Good Math, Bad Math

There's one kind of semi-mathematical crackpottery that people frequently send to me, but which i generally don't write about. Given my background, I call it gematria - but it covers a much wider range than what's really technically meant by that term. Another good name for it would be numeric pareidolia. It's been a long time since I've written about this kind of stuff, and someone just sent me a pretty typical example, so what the hell. It revolves around a mess that he put together as an image, which is pretty much a classic example of obsessive silliness.

The general idea of this kind of silliness is finding some kind of numeric pattern, and convincing yourself that there's some deep, profound truth behind that pattern. There are a couple of typical kinds of this: number/letter correspondence (classical gematria, which uses the fact that the hebrew characters are used both as letters and numbers, so a word can be interepreted as a number, and vice versa), distance coding (like the infamous "torah codes", where you find words "hidden" in a text by picking out characters according to some pattern and using them to form words), and simple numeric patterning (where you take numbers - generally some sort of constant - and find some sort of pattern supposedly hidden in its digits). Todays crackpottery is the third kind - it's written by a guy who believes that there are mystic secrets encoded into π and the square root of two that were put there by God, and that the existence of those patterns are proof of the existence of God.

This little bundle of rubbish - like all of the kinds of things I described above - are examples of pareidolia involving numbers. As I've written about before, we humans are amazingly good at finding patterns. We've got a strong natural talent for looking at things, and finding structures and patterns. That ability serves us well in many of our ordinary endeavors. The problem with it is that there are apparent patterns in lots of things. In fact, if you look at things mathematically, the odds of any text or constant not containing interesting patterns is effectively nil. If you're willing to consider all sorts of patterns, then you can find patterns in absolutely everything. The question that you need to ask is whether or not the pattern is simple the result of our ability to find patterns in noise, or whether it's something deliberate.

These numeric games are an example of that. 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. If you really want to find out if it's real, there are ways of using Bayesian analysis to work out the probability of a pattern like that occuring at random. But even there, you need to be very careful. It's incredibly easy to subconsciously set up the priors in your analysis to make your pattern appear real. Even professional statisticians have been known to foul that up when they believe that they've found something interesting.

Anyway, with that long-winded introduction out of the way, on to today's silliness! This is a supposed proof of the existence of God, demonstrated by patterns in numbers. Our intrepid author claims that he's finding patterns in three numbers. Personally, I'd argue that he's really using two numbers, and that his third isn't really disinct.

The three numbers are Π, the square root of two, and a constant that he calls S, which is the square root of Π divided by two. I don't think that it's fair to consider S to be an independent constant: it's derived from Π, so I'd claim that anything in S is really just deeply hidden in Π.

This argument is really pretty shallow, even as this sort of silliness goes. Most of his number games are played with only the first ten base-ten digits of each constant - and looks no further. He plays games with those digits, and believes that he's finding deep, profound patterns that could only have been placed there by deliberate actions of a deity.

Before getting to that, I have to point out how quickly he goes off the rails. His first statement is:

Throughout the many centuries pi (π) has been thought to be a random number. With the advent of computers which excel in mathematical analysis this concept has been reinforced.

Π isn't random. No one who understands what "random" means would say that. In fact, Π 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. Π = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11. How much less random can you really get?

Anyway, after that awful start, he goes downhill, by finding trivial patterns in π. He manages to demonstrate quite a number of the classic errors in this process.

A typical example:

1) Pi (pi = 3.141592653...) starts with 3_4_5 which is the smallest Pythagorean Triplet possible forming the Pythagorean triangle. This triplet, 3_4_5 can be recognized as the representation of the Pythagorean Theorem. Also in pi the sequence 1_ _2_ _3. In √2 early the sequence1_2_3 and 4_ _3_ _2.

This is what passes for profundity in this guy's mind: if you skip every other digit in the first five digits of Π, you'll get 3, 4, and 5. If you look at the square root of two, and you skip pairs of digits, you can find 1, 2, 3 and 4, 3, 2.

This is just meaningless. He managed to find some really trivial patterns. But they're just eyeball patterns - that is, there's no reason why skipping pairs of digits starting from the second base-10 digit should produce something odd. And if I'm allowed to look at the leading sequence of almost any irrational number, I can find a similar pattern. For example, in the square root of 31, the 2nd through fourth digits are 567. In the square root of 311, using the pattern of take one, skip 2, the in the leading section, I find 1, 3, 9 - powers of the first digit of 311!

Moving on to slightly more interesting patterns, another of his profound discoveries is, if you separate the leading digits into pairs, two of the first three pairs in Π are used in equations that are relatively good at generating small primes. Yep. Two of the first three base-10 digit pairs happen to be used in equations that are probabilistically good at finding small primes. Wow! I'm impressed, aren't you?

A great example of where a bit of Bayesian analysis could have easily showed him how trivial a pattern is comes in his example number 6:

6) Pi = 3.14159265358979323... It is very odd that a group of eight small different contiguous primes: 3, 14159, 2, 653, 5, 89, 7, 9323 are right at the start of pi. Many and possibly infinite small (numbers with five or fewer digits) and large (greater than five digits) different contiguous primes may exist after "9323". It will be interesting to see how many digits the average contiguous prime has. Perhaps more interesting may be to find how scarce are groups consisting of eight small contiguous primes of which none of the prime numbers are duplicated.

