Technology has certain traditional uses - warfare, medicine, navigation - and other, less useful, uses, such as television or the Internet. Occasionally an odd use for scientific knowledge comes up - the Clapper, tiny highway underpasses for endangered frogs, and so forth. The most unexpected use of technology of late is the use of computers and statistical methods to verify the Bible codes. Are innumerable till-now secret messages encoded in the Hebrew letters of the Pentateuch? Perhaps under its calm surface the Bible is a primordial Internet of useless information.
Or nearly useless - the vanishingly-small probability that the “codes” arose through mere chance leads many to the conclusion that God must have hidden them there. Some will not start at the allegation that God wrote the Bible, but for those who do, there are some milder conclusions also to be gleaned from the codes. This diffuse statistical phenomenon marks the entirety of the five books of Moses as the work of one hand. Attempts have been made in the past by Christian and Jewish biblical scholars to use primitive versions of stylometry, the statistical approach to verification of authorship, in order to refute the Documentary Hypothesis; the Bible code patterns, whatever their source, prove the presence of a very odd “style” in the text, which is incompatable with the postulated crowd of Pentateuch authors. Yet I doubt the Catholic Church will revise the next edition of the New American Bible with Study Helps to reflect the sudden demise of J, E, P, D and company.
The Bible codes are a phenomenon unearthed piecemeal by rabbis and mystics and lately, by a handful of Israeli mathematicians. Various Jewish legends claim that all of history is contained in the text of the Pentateuch, hidden in various encodings. Very few actual tidbits of encoded information were known before the research of Rabbi Michael Ber Weissmandl earlier this century, and most of his discoveries were lost in the Second World War. His research was revived by American mathematician Daniel Michaelson, and later the trio of Israeli scientists Witztum, Rips and Rosenberg.
The Israelis' preferred decoding method involves “equidistant letter sequences,” words encoded in the Bible text by skipping a fixed number of intervening letters. For example, take out your Hebrew Bible, and start with the second occurrence of the letter aleph in the second verse of Leviticus. Skip eight letters and the ninth is a he. Skip another eight and the ninth is a resh, and the ninth after that is a nun. This is (unvowelled) Hebrew for Aaron. In fact, the number of such encoded Aarons (with different starting points and skip values) in the first half of this chapter of Leviticus is 25. Considering the frequency of the letters aleph, he, resh and nun in the text, one would expect, statistically speaking, far fewer encoded Aarons - by one estimate, the occurrence of 25 of them together is a one in two million chance.
Finding these things has become a new theological sport, but from the statistical standpoint such practices are known as cherry-picking. To add scientific rigor to the codes, the Israeli mathematicians designed a formal experiment. They opened the Encyclopedia of Great Men of Israel and listed all those notables who occupied between 1.5 and 3 columns. This list of “sages” with a birth or death date each formed the statistical sample. Using a special notion of distance between encoded words, they measured the distance between a sage's name and his date (traditionally written with letters, as Roman numerals are) in the text of Genesis.
Next, they scrambled the list, putting a sage with a random date from the list, in 999,999 different ways, to total a million samples including the original, correct list. Here, of course, a computer gets involved. For each of the million lists, the distances between the coded sage's name and purported date in the text was calculated and an aggregate distance was determined. The “null hypothesis”, what one would expect if nothing were going on, is that the true list would fall somewhere in the middle of the million lists when they are ordered by this distance between coded words; about half of the fake lists should have pairs closer together, and half should have pairs which lie further apart. In fact, the true list came in fourth out a million according to one of the distance measures, meaning that the hidden names of the sages occurred suspiciously close to the correct coded date. Since there were several candidate distance measures, the official result was not quite 4 in a million but only 1 in 62,500. This is well beyond statistical significance - it is shocking.
A significance level of 1 in 20 (corresponding to a 5% chance that the results are due merely to chance) is usually considered sufficient for publication in academic venues, but the Israelis' paper was refereed for six years before being passed along by the relucant editors of the Proceedings of the AMS to a more open-minded statistical journal. At the suggestion of the referees, the experiment was run on several other texts, including Isaiah, a Hebrew translation of War and Peace, and various random permutations of the book of Genesis, but no other text yielded statistically significant results. The failure of these alternate texts show that the codes phenomenon is not a quirk of biblical Hebrew or the Hebrew language in general.
Since the results were published in 1994, many debunkers have tried to refute them, but have failed on several counts. After so many years of peer review, no obvious flaw in the paper itself is likely to appear now. Attempts on similar data sets have turned up more positive results. Also, since no one knows what exactly God intended to hide in the Pentateuch, failure with a new data set (e.g., a list of the great generals of Israel and their battles) does not contradict the original research. The generals could be coded in a different manner, or not encoded at all - yet the codes that are there persist. Also, the researchers have conducted at least seven new experiments; none have yet been published.
