The subject of this article is a story which is both comic and sad. It is a story of how two people, of which one (Eliahu Rips) was a brilliant mathematician with a record of excellent achievements in modern mathematics and the other (Doron Witztum) although without any scientific credentials, was obviously very smart and ingenious, started what looked like a serious scientific research, and very soon, motivated by obviously non-scientific goals, went astray so their activity degenerated to propagating absurdity. The history of science knows more than one example of alleged discoveries which turned out to be the results of experimental errors. The authors of these pseudo-discoveries reacted to the disproval of their claims in various ways. Many of them promptly admitted their errors. Of course, this is the most respectable way to deal with the situation and the best way to preserve a good reputation. The heroes of this story, however, continue to fight tooth and nail trying to assert the validity of their theory despite all the proofs against it. This is also a story of how an allegedly-scientific discovery, despite being shown to be unsubstantiated, was appropriated with enthusiasm by a crowd of people who did not understand the essence of the work by the theory's originators but were delighted by its religious implications and vulgarized it to such an extent that even its original authors had to assert the absurdity of the claims of their epigones.

2. A bit of history

Statistical Science is a prestigious peer-reviewed scientific journal published by The Institute of Mathematical Statistics. It prints papers on a variety of subjects, dealing with the application of statistical methods to any area of research. Its readership is limited mainly to faculty and researchers interested in the use of mathematical statistics in various fields of science and technology. Unexpectedly, though, one article [1] printed in that journal in 1994 found a much broader audience since it was reprinted in full in a best-selling sensational book [2]. Its three authors were Doron Witztum, Eliahu Rips and Yoav Rosenberg (to be referred as WRR.)

The essence of WRR's paper was as follows. First, these writers showed that in the Hebrew text of Genesis there are large numbers of what they called Equidistant Letter Sequences (ELS.) This term denotes words formed in a text by letters separated by equal intervals ("skips"). For example, look at the word "DenOteS" in the preceding sentence. Letters "D," "O," and "S" are separated by equal intervals of 2 letters (en between D and O, and te between O and S). The three equidistant letters form the acronym DOS, which stands for Disk Operating System and constitutes an ELS with a skip of 3. According to WRR, the ELS that run along the text (which in Hebrew is from right to left) and against text's flow (in Hebrew from left to right) are equally valid. In the latter case the skip is negative. For example, in the same word "denotes," if read from right to left, letters s, o and d constitute an ELS spelling SOD with a skip of -3. Naturally, every word of the text itself is also an ELS with a skip of 1.

Furthermore, WRR claimed that in the text of Genesis there are multiple pairs of ELS, with the words of the pair related by meaning and situated within the text in close proximity to each other. WRR claimed that the proximity in question far exceeds what could be expected if those pairs of ELS appeared in the text by chance alone.

To prove their point, WRR conducted a computerized statistical experiment. They first compiled a list of 34 "famous rabbis" who lived between early medieval times and the 18^th century and whose biographies in the Encyclopedia of Great Men of Israel [3] each occupied at least three columns. Later, following advice from a referee, they compiled a second list of 32 "less famous rabbis," whose biographies each occupied between 1.5 and 3 columns. Then, with the help of an expert in Judaic bibliography, Professor S. Havlin of the Bar-Ilan University in Israel, they matched the rabbis' various appellations, with the dates of those rabbis' birth and/or death, according to the Responsa database maintained by the Bar-Ilan University. In Hebrew the dates are written using letters of the alphabet, so WRR searched in the text of Genesis for ELS both for the appellations and for the dates.

They suggested a formula which supposedly estimated the "distance" between any two ELS and used it to "measure" distances between ELS for appellations and ELS for dates. Then they shuffled the lists of appellations and of dates, creating 999999 scrambled lists, where the rabbis' names and the pertinent dates were randomly mismatched. Their computer program measured the "distance" for each pair of appellations/dates, both in the correct and in the mismatched lists. Then the program calculated four aggregate measures of "proximity" between appellations and dates for each one of the above one million of lists. They denoted the aggregate criteria of proximity as statistics P₁, P₂, P₃, and P₄. The values of these proximity measures were then ranked, assigning rank 1 to the appellations/dates list which displayed the "best" proximity (that is the lowest values of the above statistics), rank 2 to the list with the next higher value of P etc. Somewhere on the ladder of ranks thus created was the original "correct" list of appellations/dates.

The results claimed by WRR were nothing less than astonishing. WRR claimed that the ranks of the "correct" (unscrambled) list of appellations/dates were always very low, never exceeding a few thousand out of one million competing scrambled lists, and often much less than that. WRR concluded that the ELS for the rabbis' appellations and ELS for their correct dates of birth/death, are situated in the text of Genesis at an unusually close proximity, with the level of confidence 1 in 62,000. WRR finally stated that the proximity of the ELS for the rabbis' appellations to the ELS for their correct dates of birth/death in the Book of Genesis was "not due to chance."

The meaning of such conclusion was quite far reaching. Since the Book of Genesis was written many centuries before the rabbis in question were born, its creator must have known the future. Then the results by WRR, in their opinion (shared also by many others who were impressed by the amazing outcome of WRR's experiment) prove the divine origin of the Book of Genesis. Hence the results of WRR's experiment were touted as the alleged scientific proof of the existence of God, and more specifically, of the God of the Torah.

The editor of Statistical Science, R. Kass supplied WRR's paper with a comment saying that the paper was being published as a puzzle offered to readers. In subsequent additional comments posted on the Internet Kass made it clear that the editors did not accept WRR's claims, and hoped that some readers would be willing to invest enough time and effort to unearth the hidden flaws in WRR's methodology.

The publication of WRR's paper had a number of unexpected consequences. One consequence was the appearance of a large number of papers, lectures, Internet postings, etc, whose authors, most of them not capable of comprehending the mathematical apparatus in WRR's article, vulgarized WRR's results by searching for multiple ELS in the Bible without any regard to a statistical verification of their claims. A number of computer programs enabling anyone to search for ELS in the Bible have been peddled all over the Internet. Some other fast entrepreneurs offer to find, for a fee, the information allegedly "encoded" in the Bible about anyone's personal matters. Many other writers and authors of Web postings, referring to WRR's paper as "the proof" of the Bible code, utilize ELS in the Bible to promote various agendas. One that seems to be most widely spread is that of Christian preachers.

There is an organization named Aish Ha Torah, whose aim is to attract those Jews who have either lost faith or have never been observant back to the fold. This is done by conducting seminars where trained lecturers discuss various facets of Judaism. Even though various topics are presented in these lectures, the subject of the "codes" in the Torah is strongly emphasized there, as one of the main tools for convincing the doubting participants to re-embrace the faith. That organization maintains a website (Torah Codes) in which its creators state that their goal is to provide a forum to Witztum and to promote his work.

At least nine books about the "Bible code" have recently appeared, at least two pseudo- "documentaries" have been broadcast, and at least one dramatic film has been released, all of them promoting and supporting the Bible code. The largest splash was probably made by the sensational book [2] published by the journalist Michael Drosnin and titled "The Bible Code." The book appeared on the New York Times list of best-sellers and was translated into many languages, while its author was featured in Newsweek and Time, appeared on a number of TV shows, sold movie rights for the book to Warner Bros., and set out on a world wide promotion tour, in the course of which he did not spare some not quite polite words regarding those people who dared to doubt the validity of Drosnin's sensational claims.

Drosnin claimed to have discovered in the Bible multiple predictions of future events "encoded" as clusters of ELS. The most loudly advertised was the alleged prediction of the assassination of Israeli Prime Minister Itzhak Rabin.

In Fig.1 a table from Drosnin's book is shown with the letters of an ELS that spells the name Itzhak Rabin circled. The skip of this ELS is 4772. In order to display this ELS on a vertical, the text of the Bible was converted into a table 4772 letters wide (the picture shows a "window" into that table, which contains the ELS in question as well as a set of letters of the plain text, surrounded by rectangles, spelling words Rotzeakh Asher Irtzakh, meaning "killer who will kill"). The table in fig.1 also contains a number of ELS for words seemingly related by meaning to Rabin's assassination, which we will discuss later. According to Drosnin, the appearance of the ELS for Rabin's name close to the above phrase about a killer means the prediction of Rabin's assassination was "encoded" by the Bible's creator by means of the ELS.

