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Critique of Intelligent Design

Evolution vs. Creationism

The Art of ID Stuntmen

Faith vs Reason

Anthropic Principle

Autopsy of the Bible code

Science and Religion

Historical Notes


Serious Notions with a Smile


Letter Serial Correlation

Mark Perakh's Web Site

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Some Bible code related experiments and discussions

By Mark Perakh

First version - January 10, 1998
Last updated July 10, 1998
(Appendix 2 added on December 6, 1998)


  1. Preface
  2. Introduction
  3. How common are ELS?
  4. About ELS regarding Jesus
  5. Was a superhuman mind necessary to create code?
  6. About "simple" statistical proofs of "code" authenticity
  7. About the meaning of ELS phrases in the Bible
  8. About the "amazing" discoveries by Drosnin, Satinover, etc.
  9. About the "scientific" proof of "code" by Witztum, Rips, and Rosenberg
  10. Conclusion
  11. Appendix 1 (Discussion of "simple" calculations of probability)
  12. Appendix 2. About WRR's formula for the calculation of the expected number of ELS
  13. Endnote

1. Preface

I would like to report on some simple experiments I conducted in order to see for myself whose claims in the ongoing war between the proponents and the adversaries of the authenticity of the Bible codes conform to facts. Also, I would like to offer some conclusions I have formed in regard to the question whether the Bible codes are real or imaginary.

I am a Professor of physics (Emeritus). I have taught physics and related disciplines at several universities, first in the former USSR, then in Israel and in England, and finally in the US, at University of California, California State University, and University of San Diego. Among the courses I have the privilege to teach is statistical physics, which I have taught for both undergraduate (physics majors) and graduate students. I have published nearly 300 scientific articles, several books, and was granted a number of patents. Although I never did any research per se in the field of mathematical statistics, I used quite extensively statistical methods in my work in the areas of magnetic Phenomena and of electrochemistry.

I have acquired the information about the Bible Code controversy from the following sources:

  1. A book by Grant Jeffrey, titled The Signature of God;
  2. A book by Michael Drosnin titled The Bible Code;
  3. A book by Jeffrey Satinover titled Cracking the Bible Code;
  4. A number of letters posted on the Internet, including the exchange of criticisms between Rabbi Daniel Mechanic, Guy Cramer and Lori Eldridge, and some other people (see http://www.discoveryseminar.org/response.html; http://jewsforjudaism.org/codes/jesuslb.htm; http://www.yfiles.com/;
  5. Excerpts from the book by Pastor Y. Rambsel, titled Joshua, The Hebrew Factor, as posted on Internet. (See http://home.cvnet.com/crm80/)
  6. An article by Doron Witztum, Eliahu Rip, and Yoav Rosenberg as reprinted in the book by M. Drosnin.
  7. A set of articles by Dr. Brendan McKay of Australia as well as papers by Dr. B. McKay with Dr. Dror Bar-Natan, Alec Gindis, and Aryeh Levitan. (see http://cs.anu.edu.au/~bdm/dilugim/).
  8. Articles by Dr. Barry Simon of Caltech, by Dr. Gil Kalai of the Hebrew University of Jerusalem, and by Dr. James D. Price of Chattanooga (see, for example http://www.math.gatech.edu/~jkatz/Religions/Numerics), an article by D.E. Thomas (see http://www.cscop.org/si/9711/bible-code.html); articles by Dr. R. Haralick (see, for example http://www.prophezine.com/tcode/Obj15.html) and some other publications on the Web.

2. Introduction

Whatever views various participants of the dispute adhere to, the focal point of discussions is the so called ELS, which stands for Equidistant Letter Sequences. This term denotes any meaningful words that are formed by characters, which happen to appear in the text at equal "skips." For example, look at the word "denotes" in the previous sentence. Letters d, o, and s in that word, which are separated from each other by the same skip of 2 characters (e and n between d and o, and t and e between o and s) form the word DOS which is an acronym for Disk Operating System, and therefore it constitutes an ELS.

The dispute between pro-codes and anti-codes camps is whether ELS appear in the text of the Bible just by chance, or they constitute a deliberately inserted code. While all pro-code people believe that the creator of the code cannot be a human being, some of them (as, for example, Dr. J. Satinover) attribute the authorship of the codes to God, while some others (for example, M. Drosnin) attribute it to extra-terrestrial visitors to our planet.

Various participants of the dispute who do not believe that the "codes" are deliberately inserted in the Bible, approach the problem from different angles and use different sets of arguments. However, they usually complement rather than contradict each other. They consider each otherís efforts as contributing, from different viewpoints, to the same goal, namely to the assertion that the "codes" are fictitious and do not testify to the authorship by a non-human mind.

On the other hand, there is no unity whatsoever in the "pro-codes" camp. Indeed, Dr. Satinover, who, like M. Drosnin, believes that "codes" are real, expresses in his book the opinion that M. Drosninís book "presents the Bible code in a most unfortunate light... It has the potential to discredit the serious research..." (Page 261 in J. Satinoverís book). One of the most prolific defenders of the codesí authenticity, L. Eldridge relates to M. Drosninís book with the utmost contempt. Dr. Eliahu Rips, to whom M. Drosnin makes numerous appreciative references in his book, had disavowed Drosninís book in quite resolute terms. Rabbi D. Mechanic has a high praise for the work of Rips, Witztum, and Rosenberg, but dismisses with disdain the books by Y. Rambsel, and by G. Jeffrey. In their turn, G. Cramer and L. Eldridge value the books by Y. Rambsel and G. Jeffrey, but vigorously repudiate Rabbi Mechanicís articles, and so on.

Studying the publications of the "codes" proponents makes it rather obvious that the disagreements between them are due mostly to different agendas underlying their efforts.

In this paper I shall try to clarify the controversy, supporting my conclusions by observations of facts rather than pursuing any agenda.

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 people who are "pro-code" usually answer the first question with the assertion that the probability in question is extremely small. To the second question the "pro-code" people usually answer with "No." Likewise, the people who believe in the Bible code usually also believe that the answer to the third question is unequivocally "No".

I will try to test all three answers in this article.

3. How common are ELS?

The numerous examples of the ELS found in the Bible and demonstrated by Witztum, Rambsel, Drosnin, Satinover, and others, have made a strong impression on many people. Indeed, it seems hard to imagine that so many coincidences could've happened by chance. Therefore, after having read the books by Jeffrey, Drosnin, and Satinover, I decided, as a beginning of my exploration of the matter, to test how often various ELS appear in any randomly chosen texts, in various languages.

For my first test I chose one page from a letter I wrote in Russian to my friend in Moscow. That page contained the total of 2025 Cyrillic characters. Using WordPerfect word processor, I removed from the text in question all spaces between the words, as well as all commas, periods, etc. The text thus converted into a continuous string of characters. That is how it has been done in the Bible code studies. Then, without using any computer program, I started looking for ELS that would simply pop up from the text. I discovered in that sole page over 30 ELS, with skips ranging from 15 to 60 characters. At that point I stopped looking for more ELS. Most of those ELS, as it could be expected, turned out to be just three letter words, but a few four-letter words jumped out of the text as well. Certainly, by using a computer and extending the size of the skip to larger values, many more ELS could be found in that page of my letter, and much more in a larger text.

Then I switched to English. I conducted tests with three texts. One text (let us call it E1) was one page from a letter I wrote to the USA embassy in Kazakhstan. The other (E2) was a part of the first page of a story I wrote years ago in Russian. I translated it into English. The third (E3) was the first page of another story of mine, also first written in Russian and then translated into English. Again, I stripped the texts of commas, periods, spaces, etc.

Text E1 contained 2107 characters. I identified in that text over forty ELS, and after that I stopped searching for them. While three-letter words appeared most often, there were a number of 4-letter words ELS (for example rest, male, root, the name of Vice-President Gore, and others.)

In most of the sources dealing with ELS in the Bible, a special significance is attributed to the simultaneous appearance of several words related by meaning, in a proximity to each other. I was curious to see, if such occurrences can pop up in the tested page. At one location, the following four words appeared, three of them having a common letter (T), while the fourth word (ARE) was situated across the other three. Of the three words that had a common letter T, two were oriented symmetrically along the sides of an isosceles triangle. The third word (TEN) was aligned with the triangle's diagonal. This "array" of ELS spelled BRIT TITS ARE TEN. The skips were 47 (for TITS), -43 (for BRIT), and 45 (for TEN). Of course, one may either agree or disagree with the quoted sentence, but its appearance hardly could be attributed to any non-human entity. (A copy of the text in question can be viewed at brit.cfm).

Text E2 contained only 520 words. Even in that small sample 34 ELS popped up at once, at first glance, after which I stopped looking for them. Among those ELS three words happened to be 4-letter ones, the rest were three-letter words.

Text E3 contained slightly below 2000 characters. As in the previous examples, ELS were abundant. There were some 5-letter words, a little more of 4-letter words, and a whole enchilada of three-letter words. Right in the middle of the page, there was a 6-letter word TORVIL. Parallel to it, in the adjacent vertical, there was word ICE with the same skip of 45. Across the page, there was word DEAN, with the skip of 47. Close to both Torvil and Dean the word WIN appeared twice. Of course, Torvil and Dean used to be famous champions in figure ice-skating. Since the story in question was written years before TORVIL and DEAN demonstrated their skills on ICE, having twice WON championships, one may suggest that I predicted 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. Sorry, I did not possess such abilities. Among the 4-letter words were LAND, (not far from a 3-letter word SEA), LULL, TILT, ODOR, etc. (A copy of the text in question can be viewed at torvil.cfm).

Finally, I had written. especially for this Web page, a short poem whose contents relate to the "code" dispute. The slightly shortened version of that poem can be viewed at Poems.cfm. The full length of that poem is only 558 letters. Then I stripped its text of spaces, punctuation marks, etc, converting it into a continuous string of letters, in a conventional way used by all ELS researchers. It took about 50 minutes to compose the poem. It took only about 15 minutes to locate in it 37 ELS, after which I stopped searching for them. I found a funny feature in that text, namely three symmetrically situated appearances of the same two words, as it is shown and described at trihen.cfm.

