Some remarks in regard to D.Witztum's writings concerning the "code" in the Book of Genesis

By Mark Perakh

First posted on April 27, 1998

Last update - July 31, 1998


  1. Introduction
  2. General remarks in regard to D. Witztum's writings
  3. About analogies as a tool of discussion
  4. Remarks about WRR's experimental procedure
  5. Discussion of the table of "ranks" in WRR's paper
  6. Alternative explanations of the table of "ranks"
  7. Final remarks
  8. Conclusion
  9. Appendix (Would an alternative "straightforward" method be preferrable to the permutations of the data list?)


The paper by D.Witztum, E. Rips, and Y. Rosenberg (WRR) was published, in 1994, in the journal Statistical Science. The three co-authors had, reportedly, contributed different parts to that paper. Apparently, the role of Y. Rosenberg was limited to the development of a computer program (or programs) used in WRR's experiments. Indeed, besides being indicated as a co-author of the paper in question, and of some subsequent papers (which have not appeared in print) Mr. Rosenberg did not seem to publicly offer opinions in the discussion that followed the publication of the paper by WRR. Dr. E. Rips, was, reportedly, the main force in developing the mathematical algorithm for the estimation of the quantity WRR introduced under the name of "proximity" (and of what they denoted "c-value"). WRR considered these quantities to be the criteria enabling one to distinguish between the God-inserted "code" and accidentally appearing ELS. Dr. Rips has expressed certain views and opinions in response to criticisms of the WRR' work. Dr. Rips stressed, though, that the main contribution to the "code" research was made by D. Witztum. D. Witztum, in a paper printed in the Jewish Action magazine, also asserted himself to be the main contributor to the "code" project. Therefore, everybody involved to some extent in the discussions of the Torah "code" was interested to hear from Mr. Witztum in regard to the critical remarks directed at WRR's theory.

Some of the papers in which D. Witztum has offered his rebuttals of criticisms of WRR's work and, in his turn, countered his detractors with his own criticisms, have been gathered in a Web site (see ) which seems to be connected in some way to AishHaTorah organization  and to its seminar's Senior lecturer Rabbi D. Mechanic. Some additional material by D. Witztum can be found in the Jewish Action magazine. The Web site in question had been established by, as they have themselves defined it (I am quoting): "friends and supporters of Doron Witztum's research into the hidden codes in Genesis in order to provide a forum for him to answer some of the questions and criticisms which have arisen during the past three years. There are a great number of misconceptions which have been perpetuated about this research, and the general public is unaware as to what is fact or what is fiction." (End of quote).

As it has been indicated in the web page in question, Mr. D. Witztum had written all that material in Hebrew, so the items in that web page are translations into English of D. Witztum's original writings. Since I will be referring in this article only to the English version of D. Witztum's material, I have to assume that the translation faithfully represents the authentic writings by D. Witztum.

General remarks in regard to D. Witztum's writings

If the English text is the faithful reproduction of the original, then one may conclude that D. Witztum has a way with words. His writing is quite eloquent, displays a rich vocabulary and the ability to clearly express his views and opinions. Moreover, one can see that D. Witztum has quite a nimble mind and a good grasp of details and of their relations to the overall picture.

On the other hand, regretfully, D. Witztum, rather than concentrate only on the substance of criticisms directed at WRR's publications, devoted considerable effort to remarks of rather personal character. Some of these remarks are of a general type, like dividing the people involved in the "code" controversy, into two groups, whose unnamed participants, in D. Witztum's view, all have some not kosher motivations, etc. Some other remarks by D. Witztum are more direct, for example, accusing certain opponents of his views in lying, etc. I have no intention to discuss in detail these remarks by D. Witztum, because, first, I do not wish to engage in irrelevant matters which have no bearing on the controversy in question, and second because I am sure the targets of D. Witztum's personal negative references are perfectly capable of defending themselves should they choose to do so. Of course, I fully expect to also become a target of some personal unflattering comments on the part of some "pro-code" people. Indeed, as soon as my earlier article was posted (see ), I received an unsigned e-mail message from Aish Ha Torah organization in which its author said that they, at Aish Ha Torah, decided not to read my paper because it "is meaningless and intellectually dishonest." I wonder, how could the writer (or writers) of that message have formed an opinion of a paper, which they did not read?

I believe that airing doubts about moral standards of one's opponents is counter-productive. It is also demeaning first and most for those who indulge in such personal remarks, and more so if expressions like "intellectually dishonest," or "meaningless" are employed in a general way. We all will be better off concentrating only on the essence of the controversy. The best way of discussion would be to weigh all the evidence in a dispassionate way. Unfortunately, the matter has become too emotional, hence the dispute necessarily will become quite heated from time to time. It can be acceptable, with regret, as long as no personal insults and hints are being circulated.

In this paper my goal is to show that, despite the detailed rebuttals forwarded by D. Witztum's in defense of his views, D. Witztum's theory is, in my view, flawed both at the very fundamental level, and in its experimental implementation. Therefore, in my view, the conclusions of WRR in regard to the "code" in the book of Genesis have not been sufficiently substantiated.

About analogies as a tool of discussion

In one of his papers (see ) D. Witztum uses an analogy to explain the alleged vast difference between his own, scientifically impeccable research, and the allegedly faulty attempts by Dr. McKay, Dr. D. Bar-Natan, and others, to show that in certain non-Biblical texts the "proximity" as defined by WRR, can have "ranks" as extreme as in the text of the Genesis. In that analogy, D. Witztum compares the search for ELS in texts to a bombardment of a target.  In a "good" bombardment (if we resort to the use of sucn an oxymoron) the bombs land in close proximity to a target. Similarly, in a "good" search for ELS (such as the one allegedly employed by WRR's method) the conceptually related ELS are found within a small segment of a text, and are in close proximity to each other. In a faulty search for ELS (such as allegedly performed by Dr. B. McKay et al) the conceptually related ELS are found within a chunk of text much larger than that in WRR's experiment, which, according to D. Witztum, is analogous to a "bad" bombardment where the bombs land over an area much larger than the target's close vicinity.