The book by Drosnin was soon followed by a book [3] authored by a psychiatrist Jeffrey Satinover. Strangely, whereas Drosnins's book was repudiated by Rips and some other adherents of WRR as an invalid vulgarization of WRR's results, Satinover's book met their apparent approval, even though both books contain very similar examples of ELS arrays, whose interpretation is in no way supported by any statistical considerations.

Almost immediately after the publication of WRR's paper, some scientists started expressing skeptical views of that paper. With time, more and more people got involved in the controversy, joining the dispute on both sides.

The dispute in regard to the authenticity of codes essentially revolves around three topics, to wit:

1. What is the statistical probability that the ELS occur in the Bible by chance?

2. Do similar ELS occur in texts other than the Bible?

3. Are human beings capable, with or without computers, of creating complex arrays of ELS such as found in the Bible?

The "code" proponents usually maintain that the probability in question is extremely small. To the second and third questions the "pro-code" people usually answer "No."

All three assertions have been disputed by the code opponents.

3. How common are the ELS?

The numerous examples of the ELS found in the Bible and demonstrated by Witztum, Drosnin, Satinover, and others have made a strong impression on many people. Indeed, it seemed hard to imagine that so many coincidences could have happened by chance. Therefore I decided to first test how often various ELS appear in randomly chosen texts in various languages.

For my first test I chose several short texts in Russian, English, and Hebrew. Following the method used by all Bible code searchers, I removed from the tested texts spaces between the words and all punctuation marks, so the texts converted into continuous strings of letters. As soon as this was done, in all the tested examples, numerous individual ELS popped into sight from the text. As expected, most of those ELS happened to be three-letter words, but some four-, five-, and six-letter words could be identified on almost every page as well.

The code proponents often demonstrate "arrays" of semantically related ELS they find in the Hebrew Bible. I could easily find many similar "arrays" of semantically related words in the form of ELS in every text I tested. Here is an example. One of the texts I tested was taken from a short story I wrote in English many years ago.

Fig. 2 shows the text's segment in question. Its length is 1460 letters. This segment was chosen without any preliminary trials, right from the first page of the story.

Among the abundant ELS found in that segment, there were many three-letter words, fewer four-letter words, still fewer five-letter words, etc. In fig. 2, only a few of those ELS are marked by straight lines along those ELS, with arrows indicating the direction of reading. (If the skip equals the width of the page, the ELS appears on a vertical line. If the skip differs from the page width, the corresponding ELS' appear on straight lines which are at some angle from the vertical.) In the middle of the page, there is an ELS for the six-letter word TORVIL (marked 1) situated vertically with the skip of 68 (since the page's width is 68 letters.) Parallel to it, in the adjacent vertical, there is an ELS for the word ICE (marked 2) with the same skip of 68. Across the page, there is an ELS for word DEAN (marked 3) with the skip of 70. Close to both TORVIL and DEAN, ELS for the word WIN appear twice, one with a skip of -70 (marked 4) and the other with a skip of -140 (marked 5.) Of course, Torvil and Dean used to be famous champions in figure ice-skating. The story in question was written about 10 years before TORVIL and DEAN demonstrated their skills on ICE, WINning championships twice. Using the "logic" usually applied by Drosnin and other code proponents, one may suggest that I predicted an event which would occur later, and had "encoded" my prediction in the form of a combination of several ELS placed close to each other in the text of my story. I did not.

Among the ELS for 4-letter words in that page were LAND (not far from a three-letter word SEA), LULL, TILT, ODOR, etc. I discovered similar occurrences of "arrays" of ELS, seemingly related by meaning, on almost any page of any texts I tested in Russian, Hebrew or English (many examples can be seen at B-Codes Page). These simple tests have shown that the phenomenon of ELS is very common, and that many individual ELS and "arrays" of ELS appear in any text. The reason for that is, of course, the fact that any language consists of a vast number of words.

I performed the described non-computerized tests before I came across the Web publications by Dr. Brendan McKay (In Search of Mathematical Miracles and Dave E. Thomas (Hidden Messages and The Bible Code.) I was gratified to find that my conclusion turned out to be well in agreement with the multiple examples of ELS clusters found by McKay in a number of non-biblical texts, most notably in Moby Dick, as well as with many examples of similar clusters of ELS in English texts demonstrated by Thomas (they used computer programs.)

4. Was a superhuman mind necessary to create the "Codes"?

The proponents of the Bible "codes" claim that the alleged creator of the "codes" must have possessed superhuman abilities, since neither a human mind nor the best computers available to us are capable of creating such a complex web of ELS. I would like to refer to a book [4] by David Kahn, who is one of the foremost experts in cryptology. In that book one can find a plethora of information about the ability of men and women to both encode and decode information using methods whose complexity and sophistication make the alleged Bible codes look simply primitive.

Indeed, let us recall one of the simplest tools used for encoding secret information. It was apparently invented by the renowned Italian mathematician and writer Girolamo Cardano in the 16th century.

The method is as follows. In a sheet of paper, a set of holes is cut forming the Cardano grille. The grille is placed over a sheet of blank paper and the message to be enciphered is written through the holes, one letter per hole. Then the grille is removed, and the blank spaces between the letters of the secret message are filled with a text that has some innocent contents. The letters of the encoded message thus become parts of the overall text, but now are separated by "skips." To decode the message, a grille identical to that used for encoding is placed over the text, and the secret message is read through the holes. When the holes are cut at equal distances, it constitutes the "simple" Cardano grill. It, of course, produces ELS exactly like those discovered in the text of the Bible.

When I was a kid of about 12, I and a few of my friends used to send each other secret messages in the classroom right under the nose of our teacher. To encode a message, we used several techniques, including the Cardano grille. We had no idea that it was invented in Italy in the 16th century. Sometimes we used a grille with equidistant holes ("simple" Cardano grille) thus creating sets of ELS not unlike those found in the Bible. On other occasions we used a grille in which the distance between the holes would either increase or decrease from letter to letter. It thus produced encoded messages where the "skip" changed from character to character. As I will show later in this article, similar "codes" with regularly increasing or decreasing "skips" can be easily located in any text, as well as can ELS.

Especially for this article, I decided to try to encode again (like I did as a 12 year old) some simple phrase, being willing to spend on that task no more than 10 to 15 minutes. First I wrote the following phrase:Rabin Will Die, which, in the parlance of cryptology, would be my plaintext.The choice of this particular expression was due to the widely publicized alleged prediction of Rabin's assassination given in the already mentioned book by Drosnin. I wrote the letters of the above expression on a piece of paper, leaving spaces between letters which would enable me to insert 9 other letters between any two consecutive letters of the above expression.Then I wrote between the letters of that expression a text, which, even if it is not very sophisticated (and slightly imperfect grammatically) is nevertheless meaningful. The entire exercise took 9 minutes. Here is my resulting ciphertext: Rivers are dAmningly roBust in some Indian coloNies, moving Water unerrIngly, thus aLleviating Lust for the Drinks, forcIng people rEconcile with the otherwise harsh climate. (The hidden plaintext is shown in capitals.)

The intended recipient of that message, knowing the skip (10 in this case), would have no problem in decoding the message. If a serious need existed for me to prepare a secret message using Cardano grille, doubtlessly I would be able, by spending more time, to write a much more sophisticated ciphertext.

The conclusion: a human mind is quite capable of creating arrays of ELS not unlike those found in the Bible.

An impressive confirmation of the above conclusion was demonstrated by Mr. Gidon Cohen of York, Great Britain. The Canadian Christian preacher Grant Jeffrey who propagandized the alleged biblical codes in his publications, was especially impressed by a segment of a biblical text wherein the names of 25 trees happened to occur as ELS. In Jeffrey's opinion, such an array of 25 trees' names "encoded" in the form of ELS within a relatively short text, could neither happen by chance nor have been created by a human mind. Confident that the array in question can only be of divine origin, Jeffrey challenged anybody doubting the reality of the "code" to compile an English text of comparable length in which the ELS for the names of any 25 trees would be found. Jeffrey offered one thousand dollars to anybody who will successfully perform the task. It did not take long until Mr. Gidon Cohen of York, England, submitted an English text of about 300 words within which he successfully inserted ELS for the names of 29 trees. (The text in question can be obtained directly from Mr. Cohen).

By refuting the claims about the necessity of a superhuman mind for the creation of the alleged codes in the Bible I am not suggesting that the ELS in the Bible had been created by men or women. The explanation best compatible with the factual evidence is that arrays of ELS appear in the Bible not by design but are chance coincidences.

