Squaring Off with Bible Code Skeptics

How the Mathematicians’ Statement is Mistaken

By Ed Sherman

Dr. Barry Simon is the IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology (Caltech), where he is also Executive Officer (chairman) of the Mathematics Department. He has spearheaded the circulation of a petition providing a “Mathematicians’ Statement on the Bible Codes.” By the end of May 2000 this petition included the names of 54 Ph.D.s in either Mathematics or Statistics.

The Statement makes the following assertions:
  • “The almost unanimous opinion of those in the scientific world who have studied the question is that the theory is without foundation.”
  • The WRR study “suffers from major problems concerning both its execution and the interpretation of its conclusions.”
  • “A vastly more systematic and thorough investigation” would be needed to support any substance to the codes.
  • “Word clusters…will be found in any text of similar length. All claims of incredible probabilities for such clusters are bogus.”

Skeptics are like arteries. They can either be healthy or hardened. Healthy skeptics can serve many constructive purposes in society. They can help us to question things that on the surface may appear valid or true and perhaps point out why something seemingly credible is not. Their observations can refine and enhance the quality of scientific investigations into various matters. That is all very well and very good.

Hardened skeptics, on the other hand, are a regressive influence on society. They can close minds that otherwise would be open and stifle inquiry that should not be actively discouraged.

The Mathematicians’ Statement is, without question, a succinct expression of skepticism. Is it, however, an expression of healthy or hardened skepticism? It would take only a small number of changes in form and wording to transform the petition from a document that appears to bear the marks of an unhealthy skepticism to one that could be accepted as an expression of healthy skepticism -- even by the most ardent of code supporters. Is there a willingness to improve the petition’s wording in order to remove such questions? We will be contacting Dr. Simon to explore that possibility.

Surprising Leaps of Faith?

An explicit purpose of the statement is to testify to the “fact” that the scientific community is almost unanimous in its denial of any substance in the phenomenon of Bible codes. In attempting to do so, its author(s) may have employed one or more surprising leaps of faith in the name of science. We will delve into the awkwardness of these leaps in this and at least one subsequent issue of the Digest.

That a rush to judgment has apparently been slapped onto a clearly embryonic field of inquiry by “open-minded” scientists is quite disturbing. Furthermore, the presence of 54 Ph.D.s on such a statement carries with it a sense of intimidation against any scientist who would dare to take a different position. Such an atmosphere is clearly not conducive to the dispassionate search for truth. This could have easily been avoided by toning down the petition’s language to more accurate statements. It would then represent a truly respectable document worthy of the distinguished scholars who have signed it.

The Petition makes the very bold statement that “the almost unanimous opinion of those in the scientific world who have studied the question is that the theory is without foundation.” Was a scientifically-designed poll of scientists taken in order to arrive at this conclusion? If so, why weren’t the specifics of the poll published rather than implicitly suggesting such a possibility within the petition? One gets the impression that the real situation is that the petitioners are of the opinion that their opinion is the almost unanimous opinion. Perhaps they arrived at this conclusion by quickly checking around among their like-minded peers? How statistically valid a poll would that be? Such a strong statement should be backed up by some sound sampling. Wouldn’t this be especially true since the tenor of their remaining assertions is that it is critical that everything should adhere to high standards of statistical testing? Or are the petitioners guilty of falling well short of such standards and of thereby doing the very thing that they accuse other code researchers of doing?

Wording Could Be Improved

If no statistically sound poll of scientists was taken, we would suggest that the petition’s statement on this matter be changed as follows:

CURRENT WORDING: “On the contrary, the almost unanimous opinion of those in the scientific world who have studied the question is that the theory is without foundation.”

SUGGESTED WORDING: “It is the belief of the undersigned that the great majority of those in the scientific world who have studied the question are of the opinion that the theory is without foundation.”

ALTERNATIVE SUGGESTED WORDING: “It is the opinion of the undersigned, after studying the question, that the theory is without foundation.”

A quick scan of the Statement reveals a major difference between it and the nature of most petitions: it carries no date as to when its wording was finalized. Furthermore, though the petition includes 54 names (as of May 31, 2000), no date of signature is shown in conjunction with any signatory. Is this omission an oversight? Or was it intentional? If it is the latter, it would be a strong, implicit statement that its conclusions are final and utterly conclusive. It will never need updating based on the examination of new evidence. If so, is this based on the notion that the research done to date has been extensive enough to be a clearly representative sample of all the possible research that could be conducted on Bible codes? Certainly not. If anything, the petition claims that the quality of the research done to date has been lacking. If that is true, then the only proper conclusion to be drawn is that the whole matter has not been appropriately researched and tested. Therefore, it is much too early to attempt to put a damper on this area of investigation and inquiry.