Jeffrey Satinover, who wrote a book [3] on the subject, takes the stance that the codes are a purely statistical phenomenon, a sort of “watermark” proving that a superhuman intelligence wrote the Pentateuch. This watermark persists up to deletion of 77 letters from the total text, according to the Israeli experimenters. One is tempted to use the strength of the codes statistics to judge textual transmission errors. For instance, the Biblia Hebraica Stuttgartensis differs from the standard Jewish text by 130 letters (not all deleted). When the codes experiment was run on it, the results were poorer by a factor of 100, but still statistically significant. The Samaritan Bible differs much more (6,000 letters) from the Jewish text used in the experiment, and produces no statistically significant results at all. There are, in fact, several Jewish texts and a tradition of speculation about possible errors and corrections; though the texts differ among themselves by only 9 letters, perhaps the original text could be recovered using the Bible codes.
It is mathematically conceivable that the whole history of the universe is encoded in this particular five-book Logos, substantiating the traditional view that the Torah the preexistent blueprint of creation. It is generally acknowledged to be humanly impossible to write a sensible text of that length yet word it just so that countless secret codes are hidden in the arrangement of the letters. Space aliens have not been ruled out; for logical reasons it is impossible to prove conclusively that God Himself wrote the codes, spoke to Moses, etc.; He is merely the most likely cause. Similarly, the codes cannot decide most theological issues; although many of the Jewish researchers have turned to Orthodox Judaism, the codes are not in any sense anti-Christian. A list of Church Fathers could be hidden in the Pentateuch beside the rabbis the Israelis chose to seek, and indeed some enthusiasts have published books about Jesus-in-code.
Given a text G containing t letters, an equidistant letter sequence (ELS), denoted (n,d,k), is the unique k-letter “word” beginning with the nth letter of the text and ending in the (n+(k-1)d)th letter, and containing, in order, every |d|th letter in between. Note that the value n and the sum n + (k-1)d must both lie between 1 and t and that d may be negative, but to avoid trivial cases (e.g., of the superficial words of the text) its absolute value of d will usually be assumed to be greater than 1.
Instead, the authors consider the text G as a three-dimensional space where the third dimension is “row length”; the other two are the Euclidean dimensions on a two-dimensional surface variously described as a cylindar or helix. Given a row length h, the text G becomes a two-dimensional surface in which the nth letter is at a distance of 1 from the (n-1)th,(n+1)th,(n-h)th, and (n+h)th letters, where such letters exist in the text. The letters form a grid upon which the usual Euclidean distance can be calculated. As on the surface of the Earth, there are two possible straight-line distances between any two points and the shorter value is the official distance.
Thus far, only the distance between two letters in a particular h-slice of the three-dimensional space has been defined - call it the h-letter distance. Given h, a notion of distance between two ELS's e = (n,d,k) and e¢ = (n¢,d¢,k¢) is now easily obtained; call such a thing an h-ELS distance. The authors define a weighted h-ELS distance d(e,e¢) = f2 + f¢2 + l2, where f is the h-letter distance between consecutive letters of e, likewise with f¢ for e¢, and l is the minimum h-letter distance between a letter a of (n,d,k) and a letter a¢ of (n¢,d¢,k¢).
Next, h must be taken into account - what is the distance between e and e¢ in the three-dimensional space? Here the authors make their first approximation - they take twenty values of h and make a weighted average of the associated h-ELS distances dh(e,e¢). One question of interest is whether there is a more natural metric for this odd text-space. To obtain the twenty values, ten values apparently favorable to e and ten for e¢ are chosen, according to what would seem to minimize the respective factors f and f¢ in the definition of h-ELS distance. Specifically, they are the row lengths h1, ..., h10, such that hi is the rounded value of |d|/i, and likewise for h¢i. This approximation insures that either the word e or the word e¢ will be relatively compact in the hi or h¢i slices, respectively, but no similar effort is made to minimize the factor l in h-ELS distance. The ELS-proximity s(e,e¢) is defined thus:
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The next difficulty to be faced is the presence of more than one ELS in the text G for any particular word, say “hammer”. The authors come to the text wondering whether “hammer” is close to “nail” in general, but so far only the ELS-proximity between a particular occurrence e of “hammer” specified by (n,d,k) and a particular instance e¢ = (n¢,d¢,k¢) of “nail” has been calculated. A new notion of proximity, “word-proximity”, is desired.
Let e and e¢ range over all ELS-occurrences of two given words w and w¢, respectively. Giving each such ELS a weight, one may compute a weighted average of the ELS-proximities s(e,e¢). The authors consider ELS's e = (n,d,k) with minimal skip value |d| where |d| ³ 2 to be of particular significance, so they define the domain of minimality Te to be the maximal contiguous portion of the text G in which |d| is minimal; that is, there is no other ELS in that subtext spelling the same word but having a smaller value of |d|. The weight w(e,e¢) associated with a particular ELS-proximity s(e,e¢) is the percentage of the text represented by Te ÇTe¢, that is, the ratio of the length of the intersection of the domains to the length of the full text. The word-proximity is denoted
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The final notion of distance is the “relative distance”, c(w,w¢), between two words. This value will lie between 0 and 1, meaning that w is unusually close (respectively, far) from w¢. This is the tersest part of the paper; the casual reader could easily miss this first application of a sort of Monte-Carlo simulation to the data.