This is a case where even a moment of thought should have shown him how stupid this one is. Take a set of irrational numbers with roughly uniform digit distributions. Now, you're allowed to choose any length sequences of digits, without any reason behind the lengths. What's the probability of finding primes in the leading sequence? It's not a trivial analysis - but if you take an hour (which is a lot less than this guy spent on this nonsense), and work it all out, you can find the probability of that occurring. An admittedly sloppy back of the envelope calculation gives me a rough estimate that 1 in 5 will contain a sequence of primes varying between 1 and 4 digits in the first 10. And Π isn't one of them! To make that work, he needs a 5 digit prime.

Another one - in the sequence for Π, starting at digit 7, you find 2653. In the sequence for the square root of 2, starting at digit 6, you find 3562. Oooh, look, Π and the square root of two contain 4-digit mirrored sequences, occurring at different positions! That one would only take a 5 minute Bayesian computation to show that it's not an unlikely phenomena.

He goes on - and his patterns start getting even more silly:

10) √2 starts with the Fibonacci sequence 1_1_2_3 in which 11, 23 and 1123 are primes. Also note: √2 = 1 .4 1 4 2 1 3 5 6 2 3 7 3... Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13... In √2 the Fibonacci sequence is included: 1, 1, 2, 3, 5, 8(6+2), 13(3+7+3). Very odd that seven numbers of the Fibonacci sequence are found at the start of the square root of 2.

So... If you take the square root of two, start by skipping every other character, then after the first three pairs, stop skipping, and adding arbitrarily sized groups of digits together you can get first seven members of the fibonacci sequence.

This is supposed to be profound, deep truths hidden in the numbers. God deliberately created the universe so that when represented using base-ten digits, the first seven numbers in the fibonacci sequence are hidden as the first, third, fifth, seventh+eighth, and ninth+tenth+evelenth digits of the square root of two. And of course, the "pattern" stops there.

Of course, our brilliant author has anticipated my objections. From his main page, he has a link to another page, "Your questions answered by Vas"."

Question: If you are clever enough you can find oddities in all numbers. What do you have to say about that?

Vas: Perhaps I am not so clever. I have tried several random numbers and have only limited success in finding oddities. One could successfully write countless unique equations to fit every constant or random number and call it an oddity. It is difficult however to find oddities using only the basics such as prime numbers, Fibonacci sequence and the Pythagorean triplet. No equations or algorithms are used in Document A. What makes it even more odd and interesting is that pi, the square root of two and S have several interwoven oddities. There are also two interesting blocks, one at E-3 and the other at E-4 (see alphanumeric locators on the sides of Document A).

If you are correct all numbers lets say from 000000001 to 999999999 should have oddities similar to pi. Why does 000000001 and 999999999 lack oddities which includes Fibonacci, geometric and prime numbers and the like which are easily found in the first nine digits of pi?

You are welcome to try your luck at finding oddities in random numbers. See random number generator: http://www.randomnumbers.info (enter 10 and 9). Good luck. Let me know how you did.

Well... First, as I showed above, I spent about thirty seconds eyeballing the leading digits of the square roots of 31 and 311 - and in both cases, I was able to find "interesting" oddities. If I were to spend several months obsessing over those numbers, I'm pretty sure I could come up with a list at least as compelling as his.

I also find his claim that "no equations or algorithms are used" hysterically funny. Skipping every other digit is an algorithm. He explicitly talks about the role of the numbers 41 and 59 in equations that probabilistically generate likely primes. His entire argument consists of random eyeball patterns, trivial algorithms, and silly equations.

One last thing, and then I'll stop picking on this idiot.

In general, this kind of numeric pareidolia is Π is silly. But if you want to find messages from God encoded in constants, you should at least pick constants that are manipulable by a deity. One of the interesting things about Π is that it isn't a constant of our universe; it's an inescapable constant. It's also a number that doesn't really exist in our universe. Try as much as you want - you can never find anything in this universe which reifies Π. There's no perfect circle that you can measure to find a perfect value of Π. Π is an idea, defined by a axiomatic system - and given that axiomatic system, its value is a fixed and unchangeable result of that axiomatic system.

If you imagine euclidean plane geometry, you have to have this specific value for Π. You can't have a different value for Π in a plane. If you were to use a measured Π, that might be interesting - because it would vary from the ideal Π in a way that measured the curvature of the universe, which is something that a creator deity could change. But you can't change the fundamental, mathematical Π - because that's the product of the axiomatic system called Euclidean geometry. You could make an argument that God created us in a way where we'd naturally devise Euclidean geometry; but he can't have chosen the digits of Π in that system.

If you want to waste your time on searching for messages from a Creator encoded in its Creation, you should at least try looking at something that is a real part of that creation. The ratio of masses of the primitive particles, the curvature of space, the strengths of the basic forces, Planck's number, etc. Things that are conceivably variable between different universes, and that therefore say something about what your deity created.