Bible Codes Too Recent a Field of Inquiry

Is the petitioners’ apparent rush to judgment due to some kind of a priori reasoning? If it is, the statement does not reveal this as the basis for such finality. It would be most helpful if such reasoning would be disclosed.

Since the study of Bible codes is a very recent field of inquiry, it is only fair to ask how such certainty could possibly have been reached in so incredibly short a period of time. Could it be that this statement is as premature as the Catholic church’s medieval dictum that the earth is the center of the universe because no clear evidence to the contrary had yet emerged?

Throughout the ages, scientific theories have come and gone, or have at least been modified, as new evidence has been discovered and/or new experiments have been conducted. How is this different? What if someone were to discover a word for word equivalent in Hebrew of the Gettysburg Address as one continuous ELS with a skip of +3 in the Book of Deuteronomy? That kind of ELS might be 500 letters long. How many names would drop off Dr. Simon’s petition if that occurred? Or has the final verdict been rendered, never to be appealed?

In all fairness, the statement does contain elements that allude to the possibility (presumably highly remote) that it could be subject to future revision. It states that “the work so far” [emphasis added] has not “established a prima facie case.”


The Statement also refers to the conducting of “a vastly more systematic and thorough investigation” as the kind of event that would be necessary to cause a change in position by most of the petitioners. WE WOULD RECOMMEND THAT THE ADVERB “VASTLY” BE REPLACED BY “SIGNIFICANTLY.” The adverb “vastly” bears an implication that has the same feel as what it would be like to add leaping over the Empire State Building as a new event in the next Olympics. Is the “vastly” adverb an indication of a degree of hardness to the skepticism expressed in the petition? Or is it an expression of frustration on the part of the petitioners that all of the time they have devoted to investigating this topic has been, in their, opinion, a waste of time?

Imagine a group of paleontologists in the 1700s issuing a joint statement to the effect that no fossils would ever be found substantiating the existence of any life forms that have since become extinct? Making such a statement would be a bold and risky venture. Whoever made it would be subjecting their reputations to the possibility that some new turn of the shovel would unearth the bones of a creature quite unlike any alive today. All it would take to transform that kind of fossil scientist into an endangered species would be for someone to unearth a pile of bones from one Tyrannosaurus Rex. Is this perhaps analogous to the Mathematicians’ Statement on the Bible Codes? Is there a need for this bold Statement to be refined by adding some qualifiers?

Orthodox Judaism

From the point of view of orthodox Judaism, there is a theory that has been widely held for a long time. It is based on the belief that God dictated the entire Torah to Moses, letter by letter, without spaces. Many who hold to this belief also believe that the whole history of the world is encoded within the Torah. Obviously, for this to be true, it would be necessary that the entire Torah be utterly thick with codes. Is this “the theory” that Dr. Simon is referring to? It would appear so.

The “simple Simon” view of things is that they are either totally black or white. No ELSs in the Bible are real or most of them are intentional, and they are everywhere dense in the text. Therefore, we can test the latter theory by conducting only a small number of experiments. We do that by pre-selecting a topic and a set of words judged to be relevant to that topic, as well as a section of text. We search for codes and compare what we find with what we would expect to be there due entirely to chance. If there isn’t a significant difference between actual code appearances and expected, the Bible code theory is bogus. If there are substantial differences, we have evidence that the theory has merit.

What if reality is more complicated than that? Suppose there is a third alternative—that if any codes are real, they are fairly rare. That, in fact, is the finding of Dr. Randall Ingermanson in his book, Who Wrote the Bible Code? If this theoretical physicist is right, the simple Simon approach to testing the hypothesis that some Bible codes might be real could easily produce a false result. The experiment will indicate there are no real codes when some might exist.

A more realistic view of the range of possible theories would be that we have these options:

  1. No real Bible codes exist.
  2. Real Bible codes are fairly rare.
  3. Real Bible codes are fairly common but are not dense within the text.
  4. Real Bible codes are everywhere dense within the text.

As we saw in the discussion above, our assumptions about which group of these options could be true will affect the type of tests we would include in our experiment. If we are not careful we can fall into the trap of false presumption that typifies code skeptics today. They design and conduct experiments that implicitly based on the notion that only 1 or 4 could be true.