Analogy is akin to modeling. Modeling is quite common in Physics and other sciences. Without using a model, many problems would be too complex to crack. Here are a few examples. After J. Kepler had formulated, on the base of the empirical evidence, his three laws of planetary motion, Newton set out to derive equations which would explain Kepler's laws from a single postulate, namely from his universal law of gravitation. To do so, Newton had to adopt a model of a planet. His model of a planet was simplicity itself. It was a point mass. Was it a good model? Surely it was. This model had only one common characteristic with the object it represented, namely the planet's mass. All other properties of a planet, whose number is enormously large, were ignored in the model. This allowed Newton to simplify the problem to the extent making the calculation feasible. All those other properties of a planet could be ignored as long as only the Kepler's laws were to be derived. How did Newton make his choice of the properties to be ignored and the properties to be accounted for (in his case, only the planet's mass)? He did it based mainly on his intuition.

The choice of a model is art. The talent of being able to choose a good model is what distinguishes a good scientist from a bad one. Let us see what is the model of a planet, more specifically of the earth, in some other physical problem. In the science of seismometry, which deals with the propagation of seismic waves in the crust of the earth, the model of the earth is an elastic half-space. It has only three properties, namely Modulus of Elasticity, Sheer modulus, and Poisson coefficient. All other properties of the earth are ignored in that model (including the planet's mass!) Is it a good model? Yes, it is good, because it simplifies the problem to the extent making the calculation feasible, while the results of that calculation are reasonably close to the actual characteristics of wave propagation.

So, we have here two models of the same object, earth, which do not seem to have anything in common, while representing the same object but in different problems. What makes both models good? In both cases, when choosing a model, its creators guessed successfully which properties of the planet are important for the problem to be solved, and which are secondary and could safely be ignored.

What if the models were chosen differently? For example, what if in calculating planetary motion, Newton were to choose a model of the planet as that of an elastic half-space? It is after all also a model of earth. Obviously, it would be a bad model as it would account for a property of no importance for the problem, but ignore the property which is crucial. Equally, a model of earth as of a point mass would be completely inadequate for the calculation of seismic wave propagation.

Models in Physics are usually those of objects. Analogy is also a model, but usually that of a situation. Analogy can also be good and bad. A good analogy considers a substitute situation (model) which retains all the crucial features of the actual situation, and ignores its features of secondary importance. Such substitution is designed to simplify the situation under scrutiny to the extent enabling one to think through all of its implications and possible outcomes. A bad analogy retains secondary features of the actual situation but ignores its important features. It is useless and often simply misleading.

The analogy used by D. Witztum substitutes, for the situation with ELS in a text, another situation, that of a bombardment of a target. This is a good example of a bad analogy. The situation with the bombardment of a target is superficially similar to that of locating ELS in a text, as it seems to have some common features with the latter. However, those common features are the secondary ones and have no bearing on the comparison between the results of Witztum's vs McKay's searches for ELS. At least one crucial feature is different in the two situations. Analyzing that crucial feature reveals the fundamental oversight in Witztum's reasoning which undermines his arguments.

When the situation with bombing a target is considered, there is an indisputable criterion of what is to be considered a successful bombing. In a "good" bombardment  bombs are expected to hit locations which are in closest proximity to the target. D. Witztum believes that his analogy is good because in the search for ELS the successful outcome is to locate pairs of ELS at minimal distances from each other, in a portion of the text as small as possible.

This is the fundamental flaw in Witztum's reasoning.

Assume that there was a superhuman creator of "code." How could WRR know the specific thoughts of that creator of code in regard to the ELS placement in the text of the book of Genesis? There is in Isaiah 55:8 the following passage: "My thoughts are not your thoughts, Neither are your ways My ways, saith the Lord. For as the heavens are higher than the earth, so are My ways higher than your ways." (Quoted from The Holy Scriptures according to the Masoretic text, The Menora Press, Chicago, page 362). This saying seems to be relevant to my question, especially since this question is directed to D. Witztum.

What criterion may we choose to distinguish between the deliberately inserted "code" and the ELS occurring by chance? This is the crucial question. Before this question has been answered, all the subsequent actions, such as developing a mathematical algorithm for calculating the chosen criterion, or choosing the proper list of appellations and corresponding dates of birth/death, etc, are aimless exercises. In the bombing analogy, this criterion is easily defined as the "proximity" of hits to the target. There is no such simple criterion for the problem of a "code" in a text.

Witztum and his colleagues, if they wanted to employ a scientific approach to the problem they faced, should have first discussed the question as to how to choose a criterion. Instead, they simply accepted that, if the ELS had been deliberately inserted into the Torah by design, then pairs of conceptually related ELS must be located in close proximity to each other. They have never discussed this choice, as if such a choice of a criterion was self-evident. As they did not discuss it, introducing it instead as something self-evident, they of course did not provide any justification for their choice. In my view, this oversight invalidates their subsequent actions, rendering immaterial their discussions which famous Rabbis had to be included in their list, how many permutations of the data list to perform, etc, etc.

I can foresee the following counter-argument by WRR. They may claim that they encountered by accident pairs of conceptually related ELS, being situated close to each other. Impressed by that discovery, they designed an experiment to investigate the phenomenon statistically. The results of such an experiment proved that indeed pairs of conceptually related ELS appear in the book of Genesis in close proximity to each other. From that they concluded that the effect is "not due to chance."

Maybe WRR are satisfied with such a justification, but in my view it runs against logic. It does not matter in which order to place the notions. It still means that they assumed, even if after the fact, that the alleged creator of the "code" for some reason decided to place the related ELS close to each other. WRR did not offer any justification for such a hypothesis, either on logical, or factual, or religious grounds.

Remarks about WRR's experimental procedure

As I have mentioned, WRR did not provide any justification for why the extreme values of the "proximity" (or certain values of what they denoted "c-value") should indicate the superhuman origin of the "code." I submit that this omission alone seriously undermines the entire presentation by WRR.

I further submit that such a justification would encounter very serious, perhaps insurmountable difficulties.

Indeed, let us look at the method used by WRR to test their hypothesis. This method involved multiple permutations of the list of "names vs dates" and calculating the value of "proximity" (or "closeness") for each permutation. I submit that there is a serious argument against the validity of all those numbers for statistics P1, P2, P3 and P4 calculated by WRR.

This argumentation will be offered in the following sections. First, though, let us take a look at the experimental procedure employed by WRR.

WRR have performed the statistical verification of their results using two types of control texts. Out of the total of six control texts, four were permuted versions of the text of the book of Genesis. Two more control texts were taken from sources other than the book of Genesis. One of those two texts was taken from the book of Isaiah, and the other from a Hebrew translation of L. Tolstoy's War and Peace. Testing all six control texts alongside with the actual text of the book of Genesis, WRR checked in each case one million of "appellations vs dates" lists. One list was the "correct" one and 999999 were permuted versions of that list where the "names vs dates" pairs were deliberately mismatched.