These simple tests have shown that the phenomenon of ELS is very common, and a very large numbers both of individual ELS and of ELS clusters necessarily 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. B. McKay and D.Thomas. I was gratified to find that my conclusion turned out to be well in agreement with the multiple examples of ELS clusters found by Dr. 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 D.E. Thomas (they both used a computer program to locate the ELS).

4. What about ELS regarding Jesus?

Now let us take a look at the works by Grant Jeffrey and Pastor Y. Rambsel and at their criticism by D. Mechanic. Grant Jeffrey first refers favorably to the work done by Witztum, Rips, and Rosenberg. Then he switches to the results reported by Pastor Rambsel. In Mr. Jeffrey's view, the article by Witztum et al should be accepted as a proof that numerous occurrences of the ELS in the Torah indicate a non-human authorship of the Bible. Mr. Jeffrey believes that God had deliberately inserted into the text certain codes in the form of those ELS. In this point, there seems to be an agreement between G.Jeffrey and Y. Rambsel, on the one hand, and D.Mechanic, on the other. However, soon they drastically part their ways.

Jeffrey, following Rambsel, insists, that the Torah contains numerous references to Jesus in the form of ELS which prove that Jesus was the Messiah.

Rabbi D. Mechanic suggests, on the other hand, that there is a significant difference between the works of Witztum et al, on the one hand, and those of Rambsel and Jeffrey, on the other. According to D. Mechanic, the work of Witztum et al represents what D. Mechanic refers to as a legitimate Bible code research, based on scientific methods involving mathematical statistics. On the other hand, says D.Mechanic, the works of Rambsel and of Jeffrey represent an effort to use the legitimate results of Witztum et al in order to validate Rambsel's unfounded, and sometimes even fraudulent claims.

Guy Cramer and Lori Eldridge, in return, accused Rabbi D. Mechanic of disinformation.

Let us look at the arguments by D. Mechanic and his opponents. Rabbi D. Mechanic indicates in his article that the ELS that spells the four letter Hebrew word Yeshua, which is considered to be the shortened version of Yehoshua, the Hebrew name of Jesus, is mathematically expected to appear in the Torah by chance, over 10,000 times. It is also expected to appear as many times in any other Hebrew text of comparable size. Therefore locating this word as ELS, as it was done by Rambsel, in the Torah, has no meaning.

Out of curiosity, and to see whether R. Mechanic's assertion can be verified by some simple tests, I decided to try to locate word Yeshua in some non-Biblical Hebrew texts. 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. (The cover of that book is reproduced at amotzcovr.cfm and its inside title page in English, at amotinsd.htm). I decided to look for occurrences of ELS that would spell the 4-letter word Yeshua, as well as some combinations comprising both Yeshua and some other words, such as Yeshua Shmi, Yeshua Moreh, Yeshua Khali, Yeshua Iakhol, Dam Yeshua, etc. These phrases are examples of Pastor Rambsel's findings in the Bible, which, in his opinion, constitute the "codes" proving that Jesus was the Messiah. Sometimes Y.Rambsel uses instead of a four-letter form of Yeshua (Yud-Shin-Vav-Ayin) a shorter, three-letter version (Yud-Shin-Ayin). Likewise, I decided to look for such three-letter version occurrences as well. Since I did not have the text of the book in question in my computer, I could not remove from it the spaces between the words, the commas, the periods, etc. Obviously, if the spaces, commas, etc, are preserved in the text, and counted as meaningful characters, it makes locating ELS more difficult. In such a case some of the sites in the characters strings are occupied by meaningless spaces etc. So, if I could locate ELS, counting the spaces etc as characters, then it would mean that indeed the ELS in question are quite common. Then I would repeat the search for ELS spelling Yeshua, Yeshua Shmi and the like, this time ignoring the spaces, commas, periods, etc.

Including the spaces, commas, etc, into the string, I discovered, on the very first page of the Ben-Amotz's book, ELS spelling Yeshua with a skip of -68, as well as Shmi (My name), right next to Yeshua, with a skip of -60 (all these skips included the spaces between the words).

Then I switched to a search for Yeshua Shmi by counting only letters, and ignoring the spaces between the words, the commas, hyphens, etc. I leafed randomly through the book, and soon I located, on page 47, right in its middle, ELS spelling the words Yeshua Shmi one right after the other, both with a skip of only 2. Then I leafed more, randomly, through Ben Amotz's book and found the following clusters (or arrays, if we use the term used by Satinover) of ELS.

On page 23, within only three lines of text, words Dam (Dalet-Mem) and Yeshua (Yud-Shin-Ayin, which Rambsel and Jeffrey consider a legitimate spelling for Yeshua) meaning Blood of Jesus, occurred one right after the other, both with a skip of only 3.

On page 27, within only three lines of text, the same words Dam Yeshua, appeared, both with the same skip of 4.

On page 63, within only 2 lines of text, word Yeshua appeared twice, once as a three-letter version (Yud-Shin-Ayin) with a skip of 3, and once as a four-letter version (Yud-Shin-Vav-Ayin) with a skip of -1. In the same two lines of text word Moreh (Mem-Resh-Hey, meaning teacher) appeared 3 times with skips of 3, 4, and -6. In the same lines word Mori (Mem-Resh-Yud) meaning My teacher appeared with a skip of -5. The characters of words More and Mori appeared interspersed with the characters forming ELS for Yeshua.

On page 164, within three paragraphs, ELS for Yeshua appeared 4 times, three times as a three-letter version (with skips of 2, 4 and -6) and once as a 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 Y. Rambsel in the Bible, was interpreted by that writer as one of the proofs of his claims.

A copy of the above paragraphs on page 164 of Ben Amotz's book, provided here as an example, can be viewed at yeshyakh.cfm.

On page 315 an ELS for Yeshua (as a four-letter word) appeared with a skip of -6. Overlapping that word, word Khali (Khet-Lamed-Yud) appeared with a skip of -3, letter Yud being a part of both ELS for Yeshua and Khali. Word Khali was translated by Rambsel as "Polished Jewel" and the phrase Yeshua Khali, when found by Rambsel in the Bible, was interpreted by him as another proof of his views on the codes and Jesus.

All the above ELS have been listed by Cramer and Eldridge as those for which the "significance index" (see the explanation below) was found to be extremely small and therefore the ELS in question, in Cramer & Eldridge's opinion, must have been deliberately inserted codes in the Bible.

At this point I decided to try some other Hebrew text. This time it happened to be a book by an English-speaking writer Michael Moorcock, titled in English The Stealer of Souls. It was translated into Hebrew and published in Tel-Aviv in 1978 by Am Oved Publishers. On page 13 of that book, two clusters of ELS appeared, both anchored around word Yeshua. One cluster, which all was within only three lines of text, contained the four-letter version of Yeshua (Yud-Shin-Vav-Ayin) with a skip of 5. Following it, ELS spelling Shmi (My name) appeared with a skip of 7, and right after that word, another ELS appeared spelling Khali with a skip of -18. (Khali was translated by Rambsel as Polished Jewel). The other cluster on the same page contained the three-letter version of Yeshua (with a skip of 5) followed by an adjacent ELS for Shmi with a skip of 11.

At this point I pulled one more book from a shelf. This one happened to be a translation into Hebrew of Ernest Hemingway's The Old Man and the Sea published in 1977 in Tel-Aviv by Am Oved Publishers. On the second line of the first page of that book, the three-letter word Yeshua with a skip of -4 popped out at once. Having leafed randomly through the famous Hemingway's story, I saw, on page 17, the four-letter version of Yeshua with a skip of -7. Of course, I looked around that word to see if there are there any phrases like those touted by Rambsel. I don't know why, but besides the two-letter word (Ayin-Zayin, skip of -11) meaning strong, which probably would delight Rambsel, if located in the Bible next to Yeshua, I found, next to the Yeshua sequence, an ELS which had a common letter (Ayin) with Yeshua, and spelled (with a skip of -11) a Hebrew word (Ayin-Kaf-Bet) meaning crook. I am sorry, I had no intention to find an ELS with a negative connotation. My conclusion though had become unambiguous, namely that a variety of ELS, including combinations of word Yeshua with various other words occur quite commonly in any text, 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.

To locate all the listed ELS and their phrasal combination, I did not use any computer program and did not rearrange the text in the way it is routinely done in the Bible codes studies. There is little doubt that by applying a computer program and by spending more time on that effort, many more examples of ELS would be located in the books I tried, those ELS seemingly related to whatever topic one would choose.

There seem to be at least two possible interpretations of the facts presented. One is that God (according to Rambsel, Jeffrey, Cramer-Eldridge, Satinover, and others) or extraterrestrial visitors (according to Drosnin) not only had dictated the text of the Bible to Moses but also continues to dictate, character by character, every book, and even every piece of text anybody endeavors to write. The other interpretation is that all those allegedly amazing ELS in the Bible have not been deliberately inserted but rather occur there by chance. Of course, the burden of proof is on the shoulders of the "pro-codes" people. Those who do not believe that codes have been deliberately inserted in the Bible, do not need to prove anything as they do not make any extraordinary claims. If the people choose to believe in the authenticity of "codes" they are expected to present convincing explanation as to what is the difference between the ELS found in the Bible and the same ELS found in non-biblical texts. Given the extraordinary character of the claims about the "codes," the explanation in question is expected to be factual, unambiguous, and not based on vague concepts such as "relevance," "proximity," etc.

5. Was a superhuman mind necessary to create "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.

This statement betrays a plain ignorance on the part of those who offer it. I would like to refer here to a book titled "The Codebreakers" by David Kahn (Weidenfeld and Nicolson Publishers, London, 1967). 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 as a children game. Even the crossword puzzles that are being printed daily in local newspapers look as pinnacles of sophistication as compared to the "codes" in the Bible reported by Rambsel, Jeffrey, etc.

Indeed, even in the book by J. Satinover, who believes that the "codes" are real, there is an example of a method of encoding/decoding which is one of the simplest tools that had been used for encoding secret information. Nevertheless it can easily produce "arrays" of ELS quite similar to those allegedly "encoded" in the Bible. The method in question was apparently invented by the renown Italian mathematician and writer Girolamo Cardano in 16th century. It has since been used many times for encoding moderately important secret messages. The method in question is as follows. A grille, which is a sheet of a stiff paper, is used in which a set of holes has been cut forming a template. Each hole has a size enabling one to write one letter through it. The template is placed over a sheet of blank paper and the message to be encoded is written through the holes, letter by letter. 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. In those uncounted cases when this method was successfully used, no superhuman mind was ever required.