5. About the Codes related to Jesus

Among the publications about the Bible code there are several books by a Messianic pastor Yacov Rambsel, and by the already mentioned Jeffrey. For example, let us look at Jeffrey's book Signature of God[5]. In this book, Jeffrey shows six examples of ELS found in the Hebrew Bible, which, in his opinion, spell word Yeshua, the Hebrew form for Jesus. The examples of the passages from the Hebrew Bible are printed in Jeffrey's book, preserving what is called in Hebrew "nekudot," meaning diacritical marks (dots and short bars) placed either under or above or within letters (which in the Hebrew alphabet all are consonants) to indicate the accompanying vowels. The presence of "nekudot" forces a definite reading for each syllable. Accounting for the printed "nekudot," not a single example in Jeffrey's book does indeed spell Yeshua, but rather meaningless combinations of letters spelling non-existing words like Yasvei, Yashaua, and the like. Of course this observation is only a testimony to Mr. Jeffrey's ignorance of Hebrew rather than an argument against the existence of the "codes." Despite the errors by Jeffrey, the existence of numerous ELS in the Hebrew Bible which spell not only the word Yeshua, but also combinations of words like Yeshua Shmi (My name is Jesus), Yeshua Yakhol (Jesus can), Dam Yeshua (Blood of Jesus), Yeshua More (Jesus teacher) etc, can be indeed easily demonstrated. The question is, though, whether such ELS are unique for the Bible or they occur commonly in all texts.

I decided to search in non-biblical Hebrew texts for ELS spelling word Yeshua and its combinations with some other words, such as those listed above, all of which have been found by Rambsel in the Bible and touted as proofs of his beliefs. I randomly pulled from a shelf a few Hebrew books. The first one happened to be a book by a contemporary Israeli writer Dahn Ben-Amotz, published in Tel-Aviv in 1979 by Metziuth Publishers. The title of the book is Ziunim Ze Lo Ha Kol, which translates as Screwing is Not Everything.

In Hebrew, word Yeshua is a four-letter one (Yud-Shin-Vav-Ayin.) Sometimes Rambsel used, instead of the four-letter form of Yeshua, the shorter, three-letter version (Yud-Shin-Ayin.) Likewise, I decided to look for the three-letter version's occurrences as well.

I leafed randomly through Ben Amotz's book, and soon I located, on page 47, ELS spelling the two words Yeshua Shmi (My name is Jesus) one right after the other, both with a skip of only 2. On page 23, within only three lines of text, ELS for words Dam and Yeshua, meaning Blood of Jesus, occurred one after another, both with a skip of only 3. On page 27, within only three lines of text, the same ELS for the words Dam Yeshua, appeared, both with the same skip of 4. On page 63, within only 2 lines of text, an ELS for the word Yeshua appeared twice, once in the three-letter version (Yud-Shin-Ayin) with a skip of 3, and once in the four-letter version (Yud-Shin-Vav-Ayin) with a skip of -1. In the same two lines of text the word Moreh (Mem-Resh-Hey, meaning teacher) appeared 3 times with skips of 3, 4, and -6. In the same lines the word Mori (Mem-Resh-Yud) meaning my teacher appeared with a skip of -5. The characters of the words More and Mori appeared interspersed with the characters forming the ELS for Yeshua. On page 164, within three paragraphs, an ELS for Yeshua appeared 4 times, three times in the three-letter version (with skips of 2, 4 and -6) and once in the four-letter version, with a skip of 7. On the same page, within the same three paragraphs, the four-letter word Yakhol (Yud-Khaf-Vav-Lamed) appeared with a skip of 51. The characters of the ELS for Yakhol were interspersed with those for Yeshua. The phrase Yeshua Yakhol means Jesus Can or Jesus is Able, and, when found by Rambsel in the Bible, was interpreted by that writer as one of the proofs of his claims.

I found in Ben-Amotz's book many more similar groups of ELS spelling every single phrase Rambsel found in the Bible, which included word Yeshua, I found also, just as easily, many similar phrases including word Yeshua in a number of other Hebrew books. One of the books in question was the textbook Geography of the Land of Israel, published in 1975 in Tel-Aviv by Am Oved publishers. In fig. 3, page 91 from that book is shown.

Fig.3

On the very first line of that page (right under the section's heading) there is an ELS with a skip of 6 which reads Tora. In the second line from the top, the third letter from the right is Yud. It starts an ELS with a skip of 4 which reads Yeshua. Right after that ELS, in the same line, there is another ELS, with a skip of 2, which reads Mori (my teacher). One line down, starting from letter Mem in the fourth word from the right, there is an ELS with a skip of 2, which reads More (teacher). Finally, in line 8 from the top (not counting the heading) letter number 9 from the left is Yud. It starts an ELS with a skip of -14 (i.e. from left to right) which reads Yakhol (can, or is able). Rambsel and Jeffrey touted occurrences of similar ELS in the Bible as miraculous confirmations of their beliefs.

Is page 91, shown in Fig,. 3, an exception? In Fig. 4, page 140 from the same textbook is shown.

There are two groups of ELS on that page. Let us look at the group of ELS at the top of the page. In the third line from the top (not counting the heading) the second letter from the right is Shin. It starts an ELS with a skip of 3, which reads Shmo (his name). In the same line, to the left of that ELS, there is another ELS with a skip of -3, which reads Yeshua. Two lines down, there is one more ELS with a skip of -4, which reads Shmi (my name). Finally, one more line down, there is an ELS with a skip of -1, which once again reads Shmi. Of course, if all the above ELS were found in the text of the Bible, then Rambsel and Jeffrey, and with them their gullible readers, would be announcing, with delight, a miracle confirming their beliefs.

My conclusion was obvious, namely that a variety of ELS, including combinations of word Yeshua with various other words occur quite commonly in non-biblical texts as well as in the Bible. Therefore their appearance in the Bible does not constitute a proof of anybody's views or beliefs.

When I was done with Jeffrey and Rambsel, I learned about some other books also discussing the Bible code, of which two best known were those by Drosnin and Satinover. Could it be that these writers have done a better job than Jeffrey and Rambsel and provided some stronger evidence in favor of the alleged code in the Bible? Let us look at these two best sellers.

6. About the "amazing" discoveries by Drosnin and Satinover.
Were the assassination of Rabin and the AIDS epidemic predicted?

Drosnin claims to have made amazing discoveries in the Hebrew Bible, where many predictions of future events have been allegedly "encoded" as arrays of ELS. The most acclaimed one of those predictions is probably that of Yitzhak Rabin's assassination. The array of ELS in question, reproduced as a table in Drosnin's book, comprises the name of the late Israeli Prime Minister (found with a very large skip of 4772 characters) as well as the phrase Rotzeakh Asher Irtzakh, meaning killer who will kill, the sets of letters denoting the year (in the Hebrew calendar) when Rabin was assassinated, and also the name of the killer, Amir.

Looking at the table in question reveals that of those words only the name of Yitzhak Rabin is an ELS (with a very large skip.) All the rest of the words in the set are just parts of the regular text, so are not codes at all. Moreover, the ELS for Rabin's name extends over more than 30,000 letters of the text. Hence, the concept of "proximity" of all the mentioned words to the ELS in question is rather ambiguous. The illusion of proximity has been created by presenting the text of the Bible as a table 4772 letters wide. Otherwise, all those words allegedly situated close to the ELS in question, would actually be found scattered over the length of the text.

Since many people remained skeptical in regard to Drosnin's claims, that author, in an interview, published in Newsweek (on June 9, 1997) said that if somebody found a prediction of a Prime Minister's assassination in Moby Dick, then he, Drosnin, would admit being wrong. I assume that by saying Moby Dick Drosnin actually meant any book other than the Bible.

Not to look too far, I turned again to the book by Ben-Amotz, mentioned earlier in this article. Now I decided to look for some ELS related to Rabin. Again, I did not use any computer program, so I automatically limited myself to relatively short skips and only those ELS located close to each other. Following Drosnin's practice, I counted only the letters, ignoring the spaces between the words, commas, periods, etc.

