Nine years ago, Jewish Action published a fierce debate on the validity of the Torah Codes. Emotions ran high as the debate touched a raw nerve for many, though not always for the same reason. Although things have quieted down since then, the issue is far from resolved, and indeed, codes research is continuing unabated. Last year, several scientific papers relating to the Torah Codes were presented at the 18th International Conference on Pattern Recognition held in Hong Kong. At the conference, which drew over 1,000 participants, retired senior cryptologic mathematician and Torah Codes researcher Harold Gans presented a paper with important ramifications to the validity of a famous code known as the Great Rabbis Experiment. Summaries of the paper, including work on Torah Codes that is still under development, can be found at www.torahcodes.net.
An intimate conversation with Mr. Gans follows. We welcome letters from our readers.
Jewish Action: Can you tell us about your work in intelligence?
Harold Gans: For twenty-eight years I worked as a cryptanalyst at the National Security Agency [of the US Department of Defense] in Fort Meade, Maryland, outside Washington, DC. My first job at the Agency was to use techniques developed by other mathematicians to break codes. Later I became a senior cryptologic mathematician, responsible for developing mathematical techniques used to break codes. Eventually I moved into a management position, supervising a staff of twenty-five mathematicians, computer programmers and engineers.
JA: As a member of the group of Orthodox mathematicians conducting research into Torah Codes, could you explain (for those of us who failed algebra) the relationship of math to Torah?
Gans: We are constantly encountering situations which we evaluate instinctively as being either expected or unexpected. If you’re walking down the street and everyone you see on the sidewalk is staring up at the sky, this is something unexpected. So you might look up, too, to see what they’re looking at. If only one person is staring up at the sky, or perhaps two people, you probably would not be as curious.
We know instinctively that some events are ordinary, some extraordinary, and that there’s a whole spectrum in between.
Another example, based on a gemara. You’re walking down the street and notice two ten-shekel coins on the sidewalk stacked neatly, one atop the other.
Could it have happened by chance?
What about if you’re walking down the street and notice three coins stacked up neatly, one atop the other.
We know instinctively that it’s unlikely that [these occurrences] happened by chance. A statistician would view these situations in terms of probability, and could come up with a mathematical formula to describe the fact that the probability of three coins neatly falling on top of each other is very small. Probability is a precise mathematical way of measuring just how unexpected an event is. A small probability means an event is highly unusual or unexpected. This, in turn, implies that it is highly unlikely to have happened by chance.
The idea of concealed codes in the Torah is found in the mesorah. In the sixteenth century, Rabbi Moshe Cordovero wrote [in his work Pardes Rimonim] of the hope that in the end of days, the hidden codes in Torah would be revealed. Most people consider the idea of Torah codes so extraordinary that they are not inclined to give it any credence whatsoever without extraordinary evidence.
JA: According to the mesorah, what do these hidden codes consist of?
Gans: The Torah is structured in such a way so as to contain hidden information encoded in its letters in a variety of ways (Pardes Rimonim, sha’ar 30). This information was purposely incorporated into the Torah by its Author, and given to Moshe Rabbeinu on Har Sinai.
Rabbi Cordovero listed the various kinds of patterns, one of which is known as Equidistant Letter Sequences, commonly referred to by modern researchers as ELS.
An ELS code is a sequence of letters that are equally spaced in a text. In this case, the text is the Torah with all spaces deleted.
We usually visualize the Torah as letters arranged in two dimensional blocks for easy reading, as on a page. What we have observed in our research is that the equations that describe the proximity effect between ELS codes in the Torah are best visualized in the geometrical shape of a single helix, with a sequence of letters spiraling down the surface of a cylinder, much like a spiral staircase.
The Torah was given to Moshe Rabbeinu letter by letter in an unbroken line. If we visualize the Torah as an unbroken line of 304,805 letters wound around like a spiral staircase, we can imagine looking at it from different points. If we loosen or tighten the cylinder, letters from different points in the plain text will become aligned. In this configuration, previously unseen phrases, sentences and clusters of related ELS words become visible.
In the modern era, ELS patterns in the Torah were first researched by Rabbi Michoel Ber Weissmandl during World War II, who searched for codes by hand. The invention of the computer has made research possible on an unprecedented scale. In the 1970s, Dr. Eliyahu Rips, a professor of mathematics at Hebrew University, was the first to study the codes using a computer and to apply scientific methodology and statistics in such research.
JA: How did you get involved in Torah Codes?
Gans: In the mid-1980s, a cousin of mine—also a frum mathematician—attended a conference in Jerusalem, and while there, he met a professor of mathematics from the University of California. My cousin had remembered this man as an atheist but he was now wearing a yarmulke. He told my cousin that he became frum as a result of examining hidden codes in the Torah.
When he got back to the States, my cousin told me about this encounter and asked what I thought [about Torah Codes]. I told him it didn’t interest me.