When I was a kid of about 12, in the city of Odessa in Ukraine, myself and a few of my friends had fun playing a game in which we sent to each other secret messages in the classroom right under the nose of our teacher. To encode a message, we used several techniques. One of them was the use of the Cardano grille, even though we had no idea that it was invented in Italy in 16th century. I do not remember now, how we came across the idea of the grille. 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 in a regular way from letter to letter. It thus produced encoded messages where the "skip" changed uniformly from character to character. As I will show later in this paper, similar "codes" with regularly increasing or decreasing "skips" can be easily located in any text as well as the ELS. I don't believe any of my 12-year-old friends possessed a superhuman mind.

Especially for this paper, I decided to try to encode again, like I did it as a 12 year old, some simple phrase, being willing to spend on that task not more than 10 to 15 minutes.  I did not make any Cardano grille, but simply counted the letters manually. 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 publisized alleged prediction of Rabin's assassination given in the book by M. Drosnin, which will be discussed later in this paper. I wrote the letters of the above expression on a piece of paper, leaving spaces between letters which would enable me to insert 10 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, is nevertheless meaningful.  The entire exercise took 9 minutes. Here is the result:

Rivers are dAmningly roBust in the Indian coloN ies, moving Water unerrIngly, thus aLleviating Lust for the Drinks, forcIng people rEcognize.....

This is how the ciphertext looks:  Rivers are damningly robust in the Indian colonies, moving water unerringly, thus alleviating lust for the drinks, forcing people recognize.... etc.

The presence of a plaintext encoded in the above two lines as an array of ELS is not evident to an uninitiated reader.   However, the intended recipient of that message, knowing the skip (10 in this case) would have no problem in decoding the message.

Of course, if a serious need existed for me to prepare a secret message using Cardano grille, doubtlessly I would be able, by spendng more time, to write a much more sophisticated ciphertext. The conclusion is inevitable, namely that a human mind is quite capable of creating arrays of ELS not unlike those found in the Bible.

A quite impressive factual confirmation of the contention that ELS clusters like those found in the Bible can quite easily be produced by regular men and women, came from Mr. Gidon Cohen of York, Great Britain. The story is as follows. Grant Jeffrey, who authored several books about the Bible code and who maintains that the "codes" are real, has suggested that the sheer complexity of the ELS in the Bible points to its creator being God. As one of examples of supposedly amazing arrays of ELS in the Bible, Mr. Jeffrey pointed out the array which contains, within a segment of the text of a relatively small length, names of 25 trees "encoded" in the form of ELS. Mr. Jeffrey was amazed by that array to such an extent that he issued, on radio, and in several conferences, a challenge to anybody to compile a text, of about the same size, in English, which would also contain ELS spelling the names of any 25 trees. Being confident that simple men or women, and even Moses himself, were not capable of performing such a task, which only could be done by God, Mr. Jeffrey offered to pay $1000 to anybody who would successfully fulfill the above requirement.

It did not take long until Mr. Gidon Cohen presented a piece of a meaningful text, in English, consisting of less than 300 words (the total of 1139 characters) in which the names of 29 trees had been encoded in the form of ELS. Mr. Cohen performed the task without using a computer program, by manually counting characters. When a computer program was used (by Dr. McKay and Mr. D. Thomas) they discovered in Mr. Cohen's text additionally ELS spelling the names of at least 5 more trees plus a number of ELS for various related words such as forest, copse, bark etc, and plus a dozen ELS for plants other than trees. (Any reader who wishes to see the text by G. Cohen, can request it via e-mail: gdc101@york.ac.uk).

To Mr. Jeffrey's credit, he admitted that his challenge was met and paid the promised amount. However, it did not make Mr. Jeffrey retract his statement about the alleged Bible codes.

Lest I will be misunderstood, I would like to point out that by refuting the claims about the necessity of a superhuman mind for the creation of the alleged codes in the Bible I was not at all 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 rather chance coincidences. One more argument in favor of the above explanation is the fact that, after Mr. G. Cohen had compiled his short text, in which he "encoded" deliberately the names of 29 trees, a computer analysis of his text revealed there many more ELS spelling names of more trees, as well as of other plants, and also 35 names of animals, as well as various names of people, including that of Dr. Rips. Mr.Cohen did not place all these additional ELS in the text deliberately. They appeared there by chance.

6. What about the "simple" statistical proofs of the Codes' authenticity?

As I have mentioned earlier, many participants of the dispute about the "codes" try to calculate the probabilities of the appearance, by chance, of various ELS in the text of the Bible. Indeed, the most highly acclaimed publication which has become the pivotal point of the recent Bible code controversy, namely the paper by Witztum, Rips, and Rosenberg, was published in 1994 in Statistical Science magazine. This fact underscores the role ascribed to statistics for the problem in question. There are several good publications, mainly on the Internet, providing criticisms of the paper by Witztum et al. I would like to indicate, in particular, articles posted on the Web by Gil Kalai, Brendan McKay, and Barry Simon.

Unfortunately, many of the people who are interested in the Bible code controversy, either could not or would not read either the paper by Witztum at al or the criticisms of it by G. Kalai, B. McKay, and B. Simon, since much in this material is not intended for laymen. On the other hand, more people read and are influenced by some other publications, which pretend to treat the problem from the scientific viewpoint. I believe that even if Witztum at al announced that they were retracting their statements about the codes (I don't expect this of course) many other proponents of the codes, such as Rambsel and Jeffrey, would not abandon their efforts to prove the codes' reality. (Recently, Guy Cramer and Lori Eldridge, who used to be among the most prolific defenders of the "codes" have announced, commendedly,that, in view of the evidence, they do not any more support the beliefs in the codes' authenticity. Very courageous and honest of you, Guy, and Lori ).

Scientists such as G. Kalai, B. McKay, and B. Simon, while being involved in a discussion with Witztum, Rips, and Rosenberg, evidently do not consider it worth their time to argue with writers of allegedly statistical treatises that are not on the same scientific level. Hence the claims of many proponents of "codes" other than Witztum et al remain largely unchallenged.

Unfortunately, many of these claims are characterized by errors and misconceptions and produce unreliable values of probabilities. I believe that, for the sake of many unsuspecting readers, it is desirable to explain the errors in those claims.

An example of such a faulty calculation is the one which was once offered by two code proponents, whose names I would like to omit because, since the publlication on the Web of their calculation, they both changed their views and do not support the "codes" any longer.  I will refer to them as XY. It may be instructional to analyze in detail those old calculations of XY as a warning example.

XY first agreed that some of the ELS found by Rambsel can indeed happen in the Bible by chance, because the probability of their occurrence as they have calculated it, turned out to be not very small. On the other hand, XY had singled out eleven ELS found in the Bible by Rambsel, and insisted that the appearance of those ELS by chance has such a small probability that these 11 ELS must be deliberately inserted codes. To demonstrate their point, XY suggest a certain procedure (they call it formula) for the calculation of the probability of an ELS appearing in the text by chance. For certain ELS found by Rambsel, that alleged probability turns out to be very small (between 1/99 for some ELS, and down to about 1/58,507,687 for some other ELS).

XY calculated for each of the 11 ELS they believe to be a genuine "code" a quantity they call "significance index", which is the same as the probability of a sequence of characters to appear in a text by chance. A detailed discussion of XY's caculation of probabilities is given in an Appendix to this article. It is shown in the Appendix that the calculation of probabilities was conducted by XY in a way contradicting some basic rules of the Probability Theory and therefore the values of "significance index" calculated by those writers are meaningless numbers.

Now, let us consider the question: what if, regardless of the values calculated by XY, the actual values of probabilities in question are indeed very small? Does it mean the occurrence of corresponding ELS must be attributed to a conscious design? Not at all.

Again, as an example, let us turn to XY's treatment of the probabilities in question. As we shall see, XY's interpretation of those probabilities is based on a misconception. It invalidates their conclusions that some ELS found by Rambsel must be deliberately inserted codes.

The error in question stems from XY'se misinterpretation first of the meaning of probabilities in general, and second of the meaning of the quantities they calculated, in particular.

Let us see how XY explain the meaning of probability. What they say is as follows. If the calculated probability of a certain ELS in a given text is, for example 0.001, then, according to XY, one has to look through 1000 texts of that size to encounter an ELS in question. This explanation is wrong. The actual meaning of probability is as follows. If it is 0.001, it means that if a very large number of tests has been performed (that number being much larger than 1000) then the ELS in question will appear, on the average, approximately once per every thousand tests. In order for that to happen, the number of tests must be very large indeed. The more tests are performed under identical conditions, the closer will be the number of occurrences of ELS in question to the calculated probability (in this example, one occurrence, on the average, per one thousand tests). If the number of tests tends to increase indefinitely, the number of occurrences of ELS in question will approach the calculated value of the probability. However, nothing can be asserted as to what would happen in any particular thousand of tests.

Theory of Probability 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 mean at all that that event will not happen. All what Theory of Probability 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 along with the increase in the number of tests. Not more and not less than that.

Consider an example. Nearly 100 times a year, a lottery is played in California. I, and you, and your cousin, and every John Doe who buys a ticket, have been told more than once that his/her chance to win the big prize in the California lottery is about 1 in eighteen million! (The precise number, as it can be easily calculated, is actually 1/15,890,700). Nevertheless, millions of tickets are sold every three or so days. Why? Do those results of the lottery game contradict the theory of probability? Not at all, if one understands properly the meaning of the quantity called probability.

The value of probability does not predict how many times one must play to ensure winning. One may well win having bought just one ticket (as it had happened a few times) and one as well may never win even buying tickets millions times in a row.