The word Rabin (four Hebrew letters, RESH-BETH-YUD-NUN) popped up on page 33, in the uppermost paragraph on that page, with a skip of -35. Furthermore, within the two uppermost paragraphs, consisting of only about 600 characters, the following ELS appeared, one following the other: 1) An ELS consisting of four Hebrew letters, with a skip of only -3, YUD-RESH-TZADE-KHET, which reads Irtzakh (meaning he will kill). 2) Three Hebrew characters, with a skip of -15, RESH, HEY, MEM, which is the Hebrew abbreviation for Rosh Ha Memshala, meaning Prime Minister (this abbreviation is commonly used in Hebrew, and was also used as such by Drosnin.) 3) The following Hebrew characters with a skip of -8, GIMEL, BET, RESH, which is normally read as Gever (meaning Man) but is interpreted, for example, by Satinover as Gibor (meaning Hero or Mighty man). 4) As mentioned before, the word Rabin with a skip of -35. So, in just two short paragraphs in a randomly opened page, in a randomly chosen Hebrew book, without employing any rearrangement of the text like that usually performed by Drosnin, a set of 4 ELS, all words appearing in one chain, in a very close proximity, reads: he will kill, Prime Minister, hero, Rabin. If we wished to calculate the probability of the occurrence of these four ELS within those short paragraphs, using methods of calculation routinely employed by the code proponents, we would arrive at an astonishing number of about 1 in several billions. (As is shown at Improbable probabilities, as well as in the next section of this article, the calculation of probabilities performed by the code proponents actually provide meaningless numbers.) What seemed to be still missing in those two paragraphs, was an ELS for the name of Rabin's killer, Amir. So I recalled my experience with Cardano grille, where I and my 12-year old friends sometimes used a grille with gradually increasing or decreasing skips. Indeed, if the alleged creator of codes hid information in the form of ELS, couldn't he/she use also, for example, DSLS (Decreasing Skip Letter Sequences) as I and my friends did at the age of 12? It would be even a better code, wouldn't it?

I looked for any occurrence of such a "code." In the same chain of words mentioned above I found the following 4 Hebrew letters: AYIN, MEM, YUD, RESH, which read Amir, with the skip between the first and the second letter being -8, between the second and the third letter being -9, and between the third and the fourth letter being -10. What a nice regularity, and what a beautiful code, isn't it? Now all "encoded" words, all within the same two paragraphs, with small skips, in a chain, read Amir Will Kill Prime Minister Hero Rabin.

Following Drosnin, one may say, "What an amazing discovery!" Obviously, the probability of those words appearing within two short paragraphs, in such a close proximity to each other, must be exceedingly small. Then shall we conclude that in a book published in 1979 a message was deliberately encoded predicting the assassination or prime Minister, the hero, Rabin, by Amir, some 16 years before it actually happened?

The above example from Ben-Amotz's book in regard to Rabin's assassination was posted on the Internet in January of 1998. So far, despite his promise, Drosnin did not reply in any form.

While I found the "prediction" of Rabin's assassination in a Hebrew book, what about Moby Dick, specifically mentioned by Drosnin in his interview?

Responding to Drosnin's challenge, an Australian mathematician and computer expert, Dr. Brendan McKay, applied a computer program searching for ELS to the text of Moby Dick. He discovered there numerous arrays of ELS "predicting" assassinations of many prime ministers and other known personalities, including Chancellor of Austria Dolphus, Leon Trotsky, Indira Gandhi, Abraham Lincoln, Martin Luther King, and, of course, Itzhak Rabin.

The book by Satinover also contains similar examples of alleged amazing finds in the Bible. Although Satinover uses an additional criterion of nearly minimal skips, his finds still are almost certainly results of a random chance.

Here is one example. In chapter 10 of his book Satinover presents a number of "arrays" of ELS found in the Hebrew Bible by himself as well as by Witztum, by Rips, and by Moshe Katz. These arrays supposedly contain encoded information about Satinover's ancestor, Rabbi Abraham nicknamed The Angel, as well as about Emperor Franz Joseph of Austria, about diabetes, about the AIDS epidemic, and about the assassination of Anwar Sadat.

I decided to look for similar "arrays" in a text other than the Bible. I chose the array from page 164 of Satinover's book, attributed to Witztum. It deals with the AIDS epidemic. The array in question contains ELS which spell the following words: AIDS (Aleph-Yud-Dalet-Samekh), Mavet (Mem-Vav-Tet, meaning death), BeDam (Bet-Dalet-Mem, meaning in blood), The HIV (Hey-Hey-Yud-Vav) etc.

Again, I leafed randomly through the book by Ben-Amotz, mentioned before. On page 67 I found the following ELS, all situated within the two uppermost paragraphs, which contained a total of about 600 characters: BeDam (Bet-Dalet-Mem, meaning "in blood") with the skip of 15. Mavet (Mem-Vav-Tet, meaning "death") with the skip of 10. HIV (Hey-Yud-Vav) with a skip of 18. Additionally, in the same paragraphs there was word Kholi (Khet-Lamed-Yud) meaning "disease."

There seemed to be no ELS for AIDS in that short text. Again, remembering the Cardano grilles I used as a kid and realizing that the "code" might not necessarily be limited to ELS, I looked for word AIDS (Aleph-Yud-Vav-Samekh) "encoded" with a regularly increasing or decreasing skip. I found that word in the same paragraphs, the skips being -18 between the first and the second letter, -17 between the second and the third one, and -16 between the third and the fourth letters. How small is the probability of the described occurrences expected to be?

I easily found a very similar group of ELS, again without using a computer, in the above mentioned textbook "Geography of the Land of Israel." Let us look again at page 140 of that book, shown in Fig. 4. This time look at the group of ELS in the lower part of that page. In the sixth line from the bottom, the seventh letter from the right is Khet. It starts an ELS with a skip of 4, which reads Kholi (disease.) One line down, the first letter from the right is Hey. It starts an ELS with a skip of 3, which reads HIV. In the same line, to the left from the above ELS, there is an ELS with a skip of 3, which reads BeDam (in the blood.) One more line down, in the third word from the left, there is letter Mem. It starts an ELS with a skip 9, which reads Mavet (death.) Is there any difference between these ELS and those found in the Bible? None whatsoever.

7. About the "simple" calculations of probabilities

Many proponents of the "code" tried to calculate the probabilities of the appearance, by chance, of various ELS in the text of the Bible. Unfortunately, most of these calculations are characterized by errors and misconceptions and produce unreliable values of probabilities. In such calculations the very small probabilities (like "one in quadrillions" in Jeffrey's book) are usually obtained by multiplying many not very small probabilities. For example, the probability of a particular letter appearing at a certain location in the text is calculated by dividing the total number of occurrences of that letter in the text, by the overall text's length expressed in the number of all letters in it. Even this first step is actually wrong as it only applies to a perfectly random conglomerate of letters. Meaningful texts are very far from being random. On the contrary, the meaningful texts possess a high degree of order in various forms (see, for example, several articles by myself and B. McKay at A study of certain statistical properties of meaningful texts as compared to randomized conglomerates of letters). Therefore the probability of a particular letter appearing at a certain site in a meaningful text may be very different from the probability calculated for a random text.

However, there is a much more serious error in the calculations in question. Having calculated (erroneously, as explained above) the probabilities for various letters occurring at various locations in the text, the code proponents then usually multiply the probabilities found for each letter in the ELS in question and thus arrive at extremely low numbers. Multiplication of individual probabilities is routinely employed in the theory of probability, but the necessary condition to make this multiplication valid is independence of the multiplied probabilities. If the individual probabilities are not independent, probability theory treats them in a different way, as the so called "conditional probabilities." To determine whether the individual probabilities are independent or not is by no means a trivial task. In particular, the individual probabilities multiplied by the code proponents are definitely not independent (see Improbable Probabilities). Therefore the extremely small probabilities of the occurrence of that or this ELS, or of ELS arrays, which are so triumphantly claimed by the code proponents like Jeffrey, Rambsel, Drosnin, Satinover, etc, are meaningless numbers.

Now, let us consider the question: what if the probabilities in question are indeed very small? Does it mean the occurrence of corresponding ELS must be attributed to a conscious design?

As is shown in my article at Improbable probabilities, probability theory can not and does not predict whether an event will or will not occur. Even if the calculated probability of an event is very small, it does not at all mean that the event in question will not happen. All the probability theory can assert is that in a very large number of tests the average number of occurrences of a specific outcome will approach the calculated value of probability for this outcome if the number of tests increases. No more and no less than that.