WRR have presented their results as a table of "ranks" assigned to every combination of each text with 999999 permutations of "names vs dates" pairs.  The idea of "ranking" can be explained as follows. For every text, including the actual text of Genesis, all of its permutations, and also two non-Genesis control texts, the "proximity" of pairs "name vs date" was calculated one million times. This calculation was repeated using four different formulas (referred to as statistics P1, P2, P3 and P4). Each time it was done both for the "correct" list and for all the scrambled lists. Then, for every text, the ascending order of the "proximity" values was determined for all lists, including the correct one and all the scrambled ones. If, for example, for a text X it was found that the correct list of dates produced proximity whose value occupied position number r in the ascending series of proximities, then this text was assigned rank r . In other words, if a specific text has rank r, it means that there are, among the explored texts, r-1 texts in which the "name-date" pairs are, on the average, situated  closer to each other than in the text ranked r. Hence, for each text, four ranks have been determined, by using the four formulas suggested. The lower is the value r of the rank, the "better" is the overall proximity of the "appellation-date" pair in the given text.

The list of appellations contained 32 personalities juxtaposed to their dates of birth and/or of death. The number of all possible permutations of 32 names is an expression with 36 digits. As mentioned before, out of this multitude of permutations, WRR chose at random a subset of 999999 permutations.

One of the permuted versions of the Genesis text was created by permuting all the letters in the Genesis (control text R), the second one by permuting all the words in the Genesis without permuting letters within the words (control text W), one more permuting the verses without permuting letters within the verses (control text V), and finally one more permuting words within verses without permuting the letters within each word and the verses themselves (control text U). (It is obvious that the listed permutations actually produce not four but only one type of the control text. The R-type permutations produce all possible permutations of 78064 letters constituting the text of Genesis, including the versions resulting from W, V, and U-types of permutations. Hence, all four permuted texts were actually just four particular permutations of the R-text. Hence the classification of control texts by WRR into four allegedly different types hardly had any consequence). .

Apart from discussing the results of the tests presented by WRR, which were based on the use of permutations of the data lists, we can also consider an alternative way of investigating the texts.  This alternative way is discussed in the Appendix to this article.

Discussion of the table of ranks in WRR's paper

Look at the table of ranks reported by WRR. While the values of "proximity" can be expected to be nearly extreme for the actual text (see the Appendix) they have no a priori reason to be sensitive to the permutations of pairs of related ELS. Therefore, the reported by WRR almost extreme ranks of the lists they considered to be "correct," among all the scrambled lists of name-date pairs, is not explained by the considerations offered in the Appendix. To account for the composition of the table of ranks, we will have to look for a different way to interpret the reported results.

WRR, on the base of their table of ranks, concluded that  the close proximity of conceptually related ELS does not happen by chance. In the subsequent comments, including D. Witztum's writings, the above conclusion had been elaborated by asserting that the "code" had been designed by God.

I believe this conclusion was premature. Before offering such a notion, WRR, if they wanted to adhere to scientific standards, must have first explored all possible explanations not requiring extra-rational arguments. This requirement has nothing to do with the scientists' personal beliefs. Many outstanding scientists had been and are religious people and it is not in any way contrary to their being also very cautious in seeking an extra-rational explanation for results of a research. Finding a rational explanation in no way diminishes genuine faith as religious people see God's design in all things that can be explained rationally. As to the table of ranks in WRR's work, it creates more questions than answers.

Let us divide these questions into two categories. First consider those questions which seem to be so crucial that the absence of an attempt to find answers to them on the part of WRR cast serious doubts in regard to WRR's overall attitude to the verification of their results.

1. The first question arises immediately when taking the first glance at the table of "ranks" in WRR's paper. The ranks for the text of Genesis were found to be, for four "statistics" used, between 4 and 570 (out of one million). Recall that if for a certain text  the "correct" list of appellation/dates has rank 4 it means that there are only 3 permuted lists (out of one million) that produce "closer" proximity" for the text in question. Hence, the rank found for the actual text of the Book of Genesis turned out to be close to the minimal value among all one million of data list permutations. This was construed by WRR as the indication of "close proximity" not being due to chance. Indeed, for all the control texts the ranks were quite far away from the minimal values. Among the six control texts, there was the Book of Isaiah. Its ranks, found for four statistics, were between 899830 and 946261, which were the worst results of all six control texts. In other words, the test proved to be a complete failure for the Book of Isaiah, where, unlike the Book of Genesis, the criterion chosen by WRR showed the complete absence of the "proximity" between the dates and appellations on WRR's list. This fact obviously could not be left unnoticed by the people who conducted their experiment. This fact must have evoked an inordinate curiosity on the part of the experimentalists. Strangely, this did not seem to happen.

The driving force of science is curiosity. I find it very hard to imagine that WRR simply ignored the exceptionally poor performance of the Book of Isaiah as if it was some insignificant detail. Generally speaking, even small details of the experimental results are normally subjected to a thorough scrutiny. In the case under discussion, the Isaiah outcome was not an insignificant detail, but a result that related to the very core of the investigation. How could the three researchers not to get curious about it and not to set out to study it deeper?

The natural step would be first of all to apply the method they developed to the four other books of the Torah, and to see if in those four books the "proximity" will again be nearly extreme for the actual list of Rabbis as compared with the scrambled lists. Instead, WRR hurried to form a conclusion, based only on the tests of Genesis plus 6 control texts.

There can be imagined two possibilities. One is that WRR announced their conclusion to the world and simply ignored the Isaiah result, as something inconsequential. In this case, they displayed the lack of curiosity quite amazing for scientists. The other assumption can be that WRR had noticed, and explored the strange result, and had actually applied their method to the books of the Torah other than the Genesis, but withheld the results of such exploration.

Fortunately, some other scientists have filled the void left by WRR's paper. Dr. A.M. Hasofer, who is a Professor of Mathematical Statistics, performed the experiment, precisely reproducing WRR's method, on the books of the Torah other than the Book of Genesis. Prof. Hasofer, who, also, is a religious Jew, used the same list of appellations and dates as WRR did. He used the program provided by WRR. For the Book of Genesis, Prof. Hasofer (who used only 2 out of four "statistics" suggested by WRR) found the "ranks" to be 2 and 716, which is in agreement with WRR's result for the book of Genesis, as these ranks were quite close to the minimal possible values of ranks among one million. (The difference between the exact values of ranks for Genesis in Hasofer's and WRR's tables was due to the use by Hasofer of an updated program provided by WRR). If the results for Genesis are to be trusted, then also those for the other four books of the Torah have to be trusted to the same extent. A. Hasofer found for the other four books of the Torah the rank's values between 135735 and 947381. Hence, the experiment utterly failed to show any "close proximity" of "appellations vs dates" pairs for the books of Exodus, Leviticus, Numbers, and Deuteronomy.

