I wrote (in answer to John Burgeson):
> >
> > The agreement with the "interesting number" was to five places of decimals,
> > so the odds of any given number on any particular text variant with any
> > particular mathematical transformation is 1e-5.
>
The rest was a careful probability calculation demonstrating that John's claim that you were "virtually certain" to find a hit, was in fact the exact opposite. Even given a generous selection of key numbers, all of which were intrinsically interesting from a mathematical point of view (or at leas famous in maths), you were in fact virtually certain not to get a hit.
George has chosen to ignore this point completely, but instead change the subject by latching on to my use of the word "Interesting" , - which I was re-quoting from John in the first place:
> Is it superfluous for me to again point out the fact, easily provable by
> mathematical induction, that there is no "uninteresting number"?
>
There are of course many such "proofs" that could be constructed, and none of them make any difference to the argument. I do not know which one you were referring to, George, but here's one example, and why I think it is fallacious:
This is the essence of the induction proof that all numbers are "interesting".
Step 1:
Given an infinite set U(n) of numbers some of which are "interesting" and some
of which are "uninteresting", construct the two subsets I(n+1) and U(n+1)
containing respectively the interesting and uninteresting numbers from U(n).
Arrange the members of U(n+1) is some specified order (e.g. ascending).
Note that element j of U(n+1) might be termed "interesting" if j is a member of
I(n+1).
Therefore it is possible to partition an uninteresting set of numbers into an
interesting and an uninteresting set by adding in the extra definition of
"interestingness".
(*) Therefore given an initial set that contains a mixture of self-evidently
interesting numbers and uninteresting ones, it is possible to prove that all
members of that set are interesting by induction.
Step 2:
Consider the set U(0) of all positive integers. Self-evidently the number 1 is
interesting because it is the first and unity has unique properties in
arithmetic.
So the set S has the required properties for (*) to be true.
Therefore all numbers are interesting.
The fallacy of the argument, I believe, is that at every step in the induction,
you have to introduce a new (and subtly different) criterion for
"interestingness". Each one is dependent on the last, and in the whole argument
there is only one number that is intrinsically interesting, rather than interesting by
virtue of an artificial criterion you have introduced.
An infinite number of such criteria are necessary in order
to complete the proof.
This reminds me of the old joke about the apprentice Zen monk asking the head monk
what it is that holds the earth up in space. The head monk replies that the
earth rests on the back of an enormous elephant that in turn rests on the back
of an even bigger turtle. When the apprentice asks what holds the turtle up,
the head monk replies "it's turtles all the way".
In essence, the induction proof that all numbers are interesting is an infinite
stack of turtles and doesn't demonstrate anything mathematically interesting.
It is perfectly clear that the above "proof" allows no mathematically
interesting or useful computations to be made.
-------------------------
By contrast, let me try and put in a reasonably short space of time why I
believe the mathematical patterns presented by Jenkins are of a genuinely
interesting kind, and therefore may be said to be part of a deliberate design.
Unlike Jenkins, I will not discuss the identity of the designer; that is up to
the you to decide. This is part of the original research on Jenkins' web-site; the
"e" and "pi" stuff are quite different; almost an added extra.
First, let's point out that the attachment of numbers to Hebrew letters is
historically verifiable - and was probably influenced by the Greek system,
which was introduced in the 5th Cent. BC. My previous posting gave the
references.
Summing the letters in each word allows a sequence of seven integers to be
derived from the first seven words in the bible, which are often (though not
always) treated as a complete sentence.
The sum of the seven integers is 2701, the product of two primes, 37 and 73,
which are self-evidently digit reversals in base 10 arithmetic. What has not been
pointed out before is that it is easy to demonstrate that the digit reversal is connected
to the numerical geometry that forms the basis of Jenkins's thesis.
Jenkins has pointed out that the pair 37 and 73 may be represented in figurate
geometry as a hexagon and a hexagram (six-pointed star). This is shown below:
*
* *
* * *
* * * & & & & * * *
* * & & & & & * *
* & & & & & & *
& & & & & & &
* & & & & & & *
* * & & & & & * *
* * * & & & & * * *
* * *
* *
*
(note that this will only come out symmetrically if your e-mail program can display Rich Text Format, and has the fixed spacing Courier font. Otherwise the difference in width between a space and an asterisk character will make it seem lop-sided - apologies for the crudeness of the diagram - I did not wish to waste bandwidth by sending an image file to everyone).