The chance of finding in the Bible a particular ELS may be small (but still much larger than the one usually claimed by the code proponents) but the chance of finding some ELS, for example combining the word Yeshua with certain other words, is much larger. The reason for that is that the Hebrew language (and any other language as well) contains so many different expressions and phrasal constructions that the chance of coming across some seemingly meaningful combination of Yeshua with some other words in the Torah is quite large. The fact that many expressions found by the code proponents in the Bible were also located in non-Biblical texts may serve as a confirmation that calculations of probabilities, like those often used by the code proponents, are meaningless and cannot be used to justify any conclusions about the reality of "codes" in the Bible.

8. A general look at WRR's paper

When I felt confident that I was done with such Bible "code" epigones like Drosnin, Satinover, Jeffrey etc, I turned to the paper [1] by WRR themselves.

All the Bible code proponents and defenders refer to that paper with the utmost esteem and repeat time and time again that WRR's paper had "scientifically proven" the genuineness of the "Bible code." Among the arguments in favor of the "codes" authenticity, the WRR's epigones usually emphasize that Witztum, Rips, and Rosenberg are genuine scientists, that their paper was published in a prestigious refereed magazine, and that no skeptical scientists have yet found any errors in WRR's calculations.

All the above assertions are without merit.

First, out of the three authors only Rips is indeed a scientist, a highly qualified mathematician in the field of group theory (but, unlike many of the code opponents, not in mathematical statistics). Rosenberg developed a computer program used by WRR. Many similar programs have been offered, some for free and many more for sale, and none of those programs' creators (such as Kevin Acre, Dave Thomas, Randall Ingermanson etc., all of whom maintain web sites) pretends that their programs qualify as a scientific achievement. Witztum, who has characterized himself, in an article published in the Jewish Action magazine (March 1988) as "the foremost code researcher in the world" has no scientific credentials (although he has a Master degree in physics).

WRR's followers have a habit of constantly exaggerating scientific qualifications of anybody who is or could possibly be in favor of WRR's work. The epithets such as "world-class" mathematician, probabilist, etc., are routinely applied to whoever might, often just in passing and often in ambiguous terms, say anything that WRR's defenders choose to construe as a support of WRR's conclusions.

Of course, the question of whether WRR are genuine scientists or amateurs is irrelevant and I have touched on it only as a response to WRR's followers' persistent misuse of that argument.

Furthermore, if one wishes to use the argument based on the credentials of WRR, then one has to account for the opinion of many real experts in mathematical statistics who signed a letter (at Mathematicians' Statement on the Bible Codes) stating that they had personally analyzed WRR's paper and categorically rejected WRR's conclusions. At the time of the writing of this book more than 50 mathematicians have signed the letter in question. Only experts in mathematical statistics were offered a chance to sign the letter in question. Were it offered also to physicists, linguists, Bible scholars, etc, the number of signatures would certainly be much larger. There is no document signed by any number of scientists who would support WRR's conclusions. It can be therefore stated that the scientific community, and more specifically mathematicians and specialists in statistical science, have overwhelmingly rejected WRR's work.

To thoroughly analyze WRR's work and to determine where WRR were at fault took a considerable time. A number of Internet postings appeared since 1994 in which various facets of WRR's work have been criticized from various viewpoints. Therefore the WRR's followers' argument that no scientist could find errors in WRR's paper, is a blatant distortion of the actual situation. For example, let us mention just one printed article [7]. Postings on the Internet by scientists criticizing WRR are numerous (see for example In Search of Mathematical Miracles or B-Codes Page, both containing many links to relevant websites.) Finally, in May 1999, in the same Statistical Science journal, a paper [8] by three mathematicians and one psychologist (to be referred to as MBBK) offered the most comprehensive refutation of WRR's work. The same Dr. Kass who was the editor of the journal when WRR's paper was printed, supplied a comment to MBBK's paper, stating that the puzzle offered in 1994 has been successfully solved and the allegedly amazing results of WRR's experiment have been given a convincing rational explanation.

While the excellent paper by MBBK provides a devastating criticism of WRR's procedures and conclusions, it does not encompass all facets of the controversy. In particular, MBBK actually accept some rules of the game employed by WRR, concentrating mainly on the demolition of the statistical evidence provided by WRR, but neither rejecting the fundamental hypothesis by WRR nor analyzing some of the details of WRR's basic approach to the statistical procedures. While MBBK's argumentation is more than sufficient to completely destroy the credibility of WRR's conclusions, a look at some other fundamental weaknesses of WRR's work is also of interest.

In the following sections I will provide both a brief report on the MBBK formidable analysis of WRR's work and an equally brief consideration of some additional aspects of WRR's basic approach to the problem in question.

9. The basic hypothesis by WRR

At the beginning of their paper WRR offer an example. If one is dealing with any meaningful text, then, as WRR suggest, it is reasonable to expect that words which are semantically related, for example, hammer and anvil, occur in the text in close proximity to each other. Likewise, they contend, if the ELS in the Bible constitute a "code" inserted there by a superhuman mind, then the ELS which are related by contents are also expected to occur in close proximity to each other.

WRR's hypothesis is doubtful even in regard to human-created meaningful texts. It is easy to imagine a text where the word hammer occurs very frequently whereas the word anvil is completely absent. However, for the sake of argument, let us accept the above hypothesis for human-created texts. WRR's extension of that hypothesis to the occurrences of semantically related ELS in the Bible is a jump across a logical pit. If the "code" has indeed been created by a superhuman mind, then, by definition, we have no way to know how that mind works. What is the reason to assume that the alleged superhuman creator of the "code" wished to place the ELS "related by meaning" close to each other?

All ELS reported by WRR and their followers are just individual words without any grammatical connections between them. Despite a number of attempts, so far neither WRR nor any of their followers succeeded to prove that sets of ELS' in the Bible form meaningful phrases not to mention whole sentences. Then what could be the reason to haphazardly place various ELS in a text "close" to each other? Moreover, it seems more logical (or, better, no more illogical) to guess that the alleged creator of the "code" would place related ELS as far from each other as the text would allow. Indeed, as it was shown in one of the preceding sections, the human mind is quite capable of creating ELS in any texts. This task is relatively easy if the skips are short. If the skip exceeds the length of one page, the creation of an ELS becomes difficult. Hence, if the alleged superhuman creator of codes wished to "encode" some information using ELS, it is equally plausible (or, rather, no more implausible) to hypothesize that "he/she" would rather opt for placing parts of the code at long distances from each other thus making clear that "his/her" code is not of human origin.

The hypothesis by WRR which is at the core of their entire effort, seems to be arbitrary, as it has no foundation in any factual evidence and is not supported by any argumentation. In other words, even if WRR managed to prove that semantically related pairs of ELS indeed occur in the Book of Genesis in an unusually close proximity, this in itself cannot serve as a proof of the superhuman origin of those ELS.

The proximity of ELS may have quite non-miraculous origin stemming from the natural properties of texts. Indeed, research conducted by myself and Dr. B. McKay (see for example our postings at A study of certain statistical properties of meaningful texts as compared to randomized conglomerates of letters revealed some specific pattern of strong organization in all meaningful texts which are absent in randomized texts. Because of the high level of order in meaningful texts, every mathematical function reflecting properties of a text has nearly extreme values for meaningful texts as compared, for example, with most randomized texts. The complex mathematical quantity WRR use as a measure of "distance" between ELS is no exception. Hence, even if the unusually close proximity of semantically related ELS in the Book of Genesis were proven, it would not constitute a proof of a supernatural origin of those ELS. Moreover, as we will see, the unusually "close proximity" claimed by WRR has by no means been proven in their work.

10. Unanswered questions in WRR's paper

To verify the results observed for the Book of Genesis, WRR also conducted their experiment on six control texts. Four of those control texts had been obtained by permutations of various elements (words, verses, etc) of the Genesis text. One more control text was the Book of Isaiah, and one more was the Hebrew translation of Leo Tolstoy's novel War and Peace.

It is of interest that among all the texts explored by WRR the worst results (that is, the highest rank of the "correct" appellations/dates list among all the competing shuffled lists) was observed for the Book of Isaiah. In other words, the experiment which, as WRR maintain, was successful in the Book of Genesis was a complete failure in the Book of Isaiah.

Since many arrays of ELS found in the Book of Isaiah are often demonstrated by the "code" proponents, it seems puzzling that WRR did not provide any discussion of their failure to get a positive result in the Book of Isaiah.

Moreover, how can the complete absence in WRR's work of any reports on attempts to apply their method to the other four books of the Torah be explained?