Why some lucky guy wins almost every time the lottery is played, despite his chance to win being that small? The explanation is quite simple. The chance for a particular ticket to win is indeed 1 in about 16 million. However, the chance for some ticket out of all the tickets sold, to win, is much larger. If several million tickets are sold, then the probability of some ticket (and we never know in advance which one) to win becomes so large, as to approach certainty. So, we witness about 100 times a year how an event whose particular probability per se is only about 1/16,000,000, actually happens with the utmost regularity! This fact in no way contradicts the laws of probabilities. It fully conforms to the mathematical expectation for some ticket (but not known in advance which one) to win, whereas the mathematical expectation for the particular ticket to win still is very small.

How is all this applicable to the ELS found in the Torah and spelling Yeshua and some combinations of Yeshua with other words, such as Shmi, Yakhol, or Moreh? The chance to find in the Torah a particular ELS may be small (but still much larger than calculated by Cramer & Eldridge) but the chance to find some ELS, 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 to come across some meaningful combination of Yeshua with some other words in the Torah is quite large. Indeed, when Rambsel sets out to locate meaningful phrasal combination in the Torah, he is not after any particular expression. Were it the case, the probability to locate such an expression could be small indeed (which still does not mean at all it would not be found by chance). But since he is looking for any combination of words which seems to support his contention that the creator of the Torah encoded in it a message about Jesus being Son of God and the Messiah, then the probability of coming across many such suitable phrasal sets is not small at all. Like in the California lottery, where an event whose individual probability is only 1/16,00,000, happens, on the average, twice a week, so Rambsel's quest for suitable ELS in the Torah is expected to reveal many such combination of words almost on every page. Obviously, also many expressions, which contradict Rambsel's beliefs, would necessarily be encountered. However, either consciously, or subconsciously, such undesired phrases would be ignored, most often before their full length has been revealed. This is a phenomenon that happens in every research. Scientists know very well how difficult it is to accept anything contradicting their expectations and how easy it is to see a confirmation of a preconceived view where no such confirmation really exists.

The fact that many expressions found by Rambsel in the Bible, were also located, and quite easily, in non-Biblical texts, may serve as a confirmation that calculations of probabilities, like that offered by XY, are meaningless and cannot be used to justify any conclusions as to the reality of "codes" in the Bible.

7. What about the meaning of the ELS phrases found in the Bible?

In view of the above considerations, what is striking is how actually poor Rambsel's catch has been. While, on the one hand, people like Drosnin, Satinover, Witztum, etc, have found a big number of what they believe are detailed predictions of many events, such as Rabin's and Sadat's assassinations, AIDS epidemics, and the like (we will discuss those finding a little later), on the other hand, for such pivotal event in the history as the emergence of Christianity, all what Rambsel and his followers could discover in the Torah is only a number of grammatically lame, ambiguous phrases.

Let us recall that the name of Jesus of Nazareth does not appear even a single time in the open, not-coded text of the Hebrew Bible. If the creator of the Bible wanted to send a message to us, humans, about Jesus Christ, why would he resort to a code while avoiding any direct reference to Jesus?

Of course, the name of Jesus appears overwhelmingly in the text of the New Testament, whose original, unlike the Old Testament, was written in Greek rather than in Hebrew. When the New Testament was translated into Hebrew, the name of Jesus was transliterated by the translators as a four-letter Hebrew word Yeshua ( in Hebrew Yud-Shin-Vav-Ayin). Actually, though, Yeshua is a shortened version of the five-letter Hebrew name Yehoshua (Yud-Hey-Vav-Shin-Ayin).

As the name of Jesus in the Hebrew translation of the New Testament has been traditionally spelled as Yeshua, Rambsel was looking for and has found in the Old Testament many occurrences of the four-letter version (Yud-Shin-Vav-Ayin) and of the even shorter version, the three-letter word (Yud-Shin-Ayin), but not the five-letter version spelling Yehoshua.

Of course, if one believes that the ELS in question occur in the Bible just by sheer chance, then it is natural that a five-letter ELS would be encountered about twenty times less often than a four-letter ELS. It would be even more true for phrases containing, besides ELS for Yehoshua, also such accompanying ELS as Shmi, Khali, Rimon, (see below the discussion of these ELS) etc. On the other hand, if one believes that the ELS are deliberately inserted codes, then unavoidable question is, why Yeshua rather than Yehoshua?

While in the Hebrew language Yeshua and Yehoshua are indeed two forms of the same name, there is a semantic difference between the uses of the five-letter and of the shortened four-letter form. Letter Hey which is present in the five-letter version but is absent in the shortened one, is considered by the Bible scholars to be a link to the name of God (which is in Hebrew Yud-Hey-Vav-Hey). Indeed, that letter, according to the Bible, was added by God to the name of Abram, converting it into Abraham, thus linking the name of the patriarch to the name of God.

Let us imagine that in a historical book about Franklin Delano Roosevelt he is consistently referred to as Frankie. Of course, Frankie is a shortened form for Franklin, but obviously referring to the late President as Frankie is not the same as referring to him as Franklin Delano Roosevelt. It is even more true in the case of Yehoshua's name, given the special significance of letter Hey which is omitted in the four-letter version, Yeshua. The four-letter ELS for Yeshua, and even more so its three-letter version, are diminutive forms. They have in Hebrew the connotation of either familiarity, or even of a certain lack of respect for the person bearing that name. Does it not look strange that the alleged creator of the "codes," if he wished to encode references to Son of God and the Messiah, would use in those codes, so often, a diminutive form of Messiah's name? Obviously, the alleged creator of "codes" would have no difficulty to encode the full five-letter version Yehoshua as many times, instead of, or at least along with its diminutive form.

Now, if the author of the Torah wanted to encode in it a message about Jesus, would it not be expected that Jesus would be at least somehow identified in the "code"? For example, the encoded words could have spelled something like Yeshua Ha-Notzri (Jesus of Nazareth), or, perhaps Yeshua ben Yoseph (Jesus son of Joseph), or maybe Yeshua ben Miriam (Jesus son of Mary), or, say, Yeshua Nolad beBetlechem (Jesus born in Bethlehem), or any other of numerous possible identifications. Nothing of the sort has been discovered by Rambsel. So, if any phrases found by Rambsel has indeed been deliberately encoded (which, of course, is not proven at all), then who is Yeshua these "codes" refer to? For example, it could very well be Yehoshua (equivalent of Yeshua) Bin Nun (Joshua the son of Nun) who led the Israelis into the promised land after the death of Moses. (While in English the name of Yehoshua Bin Nun is transliterated as Joshua, and that of Yeshua of Nazareth as Jesus, in the original Hebrew it is the same name Yehoshua or, in a diminutive form, Yeshua. In some other languages both Joshua Bin-Nun and Jesus are referred to by the same name, which, for example, in Russian is transliterated in both cases as Yisus). At least the not coded, "surface" text of the Torah speaks a lot about that Yehoshua. It could be any other Yeshua, as this was not an uncommon name in Israel.

Now, what is the meaning of those expressions Rambsel and Jeffrey attach so much significance to? For example, what is really the meaning of words such as Yeshua Shmi (Jesus is my name)? What can this possibly mean except of some Yeshua (and there is no indication as to which Yeshua is meant) to announce that his parents named him Yeshua? What hidden meaning has the expression Yeshua Rimon (Jesus pomegranate)? To mean "Jesus is a pomegranate" that expression in Hebrew should've been Yeshua Harimon, or Yeshua Hou Rimon. As it spells, it really has no meaning by itself. Maybe some Yeshua likes pomegranates? Or grows pomegranates? Or has red cheeks and is nicknamed pomegranate? Obviously, the expression is ambiguous.

One more expression touted by Rambsel is Yeshua Yakhol (Jesus can, or Jesus is able; or Jesus is allowed. The word Yakhol means will overcome only in conjunction with another word, a noun or a pronoun, denoting somebody or something being taken over, and with prefix l' before that other word. If the word following Yakhol is a verb preceded by prefix l', it means can or is able or is allowed. The meaning of Yakhol standing alone is ambiguous, and its exact meaning becomes clear only in context). Moreover, again, which Yoshua can, or will overcome, or is able, etc? And can what? Yehoshua Bin Nun obviously was able to perform, and did perform, according to the not-coded text of the Bible, many feats. To assert that the phrase in question refers to Jesus of Nazareth is an example of an arbitrary interpretation of the text, which by itself has no clue as to its possible meaning. And, most telltale is the fact that practically all expressions found as combinations of ELS in the Bible and discussed above, have been found as well in texts other than the Bible.

As mentioned earlier, some of examples given by Rambsel employ imprecise translation of Hebrew words, or choose out of several possible meanings the one fitting their goal. For example, consider the expression Yeshua Moreh. The word Moreh, which has several meanings in Hebrew, does not mean at all "Teacher of Righteousness", as they claim. The most common meaning of that word is simply teacher, and it can be a school teacher of any subject, or even driving instructor, tourist guide, etc. but the word Righteousness was added by out of thin air.

Another Rambsel's example is the occurrence of the combination Yeshua Khali, which allegedely translates as Jesus polished jewel. First, if the meaning were to be Jesus is a jewel, it must be Yeshua Hakhali, or Yeshua Hou Khali. Moreover, while one of the meanings of the word Khali (in Hebrew three characters Khet, Lamed, Yud) is embellishment (any bauble, not only a polished jewel) its other, quite common meaning is disease. So, why did Rambsel choose the meaning fitting his purpose, but chose to ignore another, quite common meaning, which obviously would be working against his contentions? I am curious, how many more words similar to disease, situated close to Yeshua, Mr. Rambsel did not care to report about?

But the most important fact is that, whatever those expressions may mean, there is no proof whatsoever that any of them have been deliberately encoded rather then appear by chance. Indeed, similar phrases occur in non-biblical texts as well, as I have demonstrated above.

The claim that those ELS have been deliberately encoded in the Torah has no more validity than the claim that John Doe had won big in the California lottery not by chance but because he was deliberately chosen (by God? By extra-terrestrials? By the lottery's management?) Reportedly, some people who had won in the lottery believe indeed that they had been personally chosen by God for a reward. Well, if you believe in God who encourages gambling, then you have to believe that Las Vegas and Monte-Carlo are his holiest cities, casinos are his temples, and roulette tables are his altars.