Comment. Let us take a look at the recent "new claims" about the Torah code, namely at the results reported by Dr. A. Rotenberg at A. Rotenberg indicated that the resullts allegedly proving the existence of a "code" comprising the names of the 10 sons of Haman, their date of deaths, and the words denoting the Purim holiday, improved when he extended his analysis from the Book of Genesis to the entire Torah. In view of the failure to find a "close proximity" according to WRR, between ELS for names and dates of the "famous Rabbis,"  in four books of the Torah other than Genesis, this statement by A. Rotenberg sounds quite contradictory, and hints at his results being just a concidence without a statistical meaning.

A few sub-questions seem to be in order. Did WRR try their method on any of the four books in question? It would take quite a short time and a minimal effort to do so, as they had the program ready, the text of the Torah ready, and the computer working. If they did not, it would be an incomprehensible omission on the part of researchers pretending to approach the problem in an unbiased way and being interested only in facts. If they did, why did they not report the results?

It seems worth to note that it would be quite natural for WRR to test at least the book of Leviticus, probably even before the book of Genesis. I base this statement on the following information. The book by Dr. J. Satinover provides a history of WRR's approach to the Torah "code" problem. Supporters of WRR, in particular Rabbi D. Mechanic, have only nice words in regard to J. Satinover's book. So far, neither WRR nor their supporters, who quite vigorously denounce the book by M. Drosnin, have ever contradicted J. Satinover's account. J. Satinover himself claimed to get all that information directly from WRR. According to J. Satinover's account, out of the three authors of the paper in the Statistical Science, Dr. E. Rips was the first to get involved with the "codes". As J. Satinover tells us, Dr. E. Rips was approached by one Abraham Oren who asked Rips to investigate ELS that spelled the name of Aharon (in Hebrew four letters Alef, Hey, Resh, Nun) which Oren found in the book of Leviticus. That was, according to J. Satinover, the beginning of E. Rips involvement with the "code." The first discoveries of clusters of ELS in the Torah made by E. Rips were the occurrences of word Aharon with a frequency that, according to E. Rips, considerably exceeded the mathematically expected one.

In view of that history, is it not highly unusual that WRR, if not to start with the book of Leviticus, at least should have studied it along with the book of Genesis? Today we know that, if WRR started with Leviticus, their result would be negative. Or would it?

As many critics of WRR have indicated, the list of "appellations vs dates" used by WRR had many variables that could be altered at will. Could the list in question be compiled in a slightly different way, so it would ensure a positive result with the book of Leviticus? Of course, then the experiment would fail in Genesis.

In his writings, D. Witztum has not uttered a single word which would provide an explanation in regard to WRR's failure to investigate the four books of the Torah other than the book of Genesis.

Comment: WRR very strongly reject some remarks by their critics, in which the critics imply that WRR could, maybe inadvertently, slant their data list to get the desired result. I sympathize with WRR in this respect, as such a suspicion is highly destructive to any scientist's reputation and should cause a lot of distress. I prefer to believe that WRR have performed their study in a perfectly honest way. They must admit though that, by failing to investigate the other four books of the Torah, and more so in view of the test's subsequent failure with the other four books, WRR have exposed themselves to those, maybe unfair, but understandable doubts.

2. Besides question 1, another, concomitant question seems to be: if the "code" is real, why did the alleged creator of the "code" insert it into the text of the Genesis, but not in the rest of the Torah? This question is quite natural if one remembers that, according to the tradition in Judaism, whose adherents E. Rips and D. Witztum claim to be, the Torah was given on Mount Sinai to Moses as one piece, without being divided into words, sentences, or books. The division occurred later, even though it is not known when and in which circumstances. The boundaries between the five books of Moses have no absolute meaning, and some of those boundaries had been subjects of various interpretations by prominent Rabbis (for example, by Vilna Gaon). Then why is there such a vast difference between the "proximity" in the book of Genesis, and the other four books? I can hypothesize that WRR could reply that they could not know the reasons the God chose to do or not to do something. Recall though that WRR claimed to know that God placed the conceptually related ELS in close proximity to each other. To decide that God chose to behave in such a way, a way not dictated by a transparent logic, and without any evident reason or aim for that, seemed to be easy for WRR. So, why would WRR lose that uncanny ability to know what God wanted to do with ELS in the Genesis, when the other four books are considered?

In his writings, D. Witztum has provided no comments in regard to the possible reasons for the difference in "proximities" between the book of Genesis and the four other books of the Torah.

3. A question similar to 1, even though not as crucial, is why books of the Tanakh other than Isaiah, were not investigated? Again, the natural curiosity of a scientist should pose this question as well. Many examples of ELS found in various books of Tanakh have been demonstrated. Of course, WRR have stressed more than once, that the occurrences of either individual ELS or arrays of ELS have no meaning, as only the proper statistical analysis like that applied to the "famous rabbis" is valid. On the other hand, the book by J. Satinover contains many examples of those arrays, which have not been subjected to the statistical analysis according toWRR, but have never been repudiated by WRR. Moreover, since the tests in which WRR's method was used, failed in Leviticus, then what about those "Aharon" ELS which appear in that book in numbers exceeding the mathematical expectations? So, while WRR and their supporters (like Rabbi D. Mechanic) do not spare harsh words to denounce the communications by various people about individual ELS or clusters of ELS found in the Bible, and words like "nonsense", "meaningless" and the like are being employed casually in those repudiations, they remain silent about similar discoveries by D.Witztum or E. Rips themselves.

In his writings, D. Witztum has not said anything in regard to WRR's failure to investigate books of the Tanakh other than Isaiah.

4. One more question is as follows: if the "code" was inserted into the text of Genesis by design, then would we not expect that it should work in a non-statistical way? As a corollary to that question, why is the rank of the text of Genesis not 1? I could hypothesize a few possible answers to that question, but WRR did not raise it at all.

In D. Witztum writings there is not a single word in regard to the values of ranks in question.