Perhaps the answer can be inferred from the results obtained by some scientists who filled the void left by WRR. McKay used the computer program supplied by WRR, and the same list of the rabbis' appellations/dates as used by WRR, and tried it on the other four books of the Pentateuch (see the results of that experiment as reported by Dr. A.M. Hasofer in A Statistical Critique of the Witztum et al Paper). The results were decidedly negative. While the "correct" list of rabbi's appellations/dates used by WRR, in conjunction with their method, produced low ranks among one million shuffled lists in the Book of Genesis, in the other four books of the Pentateuch those ranks were very high (hundreds of thousands.)

The Judaic tradition holds that the Torah was given on Mount Sinai to Moses as a string of letters without divisions into words, chapters, or books. The division was performed later. The boundaries between the five books of the Torah have no real significance and have been disputed by some prominent figures in Jewish history.

WRR avoid any discussion of why their experiment produced such good results in Genesis but utterly failed in the other four books of the Torah. The natural guess which immediately comes to mind is that the list of appellations/dates was somehow optimized in WRR's study toward the positive outcome of their experiment with Genesis. As we will see later, this guess finds some confirmation when a detailed study of the above list is performed. This statement in no way implies that WRR deliberately adjusted their data to produce the desired effect. The history of science knows many examples of perfectly honest and unbiased scientists inadvertently selecting from their data sets those subsets which produced results meeting the experimenters' expectations and ignoring other subsets which might produce an opposite outcome.

11. Statistical methodology of WRR

A rebuttal of the WRR's work from the standpoint of mathematical statistics and probability theory has been proposed by the above mentioned prominent expert in mathematical statistics, Dr. A. M. Hasofer. The paper by Hasofer is available at A Statistical Critique of the Witztum et al Paper. I will briefly present here the main points of Hasofer's comments. This section will necessarily be more technical, as it deals with subtle elements of statistical procedure. The readers who are uncomfortable with mathematics, can just skip it, even though it demonstrates serious deficiencies in WRR's statistical manipulation of ELS in the Bible. '

The part of mathematical statistics which is relevant to the discussion of WRR's work is "hypotheses testing." The proper procedure in testing hypotheses from the standpoint of statistics includes several necessary steps, as follows:

1) The first step is setting the so called "null hypothesis." Usually the null hypothesis is a certain statement implying that, within the framework of a set of conditions, there is no statistically significant new phenomenon to be discovered. In other words, the null hypothesis reflects the expectations that a test aimed at the discovery of a new phenomenon will fail. In the case of WRR's work, the null hypothesis, in broad terms, is that the proximity of pairs of semantically related ELS in the Book of Genesis is such as would be expected if the distribution of those ELS in the text were purely random. When applying a statistical test, the null hypothesis normally is expressed in terms of certain quantities to be measured, whose values would indicate whether the null hypothesis must be accepted or rejected. In WRR's work, such quantities are ranks of the "correct" list of appellations/dates among one million competing scrambled lists. The lower are the ranks in question, the more likely it is that the null hypothesis is wrong and has to be rejected. As we will see, the legitimacy of the null hypothesis by WRR, is, at best, uncertain.

2) The next necessary step is setting the so-called alternative hypothesis. The alternative hypothesis implies that a new statistically significant phenomenon is likely to exist. The hypotheses testing essentially boils down to the competition between the null hypothesis and the alternative hypothesis, to determine which is more likely to be true. Unfortunately, contrary to the accepted statistical procedure, WRR never expressly formulated any alternative hypothesis, leaving it to the readers to figure it out.

3) The next step in a proper statistical test is to define the so-called critical region and the power of test. The critical region means choosing the boundaries for the set of the quantities to be measured such that if the measured quantities are found within those boundaries, the alternative hypothesis is considered as more likely than the null hypothesis. The power of test is a quantity related to the size of the critical region. WRR, contrary to proper statistical procedure, never defined the critical region and the power of test, thus rendering their conclusions statistically ambiguous.

In particular, trying to interpret what WRR meant by their hidden alternative hypothesis one comes to the conclusion that the power of test implied in their study must be 1. In plain words, it means that to produce sufficient statistical evidence in favor of their alternative hypothesis, the rank of the correct list of appellations/dates must always be 1, which of course almost never happened in their tests.

Besides the above deviations from proper statistical procedure, WRR's work suffers from many other features contrary to the rules of mathematical statistics.

One of such features is the nature of WRR's null hypothesis. As McKay et al have indicated in their paper [8], the null hypothesis adopted by WRR actually reflects not only the properties of the text of Genesis, but also those of the appellations/dates list. The ranks of the "correct" list are partially determined by the composition of the list and not only by the text of Genesis. Indeed, as McKay et al demonstrated, slight variations in the list's composition may result in drastic changes of ranks. I will return to that observation a little later.

Another serious deficiency in WRR's statistical treatment, pointed out both by Hasofer and McKay et al, is the asymmetry between the scrambled lists. This asymmetry is created by the following circumstances. For each "famous rabbi" WRR use several appellations (up to 11 appellations per person.) They also use several versions of dates of the rabbis' deaths/births. A total of 298 combinations of names with dates existed for WRR's list of rabbis. 135 of those combinations are not found as pairs of ELS in the Book of Genesis. WRR used the other 163 combinations. Let us assume that rabbi A has three appellations and two dates in the "correct" list. (For example, the dates can be one of birth and the other of death, and appellations can be in several forms, one being only the last name, the other a nickname, one more the name plus the title, etc.) Each combination of an appellation with a date provides an entry to the list of appellations/dates. Then rabbi A contributes six entries to the "correct" appellations/dates list. Let us assume that rabbi B has only two appellations and one date in that list. Hence, rabbi B contributes two entries. Then these two rabbis contribute eight entries to the "correct" list. There is, among the mismatched lists, one where the appellations for rabbi A are combined with the dates for rabbi B, and vice versa. Then rabbi A would contribute not six but only three entries to the scrambled list, and rabbi B, instead of two, would contribute three entries. Hence the two rabbis in question would contribute to the scrambled list the total of six entries instead of eight in the "correct" list. Different mismatched lists will have different numbers of entries, and this will drastically affect the values of aggregate criteria of "proximity." The symmetry of sets of numbers to be compared is a necessary condition for a proper statistical test, and such symmetry is absent in WRR's work.

One more item in WRR's work which met with criticism was their choice of the measure of the distance between the ELS. That distance has been defined by WRR using a very complicated procedure. In order to calculate that distance, in some cases as many as over 6 million arithmetic operations are needed. In one of Hasofer's example, the "distance," according to WRR's formula, between two ELS that touched each other, turned out to be much larger than the "distance" between two other ELS which were quite obviously remote from each other. Since the aggregate criteria of "proximity" and therefore the ranks of the "correct" lists were calculated by WRR based on their "distance," I believe this Hasofer's observation alone renders all WRR's conclusions meaningless.

12. Statistical vs. general scientific hypothesis.
Four statistics of WRR

So far our discussion of WRR's paper was all within the framework of statistics. This consideration, based mainly on the observations by experts in mathematical statistics, among them Dr. Hasofer and Dr. B. McKay et al, has shown that the results of WRR's statistical treatment of their experimental data are unreliable and cannot serve as foundation for their extraordinary claims. Now, however, for the sake of discussion, let us assume that WRR's claims are based on an impeccable statistical procedure, and that, therefore, their null hypothesis has to be rejected and their (not explicitly expressed) alternative hypothesis has to be accepted. From the purely statistical viewpoint this would be the satisfactory outcome of their experiment.

In this section I will offer some considerations which will lead us beyond the purely statistical procedure.

In any text on mathematical statistics (for example [9]) there are warnings against overestimation of the validity of statistical analysis. Acceptance of the alternative hypothesis completes the statistical analysis, but it is not necessarily the ultimate test of the problem in question. The only firm conclusion is that under the defined set of conditions, the alternative hypothesis is more likely than the null hypothesis. There always exists a probability that some other hypothesis, not formulated in the study, is even more likely than the formulated alternative hypothesis.

There is a difference between a statistical hypothesis and a general scientific hypothesis. A statistical hypothesis necessarily deals with random variables. Having "proven" the alternative hypothesis does not necessarily mean that it has been proven as the general scientific hypothesis.