Gans: People are always coming up with all kinds of weird claims. The Loch Ness Monster. Big Foot. Flying saucers. By nature and by training, I’m a skeptical person. When I hear about something unusual, my first instinct is to think it’s probably untrue. As for hidden codes in the Torah, I had been frum my whole life and had never heard of such a thing, so I dismissed it.
I forgot about it for around a year and a half, until one day my wife came home from a shiur and asked if I had ever heard of something called Torah Codes. I said yes, but she shouldn’t take it seriously. She asked, “How do you know? Have you looked into it?” I distinctly remember saying, “I have better things to do with my time than chase after nonsense.” And she said, “How do you know it’s nonsense?”
I saw that my wife really wanted me to look into it, and I wanted to please her.
JA: How did you go about investigating it?
Gans: I started by asking some of the frum men at the Agency if they had ever heard of Torah Codes. It turned out that one of them had been to an Aish HaTorah Discovery Seminar—this was about 1986 or 87—and had a book from the Seminar with four or five pages about the codes. He lent it to me. Most of the examples [of Torah Codes] I dismissed out of hand. I didn’t find them interesting.
Gans: We refer to something as “interesting” if we find a certain pattern to be mathematically significant, which means we can calculate a probability for it which is fairly small. Most of the codes in the [Discovery] book weren’t mathematically significant. This does not mean these particular codes were invalid, or that the hypothesis on which they were based was invalid.
What it does mean is that any well-trained mathematician would have recognized easily, as I did, how to poke holes in the methodology. The computer programming used in Torah Codes research at that point was still in its infancy, and what I saw wasn’t particularly impressive. The patterns found were simple and short, consisting, for the most part, of two related words in close proximity or intersecting each other, one or both of which could be construed to be related to the plain text of the Torah where the pattern appears.
The main problem was that no a priori methodology was established prior to searching. A priori methodology means that all the parameters of an experiment are defined before the experiment is run. If the experiment is a priori, then a valid assessment of the expectation of the outcome can be made. If the experiment is not a priori, then it is usually the case that an accurate assessment of the expectation cannot be made.
There was no way of knowing whether a priori protocols were used to find the codes in the Discovery Seminar book. In fact, there was no established protocol at all at that point. People were essentially searching in the dark. To the untrained eye, the patterns presented in the book did look as if they might be some kind of codes, and in fact, some knowledgeable people had apparently been impressed by them. But the methodology was incapable of demonstrating mathematically that the codes were not products of coincidence.
It was only years later that basic a priori protocol for Torah Codes research was developed, though until this very day, much of the supposed “research” into Torah Codes fails to adhere to basic standards of scientific methodology.
JA: When you were examining the Discovery book, did you think the data might have been “adjusted,” even unintentionally, to make it “work”?
Gans: No, because the Aish rabbis were obviously smart people with a high level of integrity. Even when I first heard about Torah Codes, I was fairly sure this was not a case of sheer fabrication.
I did think, however, that smart people had fooled themselves into seeing a significant pattern where none existed; and when I examined the methodology being used in the Discovery book, most of it was just what I had been expecting.
JA: What would have made Torah Codes seem important?
Gans: “Important” is a subjective term. If I say to you, I was just walking down the street and I saw a man with a green tie, you would probably accept my claim. If a claim is plausible, you’re likely to believe it with very little evidence. Yet how important is this information?
Most people would agree that if flying saucers exist, it would be important to know about them. Their existence could impact our lives. But it is a very unusual claim. So most people, including myself, not only do not believe in the claim but would also not bother investigating it.
They would consider it a waste of their time and resources.
The Torah Codes claim is important because the authorship of the Torah has been debated for centuries. Bible critics have persisted in asking whether the Torah was written by human beings, and if so, by one person or multiple persons. If the Codes exist, they basically answer that question.
One pattern [in the Seminar book] was mathematically interesting. It is referred to as the Aharon Code, and it appears in the first Torah reading of Vayikra. The Torah text concerns the sacrifices made in the Temple by the Kohanim, the sons of Aharon. The code consists simply of the word “Aharon,” which appears as an ELS in this section 25 times.
JA: How can you be sure that this isn’t coincidental?
Gans: There’s no such thing as being absolutely sure that a phenomenon is not coincidental. But we can speak of probabilities.
I don’t remember the exact fraction, but the claim in the Discovery book was that mathematicians had calculated a random expectation of around 8 ELSs in that paragraph. This is considerably fewer than 25.
So the first thing I did was check the calculations myself. I got the same number. I then calculated the probability of observing 25 such ELSs when around 8 were expected by chance. The probability was significantly small: less than 1 in 500,000.
It was at that moment—sitting there with my calculator in hand—that I thought, “Hmm.”
For the first time, I thought that maybe Torah Codes couldn’t be dismissed so quickly.
JA: How did you go about researching it further?