In conclusion of this section, let me state that I have not proven in this article the absence of codes in the Torah, nor did I try to do so. What I did, though, was to show that the alleged proofs of the validity of their views, as presented by Rambsel, Jeffrey, and others, actually do not prove anything. Maybe there are codes in the Torah. Maybe some of them say something about Jesus of Nazareth. However, to prove that, Rambsel, Jeffrey, etc, would need to do much better than they did so far.

G. Jeffrey and Y. Rambsel have recently published more books, in which they give more examples of ELS they found in the Bible, supposedly confirming their previous assertions. Likewise, there are additional, more recent postings on the Web, also providing more examples of "discoveries" about Yeshua, or about some "predictions" of future events, etc. All of it makes use of the same argumentation, which I have addressed in this article. None of it adds a shred of real evidence as to the authenticity of the alleged "codes" but appeals quite strongly to the gullibility of its readers. There is no point to continue addressing time and time again each of these "new discoveries," as this would be an open end game without a winner.

In this article, I have not discussed the question as to whether the ELS found by the "code" proponents in the Bible do necessarily spell indeed what they have been claimed to spell. This question is addressed in my paper at Do the ELS in the Bible indeed spell what they have been claimed to spell? where I have shown that the very spelling of the ELS in question, more often than not, is quite ambiguous and is often interpreted in an arbitrary way.

My criticisms have nothing to do with any religious beliefs, or with the question as to whether the Bible has been inspired by God, or with the question whether or not Jesus of Nazareth was Son of God and the Messiah. The scope of this article is much narrower. It deals only with the question whether or not the arguments by Rambsel, Jeffrey, etc, prove the authenticity of the alleged codes in the Bible. My conclusion is that they do not. I am not claiming though that some better proofs of the alleged codes' authenticity cannot be found one day. Such proof, if offered, must necessarily include a convincing explanation as to what is the difference between ELS found in the Bible and the same ELS found in non-biblical texts.

8. What about amazing discoveries by Drosnin, Satinover, etc.? Was Rabin's assassination predicted by "Codes" in the Bible? Was it predicted in Moby Dick? Was the AIDS epidemic predicted?

In his book Michael Drosnin claims to have made amazing discoveries in the Bible. In particular, he claims to have found a prediction, encoded in the form of an array of ELS, of Yitzhak Rabin's assassination. The array of ELS in question, reproduced as a tableau in Drosnin's book, comprises the name of the late Israeli Prime Minister (found with a very large skip of over 4000 characters) as well as the phrase Rotzeakh SheIrtzakh, meaning killer who will kill, the set of letters denoting the year (in the Hebrew calendar) when Rabin was assassinated, and also the name of the killer, Amir.

Looking at the tableau in question reveals that of those words only the name 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 not-coded text, so are not codes at all. Hence, Drosnin feels free to combine both ELS and parts of the Bible's "open" text while interpreting the meaning of his finds. Since many people remained skeptical in regard to Drosnin's claims, that author, in an interview, had said that if somebody found a prediction of a Prime Minister's assassination in Moby Dick, then he would admit being wrong. I assume that by saying Moby Dick Mr. Drosnin actually meant any book other than the Bible. As to the prediction of the Rabin's assassination, I assume he would allow us to look for such a "prediction" in the form similar to what he did, as a set of ELS that appear in some book close to each other and spell words such as Rabin, murderer, and the like.

Out of curiosity, I turned again to the book by Dahn Ben-Amotz, mentioned earlier in this article and titled, as I said before, Screwing is not Everything. As mentioned before, I found in that book, without using any computer program, ELS spelling Yeshua Shmi etc. Now I decided to look for some ELS related to Rabin. Again, I did not use any computer program, so I limited myself to relatively short skips. My intention was to see if I could find relevant ELS rather easily. Following Drosnin's practice, I counted only the letters, ignoring the spaces between the words, commas, periods, etc.

A copy of the pertinent paragraphs in Ben Amotz's book, described below, can be viewed at rabin.cfm.

The word RABIN popped up on page 33, in the uppermost paragraph on that page, with a skip of -35. Then I looked around that word to determine if there are some other ELS related by meaning to Rabin. What I found in that search was as follows. In the 2 uppermost paragraphs, consisting of only about 600 characters, the following ELS appeared, in a chain formation, one following the other: 1) An ELS consisting of four Hebrew letters, with a skip of only -2, 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 is also used as such by Drosnin). 3) The following Hebrew characters with a skip of -7, 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, word RABIN with a skip of -35.

So, in just two short paragraphs in a randomly opened page, in a randomly chosen Hebrew book, that has nothing to do with the Bible, without employing any rearrangement of the text like those usually performed by Drosnin, set of 4 ELS, with small skips between the letters of each word, all words appearing in one chain, one after the other, in a very close proximity, reads: (he) will kill, Prime Minister, hero, Rabin.

Now, what about the name of the killer, Amir? To relate to this question, let us consider the idea of a code. Obviously, if some entity planned to encode certain information, then this entity was not confined to the use of ELS. The code may be more complex, and the only requirement is that the code follows certain rule. If myself and my friends, as 12 year old boys, used a "code" in which the skips either increased or decreased in a regular manner from letter to letter, why could not such a "code" be employed anywhere else as well? So, I looked for any occurrence of such a "code". In the same chain of words, mentioned above, I found the following 4 Hebrew letters: Ain, 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 in the same one third of a page, in a chain formation, must be so small that it could not happen other than by a conscious design! Hence, we must 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.! Such a prediction could've been made only by God or by extraterrestrials who possessed an intelligence much exceeding that of humans!

Indeed? Regarding the probability of those ELS appearing in the paragraph in questions, I can only repeat what I have said earlier in this article in regard to Cramer&Eldridge statements. If the concept of probability is properly interpreted, as explained above, there is nothing amazing in the appearance of the cited ELS in the paragraph in question. Neither is it amazing in the Bible.

Since Mr. M. Drosnin has challenged anybody to find a "code" predicting the assassination of Rabin in a book other than the Bible, I would in return challenge him to admit that his condition has been fulfilled herewith. Now he is expected to retract the extraordinary claims made in his best-selling (and certainly very profitable) book. Of course, he is free to verify my claim by checking the book by Dahn Ben-Amotz. I hold my breath.

The unavoidable conclusion is that the claims by M. Drosnin asserting that the ELS in the Bible are deliberately inserted codes, many of which predict the future, have no reliable proof. Most likely those ELS are just random coincidences, which could be found in any text of sufficient size. If the codes in the Bible are real indeed (which, of course is not impossible) Mr. Drosnin would need to look for a more believable proof for his claims.

It is worth to mention that M. Drosnin, unlike Dr. J. Satinover (see below) has not used anywhere in his examples the concept of the "minimal" or "nearly minimal" skips. Therefore he hardly can resort to the argument about the skips being not minimal in the Ben-Amotzís book. The skips I found there are all very short anyway, none of them being anywhere close to sometimes very long skips in M. Drosninís book.

The book by J. Satinover contains some interesting pieces of history, popular explanations of certain aspects of cryptology etc. Unfortunately, also in Dr. Satinover's book there are examples of alleged amazing finds in the Bible. Although Dr. Satinover uses an additional criterion of nearly minimal skips, his finds still are most likely results of a random chance.

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

I decided to look if any similar "arrays" can be found in a text other than the Bible. For my test I chose the array described by Dr. Satinover on page 164 of his book. According to Dr. Satinover, it was found by D. Witztum. It deals with the AIDS epidemic.

As Dr. Satinover reported, 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(the) blood"), The HIV (Hey-Hey-Yud-Vav) etc.

Again, I leafed randomly through the book by Ben-Amotz, mentioned before. I did not use any computer program and did not rearrange the text like Dr. Satinover, D. Witztum, and other explorers of the Bible code do routinely. My goal was to see if the "arrays" in question could be discovered easily.

On page 67 of Ben-Amotzís book, I found the following ELS, all situated within the two uppermost paragraphs which contained the total of about 600 characters ( The paragraphs in Dahn Ben-Amotz's book that contain the "array" of ELS related to AIDS, which is described below, can be viewed at aids.cfm).

The ELS I found on page 67 are as follows: BeDaM (Bet-Dalet-Mem, meaning in (the) blood) with the skip of 15. MaVeT (Mem-Vav-Tet, meaning death) with the skip of Ė10. HIV (Hey-Yud-Vav) with a skip of 18. Additionally, in the same paragraphs there was word KHaLi (Khet-Lamed-Yud, meaning disease). Now, what about word AIDS? Again, remembering the codes I used as a kid, and realizing that the "code" must 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 Ė18 between the first and the second letter, -17 between the second and the third one, and Ė16 between the third and the fourth letters. How small is expected to be the probability of the described occurrences?

There remains little doubt that by using a computer, and rearranging the matrixes of text, every single ELS of Witztumís and Satinoverís arrays, as well as tight clusters of ELS related by meaning could be located in the Ben-Amotzís book, as well as in any text of sufficient length.

M. Drosnin is a journalist. J. Satinover and D. Witztum, however, supposedly are scientists. Therefore they are expected to provide a convincing explanation as to what is the difference between the "arrays" in the Bible they have described and similar occurrences in the non-biblical texts. Using the trivial argument about the "minimal" skips would be not convincing. The skips I found in Ben-Amotzís book are so small as to make the question of "minimal" or "nearly minimal" skips irrelevant. As I did not use a computer, I automatically limited myself to only very short skips. All arrays of ELS I found in Ben-Amotz's book were situated within very short segments of text.

Furthermore, in Dr. Satinover's book there are strange errors, which make one take the rest of the book with caution.

Here is one example. On page 237 of the book in question, Dr. Satinover indicated that Max Planck developed his theory (which was the beginning of the quantum physics) in 1912, and that he was at that time 19 years old. Not true. Max Carl Ernst Planck was born on April 23, 1858. So, in 1912 he was much older than 19. Moreover, he developed his theory not in 1912, but in November - December of 1900. At that time he was 42.

Another example. In the chapter dealing with the information allegedly encoded in the Torah about Dr. Satinover's ancestor nicknamed Angel, Dr. Satinover consistently refers to the town where his ancestor lived, as Fostov. There is no such town in the Ukraine. There is in that country though a town named Fastov. Not a very serious error, but such errors hardly enhance the credibility of the rest of the book in question.