Consider a trivial example. Imagine that a study has been conducted in which the frequency of cases of tuberculosis was measured in a certain country and in the course of that study it has been noticed that the cases of tuberculosis seemed to be less frequent among the people owning gold watches. To verify this observation, a systematic statistical research has been initiated. The null hypothesis was that the frequencies of TB and of gold watches' ownership are distributed in a random fashion and are independent of each other. The alternative hypothesis was that the cases of TB are negatively correlated with gold watches' ownership, since presumably rich people both more often possess gold watches and, because of better nutrition and living conditions, less often have TB. For obvious reasons, in the properly conducted statistical study, the alternative hypothesis would overwhelmingly win. From the purely statistical viewpoint, the strong negative correlation between TB and gold watches would be well established. As a general scientific hypothesis, this conclusion will be meaningless and no statistician would recommend curing TB by distributing gold watches.

Unfortunately, despite many similar well known examples, some statisticians develop a mind set that makes them sometimes lose the perspective of the general scientific hypothesis and view the statistical proof of an alternative hypothesis as the end in itself.

In the above example, the meaninglessness of the statistical result becomes obvious if some material evidence is added to the study. Namely, attempts to cure tuberculosis by distributing gold watches, whose results can easily be foreseen, will rebut the statistically valid, but nevertheless meaningless results. In some other cases, the material evidence may be not readily available, but the results of a statistical study can still be judged to be meaningless just by analyzing the behavior of quantities employed to characterize the null- and the alternative hypothesis. This was, in my opinion, the case with WRR.

To explain what I mean, consider an example.

Imagine that a study is being conducted to compare the vitamin C concentration in apples vs. oranges. The amount of the vitamin in each individual apple or orange is different, therefore this quantity is a typical random variable and hence the legitimate object of a statistical study. It is impossible to analyze every apple and orange, so a selection is made of, say, 100,000 apples and the same number of oranges from various crops and regions. The amount of the vitamin is then measured in each selected apple and orange. Then it is necessary to choose a certain aggregate measure characterizing the statistically averaged amount of the vitamin in each of the two types of fruit. Assume that in order to subject the study to a self-control, two different aggregate measures are chosen. For example, one measure can be the arithmetic mean of all the measurements, (which will be denoted P₁) while the other aggregate measure (denoted P₂) can be, for example, a product of data for all individual examples, excluding the data within the lowest five percent and the upper five percent of the measured vitamin's concentrations. Obviously, P₁ and P₂ will be different numbers, but that will not by itself invalidate the results. What, though, if the arithmetic mean P₁will be found larger for apples, while the product P₂will be found larger for oranges? In other words, if we rely on P₁ , then the rank assigned to apples will be 1 and to oranges, 2. However, if we rely instead on P₂, the oranges will be assigned rank 1, and the apples, rank 2. Hence according to P_1,we conclude that apples contain, on the average, more vitamin C than oranges, but, according to P₂, we conclude that apples contain, on the average, less vitamin C than oranges. The two aggregate criteria produced mutually exclusive results. The only possible conclusion is that at least one of the two aggregate criteria, and perhaps both of them are unreliable.

That situation is exactly what happened in WRR's research. They suggested two formulas, one for calculating what they denoted statistic P₁ and the other for statistic P₂, both being the supposed cumulative measures of the "proximity" of pairs of ELS in a text. Before WRR published their paper in Statistical Science, they made public some tentative reports on their study, in which P₁and P₂ already appeared and were claimed by WRR to represent certain probabilities. P₁ was introduced as the probability that the number of those inter-ELS distances for rabbis' appellations and dates, which is not more than 0.2 (on a scale between 0 and 1) equals a certain number m. P₂was first introduced as the probability that the product of all inter-ELS distances for rabbis' appellations and dates is not larger than a certain number X. However, when the journal's referee (Dr. P. Diaconis) had indicated that P₁and P₂did not meet the requirements for probabilities calculation, WRR changed their definition saying instead that P₁and P₂ are not probabilities but just certain cumulative quantities reflecting, in some not precisely determined manner, the overall proximity of semantically related ELS. Later WRR added two more aggregate measures of proximity denoted P₃ and P₄, calculated by the same formulas as P₁and P₂ respectively but applied to somehow modified lists of rabbis' names. The ultimate results reported by WRR were not the values of the four cumulative measures per se, but rather the ranks of the correct list among one million of competing scrambled lists. What is interesting in those data is the different ranks of the "correct" list of appellations/dates, depending on which of four P was used.

For example, in an experiment reported by E. Rips to the Israeli Academy of Sciences in 1996 (it can be viewed at Hidden Codes in Equidistant Letter Sequences in the Book of Genesis) the rank of the "correct" list was 14 if statistic P₁was used but 2724 if P₂was used. If the rank of the "correct" list is 14 it means that among the 999999 competing scrambled lists there are 13 lists where the proximity of ELS for rabbis' appellations to those of the corresponding dates is better than it is in the correct, unscrambled list. If though the rank of the correct list is 2724, it means that there are not 13 but 2723 scrambled lists with a better proximity than in the correct list. Then at least 2710 scrambled lists are believed to have a better proximity than the correct list, if we rely on criterion P₁, but the same at least 2710 scrambled list are believed to have a worse proximity than the correct list if we trust P₂. Which result do we believe?

The only possible conclusion is that either P₁ or P₂ or perhaps both, are unreliable measures of "proximity." The same relates toP₃and P_4.

In WRR's experiments, the ranks of the "correct" list of appellations/dates were consistently low (even if not equal) for all of four P (mostly between 4 and several thousand, out of one million of competing lists.) Hence, from the purely statistical viewpoint (and ignoring the above discussed deficiencies of WRR's statistical treatment) there are sufficient grounds to reject the null hypothesis and to accept the (not defined) alternative hypothesis, as the more likely one. That is what most of the critics of WRR's work do, concentrating instead on the weaknesses of WRR's data lists (which we will discuss in the next section). However, even if we were to accept WRR's conclusion as a legitimate result of their statistical study, that conclusion would still remain an unsubstantiated hypothesis under the requirements of a general scientific hypothesis. One of the reasons for that is the contradiction between their four cumulative measures of proximity. The last consideration is, of course, of purely academic significance, because, as was shown in the preceding sections, the statistical study by WRR is itself fraught by irregularities.

All that said, one more question remains, namely how to specifically explain the consistently low (even if not equal) ranks of the correct appellations/dates list WRR obtained for all four P. We will now discuss this question, which has been decisively answered by several scientists, including Barry Simon and most of all by McKay et al.

13. The wiggle room in WRR's study

A prominent mathematician, Dr. Barry Simon, the IBM Professor of mathematics and theoretical physics at Caltech, the author of many books and hundreds of scientific papers and a recipient of many awards, is just one of those many scientists who have unequivocally rebutted WRR's method and conclusions. In order to explain the paradoxical results of WRR's statistical study, Simon suggested [10,11] the concept of "wiggle room." This term relates to the large number of choices WRR could make when compiling their lists of rabbis' appellations and dates. For example, WRR inform us that the data on the famous rabbis' appellations and pertinent dates had been extracted from the Responsa database maintained by Bar-Ilan University. An examination of the Responsa database revealed that the 66 rabbis included in the two WRR's experiments had been known by at least twice as many names and appellations as WRR used in their lists. WRR seem to have made a number of arbitrary choices about which appellations and which dates to include or not to include in their list.

The natural question is, of course, whether or not the described "wiggle room," which was available to WRR, was sufficient to produce a "successful" list, and whether or not other choices available to WRR, if incorporated in their list, would invalidate their results (i.e. make the ranks of the correct list large enough to deprive them of statistical significance.)

The decisive answer to that question has been given mainly by B. McKay et al in a number of Internet postings and in their paper [8].

McKay et al have shown how small variations in the list compositions can drastically change the rank of that list among its scrambled versions. In one such example, Witztum claimed to obtain rank of 1 for a list of data where he juxtaposed ELS for a series of words related to the dreadful Nazi extermination camp at Auschwitz. McKay used a data list which differed from that used by Witztum only in minor details, and was at least equally legitimate. For that, slightly modified list, the rank was found to be 289,000 instead of 1.

In regard to the "famous rabbis" experiment, McKay and Bar-Natan similarly slightly modified the list of appellations/names. They preserved 80% of WRR's list, deleted a few names whose inclusion in WRR's list rested on a shaky foundation, and added a few names whose inclusion had at least as good reasons as all the other names in WRR's list. Using this list of names/dates, which was justified at least to the same extent as that of WRR, McKay and Bar-Natan found no statistical evidence whatsoever of unusual proximity of the pertinent ELS in the Book of Genesis.