Now, there are many other questions which had to be at least discussed. The questions in this category are of a different character as compared with the 4 above listed ones. These additional questions relate to the experimental routine employed by WRR. If WRR's results were more or less within the domains of generally expected behavior, the questions in this category could be left to rest. However, given the extraordinary behavior of the texts under study, in this case these additional questions had better to be addressed, first of all to satisfy the researchers themselves. Here are a few of such questions:

1. Will the poor performance of the Isaiah text persist for its permuted versions? (Comment: normally, this question would not be of a great importance, and the answer to it could be guessed with a good degree of certainty. However, given the odd behavior of the texts under study, in particular the astounding difference between the texts of Genesis and those of the other four books of the Torah, this question acquired the stature of an additional tool, not to verify any established rules of a statistical experiment, but to recheck their own experimental procedure by the researchers, to see if any step of it was not done in some undetected erroneous way).

  1. How will the text of War and Peace perform with alternative sets of 999999 permutations of the data list? (The same comment as for question 1).
  2. How will a permuted text of War and Peace perform? (The same comment as for question 1 ).
  3. Will the effect persist with alternative R-type permuted versions of the Genesis text? (The same comment as for question 1).

Even if all the missing information corresponding to the above questions had been collected, it still might not be sufficient to make reliable conclusions. To make conclusions before obtaining answers to the above (as well as maybe to some other) questions was so premature as to deprive the alleged conclusion of scientific significance.

The driving force of science is curiosity. (Or have I already said it before? Well, repetition is the mother of study - it is a Russian adage). I find it hard to imagine how researchers who obtained the described strange table of ranks would not be curious enough to try finding answers to all above questions, unless their subconscious agenda was to confirm a preconceived view. All the above questions must have intrigued researchers who obtained such unexpected and strange results, if they strove for a reliable understanding of their observations. WRR chose to announce their conclusions while the experiment was stopped before all the necessary information which could have been extracted by varying and expanding the experimental conditions, had been collected. During the years after the experiment was stopped in the middle of the way, WRR expanded their experimentation by exploring other data lists and adding more results of the same type they obtained in the original experiment rather than to try to answer the questions that remained unanswered in the initial experiments.

D. Witztum, in his writings, has never said anything that would justify WRR's failure to properly complete their experiment.

I have not touched here on one more, rather crucial item, namely the bizarre behavior of the four "statistics" suggested by WRR as cumulative measures of the "proximity" of conceptually related ELS. This item is discussed in my other paper at Additional critical remarks in regard to D. Witztum, E. Rips, and Y. Rosenberg "code" related publications. In that paper I submit the opinion, based on the analysis of the data reported by WRR, that the contradictory, haphazard behavior of the four statistics P is negating any validity of WRR's conclusions, and is indicative of some profound fault in WRR's approach.

Alternative explanaions of the table of "ranks

There can be many alternative explanations for the table of ranks offered by WRR.

The first explanation that comes to mind is that the table of ranks was obtained in a way that is contrary to the established rules of statistical research. Indeed, the article by a prominent expert in Mathematical Statistics, Prof. A.M. Hasofer, which has so far been circulated only as a preprint, provides a strong indication that the above suspicion has a solid foundation. The paper by Dr. A. M. Hasofer can be reviewed in this web page at A statistical critique of Witztum et al paper, where it has been placed with its author's kind permission. Dr. Hasofer's article requires for its understanding a certain background in Probability Theory and in Math. Statistics. I will provide here some points of Dr. Hasofer's expert article, trying to make them more or less comprehensible for laymen. Those readers, however, who wish to really develop a sense of Dr. Hasofer's powerful argumentation, are recommended to work through A. M. Hasofer's piece.

One of the faults of WRR's research, as shown by Prof. A. Hasofer, is WRR' failure to set an "alternative hypothesis" which is a necessary condition for a reliable statistical analysis. To set only a "null hypothesis" as WRR did, is not sufficient for the results to warrant a statistically sound conclusion.

Prof. A. Hasofer indicated a number of other faults in WRR's approach. In particular, Prof. A. M. Hasofer has thoroughly analyzed the fatal results of WRR' switching to the data lists permutations.

A.M. Hasofer indicated that WRR had improperly applied the concept of "chance" to the situation when only a unique object (the book of Genesis) was available for the analysis. Their assumption "had no frequency interpretation." Therefore, concluded Prof. A.M. Hasofer, the final conclusion by WRR in regard to the "close proximity" as happening "not due to chance," was in their context a meaningless statement.

Then, WRR had illegally applied an assumption of "equiprobability" to the items in their sample space. Prof. A.M. Hasofer indicated that, contrary to the usual routine, in WRR's case, no "symmetry argument" could be used. This comment can be understood even by readers who have no knowledge at all of Math statistics. Let us look at it.

In the data list compiled by WRR, most of the Rabbis have more than one appellation; the dates of birth/death also are given usually in more than one version; the lengths of appellations also vary. The effect of the above variations in the number of appellations and dates can be seen from the following example. Let us consider any two entries in the data list. Assume that personality X is listed with three possible appellations A1 , A2 , and A3 (for example, one of the appellations may be only the last name, the other a nickname, etc) while the dates are given in 2 versions D1 and D2 (for example one is the date of birth, the other of death, etc) etc). Then personality X contributes 6 "correct" pairs of data to the data list. Let assume Personality Y is listed with two possible appellations A3 and A4 and with only one date D4 . Then personality Y contributes 2 "correct" pairs of data to the data list. Both personalities X and Y together contribute 8 "correct" data pairs to the data list. Among the 32! possible permutations of the data list, there is one in which appellations for personality X are "matched" to dates for personality Y, and appellations for Y are "matched" to dates for X. In this combination 3 appellations for X are combined with 1 date for Y, while 2 appellations for Y are combined with 2 dates for X. Then the total number of "mismatched" pairs contributed by both X and Y to this scrambled list of data is 7 rather than 8. Hence, each scrambled list of data consists of different numbers of entries. The described situation is exacerbated even more because of various lengths of the appellations for various personalities (this affecting the "equiprobabilty" hypothesis). Hence, when the appellations/dates list is permuted, to mismatch the pairs, it creates asymmetric combinations. Prof. Hasofer then indicated that the absence of a valid symmetry interpretation is a common source of an "anthropomorphic illusion" when a statistically significant result seems to exist which has no basis in reality.