9. What about the "scientific" proof of Codes reality by Witztum, Rips and Rosenberg?

In the article by D. Witztum, E. Rips, and Y. Rosenberg (WRR) its authors have reported on some results which, although quite extraordinary, are often touted as products of a scientific approach.

G. Jeffrey, M. Drosnin, J. Satinover, and D. Mechanic all refer to that paper with the utmost esteem. As one of the arguments in favor of "codes" reality, these writers indicate that Witztum, Rips and Rosenberg are genuine scientists, that their paper was published in a prestigious refereed magazine, and that no skeptical scientists could so far find any errors in WRR's calculations.

Drosnin and Satinover also bestow on some people whose views they like, including WRR, the ranks of what they call "world-class" mathematicians, probabilists etc. It reminds me of the mores in the scientific community of the former USSR. Scientist had there ranks like officers in the army. According to that system, a full professor was meant to be automatically smarter than an associate professor ("docent"), a Doctor of Science automatically smarter than a Candidate of Science, etc. No assistant would dare to utter an opinion contradicting that of a professor, etc. Now we have a funny picture of journalist Drosnin awarding ranks of "word-class" scientists. So, now we may have a table where scientists would be ranked as "world-class", "continent-class", "country-class", "city-class", "village-class" and "no-class" experts, and the ranking would be performed by writers of sensational books who, apparently, themselves would be counted among "galaxy-class" judges.

I understand that the three authors contributed different components to their paper. In particular, the mathematical calculation was reportedly the contribution by E. Rips. Y. Rosenberg developed the computer program, while D. Witztum has been characterized as the foremost researcher of the Bible codes per se. I also understand that E. Rips is an expert in Group Theory rather than in Statistical science. I am not at all an expert in Group Theory so I can't judge Dr. Rips' scientific achievements and credentials. I am prepared to happily accept the assertion that E. Rips is a brilliant mathematician, highly qualified in his field, as well as a very nice and a perfectly honest person. Does it mean that we have to take uncritically everything WRR claim?

There had been many brilliant scientists who goofed, sometimes in a most bizarre way. One of the most outstanding Russian scientists of the last century Dmitry Mendeleev is deservedly revered for one of the most important discoveries in the history of science, namely the Periodic System of elements. Until the last days of his life he vigorously fought against the theory of electrolytic dissociation developed by Svante Arrhenius. In that, the famous Russian chemist was wrong. A brilliant Russian physicist N. S. Akulov produced in the late thirties a very elegant and powerful theory of magnetic behavior of solids based on the concept of symmetry. Later in his life, though, Akulov wrote a book on the Theory of dislocations which had become a laughingstock among scientists. Professor Fleishman, a respected electrochemist, once a President of the International Society of Electrochemistry, who authored a considerable number of good quality research papers, in the late eighties published (together with Pons) an article claiming the discovery of the "cold fusion" which turned out to be a non-existent phenomenon. There are many more such examples. Therefore an argument which makes use of a scientist's reputation as a proof that his claims are correct, be it E. Rips or anybody else, is irrelevant with all due respect for E. Rips scientific achievements.

On the other hand, if one wishes to use the argument based on the credentials of WRR, then one has to account for the opinion of many prominent mathematicians and experts in Mathematical statistics who expressed an unequivocal rebuttal of WRR's paper (see the letter of over 50 mathematicians at http://www.math.caltech.edu/code/petition.html).

One more argument often offered in favor of WRR paper is that so far no scientist had found errors in WRR calculations. This statement is not true. There are several publications on the Web in which serious doubts have been offered in regard to WRR's calculational procedure. A very serious rebuttal of the entire approach by WRR , from the standpoint of Mathematical Statistics and Probability Theory has been proposed by a prominent expert in Mathematical Statistics, Dr. A. M. Hasofer. The paper by A.M. Hasofer is available in this Web site - see A statistical critique of Witztum et al paper. Another prominent mathematician, Dr. B. Simon of Caltech has also unequivocally rebutted WRR's method (see http://wopr.com/biblecodes/TheCase.htm).

I've also read the calculations in WRR's paper in question. There is one statement in the paper by WRR, at the end of section A.1 of that paper, related to the calculation of the expected number of ELS with shortest skips, that number chosen by WRR to be 10. I believe that statement is in error. It had though no substantial effect on the final result of WRR's calculation (see Appendix 2, added on December 6, 1998). What is more important, though, is that I found in WRR's paper serious deviations from the established rules of Probability Theory and Math. Statistics, which I discuss in my other article at Some remarks in regard to D. Witztum's writings concerning the "code" in the book of Genesis. Finally, as mentioned earlier in this article, over 50 experts in Math. Statistics have signed a petition in which they denounce WRR's paper from the standpoint of Math. Science. There is no such document in existence signed by any mathematicians which would support WRR's method and conclusions.

Another argument by Drosnin, Satinover, Jeffrey, etc, is that the paper by WRR was published in a prestigious scientific journal where it was subjected to a rigorous review by experts.

I had been, for a number of years, on the Editorial board of an international scientific journal devoted to surface science. Besides, I also served as a reviewer for a number of other scientific publications. I can state with confidence that the mere fact of publication of a paper in even the most prestigious journal by no means assures the truth of the paper's claims. While papers containing obvious errors are usually rejected, in many cases an honest reviewer would not consider himself/herself the ultimate judge of the credibility of the paper's results. On several occasions, I, as a reviewer, recommended to publish some articles despite having doubts in regard to their contents. I know for fact that I was not alone with such an attitude. In 1978, an anonymous reviewer of a paper I submitted to Surface Science magazine, wrote in his (or her) review that he (or she) had doubts in regard to the physical meaning of certain kinetic coefficients in my formula. Nevertheless, that reviewer recommended the article for publication, because, as that reviewer indicated, this was a matter for discussion between the author and the readers, rather than for the reviewer's discretion. (The paper was published, and initiated some discussion in which my view finally won acceptance). The reasons for such an attitude are that most reviewers realize, first, that they can be wrong, and second, that even a paper containing errors may also contain some interesting results, some provoking ideas and some stimulating challenges. Hence, the publication of WRR's paper in Statistical Science magazine is not at all a certificate of infallibility.

The paper by WRR has already met a number of critical comments. Dr. Gil Kalai in his paper had pointed out certain features in the paper by WRR, which seem to suggest that the three authors may have used, consciously or subconsciously, certain optimization procedure that led to the desirable outcome of their experiment. In some other publications it has been shown that the data set used by WRR may not be entirely correct, etc. Those are worthy pursuits and their outcomes may serve to debunk the unfounded hypotheses in regard to the alleged "code." On the other hand, I believe that the questions, whether or not the data set used by WRR was correct or not, or whether an optimization took place or not, are not the most crucial points.

Let us assume that the data set chosen by WRR was faultless, and that their mathematical and statistical treatment was impeccable. I believe it does not matter because the paper by WRR is based, in principle, on a premise, which, in my view, is unsubstantiated.

WRR have used in their article an artificially constructed criterion of the authenticity of the codes in the form of a quantity they named proximity. I believe that the final conclusion in the paper in question, stating that (I am quoting) "the proximity of ELS with related meaning in the Book of Genesis is not due to chance" is actually an interpretation of their result by the three authors. Even if the data set used by the three authors contained no errors or arbitrary choices (this has not yet been established beyond doubts and is being disputed by a number of scientists) the only scientific conclusion could be that the quantity they introduced under the name of proximity reaches an almost extreme value when a data set chosen as the "correct" one by WRR, as well as the actual text of Genesis, are used, as compared with scrambled data sets and control texts. Anything more than that would be an interpretation open for a rebuttal.

In a scientific approach, the factual statement of result must be clearly separated from its interpretation. The authors may offer an interpretation but must not substitute it for the conclusion.

The justification WRR offered for the use of the "proximity" to distinguish between the deliberately placed "code", and coincidental ELS, is characterized by an unfounded logical jump. In the beginning of their paper, WRR discuss a situation when a text written in an unknown language is to be analyzed. They suggest that in such a text pairs of words conceptually related, for example hammer and anvil, are expected to tend to appear in "close proximity." As far as it relates to a text, with a meaningful contents, their hypothesis may seem to be fairly plausible (even if still open to rebuttal). Then, however, WRR extend their hypothesis to arrays of ELS in the Bible.

Since the words in a pair are supposed to be connected via their meaning, the described hypothesis by WRR is valid only for a) logically organized texts, and b) only if such a text is not very short. Indeed, if the text in question comprises only a few tens of letters, related words such as anvil and hammer simply may have no opportunity, spacewise, to display their tendency to appear in proximity.

Obviously neither of the above two conditions holds for ELS. All of ELS demonstrated so far do not form even simple grammatically ordered sentences, not to mention paragraphs of any length or any semantically ordered chunks of a continuos meaningful text. Even when found in "arrays" and meeting the condition of "minimal" skips, they are usually still just individual words. They appear as separate one-word islands within the framework of the grammatically ordered text. They are not connected to each other by any grammatical links. They are not arranged in any sequences that constitute segments of a continuous meaningful text.

Hence there is no reason whatsoever to assume that if ELS are products of a conscious design, then pairs of conceptually related ELS must tend to appear in a closer "proximity" than if they happen just by a random chance. Therefore, even if WRR's data set and calculations are faultless, there is no clear logical relationship between the extreme values of "proximity" and the assertion that the "codes" have been deliberately inserted into the Genesis' text by a conscious design.

By interpreting their result as they did, WRR actually attempted nothing less than to have read the mind of the alleged creator of the "code." On what grounds have WRR believed that the alleged creator of "codes" has arranged the distribution of the ELS in the text in such a way as to make their "proximity" have an extreme value? The alleged author of "codes" could have chosen any number of various ways to make his authorship recognizable, or maybe, to the contrary, to conceal it. The almost extreme values of the "proximity" found by WRR, even if they are correct, may have any number of explanations not related to any conscious design of the pattern in question. (see Some remarks in regard to D. Witztum's writings concerning the "code" in the book of Genesis).