Analyzing WRR's work MBBK found, for example, that if only four (out of 32) rabbis whose inclusion supplies the strongest contribution to WRR's low rank for the "correct" list of appellations/dates are deleted, the statistical significance of WRR's result changes from 1 in 62,000 to 1 in 30. Deletion of only one appellation (out of 102) changes the significance level by an order of magnitude. Deletion of only five names results in the change of significance level by three orders of magnitude.

In another experiment, McKay and Bar-Natan demonstrated how easily the list of names/dates can be "cooked" to achieve any desirable outcome of the experiment. Their slightly "cooked" list produced no significant result in Genesis, but produced what looked like statistically significant results in the Hebrew translation of Tolstoy's War and Peace.

In their paper [8] MBBK have shown that the "wiggle room" available to WRR was more than sufficient to compile a list of names/dates which would produce low values of rank in any selected text. Of course, the question of whether WRR made their choices deliberately or whether these choices were the result of an inadvertent optimization, is irrelevant. There is no evidence and no reason to suspect WRR in a deliberate manipulation of the data, but the existence of a wiggle room, in the traditions of scientific discourse, is in itself a sufficient reason to subject WRR's claims to serious doubts.

14. The question of the text

WRR's claim is, in particular, based on their assertion that the text of the Book of Genesis has been preserved intact, letter by letter, since it was given to Moses on Mount Sinai some three thousand years ago. Indeed, if the ELS in the Bible constitute a God-inserted "code," this code must have been present there from the very beginning. Any, even small, changes in the text would necessarily damage the code, which depends on the exact sequence of all letters.

The claim by code proponents in regard to the supposed perfect preservation of the Torah text has been convincingly refuted, for example, by Dr. Jeffrey Tigay, who is a Professor of Hebrew and Semitic languages and Literature at the University of Pennsylvania (the article by Tigay can be viewed at In Search of Mathematical Miracles, McKay's web site) as well as by Menachem Cohen who is a professor of biblical studies in Israel as well as the editor of authoritative editions of the Bible's text.(Cohen's article can be seen at my website B-Codes Page) A similar criticism has been offered also by MBBK. It has been established that the text of the Bible had undergone many changes in the course of its existence.

The code proponents sometimes respond with two different explanations. One is that the codes found in the Bible are just remnants of the original God-designed codes. To that argument, MBBK replied that deletion even of one letter in a thousand must completely destroy any ELS-based code. Given the extent of changes the text of the Bible must have experienced, there is no chance that any original "code" could have survived. MBBK have supported that claim by experimental data. In one experiment, MBBK deleted only 50 letters from the text, and this resulted in the complete disappearance of the effect claimed by WRR.

Another explanation by the code proponents is that God knew in advance which changes would occur in the text of the Bible and adjusted the sequences of letters in it in such a way as to ensure that the codes would emerge in our time in the altered text. Such explanations are at the best appropriate for theological disputes rather than for discussions involving mathematical statistics. Anyone who wishes to believe in the codes is of course entitled to that or to any other beliefs. This article is though not about beliefs but about the alleged "scientific" or, more specifically, statistical proofs of the codes' genuineness.

15. Conclusion

The arguments discussed in this article show that neither WRR nor anybody else among their supporters and code proponents have so far succeeded in scientifically proving that ELS in the Bible are anything but accidental sequences of letters which naturally occur in any sufficiently long text. WRR's argumentation has been debunked from various viewpoints. It was shown that the statistical procedures employed by WRR were in several ways contrary to the established rules of mathematical statistics. It was also demonstrated that the "distance" between ELS had been defined by WRR in an unnatural manner, depriving that quantity of a meaningless interpretation. It was indicated that the aggregate criteria of the ELS proximity behave in a haphazard way, thus destroying the credibility of their statistical conclusions from the standpoint of general scientific approach. It was demonstrated that WRR had considerable "wiggle room," i.e. sufficient freedom of arbitrary choices to fit their data to the desired outcome. It was indicated that the results of WRR's experiment, despite considerable effort, could not be reproduced by unbiased scientists. (The only other experiment which allegedly confirmed the presence of a "code" in the Bible was conducted by Harold Gans. Gans is Director of Research at Aish Ha Torah, making his living by promoting the "codes." Simon attempted to reproduce Gans's experiment. Simon slightly changed the procedure, aiming at the elimination of a possible ambiguity caused by the "wiggle room." Namely, Simon included into consideration the names of all cities where the "famous rabbis" were born, died, worked or studied, without any changes in spelling or any addition of prefixes. Thus the "wiggle room" was substantially reduced. The result? No trace of any code found in the Book of Genesis. Conclusion? Gans' results are most likely due to the "wiggle room" and therefore they cannot be considered a confirmation of WRR's results. MBBK have also tried to reproduce Gans' experiment, using a slightly different version of the data lists, and found no code in Genesis either. (The details have been described in MBBK's paper [8].)

As to WRR's epigones of all persuasions, such as Drosnin, Satinover, Jeffrey, etc., their work simply does not meet even the minimal scientific requirements to be considered seriously.

It is therefore not surprising that in 1997 WRR and Drosnin were awarded the "Ig Noble Prize" at a ceremony at Harvard (see The Ig Nobel Home Page) The Ig Noble Prize is awarded annually by a committee which includes real Nobel price winners. It is given for "the discoveries which cannot and must not be reproduced."

The publication of MBBK's paper [8] deprived the code proponents of one of their most frequently used argument, which suggested that the publication of their paper in 1994 in Statistical Science meant the approval of their claims by the scientific community.

However, rather than admitting, at least silently, the faults of their study, WRR organized instead the so-called International Torah Code Society. They conduct annual meetings of that society whose status in the scientific community is not much different from that of the Flat Earth Society. To give a talk at these meetings, the potential presenters must claim adherence to the Halakha, i.e. to the tenets of the Orthodox Judaism.

On the other hand, the Internet is full of websites where endless examples of alleged codes in the Bible are demonstrated, mostly from a Christian standpoint, many of them blatantly absurd, often revealing their authors' ignorance both of mathematics and of Hebrew. Of course, Rips, Witztum, and the International Torah Code Society as a whole disdainfully dismiss all these publications as nonsense, while stubbornly adhering to their own "discoveries." This gives the entire story a distinctively comic hue.

It seems reasonable to conclude that the errors by WRR's and by their followers invalidate their claims in regard to the alleged codes in the Bible and make their work an epitome of what initially was bad science but eventually became pseudo-science.

16. References

[1] Doron Witztum, Eliahu Rips, and Yoav Rosenberg. Statistical Science, v.9, No 3, pp. 429-438, 1994.

[2] Michael Drosnin, The Bible Code, Simon and Schuster, New York, 1977.

[3] M.Margaliot. The Encyclopedia of Great Men in Israel. Joshua Chachik Publisher, Tel-Aviv, 1961.

[4] David Kahn. The Codebreakers, MacMillan, London, 1967.

[5] Grant R. Jeffrey, The Signature of God, Frontier Research Publication, Toronto, 1996.

[6] Yacov A. Rambsel, Yeshua – the Hebrew Factor. Messianic Ministries, Inc., San Antonio, 1996.

[7] Maya Bar-Hillel, Dror Bar-Natan and Brendan McKay, Chance, No 11, 1998.

[8] Brendan McKay, Dror Bar-Natan, Maya Bar-Hillel and Gil Kalai, Statistical Science, v. 14, No. 2, pp.150-173, 1999.

[9] R.J. Larsen and M.L. Marx. Introduction to Mathematical Statistics. Prentice Hall, 1986.

[10] Barry Simon, Jewish Action, March 1998.

[11] Barry Simon, Vremya Iskat, No 2, 1999 (In Russian).

Discussion

the methods of the Bible code, by yongkiat, sim [14-Aug-03]

the methods of the Bible code, by McKay, Brendan [14-Aug-03]

the methods of the Bible Code, by TalkReason , [04-Apr-04]

the methods of the Bible Code, by Qasim, Adib [04-Apr-04]

the methods of the Bible Code, by TalkReason , [04-Apr-04]

the methods of the Bible Code, by Norowitz, Avi [05-Apr-04]

world, by X, [28-Dec-05]

world, by Y, [29-Dec-05]

Bible Code, by Way, No [01-Feb-06]

The Rise and Fall of the Bible Code

By Mark Perakh

Contents

1. Introduction