One more comment by Prof. A.M. Hasofer relates to WRR's choice of the measure of the "distance" between the ELS. Prof. A.M.Hasofer has convincingly showed that the choice made by WRR led to the values of "distances" between ELS which often defied common sense. Prof. Hasofer offered examples when the distance between two obviously close ELS, when calculated by WRR's method, yielded larger values that the one calculated for two ELS which were situated obviously much farther from each other. (Similar criticism of the "distance" measure has been offered by another prominent mathematician, Prof. Barry Simon of Caltech). Prof. A.M. Hasofer provided a number of other critical comments, which are too technical to be discussed in this article but which are well founded in Mathematical Statistics. A.M. Hasofer concluded finally that WRR's work did not meet the requirements of Mathematical Statistics and therefore WRR's conclusion had no scientific significance.

The views expressed by Prof. A.M. Hasofer, are well in agreement with the opinions of 45 prominent specialists in Math, Statistics and Probability Theory (see

Finally, as it is discussed in my other article at Additional critical remarks in regard to D. Witztum, E. Rips, and Y. Rosenberg "code" related publications, the contradictory behavior of the four cumulative measures P1, P2, P3, and P4 of "proximity" suggested by WRR can alone explain the strange values of "ranks" they reported.

In D. Witztum's writings there is not a single word in regard to the faults of WRR's research analyzed by A.M. Hasofer, which WRR must have foreseen themselves if they wished their work to meet the standards of a statistical approach. Neither has D. Witztum mentioned anywhere the reasons for the haphazard behavior of the four measures of "proximity" suggested by WRR.

Comment. <Consider again the recent "new claims" by A. Rotenberg in regard to the alleged "code" in the book of Genesis, that "code" comprising the names and the date of deaths of 10 sons of Haman etc. ( The work done by A. Rotenberg is characterized, as well as that by WRR, by his ignoring the basic rules of a statistical test. There is no mention in A. Rotenberg's page of a null hypothesis, or of alternative hypotheses, or of the sample space, or of the power of test, or of the critical region, etc, which are necessary components of a legitimate statistical test. This fact alone invalidates any conclusions one may try to derive from the tables of "c-value" and P1 and P2 statistics provided by A. Rotenberg. >

One more point is that, since the values of "proximity" are expected to be nearly extreme for the actual text (see the "entropy" argument in the Appendix to this article) alternative criteria other than "proximity" must have also been explored. Until such work has been conducted, the conclusion derived by WRR remains just their preferred interpretation of results, which has no sufficient justification.

In D. Witztum's writings there is nothing to justify their choice of only one criterion, not to mention the doubtful foundations for the use of the only criterion they chose.

There are also serious doubts about the WRR's choice of appellations and corresponding dates which have been discussed at length elsewhere (see, for example so that I do not need to expand on that matter here.

In publications other than the paper in the journal Statistical Science, WRR's defenders often provide values of probabilities for the effects they claimed to have observed. These values may be impressive for laymen, but scientists who dealt with statistical evaluation of their experimental data know that the value of probability itself is the least reliable criterion. It may be useful in certain circumstances. For example, if in an experiment a linear dependence is suspected between two quantities, and a least square fit is applied, then the correlation coefficient is a convenient measure to test the probability of the assumption of linearity. Still, it is just an auxiliary a posteriori measure, as probability has no power of prediction. In the case of text's analysis the cognitive value of probability is extremely limited. For example, look again at the case of Aharon ELS discussed earlier in this article. J. Satinover quoted E. Ripps who estimated the probability of 25 ELS for Aharon to appear in the pertinent paragraphs of Leviticus, and that probability was found to be very small. However, in view of the negative result for "close proximity" found for Leviticus by WRR's full-fledged method, that earlier calculated probability turns out to be of no value as a proof. There is an endless list of examples of events whose probability is extremely small, but which happen anyway.

I am not discussing here the exchange of arguments between WRR, on the one hand, and their opponents, on the other, in regard to choosing the list of names, or the spellings of those names, etc, as my argumentation lies in a different plane. There is however, one point which, in my view, weighs in favor of WRR's opponents. I believe that WRR on the one hand, and their opponents, on the other, face different levels of requirements to prove their cases. WRR try to prove a certain phenomenon and to do so they have to be extremely careful in choosing their lists of appellations, the dates, etc, The burden of proof is theirs to show that the alleged phenomenon is real and not an artifact of some sloppy experimental procedures. Their opponents, on the other hand, are not constrained in their choice of appellations, dates, etc. They do not need to prove that the phenomenon in question exists in, say, War and Peace. Given the statistical nature of the alleged effect, there is no need to adhere in those debunking experiments to lists of names or of dates that would meet some stringent criterion. It is sufficient for their task to show that a phenomenon similar to that allegedly discovered by WRR in the book of Genesis, can be also found in some other, non-Biblical text, and to do it for some lists of names and dates, not necessarily having any historical foundation. Actually if it is shown for any arbitrarily compiled list of fictitious names and dates, it would serve the purpose of disproving the claims by WRR as well.  Therefore, when opponents of WRR pinpoint imprecise data in WRR's work, it is a legitimate argument. On the other hand, when WRR pinpoint "imprecise" data in the examples of their opponents, it is inconsequential.

Overall, I consider the critical remarks in regard to WRR's experimental routines to be more convincing than WRR's counter rebuttals.

Final remarks

I submit that probably no quantitative criterion enabling one to distinguish between the God-inserted "code" and randomly occurring ELS can be plausibly postulated. I see at least two reasons for that.

One reason is as follows. If we assume that a superhuman creator had indeed inserted a "code" into the Scriptures, we still have no way to know if that creator wanted indeed to place all those conceptually related ELS in close proximity to each other. Why would the alleged creator of "codes" make such a choice? What could be the aim of such a choice?

If a "code" has been inserted, then we may expect that it should carry a message. The non-coded text of the Torah is indeed carrying a message. It is such a message, which has been fascinating millions of people for over two thousand years without any signs of this fascination to subside. There are people who accept this message as the word of God. There are people who deny it or doubt it. But there hardly are many people, at least within a half of the world population, who are indifferent to that message.

What about the alleged "code?" What message does it convey? None! The alleged "code" does not constitute any logically and grammatically arranged text. WRR themselves admit that the "code" does not reveal anything unknown without it. Then what could be the reason for the "creator" of the "code" to place conceptually related ELS close to each other if it does not convey any message anyway? If there was a creator of the "code," why would he not choose to place the related ELS, say, at equal distances distributed uniformly over the text? Or in such a way, that those ELS form certain patterns like triangles, squares, etc. Or, say, as far from each other as possible, and with skips as long as possible? The choice of "proximity" as a proof of the superhuman origin of ELS arrays is a postulate, which has no foundation either in faith, or in logic, or in facts.