Indeed, why should the alleged creator of codes have placed the ELS in close proximity to each other but failed to provide any grammatical link between them? Equally plausible (or, better, not more implausible) is the assumption that the alleged creator of "codes" would rather create ELS with as little "proximity" among them as possible. Indeed, as we have seen in previous sections of this article, a human mind is capable of creating arrays of ELS in any text. However, this task is relatively easy for a human mind only if the skips are relatively short. Likewise, it is easy to manually find the ELS in the text as long as one limits the search to short skips. For skips exceeding the length of one page, the task becomes rather difficult. It becomes much easier if using a computer. The lengths of skips contribute to the overall value of "proximity" calculated by WRR. Hence, would it be not more reasonable to guess that, if the alleged superhuman creator of codes wished to make his authorship recognizable (i.e. distinctive from what could be placed in the text by human writers) or if he wanted to delay the discovery of codes until the computer age, he would rather opt for creating ELS with very long skips? The requirement of "minimal skips," which is one of the components of "proximity," appears to be an arbitrary speculative choice without a logical or factual foundation.

There may be many ways in which the state of mind of the alleged creator of "codes" could be surmised. None of such guesses could though either be based on a factual evidence or logically bridged to any well-established concept.

Since there is no way to figure out the way of thinking of the alleged creator of "codes", there is no way to reasonably choose a reliable criterion of the code's authorship. Therefore even the most sophisticated mathematical calculation would not solve the controversy. On the other hand, given the ease the "code" quite similar to that in the Bible can be located also in other texts, the most likely explanation of the phenomenon remains so far that the ELS happen in the texts by chance rather than by design.

One feature of the alleged code is that all those arrays of ELS do not reveal any information that is not available without those arrays. So, what could be the possible motivation for the alleged creator of "codes" to play such a game with the Bible text?

I can envisage an argument in favor of the "proximity" being a valid criterion, as follows: does not the mere fact that the "proximity" has the extreme (actually it is an almost extreme) value for the text of the Genesis and for the "correct" data set, as compared with control texts and scrambled data sets, prove by itself that that quantity is indeed a meaningful criterion? I have a detailed discussion of that question provided in my paper at Some remarks in regard to D. Witztum's writings concerning the "code" in the book of Genesis. I may say here, that of course the "proximity" may be a meaningful criterion of something. The question is what is the meaning of that criterion. To that question there is so far no good answer.

I have not discussed here one more, quite crucial point, namely the very strange behavior of the cumulative criteria of "proximity" suggested by WRR. I discuss this behavior in my other article at Additional critical remarks in regard to D. Witztum, E. Rips, and Y. Rosenberg "code" related publications. I show in that other article that the four criteria P of "proximity" which WRR refer to as four "statistics" behave in an erratic and contradictory way, indicative of some profound fault in WRR's approach. This haphazard behavior of the four "statistics" alone negates the validity of their conclusion.

10. Conclusion

The primitive search for single ELS in the Bible, or for short phrasal constructions made up of two or three ELS (Rambsel, Jeffrey, etc) as well as the more sophisticated method of forming matrixes that contain arrays of ELS related by meaning (Witztum, Drosnin, Satinover etc) so far have not produced anything that definitely extends beyond coincidences which happen in any text of sufficient length. Neither is sufficiently convincing the alleged scientific analysis of the "code" by WRR. Of course, if some, better-substantiated proof could be found of the "code" being real, it would change the landscape of the field of the controversy in question. So far, though, it did not seem to happen.

11. Appendix 1

(Discussion of "simple" calculations of probability)

In this Appendix, the "simple" calculation of probabilty of a word to occur in a text as an ELS, employed by many a code defenders,  is being discussed. The following discussion will show that such calculations ignore certain basic rules of the Probability Theory and therefore produce meaningless numbers.

Let's say some word is a sequence of n characters x1, x2, x3.....xn.. What is the probability that this word appears, as an ELS, in a given text? The code proponents usually calculate the probability Pi of each of the characters in the word in question to be found at any arbitrarily chosen site in the text. They do it by dividing Ni - the number of times character xi is found in the entire text in question, by N - the total number of all characters in that text.

This operation implicitly assumes that the occurrences of the character in question are distributed uniformly over the text. In other words, they calculate the average probability of finding the character in question at any arbitrary site.

Unfortunately using average quantities in an improper way may often distort the results. The very concept of an average quantity is ambiguous. This quantity depends on the procedure chosen for its calculation. (A detailed analysis, including examples, of the effect the calculation procedure may have on the average values, has been given, for example, in my paper published in 1976 in Surface Technology, vol. 4, pages 538-564). In many cases, there are several average quantities for the same set of random quantities, each having different value and meaning. (This does not relate to the mean value, which is unambiguously defined in the Integral Calculus and in the Mathematical Statistics. The average value used by Cramer et al does not meet the definition of that mean value).

Let us illustrate the effect of the above implicit assumption. Imagine that we have a text describing tastes and properties of fruits. We choose a portion of that text, of a certain length, to see if there are some ELS in it. Suppose we discover that one half of the selected text discusses apples, while the other half of it deals with oranges. Obviously, in the first half word apple will be encountered more often than in the second half, while word orange will be met more often in the second half. Hence, the first half of the text will contain a larger number of consonants p and l (per unit of text) than the text as a whole, while the second half will contain more consonants r, n, and g per unit of text than the text as a whole.

Now we want to look for an ELS which spells, for example, word play. Obviously, we will find ELS for this word more often in the first half than in the second one. On the other hand, if we are looking for an ELS spelling word grain it will be found more often in the second half of the text in question. Finally, if we are looking for an ELS which spells, for example, word grapple, its occurrences in both halves of the text will be found with about the same frequency. Therefore, ELS for words grain and play, read both right to left and left to right, will be found more often with such skips' lengths that the corresponding ELS would cover not more than the length of a half of the text in question. On the other hand, ELS for word grapple, which require characters from both apple and orange, will typically have such skips' lengths that these ELS would extend over more than half of the text in question. Making a note of that fact, let us postpone the discussion of its consequences for probabilities' calculation until some later paragraph of this Appendix.

Having calculated Pi for every xi, thecode proponents usually assign value of 1 to P1 and multiply all other average probabilities Pi. Thus they supposedly calculate the average probability Pw of the word in question to appear as ELS, starting from any arbitrarily chosen site containing letter x1 , that ELS having an arbitrarily chosen particular skip.

Here we encounter another, much more profound fault in the calculation. The probability of a combination of consecutive events is calculated as a product of individual probabilities of those events. This is a common procedure in the Theory of Probability. Unfortunately, such a calculation ignores a crucial point. Multiplication of the "initial" individual probabilities is valid only if two conditions are met. These conditions are as follows:

  1. Each "test" must be performed under the same conditions, exactly reproduced for each test.
  2. All consecutive tests must be independent. This means that the results of each consecutive test must not depend on the results of the previous tests.

Condition 2 must be met always. Condition 1 must be met if one multiplies "initial' probabilities which existed before any test has been performed. If, though, one recalculates the probabilities for every test after the first one, then condition 1 may be waved, and the recalculated probabilities may be multiplied, as will become clear from the further discussion.

As we shall see, the "simple" calculation does not meet either of the above conditions.

Let us consider an example. Imagine that there are three balls in a box, one white, one black, and one red. Balls are pulled from the box in a random fashion. Obviously, each of the three balls has the same chance to be pulled from the box in the first test. The probability of that outcome is 1/3.

Now, for the second test we first restore the initial conditions, namely, put the ball, that was pulled out, back into the box. Then condition 1 will be met. Now, again, the probability for any of the three balls to be pulled out in the second tests is again 1/3. The result of the second test is independent of the result of the first test, so condition 2 is also met. Now the probability that after two tests any two specified balls have been pulled is indeed the product of 1/3 and 1/3, which is 1/9, and it is the same for all 9 possible combinations of consecutive results (white+black, white+red, black+red, black+white, red+white, red+black, white+white, black+black, red+red).

Now let us change the procedure. Namely, after each test, make sure that the ball, which happened to be pulled from the box in the first test, is somehow prohibited from taking part in the second test. (For example, that ball can be glued to the bottom of the box). It means that condition 1 is not met any longer. Under the new rules, again, in the first test, when all three balls are available for removal from the box, the probability for any one of the three balls to be pulled out is still the same as before, that is 1/3. However, for the second test, when only 2 balls out of three can be subjected to the test, the probability of any of those two balls to get out is 1/2 rather than 1/3. In Statistical Physics it is called "to impose constraints on the system." Now the probability of any combination of balls to wind up out of the box after two tests is 1/3 times 1/2 that is 1/6 instead of 1/9, and is the same for all 6 possible "events" (white+black, white+red, black+red, black+white, red+white, red+black). In this case we have recalculated the individual probabilities for the second test, thus making it possible to multiply these recalculated probabilities even though condition 1 is not met any longer.

This example illustrates the following rule: each time condition 1 is not met in the manner described, the probability of a combination of events to occur increases as compared with the situation when condition 1 is met.

Before discussing the effect of condition 2, let us see if condition 1 was met in code proponents "simple"  calculation. Obviously it was not, nevertheless they erroneously multiplied initial probabilities rather than the recalculated ones.

Indeed, as soon as the first letter x1 of the ELS has been chosen, all the sites in the text which are occupied by x1, are rendered unavailable for the next character in the word. The reason for that is that as soon as the identity of the first letter in the word has been chosen, this replaces some of the probabilities with a certainty. Namely, we are now certain that letter #2 can not occupy N1 sites, which are occupied by letter #1. We do not know which sites are occupied by letter #1, but we know how many! That number of sites become inaccessible for letter # 2.< Hence, the number of sites in the text that are accessible for the second letter x2 decreases from N to N-N1 . Consequently the average probability of the second letter to be found becomes N2/(N-N1) instead of N2/N. For the third character in the ELS in question the number of accessible sites decreases again etc. (There can be a rare exception when a certain character happens in a word twice in a row. In that case the number of accessible sites for character xi+1 decreases by 1 instead of Ni. Most of the time, though, it decreases by Ni in each step whose number is [i+1]). Hence, the probabilities must be first recalculated in order to multiply them. Cramer et al did not recalculate probabilities, and therefore their calculation had, already on that stage, generated underestimated values of probabilities.

