits like those of pi might in some ways seem as though it were random. Numbers are cool like that.

Now the theological point, which Mark also mentions. Omnipotence means the ability to do anything. Unfortunately language uses the same words to describe doable things as it does to generate meaningless syntactical statements. The classic example is “Can God create a rock so big he can’t lift it?” The problem is that while the question looks like sensible English, it isn’t. If one of those things exists, the other can’t. This is no ill reflection on God’s power, it’s an acknowledgment that some sentences just don’t mean anything in the first place. Mathematics works the same way. “What if pi were something else?” is a nonsense sentence, a low-rent Zen koan. The idea of God encoding messages in mathematically defined numbers simply is not coherent. If something were encoded in those numbers, they wouldn’t be those numbers anymore.

We’re thus pretty much assured that the constants of mathematics are message-free, movies and internet speculation notwithstanding.

Tags: Looking Beyond

### 3 responses so far ↓

• 1 Uncle Al // Jul 26, 2008 at 11:50 am

Pi is contained within pi. 314159265358 from the 1,142,905,318,634-th decimal digit . e is in there too, 271828182845 from 1,016,065,419,627-th decimal digit. As with the Einstein-Podolsky-Rosen paradox the trick is not data but information. Pi doesn’t contain any information (one exception noted).

Can God make a collection plate so vast that even He cannot fill it? Sure! ALL OF THEM.

Pi digit occurances are flat. For 1.2 trillion decimal digits (Kanada),

0: 119,999,636,735
1: 120,000,035,569
2: 120,000,620,567
3: 119,999,716,885
4: 120,000,114,112
5: 119,999,710,206
6: 119,999,941,333
7: 119,999,740,505
8: 120,000,830,484
9: 119,999,653,604

• 2 CCPhysicist // Jul 27, 2008 at 7:55 pm

I don’t recall the difference between pseudo random and normal, maybe because I never took statistics and learned most of what I know from listening to experimental physicists, economists, and code breakers. But I do know a fair bit about pseudo random numbers, which are algorithmically generated number sequences that (as in modern slot machines or physics Monte Carlo codes) should have as many properties of a truly stochastic sequence as possible.

Being “flat” (as Uncle Al demonstrates) is only one of those properties. Others include pairing correlations and how densely they fill space when drawn as n-digit m-tuples (triplets in 3-space). IIRC, one version of the latter is the predictable results of random birthdates in a group of N people.

Pairing correlations are easily spotted in lottery numbers. The odds that there will be some (not a particular, some) sequential pair of numbers in a lottery number is about 50%. Many people never choose such numbers because they don’t “look” random.

• 3 Maynard Handley // Aug 31, 2008 at 11:23 pm

“This is no ill reflection on God’s power, it’s an acknowledgment that some sentences just don’t mean anything in the first place. ”

Hmm. Sure there are some sentences that don’t mean anything, but I don’t think that’s the issue here. The issue, surely, is that the very concept of “omnipotence” makes no sense; in the same sort of way that “sets that contain all sets that don’t contain themselves” doesn’t make sense. Defining a new word, “heterocontainers” doesn’t change the fact that the very concept makes no sense.

In other words, this IS a reflection on god’s “powers”. Whatever those powers may be, omnipotence (as defined this way) is not one of them.

(The smart way out of this would be to redefine omnipotence, or come up with a new word, that provides for miracles and suchlike, making god something like superman. “Has the ability to manipulate spacetime and matter as he wishes”. The problem, of course, is that if you go down this road you have to be very very careful that the manipulations you allow violate “physics” but not “mathematics”, and since physicists don’t know where that boundary is, or if it even exists, theologians aren’t going to do much better.)