Since we have no way to determine what must be the choice of the alleged creator of "codes" as to in what manner to place them in the text, there is no way to choose a mathematical criterion to distinguish between the "code" and the randomly occurring ELS.

The second reason making, in my opinion, the choice of the criterion in question hardly feasible, is the high probability that the chosen criterion has an extreme value for the actual text as compared with permuted "texts", or with completely different, both Biblical and non-Biblical texts, simply because of the unique or the almost unique standing of the actual text among all the permuted versions, or because of the specific composition of a particularly chosen non-Biblical text. The extreme values of "proximity" reported by WRR, even if they are correct, could as well be attributed to that low-entropy nature of the actual text vs. most of its permutations, or, as with the War and Peace and Isaiah experiments, to accidental compositions of the control texts. Therefore the values of "proximity" cannot serve as a criterion to prove that the "code" has been inserted into the Bible by design.

In this article, as well as in the two other articles on this Web site, my goal was not to prove the absence of a "code" in the Bible, but rather to show that the existence of such a "code" has so far not been proven by its proponents. If the "code" is indeed there, it requires a much more rigorous proof.


My conclusions are that 1) The experiment by WRR was stopped before its logical completion, leaving many questions unanswered; 2) WRR failed to substantiate their choice of the criterion used to differentiate between a deliberately designed "code" and randomly occurring arrays of ELS; 3) because of the above indicated omission alone, WRR's results and assertions are not sufficiently substantiated, and 4) D. Witztum, by failing to address the crucial question of the criterion's choice, as well as many other critical comments, has so far failed to justify his position and his argumentation.


Would an alternative "straightforward" approach be preferrable to the permutations of the data list?

At the time of writing this article (April 1998), I was teaching the course of Statistical Physics at California State university. In my class I had both undergraduate Physics majors and Graduate students in the Master program. Of course, in the syllabus for that course, there was no mention of the "code" in the Torah. However, some of my students happened to see this web page, and asked question about the "code." This way, the "code" controversy came up for discussion. I explained the method used by WRR, which involved the permutations of the data list (i.e. appellations vs dates list). Some students asked, why did WRR choose such a convoluted way to compare "proximities?" Would it be not more natural as well as simpler to compare the actual text of the book of Genesis with its 999999 permuted versions, sticking to only one list of names/dates?

My students, without realizing this, had actually hit one of the most vulnerable points in WRR work. WRR, as it has been shown in the body of this article, switched from the method we have just called "straightforward" to the more convoluted one, reportedly on the advice of a reviewer at the journal to which they submitted the first version of their paper. Lest I will be misunderstood, I am not criticizing the suggestion by the reviewer (if there indeed was such a suggestion) per se. The use of a data list permutations was of course a clever (and not unusual) way to treat the problem by making the condition of the "tests" close (even if not identical) to the model of "independent" tests. What I am criticizing is how WRR had applied that method. To use the permutations of data list properly, WRR had to choose the same number of appellations and the same number of corresponding dates for every personality One way to do it would be to perform an averaging of all appellations and dates versions for each personality on the list (as suggested by Dr. B. Simon in a private communication). Then each entry would have only one appellation and one date. (Reportedly the reviewer suggested some similar manipulation of the data list). WRR failed to do so. What had actually happened as a result, was the replacement of the initial "null hypothesis" by another "null hypothesis" without first adjusting the data list as mentioned. This caused the fatal destruction of the integrity of WRR' s statistical analysis, as it is discussed in the body of this article. The initial "null hypothesis," with all its related actions, at least did not contradict basic rules of a statistical analysis. The new "null hypothesis," which corresponded to the data lists permutations approach, with its accompanying effects, when applied to the list with varying numbers of appellations and dates for different personalities, resulted in a violation of basic rules of Math. Statistics (see the body of this article).

I guess that the realization of the fatal faults inherent in WRR's "data lists permutations" implementation, may lead to suggestions that WRR would be better off sticking to the initial "null hypothesis" and to the use of texts permutations rather than those of data lists. (Indeed, such suggestions have been voiced on the Web, so far only in the form of private communications).  In this section I intend to show that the "straightforward" method, even though it would not be contrary to the fundamentals of Statistical science, has its own, probably even worse problems.

The totality of all the possible permutations of a text constitutes a statistical ensemble. In the language of Mathematical statistics, this ensemble fills up the so called sample space. The text of the Genesis contains 78064 letters. The number of possible permutations of these letters is enormously large. To understand how immensely large that number is, note that if the total number of letters in some Hebrew text is only 100 (so that every letter is encountered in that text, on the average, between four and five times) the number of possible permutations is an expression with 118 digits. For the text of Genesis this number is vastly larger.

There is also another possible way to choose the sample space. Rather than to define it as comprising all the possible permutations of the actual text, it can be chosen to comprise all the possible combinations of 78064 letters, of which there are 22 versions. This sample space would hold even more "texts" (this number would be 22 to power of 78064) of which the overwhelming majority would be meaningless collections of letters, and which would now encompass not only all possible permutations of the actual text, but also all possible control texts.

(WRR had though used in their 1994 paper a sample space containing 32! permutations of the data list).

Within the original sample space, containing the "texts," there is a hierarchy of degrees of order among the "texts". The actual text of the book is highly organized, since it is arranged conceptually and grammatically. In the parlance of the Information Theory, the actual text of the book, which has a high degree of order, has a low value of entropy. Various "texts" in the sample space possess various degrees of order, from the complete randomness (when they have a high value of entropy) to a rather high degree of order (and, correspondingly, lower values of entropy). Among these "texts" there are some with a higher degree of order than the actual text. For example, a higher degree of order has a "text" where all "alefs" are collected together, followed by all "bets" lumped together, then by all "gimels" bunched together, etc. However, the number of "texts" with a higher degree of order as compared with the actual text, is much smaller than the number of "texts" with a lower degree of order. The reason for that is that the higher is the degree of order (the lower the entropy value) the less probable is the occurrence of a "text" with such value of entropy. Since the actual text has a high degree of order (low entropy) it is highly likely that its entropy is close to the bottom of the variety of degrees of order among the "texts". The number of possible "texts" that possess specific values of entropy increases exponentially along with the increase in the value of entropy. Hence, it is highly probable that any of the control texts has a much larger entropy than the actual text of the Genesis.