If we are really open minded, however, we should also admit that 2 and 3 may be distinct possibilities. That kind of admission results in some serious inconvenience, however. If either 2 or 3 could be true, we will not have the luxury of only performing a relatively small number of experiments to determine whether or not some Bible codes could be real. In that event, even if all of our experiments produce negative results, we would not have any assurance that our painstakingly designed and implemented experiments had produced the correct result—as long as that result is negative.

If Bible codes are rare, our pre-selected set of topics/words/sections of text may have simply not included those codes that are real. This is a very frustrating circumstance for scientists whose goal is to dispense with any theory purporting that some Bible codes are real. Therefore, it is better not to mention this so that the well designed experiments will yield the desired conclusion that no codes are real, and the whole controversy can be efficiently brought to a pleasing end.


The petitioners would have us believe that the experiments conducted prior to their petition are sufficiently thorough to lead us to a decision as to whether Bible codes are real. That’s implied in their gambit of concluding that these past experiments have not had convincingly positive results, and so, there is no substance to the purported phenomenon. This is a grotesque line of implicit reasoning, however.

Since the main tack of the skeptics is to point out deficiencies in past experiments, we should consider the possibility that all they may have demonstrated is that past experiments have been deficient in either their design, implementation or interpretation, or any or all of these. And, if they succeed in proving their contention, all we really can conclude is that past experiments have been flawed, and therefore do not provide clear evidence that real codes do not exist.

The problem is that that conclusion doesn’t help us answer the ultimate question of whether some yet-to-be-discovered-and/or-analyzed Bible codes are real. All we would then know is that we don’t know much of anything. It would be much more honest intellectually to issue a statement to the effect that, to date, no investigations have yielded any convincing evidence that some Bible codes are real. Unfortunately, the Mathematician’s Statement makes giant leaps beyond that kind of pronouncement. Therein is demonstrated the awkward faith of determined skeptics.

A more objective and realistic picture of the range of possibilities regarding the nature of code experiments and cluster examples that have appeared to date in print would include the following:

  1. Published examples are representative of the types of Bible codes that are yet to be discovered.
  2. Published examples are much more improbable than most undiscovered codes.
  3. Published examples are much less improbable than many undiscovered codes.

Without doing much more research, how could anyone have any basis for deciding which of these alternatives was the correct one?

As if the situation were not already complicated enough, there is an additional difficulty. How could we recognize a real code if we had indeed encountered one? Doing this would presumably involve finding some way to calculate the probability that it could appear by chance. Yet one of the assertions set forth by Professor Simon is that calculations of that type do not conform to the rules of probability and statistics. So we are dead in the water as far as being able to recognize the real thing if it ever showed up -- if Professor Simon’s inference is correct.

Daunting Challenge

As a professional mathematician, I could not make the leap from the notion that past calculations of probabilities were done incorrectly to the implied conclusion that there isn’t any way to do it correctly. In fact, this situation served as a daunting challenge. Shouldn’t there be some way to do such calculations properly?

Even if this were an unsolvable problem, it seems that there must be some way to make relative distinctions between codes, with some being assessed as more probable and others as much less so. It ought to be possible to create a scale, or a rating system, much as exists in the field of geology, for the relative hardness of rocks. We know that diamonds are much harder than chalk. Geologists have derived tests and scales to boil that down to some kind of a number rating. The same type of thing should be done for Bible codes.

One would intuitively expect that there should be a range of possibilities regarding any given code (or cluster of codes). They ought to fall into one of these buckets:

  • Chance ELSs, which are likely to appear coincidentally. These would be of the type that could easily be found in almost any book.
  • Unusual ELSs, which are fairly improbable. Finding a similar example in another book is quite possible, but not very likely.
  • Intentional ELSs, which are extremely improbable. So improbable are such examples that it is unlikely that they could be found in any book.

To make real progress in breaking the ideological deadlock that surrounds Bible codes, developing concise tools that can clearly distinguish between these types of codes would be most beneficial. If we don’t have the tools to rank and test different codes, we end up lumping them all into one vast category. This leaves us with the choice of either accepting or rejecting all ELSs—lock, stock and barrel. Mankind would then be doomed to vague thinking about the controversy. Hunches become embellished with intuitively appealing, simplistic arguments. Quick leaps are made to hoped-for conclusions. Hunches become entrenched after being cloaked in appealing thought garb. Intellectual innuendoes haunt what could be a habitat of science.

In reality, precisely calculating the probability that a given cluster could appear by chance is enormously complex. The approach presented in Ed Sherman’s book, Breakthrough, doesn’t fully solve this problem. However, what it does do is to provide a reasonable upper bound for that probability. If we can say that we know that the likelihood of chance appearance is definitely smaller than, say, 1 in a trillion times a trillion, then we don’t really need to know what the real, smaller probability is. We will already have reached a clear-cut conclusion: the cluster is not a coincidence. And that is all that is really important.

