---- Apologies if you all get this twice; my e-mail is playing silly games
and I don't know if it went out correctly the first time!
Peter wrote:
>
> According to Iain, one would expect the closest hit to pi among 5000
> torah verses to be in the order of 10^-4, making the result of 10^-5
> found with Gen.1:1 truly extraordinary. I was not convinced by this
> reasoning, for three reasons: (a) an error (of a single measurement!) 10
> times smaller than expected might easily be a coincidence, as the tails
> of the Gaussian extend to infinity; (b) the value of pi was not given
> beforehand as a hypothesis to be tested, but was found accidentally
> after doing the calculation; (c) it is not known how many different
> formulas were tried (the more different ones you try, the higher the
> probability of hitting something "interesting").
Peter,
I think you've mis-understood what I said, so I'd like to show how the
calculations are arrived at. In fact having done it rigorously the results
come out lower (in terms of probability) that the rough (back of envelope)
calculation I did before. This is because the distribution of the
_logarithm_ of Jenkins' function is uniform, whereas the distribution of the
actual function is exponential, which accounts for the difference.
You are right that the fact that the closest to pi is 10^-5 when the
expected difference of the closest verse is 10^-4 is not that extraordinary,
and I didn't claim that it was (if it appears I did claim this, then my
apologies). (Incidentally the distribution is not Gaussian, as you imply,
it is uniform in the range zero to 1 for the logarithm of the Jenkins
formula). Here are the steps:
(1) Let's say we pick a specified number, say Y. We now pick a specified
verse, let's say Exodus 19:3, picking one at random. If you don't like the
aritrariness of the verse divisions, pick an arbitrary complete indisputable
sentence from Exodus 19, or pick words 2734-2740 inclusive from the
beginning of the Torah. The argument is the same. Now calculate the value
X, which is the result of applying Jenkins' formula to this piece of text.
Now ask the question "What is the probability that x is within 10^-5 or our
specified number Y? We know that the distribution of log(x), taken over all
the verses (or whatever division you like) is uniform in the range (0,1). So
the probability that the value of log(X) lies in the range log(Y- 0.5x1e-5)
to log(Y+0.5x1e-5) is given by the size of this range:
P( |X-Y| < 1e-5) = log(Y+ 0.5x1e-5) - log(Y - 0.5x1e-5)
(1)
(where |X-Y| denotes the absolute value, and P(...) denotes probability of
the event).
(2) Now instead of saying that our _specified_ verse (say Exodus 19:3) is
within 1e-5 of the value Y, let us calculate the chance that _any_ verse
comes to that value. The same calculation applies to any verse, so the
probabilities add up, and the probability is therefore 5000 times the
quantity in equation (1) (assuming around 5000 verses, there will be 5000
independent chances of getting a hit that close, so we add the
probabilities).
(3) Now plug the value of pi in for the number Y in equation (1). We get P
= 1.38e-6 (around 1 in 720,000). So this is the chance that a specified
verse comes this close to pi. (and Genesis 1:1 was specified by Vernon
because it is in a highly significant place in the bible -right at the
beginning), and because previously he had found other remarkable numerical
properties contained in it). Hence I claim it is reasonable to assign this
probability to the event.
(4) Now consider the chance that _any_ verse in the Torah is within 1e-5 of
pi. This is 5000x1.38e-6, or 6.9e-3 ( which is around 1 in 144).
(5) So the value computed in (4) is not that remarkable; odds of a bit over
1 in 100; enough perhaps to raise a statistical eyebrow, and be considered
"significant" i.e. worthy of further examination. However the value
computed in (3) is well beyond coincidence.
(6) But there still is the fair point that the formula is arbitrary; how
many other formulae did Vernon try out before hitting on his magic formula?
If he had laboriously tried out 720,000 different formulae, then we should
not have been surprised at the result. But if this were the case, we should
not expect a similar result for the obviously linked verse John 1:1. (If
he had tried to derive a formula that simulataneously gave pi and e for
Genesis 1:1 and John 1:1, then the expected number of formulae he would have
had to try before finding the magic one would be 4.5e11; rather more than a
lifetime's work ;-) In fact, I know this is not how it happened. Vernon
got the formula for pi in two steps from two different peoples'
observations. Later on a third independent researcher plugged the formula
into John 1:1 and got the e result.
In my field of mathematical modelling using neural networks, we would call
the independent check on John 1:1 a "validation set". The John 1:1, as we
know comes to within a similar accuracy of the value of e, the other
best-known mathematical constant. Setting Y to the value of e in equation
(1) yields the value 1.59e-6 for the probability, a similar value. To be
ultra fair in the probability calculations one should discard the
probability estimate from the verse (Gen 1:1) that was used to derive the
model (fit the data), and only take the probability that the independent
data sample (John 1:1) as the odds. We still have of the order of 1 in
100,000. However, clearly Vernon didn't dream up hundreds of thousands of
formulae to make Gen 1:1 come to an "interesting" number. [Incidentally,
one should allow for the fact that had it produced an approximation to a
famous mathematical constant, then a similar claim might have been made.
Despite the fact that pi and e are the most famous, one should at least
allow for the chance that if John 1:1 had come to pi as well as Gen 1:1 then
it would be seen as significant. This multiplies the probability up by a
factor 2. How many other numbers could one dream up and claim as
significant? There are not many obvious candidates; maybe the Golden
ratio, the square root of 2, the Fine Structure constant of the universe,
Feigenbaum's constant (in chaos theory), and ... well one is beginning to
grope around for obscure numbers that wouldn't be anything like as attention
grabbing as e and pi].
(7) Herein lies the difference. Suppose Vernon had sent me a message
saying that he had derived this magic formula, and applied it to every verse
in the Torah, and found that Exodus 19:3, when this formula was applied,
came to within 1e-5 of Feigenbaum's constant. What did I think of it? The
answer is of course "Big Deal". The chance of that happening is around 1 in
150, and the verse itself is not even a significant one; it doesn't even
contain a complete statement, as it ends "this ... is what you are to say to
the people of Israel:", and what they were to say is specified in verse 4.
But instead of this, Vernon had this formula, applied it to the one verse
that has occupied his attention all this time, and it came to Pi, a number
that almost everyone has heard of, first time, and then an independent test
showed a similar significant value for John 1:1.
I hope the above clarifies matters.
Regards to all
Iain Strachan.
This archive was generated by hypermail 2b29 : Sat Jul 07 2001 - 14:27:48 EDT