Here are some quantitative estimates. The entropy of  a fully randomized English text, as calculated by C. Shannon, is 4.76 bits per character  if spaces are counted as characters, and 4.7 bits per character if the text is stripped of spaces. A calculation analogous to that of C. Shannon, gives for a fully randomized Hebrew text stripped of spaces (which comprises 22 different characters rather than 26 as in English) the value entropy of about 4.46 bits per character. For a real English text, C. Shannon estimated empirically the entropy to be about 1 bit per character. There is no such data for Hebrew texts. An estimate can be made if we assume a certain value for the redundancy of a Hebrew text.  Since Hebrew texts contain no vowels, their redundancy is considerably smaller than that of English texts.   For the latter the accepted value of redundancy is 75%. We can then roughly estimate the redundancy of Hebrew to be close to 50%. The relationship between the redundancy R (as a fraction) and entropy S is as follows: R=1-S*/S, where S* is the entropy of the real text, and S is that of a fully randomized text. Then we find that the estimated entropy of a real Hebrew text is close to S*=2.23 bits per character. It means that the probability that a permuted text happens to be fully randomized is about ten times larger than the probability for a permuted text to happen with a degree of order close to that of the real text. Besides the fully randomized version, there are among the permuted texts a multitude of versions partially randomized, whose probabilities are also to various extent larger than that for versions with degrees of order close to the real text. The probability of a permuted version to have a degree of order larger that that for the real text (i.e. a smaller value of entropy) is quite small (but not zero, of course). Therefore the permutations with higher entropy than that of the real text necessarily are prevalent in the sample space, while permutations with a lower entropy are relatively rare.

The data by C. Shannon, namely the value of entropy of real Englsh texts to be about 1 bit per character, were obtained by Shannon empirically in tests performed on relatively short segments of texts. As Dr. McKay (see the following comment) has unearthed patterns of order extending over the entire text  it seems possible to assume that the entropy of real texts is actually even less than Shannon's estimate. Then the difference in probabilities of permuted texts to either be randomized to a large extent or to have a degree of order close to the real text, is even larger than we estimated.

Comment: The above considerations have received recently a direct experimental confirmation. (The following information in this comment is based on a private communication from Dr. B. McKay). Dr. McKay has performed a series of experiments in which he investigated certain regularities in the structure of real texts in their comparison with randomized texts. In particular, Dr. McKay studied the following features of texts: 1) Correlations between the letters situated close to each other (for example, in English letter q is usually followed by letter u, etc). 2) The uneven distribution of letters across the entire text; 3. Non-even distribution of letters within the sentences; 4. A correlation between letters occupying certain positions in one word and letters occupying the same position, or different, but fixed, position, in another, closely situated word (for example between the first letter in one word and the first letter in another word, or between the first letter in one word, and the last letter in another word, etc). 5. Variations in letters frequencies between left and right halves of verses in the Bible (and a similar phenomenon in non-Biblical texts) as compared with randomized "texts".

In all explored real texts, both in the Bible, and in non-biblical texts, Dr. McKay had found all the described effects to be present and to be well pronounced. All these effects disappeared when the texts were randomized. In his communications in regard to the above unpredicted effects Dr. McKay never used word "entropy." However, it is obvious that all these effects are manifestations of various types of order in real texts, which usually disappear in randomized texts. Presence of any types of order means lower entropy as compared to randomized texts where those types of order are absent. The considerations, preceding this comment, about the relatively low entropy of real texts among the multitude of all "texts" in the sample space, had general character and could not specifically predict what forms of order distinguish real texts from randomized versions, except for a general suggestion that some forms of order must exist there and cause the entropy of real texts to be low. The results reported by Dr. McKay reveal some specific forms of order that exist indeed in real texts, but not in the randomized versions.  The discovery of the mentioned types of order provides a direct proof of the validity of the notion about real texts having low entropy as compared with the majority of "texts" in the sample space.

Since D. Witztum, as we have seen, is in favor of analogies, we will also use one. Recall the concept of the Thermodynamic entropy. Entropy in the Information Theory was given, in the classic paper of 1948 by C. Shannon, (following some earlier hypotheses about information entropy) its name not by accident but because its behavior is analogous in many respects to its Thermodynamic namesake.

In a statistical ensemble of systems, there is one system, which had the largest value of entropy. Then, for this "most probable" system all other thermodynamic potentials also have extreme values. For example, if the entropy of a system has maximum, then its free energy has minimum. Every thermodynamic potential, starting with the most commonly used ones, such as Gibbs and Helmholtz potentials, and extending to the endless variety of combinations of the basic potentials, has an extreme value for the "most probable" system. Every "function of state" of which there is unlimited multitude, is tied to the thermodynamic potentials and therefore all of these functions also have extreme values for the "most probable system" in the ensemble, which, again, is the system that has the extreme value of entropy.

The entropy in the Information Theory behaves in a similar way. The actual text is the one with a low value of entropy, somewhere close to the bottom of the variety of entropy values among all the "texts". If for a certain system the value of entropy is close to the extreme value over the ensemble, then the multitude of other possible functions reflecting various properties of the text, very likely also have, for that system, values that are close to either minimum or maximum. This of course relates also to the "proximity" suggested by WRR

Therefore any quantity mathematically connected to the "text," including either the "proximity" calculated by WRR, or their "c-value, " is expected, quite likely, to have a value close to the extreme one over the ensemble. In some cases, even though not very often, there may be a text among the ones chosen for the test whose entropy is lower than for the actual text, and then, for that rare situation, the value of "proximity" for that permutation would happen to be even closer to the extreme one than for the actual text

Hence, the following conclusions can be deduced: a) it is natural and expected that the "proximity" as calculated by WRR's formula, has quite often an almost extreme value for the actual text of the Genesis. B) It would be rare for that value to be far from the extreme one. C) Its almost extreme value by itself has no evidentiary quality whatsoever in regard to the distinction between the "code" inserted by design and the ELS occurring at random. D) Any meaningful quantitative characteristic chosen as a criterion to test for a "code" versus random ELS, must behave analogously to functions of state in Thermodynamic systems. Therefore it quite often will turn out to have an almost extreme value for the actual text as compared with the control texts. There is no basis whatsoever to assume that the "proximity" is an exception.

If one chooses a criterion to distinguish between the "code" and the random arrays of ELS, one must prove that the chosen quantity's extreme value cannot be attributed simply to the low value of entropy for the actual text. Of course such a test had not been performed for the "proximity," arbitrarily used by WRR

Hence, I believe the above discussion has answered my students' question. If we were to use the "straightforward" comparison between the values of proximity for the actual text vs. its permuted versions the results of such procedure would hardly provide a reliable tool to distinguish between the random occurrences of ELS and a deliberately designed code.

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