A Review of Dr. Ingermanson’s Critique
of BCD Methodology

By Ed Sherman

This article provides our responses to Dr. Randall Ingermanson’s criticisms of the BCD method of calculating the combined probability of chance occurrence of a cluster of ELSs. In essence what he is saying is that our approach is inherently wrong.

When two experts differ so sharply, it is usually because of major differences between the assumptions each has made. And that is the case here.

In our opinion, Randy’s conclusions are based on intuitively appealing but incorrect assumptions about the nature of ELSs. ELSs are very strange animals that behave quite differently than what one would naturally expect. Secondly, his criticisms are out-of-date because they are based on a much earlier version of Ed’s book, Breakthrough. Consequently, Randy’s criticisms were targeted against an approach that we haven’t used for almost two years. We significantly modified that initial methodology to effectively eliminate the kinds of problems that Randy is concerned about.

In his newsletter, Dr. Ingermanson stated that “there is now an attempt to factor out wiggle room by multiplying by a large factor that accounts for some of the freedom in choosing words to search for.” That isn’t what we do. We don’t apply one huge factor to the product of all the naïve probabilities. We adjust each naïve probability (of chance occurrence of each ELS) to reflect the possibility of having searched for far more ELSs of like kind than we actually looked for, and then we multiply the probabilities together. This difference between Randy’s understanding of what we do and what we actually do is quite critical.

Subtle Distinction

In his review, Randy stated that our approach “doesn’t really deal with the main methodological problem. It isn’t valid to just multiply all those probabilities together in the first place. Probabilities are always less than 1. When you multiply them together, the product rapidly approaches zero.”

While Randy’s concern is quite valid if the probabilities are spread somewhat uniformly between 0% and 100%, it is not true if most of the probabilities are close to 100%. For example, if you multiply 0.999 times itself 1,000 times, you get 36.8%, which isn’t close to zero—even though it is the product of 1,000 probabilities. This subtle distinction makes all the difference in whether or not Randy’s assessment of our methodology has any validity.

Case in point: If you take the product of the adjusted probabilities of random occurrence of all 1,133 ELSs in the Isaiah 53 cluster that weren’t included in our comparison chart with the Hanukah cluster, that product is 39.319%. So it just isn’t true that merely multiplying together a large number of probabilities will result in a number near zero. When we apply our method to most examples in published books, we end up with cluster probabilities that are typically quite unimpressive. Many of them are greater than 50%.

Applying Our Method to a Model Cluster

We also applied our method to a model cluster of over 2,000 ELSs where each ELS appears exactly as many times as it would be expected by chance to appear. The resulting probability was 43.6%. That’s hardly close to zero—even though it was calculated as the product of over 2,000 probabilities.

There are two reasons why most of the adjusted probabilities that we multiply together tend to be close to 100%. First, we heavily adjust the initial probabilities for wiggle room. The nature of this adjustment strongly forces most of them to be quite close to 100%.

Example: The odds of the exact ELS, “It will be understood, Jesus created,” crossing the key section of text in Isaiah 52-53 are 1 in 1,247 (0.08%). However, we adjust that probability to be that of finding, not only that exact ELS, but also any one of 13,367 alternative ELSs with approximately the same chances of appearing by coincidence. That adjusts the probability of chance appearance to 99.998%. Similarly, the odds of the exact ELS, “Jesus the Gift is Master and My Lord,” appearing by chance are adjusted from 1 in 107,000 to 1 in 3. Those are huge adjustments. We have deliberately designed this adjustment process so that it clearly overdoes things. We did that so that the final probability we have calculated will be greater than what the true probability is.

A Second Reason

There is a second reason why most of the probabilities tend to be close to 100%. Most of the initial probabilities already tend to bunch near the higher end of the 0% to 100% range. Why? Short ELSs typically appear numerous times. So naturally the probability of showing up by chance is near 100%. Long ELSs usually can’t be found because their probability of showing up is near 0%. [At the end of this article we provide a short discussion for this strange behavior.] So the probabilities for the bulk of ELSs a researcher looks for will, of necessity, be quite high. [FYI: for the Isaiah 53 cluster the average probability of all events is 97.6%.] Randy’s simple model where the average probability is 50% has no relevance to the analysis of ELS clusters.

The reality is that the thing that causes some of our cluster probabilities to be so astronomically small is that several quite unlikely events converged. In the case of the Isaiah 53 cluster, numerous lengthy codes on one topic cut through only two pages of Hebrew text.