73 characters have been arranged as a six pointed star. The star can be
geometrically represented as the superimposition of two equilateral triangles of
55 objects, one an inverted copy of the other. There are 37 objects, arranged
as a hexagon that constitute the overlapping objects. (Set intersection), the 37 are represented as ampersands (&)
and 73 in total (set union). (The points to the star are asterisks).
It is noted that the triangle of 55 objects is the 10th triangular number, (i.e. the
base of the triangle has 10 objects on it) and
that 37 and 73 are digit reversals of each other in base 10. This is no
coincidence; the facts are connected. If we define the arithmetic base to be
the order of the "generating triangle", then the hexagon/hexagram pair are
always digit reversals. This fact is easy to prove, by taking the generating
formulae for the hexagon and hexagram numbers
(3x^2 + 3x +1) and (6x^2 + 6x +1)
and dividing algebraically by the arithmetic base (3x+1).
By way of example, note that the last two words of the Gen 1:1 sum produce the
total 703, which is again the product of two primes 19x37. In this case, we
note that 37, as well as its hexagonal representation, can also be represented
as a hexagram, with 19 as the corresponding hexagram. (Also note for the
record that the last two words are both individually divisible by 37; 407 and
296). This the set of seven integers exhibits self-similarity, and the prime
factors of the last two may be represented by a smaller version of the above
figure, made by the intersection of two triangles of 28 objects, which is the
7th triangular number. Accordingly, in base 7 arithmetic, 19 and 37 are digit
reversals 25 and 52 (2x7+5 and 5x7 + 2).
*
* *
* * & & & * *
* & & & & *
& & & & &
* & & & & *
* * & & & * *
* *
*
(19 &'s as a hexagon, 37 objects in total. In base 7 arithmetic 25 and 52).
The relation of course generalises for other cases. The union and intersection
of two 16 based triangles (136) objects gives the pair (91,181), which in
hexadecimal gives 5B and B5. If you work through the maths, you can also prove that the two digits always add up to the arithmetic base; 2+5=7, 3+7=10, 5+B = 16, and so forth.
It also turns out that most of the interesting base 10 properties of 37
generalize up a in precisely the same way. This is because 10^3-1 is a
multiple of 37. It can be shown easily that if the arithmetic radix is r (and
has the form 3n+1), that r^3-1 is divisible by the corresponding hexagonal
number. Hence 16^3-1 = 4095 = 45x91.
This property leads to the digit summing divisibility test for 37 (just like
the digit summing for 3). Simply divide the digits into threes and sum them.
If the total is a multiple of 37, so is the original number. This also
generalizes.
Consider the number 196560 = 2160x91. In base 16 this is 2F FD0, which sums
to FFF, or 4095, being 45x91.
So most of the above observations arise from the self-similarity of the total
for Gen 1:1 and the total of the last two words. It was Jenkins's insight to
spot the hexagonal/hexagram related symmetry, which leads in turn, not to a
collection of arbitrary facts, but to a couple of general theorems that work in
other arithmetic bases.
It is this that convinces me that there is an element of design in Gen 1:1 - I
think Jenkins's insight therefore deserves more recognition than the rather
dismissive responses I have seen so far.
There are many other numerical phenomena associated with Gen 1:1, some of them related to the geometry, and some to other matters (such as the pi approximation), but for my part, I would say that the above is quite sufficient to demonstrate evidence of a numerical design behind the text. This numerical design allows a couple of general theorems to be deduced, and for predictions from these theorems to be made for other numeric bases (e.g. the properties of base 16). By contrast, the inductive "proof" that there are no uninteresting numbers doesn't allow any predictions to be made.
As I said before, I leave the identity of the "designer" for you to consider.
Four possibilities seem to present themselves:
(1) Pure coincidence - i.e. no design. This is unlikely given the degree of
apparent design and no evolutionary process of language that would gain
selective advantage by adding up the numbers.
(2) Human meddling. Moses (or whoever) got a mathematician to design the sentence for him.
(3) A bit of physics we don't understand yet - in which case it must be
interesting to study. Indeed, there are physicists who take this kind of thing serously, and postulate that such things may be related to quantum theories of consciousness. Check out, for example Tony Smith's web page at http://www.innerx.net/personal/tsmith/TShome.html . Smith is a somewhat unorthodox mathematical physicist, who is none the less highly respected; and there are many references to his pages from reputable maths sites.
(4) God.
My personal vote is for (4), but I leave it to you to evaluate the evidence and decide for yourself.
Best regards,
Iain.
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