The authors define a “perturbed” ELS (n,d,k)(x,y,z), which would be more aptly named an almost-equidistant letter sequence. The “perturbation” alters the last three positions of the sequence, using the values of x,y,z which range over { -2,-1,0,1,2 }, so that the “perturbed” ELS consists of the letters in positions n,n+d, n+2d, ..., n+(k-3)d +x, n+(k-2)d +x+y, n+(k-1)d +x+y+z. For example,
| A | H | Q | N | G | Y | V | L | E | I | C | E | L | P | |||||
| A | N | V | I | L | x = 0 | y = 0 | z = 0 | |||||||||||
| A | N | G | L | E | x = -2 | y = 0 | z = 1 | |||||||||||
| A | N | G | E | L | x = 2 | y = 1 | z = 1 |
The ELS (n,2,5) for the word “anvil” is unperturbed, though written above in perturbed notation. Given a perturbation, say (-2,0,1), to find a “perturbed” ELS for “angle”, one must check all possible values of n and d (k = 5) for an occurrence of “angle” in the almost-equidistant progression.
The authors perturb only the last three positions for “technical programming reasons.” They also claim that for a word of length k, only the final k-2 positions may be permuted. I fail to understand this claim; in fact, it would seem possible, if fruitless, to permute even the first position n.
Just as there was a notion of word-proximity W(w,w¢) generated by ordinary ELS's, there is an independent notion of word-proximity over all “perturbed” ELS's for a given permutation (x,y,z). Although there is a natural h-ELS distance between (n,d,k)(x,y,z) and (n¢,d¢,k¢)(x,y,z), for unexplained, perhaps technical, reasons, the authors use the h-ELS distance d between (n,d,k) and (n¢,d¢,k¢) for an h-ELS distance between (n,d,k)(x,y,z) and (n¢,d¢,k¢)(x,y,z). In any event, there can be little difference between these two values.
ELS-proximity and word-proximity can now be calculated for the perturbed cases, by considering the (n,d,k)(x,y,z)-occurrences of a word w rather than the normal ELS's. The variant word-proximities are denoted W(x,y,z)(w,w¢). For any two words (w,w¢), there are 125 such values, but some are expected be zero (when w or w¢ fails to appear in any perturbed ELS's for the particular perturbation). The relative distance is defined thus:
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The distribution of a product X of independent random variables uniformly distributed over the interval [0,1] is
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The two other proximity measures, P3 and P4, are exactly like P1 and P2, respectively, but the sample is altered. The title “rabbi” was dropped out of concern that its damping presence would skew the results toward rejection of the Bible codes hypothesis.
At this point one could calculate P1, P2, P3 and P4 for the sample data, and if the authors were not so cautious on this point, one could consider these values to be p-values and use the Bonferroni inequality to obtain a final result of 4 min{Pi }, the likelihood that the smallest of the overall proximity measures would be as small as it is.
Instead, the authors permute the list of names and dates for a second Monte-Carlo approach to the problem. There are 32! combinations of personage with absolute date. They limit their calculations to only one million of these, including the original list. Let Ppi represent the overall proximity measures for the pth permutation of the list, and let Pi be the values for the unpermuted list. For each i Î {1,2,3,4}, the million values Pip are ranked, the null hypothesis being that Pi would fall around the middle of the ranking. Then 4 min{Pi} is calculated - it is 0.000016. There seems no reason, a priori, to consider the values P1 and P3 in the presence of the other, more accurate, proximity measures, so the number 2 min{P2,P4} seems a more accurate result; it is 0.000008.
The text used was the book of Genesis, G, from the Textus Receptus, a traditional Jewish text. Other texts used for comparison were R, a random permutation of the letters of G, T, the initial segment of the Hebrew translation of War and Peace of the same length as G, I, the book of Isaiah, W, a random permutation of the words in G, U, in which the words of each verse of G were permuted randomly, and V, in which the verses themselves were permuted. The experiment was run on these other texts. In no case was the value of 4 min{Pi } statistically significant; the smallest value was .847, for text V.
One can hardly imagine that anything else is related to the esoterica above, but there are several possibilities for continued work. Personal experiments with the text are a favorite with amateur codes researchers, but it is not clear what any further experiments, friendly or not, would establish. The field of stylometry, verifying authorship by statistical study of the text, seems most directly related, though finding the Earl of Oxford wrote Shakespeare is something of a letdown after showing God wrote Genesis. Another example of unexpected statistical patterns in the two-dimensional version of an otherwise one-dimensional text is the Ulam spiral, mentioned in the appendices of [3]: curl the positive integers around themselves in a spiral and the primes form diagonal lines. In his review [4], William Dembski claims that the same statistical issues arise in the field of biological design as have arisen above; he refers the reader to his book, The Design Inference [5], for more information.