An analogy: You observe two long lost friends running into one another. Conclusion: coincidence. You observe 10 pairs of long lost friends running into one another at the same hotel on the same day. Conclusion: It was planned. They call it a reunion.

Case in point: Someone finds one or two 10 letter long ELSs near one another. Conclusion: coincidence. We find 19 ELSs that are each 10 or more letters long (with many in the range of 18 – 22 letters in length) crossing Isaiah 52-53. Conclusion: Chance isn’t that capable.

Finally, as mentioned above, we performed a reasonableness test of how the BCD approach behaves. We modeled what would happen if we did a search for a typical list of ELSs and every one appeared exactly as many times as it was expected to by chance. In doing so, we simplified things slightly -- by assuming that every Hebrew letter appears just as often as every other one. That way we didn’t have to worry about variations caused by rare or common letters being part of any ELS.

Realistic Simulation

In our reasonableness test we realistically simulated what we actually do in a cluster search. A 1,000 letter long section of text (about two pages) is selected and we look for 100 different ELSs with 3 letters, 100 different ELSs with 4 letters, 100 different ELSs with 5 letters, 100 different ELSs with 6 letters and 100 different ELSs with 7 letters. We look for all occurrences with skips of 1 to 100. Each 3 letter ELS is expected to appear 20.68 times, each 4 letter ELS 0.98 times and each 5 letter ELS 0.0466 times. Since we are looking for 100 different ELSs of each letter length, if everything goes according to chance we would expect to find 2,068 total occurrences of the 3 letter ELSs, 98 occurrences of the 4 letter ELSs, 5 occurrences of the 5 letter ELSs and no occurrences of the 6 and 7 letter ELSs. So we have a pile of 2,171 ELS occurrences. Fully 95.3% of them (the 3 letter ELSs) have a probability of chance occurrence of 99.99999989557%. Another 4.5% of them (the 4 letter ELSs) each have a probability of chance occurrence of 62.6%. We assumed 80 of them were single word ELSs and 18 were two word ELSs. And 0.23% of them (the 5 letter ELSs) each have a probability of chance occurrence of 4.56%. We assumed that two of these are single word ELSs and three are two word ELSs.

Notice how the probabilities are distributed: over 95% are essentially 100%; another 4.5% of them are about 63%; and a smidgen (0.23%) are a bit below 5%. We are talking about a circumstance totally different than one where the probabilities are evenly distributed between 0% and 100%. Now we apply our procedure (in exactly the same way we did when gauging the Isaiah 53 cluster) to calculate the combined probability of chance occurrence of this cluster of 2,171 ELSs. The result? 43.6%! So even though we have multiplied together over 2,000 adjusted probabilities, our result is 43.6%. That’s hardly close to zero. Conclusion: it is not true that the extremely small probability we have indicated for the Isaiah 53 cluster is the natural result of having searched for and evaluated over 1,200 ELSs.

It is natural for someone to expect that the probabilities of random occurrence of a group of ELSs will be somewhat evenly spread between 0% and 100%. And evidently that is what Randy intuitively assumed in asserting that we aren’t doing things right. The problem is that ELSs are ornery critters whose behavior defies our natural expectations. Why is that? There is an explanation.

As mentioned above, short ELSs typically appear many times while long ELSs often cannot be found anywhere. Intuitively it makes sense that the total number of 3 letter, 4 letter and 5 letter ELSs that have at least one letter in a selected 1,000 letter long text will be about the same. And that is correct.

But then something happens when we go to calculate the probability that a given selected ELS will appear. Whenever we deal with an ELS that is one letter longer than another one, we multiply by 22 the number of alternative ELSs that could have appeared (instead of the one we have selected). For example, while there are 10,648 possible different three letter ELSs (22x22x22), there are 234,256 possible four letter ELSs (22x22x22x22) and 5,153,632 possible five letter ELSs (22x22x22x22x22). So, because the number of alternatives skyrockets, the probability of appearance of any one selected ELS drops like a rock if that selected ELS is longer. Conversely, that probability skyrockets when we shorten the selected ELS by one or two letters. That is why clusters of ELSs are heavily populated by shorter ELSs that appear numerous times and have a probability of chance appearance close to 100%.

Why does this matter? It results in a situation where the great bulk of probabilities entering into a cluster calculation are very close to 100%. Given that, Randy’s intuitively appealing objection is invalid.

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Code Skeptics' Arguments Trashed

Ever since the first Bible codes were announced, skeptics have been saying, "Oh, well, you can also find codes like that in books like War and Peace and Moby Dick."

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