This effect accumulates rapidly with every letter added to the ELS. In the example with the balls, rendering a ball, that had wound up outside the box, unavailable for the consequent tests, led to the increase of the probability of a given combination of balls to get out. Similarly, in the calculationwe discuss, the fact isignored that choosing a specific letter of the ELS necessarily imposed constraints on the number of accessible sites for every next letter. This oversight resulted in an underestimated probability of a certain ELS to be found in the Bible by chance. The longer is the ELS, the larger is the underestimation.

Now we shall discuss the effect of condition 2, which is much more profound than that of condition 1.

Let us consider a modified example with balls. Imagine that there are six balls in a box, one white, two black, and three red. The probability of any one of the balls to be pulled out in the first test is 1/6 for each of them. Since, though, there are different numbers of balls of each color, the probabilities of each of the three colors to show up in the first test are different, being 1/6 for white, 2/6 = 1/3 for black and 3/6 = 1/2 for red. (Colors in this example are analogs of various letters in a text).

Now let us see what happens if in the first test a white ball has been pulled out. As we want to model the "simple" calculation by code proponents, where condition 1 has not been met, we will discuss now a situation where that condition is not met either. It means that after the first test the white ball must be rendered unavailable for the second test. Then in the second test there are no more white balls. There are now two black and three red balls available, the total of 5 balls. Now the probability of any of the balls, either black or red, to get out in the second test becomes 1/5 instead of 1/6. Since the numbers of black and of red balls are 2 and 3 respectively, the probability that in the second test a black ball is chosen is now 2/5, while for red balls it is now 3/5.

Now let us see what happens if in the first test a black ball was pulled out. Note, again, that different colors in our example are analogs of different characters in the code proponents' calculation. According to the rules we agreed upon, and which reflect what code proponents routinely had done in their calculation, we must now make both black balls unavailable for the second test. Now there are only four balls (one white and three red) available for the second test. Then the probability of a white ball to be chosen in the second test becomes 1/4 (in the previous example it was 0), while the probability of a red ball to get out becomes 3/4.

Hence, we see that the probability of a red ball to be pulled out in the second test depends on the outcome of the first test. If the first ball out was white, the probability of a red one to be out in the second test is 3/5. If the first ball out was black, then the probability of a red one to be out in the second test becomes 3/4. It is an example of tests, which are not independent: outcome of every next test depends on the outcomes of the previous tests!

Now imagine a box containing thousands of balls, having 22 different colors (there are 22 characters in the Hebrew alphabet). It will be a model of a text for which the probabilities are calculated. The picture becomes much more complex, but its main features remain the same: if after each test the balls of a certain color that got out, are rendered unavailable for he next test, the tests are not independent, and the probabilities of various outcomes now form a tangled web of multiple variations. Multiplying individual probabilities in this case generates meaningless numbers as the probability of a combination of outcomes is not a single valued quantity any more, but is different for each possible sequence of results of the consecutive tests.

That is what happens in the code proponents' calculation. Indeed, the values of Ni (numbers of times a given characters happens in the text) are different for each character. (This is analogous to different numbers of balls of each color, in the box.). Therefore, depending on the outcome of a test number i, the number of sites accessible for character number (i+1) decreases in each case by a different number Ni . Therefore the probability of character number (i+1) to be encountered in the text varies depending on which characters were preceding it. Hence, the code proponents multiply probabilities, which are not independent! This is an elementary and quite crude error. There is usually a warning against such an error in introductory courses of Theory of Probabilities.

Let us reiterate our conclusion for this part. For each character the frequency of its occurrence (the value of Ni ) in the text is different. Hence the probability of each subsequent character to be encountered in the text depends on the order in which the characters follow each other in the specific word. In other words, the probabilities of different characters to occur in a text, in sequences, which constitute various words, are not independent. One of the basics of the theory of probability is that multiplying probabilities, which are interdependent, without first recalculating them, is meaningless. (The Probability theory has certain ways to handle the situation with interdependent probabilities. It is usually handled as the so called "conditional probabilities". Another way to cope with the problem of interdependent probabilities is to use Theory of Games. However, the simple multiplication of initial probabilities is a wrong way to go).

Having multiplied probabilities, which are not independent, the code proponents continue their manipulation of numbers, which at this stage of their calculation have already lost any meaning. Their next step is multiplying Pw (see its definition above) by the number N1 of occurrences of character x1 in the text. This way they supposedly calculate the average probability PA of finding the word in question in the entire text, that ELS having an arbitrarily chosen particular skip.

In the next step the code proponents multiply PA by the number z of skips they wish to explore (on different occasions they chose different z between 2 and 1000).

Now let us recall our conclusion that different words tend to form ELS with different lengths of skips. We had established earlier in this article that some words in a given text are more likely to form ELS with shorter skips (in our example those words were play and grain) while some other are more likely to have longer skips (in our example it was word grapple). Therefore simply multiplying PA by the number of skips z means one more averaging, this time again without accounting for the actual distribution of ELS over skip lengths (which of course is not known).

The last step of the "simple" calculation is multiplying the number calculated so far, by 2, since ELS can be read both from left to right and from right to left.

Once more, this operation would be valid only if the text in question were a chaotic conglomerate of characters. Actually it is a highly organized meaningful text. In such texts the probabilities of a certain sequence of letters to happen when reading from right to left is different from that when reading from left to right. Here is a simple example: in any meaningful English text, if read normally, from left to right, the sequence the will be met very often. When reading from right to left, the same sequence would be quite rare. In other words, by multiplying their number by 2, one more implicit averaging is performed, again without accounting for the actual distribution of the ELS, this time over the direction of reading.

Thus they arrive at a number which, the code proponents say, is the alleged total probability to find the word in question in the text in question, as ELS with skips ranging from 2 to z.

Besides the use of uncertain average values, piled upon each other at least three times, and besides the most egregious error of multiplying the interdependent probabilities, the above calculation also ignores some other factors. For example, it ignores the edge effect (which is the limitations on the possible skip values for sites that are close to both the end and the beginning of the text). It ignores different frequencies of occurrences of different characters = pairs or characters = triplets, etc.

The inevitable conclusion is that the values of probabilities often suggested by the code proponents are meaningless numbers. Generally speaking, they are, as a rule, grossly below the actual probabilities of various ELS to appear in the text of the Bible.

12. Appendix 2

About WRR's formula for calculation of the expected number of ELS

(Added on December 6, 1998)

As I have mentioned, I found WRR's calculation of the expected number of ELS to be flawed, but not substantially affecting their final result. I did not though elaborate as such an elaboration must necessarily involve certain concepts of mathematical statistics not commonly familiar to laymen. I have received though some e-mail messages whose senders requested an explanation what was WRR's error that I had in mind. Therefore I am adding now this Appendix 2, to elaborate on my assertion, with a warning that its understanding requires some, at least a minimal one, background in mathematical statistics.

In their paper WRR wrote (I am quoting): "This expected number equals the product of the relative frequencies (within Genesis) of the letters constituting w multiplied by the total number of all equidistant letter sequences with 2<=d<=D."

This statement means WRR assumed the numbers of letters in the tested text to be independent variables, because in sufficiently long texts the relative frequencies of letters actually equal the probabilities of encountering a specific letter at a site in the text, and probabilities can be legitimately multiplied only if they are independent. WRR's assumption would be correct only for a perfectly random conglomerate of letters but it was wrong for a meaningful text and it was equally wrong for any permutation of that meaningful text.   In the meaningful text and in all of its permutations the letters are not independent since the stock of letters available to fill a site is limited to the factual letter set of the text (in the case of Genesis this set comprises 78064 letters). As soon as a specific letter x has been used to fill a site, the stock of available letters loses that x.  This situation meets the condition of what is called in Math. Statistics "test without replacement," and the pertinent distribution of letters in this case is hypergeometric. WRR have implicitly used instead a multinomial distribution which pertains to a case of "tests with replacement." Their approach was principally wrong. However, the quantitative difference between the results obtained using the correct, hypergeometric distribution and those obtained using the faulty, multinomial distribution, in the case in point would be very small. One of the reasons for that is that the words' lengths are very small as compared with the total text's length. One more reason making WRR's error practically inconsequential is that their choice of the number of ELS to be 10 was completely arbitrary, and therefore if their faulty calculation led actually to a number different from 10 they chose, for example making it 11 or 9, it would hardly matter for the further treatment by WRR. I made my comment in order to dispute the assertion that, first, no scientists have found any errors in WRR's work, and, second, to show that an argument making use of WRR's reputation is irrelevant, as neither Rips, nor Witztum nor Rosenberg is an expert in mathematical statistics. (Neither am I, but noticing the described faulty assumption by WRR did no require one to be a real expert in math. statistics, as the error was both obvious and elementary).

13. Endnote

After the first version of this article had been posted on the Internet, I received several replies. Some of them supported my conclusions, while some other opposed them. Notably, the negative responses came mainly from people who claimed to have religious faith. (It does not mean that all the positive responses came only from atheists or agnostics). One of those religious respondents asserted that she is a believer because there are so many codes in the Bible obviously inserted there by God.

In my view, religious people have no more reason to believe that God inserted the codes into the Bible, than the atheists or agnostics have, and may be even less. Indeed, anybody who, like Rambsel, Jeffrey, Cramer & Eldridge, etc, believes that God had deliberately created the codes in question, actually believes in a very strange God. God, who, as they are supposed to believe, gave Moses the commandments, who created galaxies, supernovas, black holes, quarks, neutrinos, etc, who is omnipresent, omnipotent, and perfect in every sense of the word, this God, if we believe Rambsel's attribution of "codes" to him, has not even mastered the grammar of Hebrew! Rambsel's and Jeffrey's "God" appears to be rather tongue-tied, and even half-witted, as all this strange "God" managed to produce was a number of ambiguous phrases with obscure meaning, allegedly encoded in a manner much less sophisticated than a variety of human-generated ciphers and codes.

Comments are welcome ( marperak@cox.net)

M. Perakhís personal page: http://members.cox.net/marperak/

The readers who wish to explore the alleged "code" in the Bible themselves may order a computer program for that purpose, for example at http://members.xoom.com/codefinder/index.htm. I have not tested these programs myself and therefore cannot provide a judgement as to their quality, but at the above site there are posted comments of some people who had tried the programs in question.

Originally posted to Mark Perakh's B-Codes page.