Gans: Somebody put me in touch with Rabbi Ezriel Tauber, a Torah lecturer, who invited me to attend an Arachim Seminar where the codes were going to be presented. I decided to go, and to ask Dr. Andrew Goldfinger, a close friend of mine and a senior physicist at the Johns Hopkins Applied Physics Laboratory, to come along. I thought that between the two of us, we would be able to catch any mathematical flaws.
So there we were, the two skeptics at the Seminar. They would show the class one pattern after another and my friend and I would just sit there looking over at each other with blank expressions. None of the examples was demonstrably a priori.
I felt uncomfortable—it couldn’t have been easy to have us there, neither for the teachers nor for our fellow seminar participants. I remember one of the codes they presented—typical of the kind they were presenting that day—consisted of the Hebrew word for “diabetes” in close proximity to the word “insulin.” The people in the class seemed very impressed by this, and the instructor was eager to hear our response. My friend and I didn’t want to be rude. We would just look at each other and think, “Oh, no.”
There was, however, one presentation which caught our attention. This was a complicated scientific experiment, designed and executed by Dr. Rips and Doron Witztum, which later became known as the Great Rabbis Experiment.
They had found that the names of great rabbis in Jewish history were encoded in close proximity to the encoding of their dates of birth and death. The Great Rabbis Experiment is relatively complex. It appeals to the scientist, not the layman.
JA: If I, as a layman, can intuit correctly whether three coins on the street have fallen on top of each other by coincidence, without doing complex mathematical calculations, what would be wrong about my going by intuition here, too?
Gans: Any text will produce a certain number of meaningful-looking patterns and word associations, even the back of a cereal box. Your subjective judgment may lead you to imagine relationships between words and to perceive meanings, where none may exist.
The computer enables us to look through millions of configurations in search of interesting or unexpected patterns. A mathematician can then calculate the likelihood that these patterns happened by chance, while a non-mathematician would be at a loss to evaluate them correctly.
JA: So laymen really don’t have any way to judge whether Torah Codes are real.
Gans: This table that we’re sitting at, do you believe that it’s made of atoms?
JA: Because I’ve heard and read that this has been discovered scientifically.
Gans: Why do you believe what you’ve heard?
JA: Because I’ve been led to believe that the people who say so know about these things.
Gans: Exactly. And because there aren’t any scientists around today who claim there’s no such thing as the atom. In contrast, in regard to Torah Codes there are respected Torah scholars and scientists on both sides of the issue. Torah Codes research has been surrounded by controversy since 1994, when Dr. Rips and Doron Witztum published a paper describing the Great Rabbis Experiment in Statistical Science, a well-known mathematical journal. Ever since, there have been those who are convinced of the existence of Torah Codes and those who are not. So whose word is the layman to take? There are mathematicians—Jewish and non-Jewish, religious and non-religious—who believe that since chance occurrences of ELSs can be found in any text, this must be the explanation for patterns that look like codes in the Torah. Some of these mathematicians take as their starting point the firm conviction that there is no such thing as a Supreme Being, and certainly not a Supreme Being Who wrote the Torah. These individuals have made it their mission to prove that the apparently coherent patterns we claim to have discovered in the Torah are a matter of coincidence, or the result of either self-deception or falsification. The most prominent of these critics, Dr. Brendan McKay of the National University of Australia, claims to have found patterns comparable to Torah Codes in the novels War and Peace and Moby Dick.
JA: I remember reading an article about that in the mid-1990s, in Jewish Action. It convinced me that there is no such thing as Torah Codes.
Gans: The article you are referring to established in the minds of many in the Orthodox community that Torah Codes do not exist. The impression remains among the Orthodox public that “codes” similar to Torah Codes were discovered in non-Torah texts.
While the mathematical issues are difficult for non-mathematicians to comprehend, I can summarize as follows: Professor McKay and his colleagues never claimed to have discovered real codes in those non-Torah texts. Their only “successful” results were obtained by deliberately rigging the experiment in such a way that the layman wouldn’t recognize the mathematical flaws.
To their minds, there is no other possibility, because they took it as a given that the Torah is not a Divine document, and that therefore the whole idea of Torah Codes is intrinsically absurd. The “discovery” of “codes” in Moby Dick or War and Peace to them is no more absurd a claim than the “discovery” of “codes” in the Torah.
The public does not understand that these critics openly admitted that their research did not result in the discovery of even a single valid code in any text other than the Torah.
Doron Witztum, Professor Robert Haralick of City University of New York and I have delineated the mathematical flaws of Professor McKay’s experiments. To someone knowledgeable about Torah Codes, the difference between Torah Codes and McKay’s “codes” is self-evident in terms of coherence and linguistic and statistical methodology.
As I stated, however, looks are deceiving. ELS word patterns can be found in any text. Any passage in a newspaper can produce real words with equal letter spacing. But these are random occurrences; they are not codes.
To find a coherent phrase or even one meaningful sentence in this manner would be rare, indeed. There are hundreds of Torah Codes, however, consisting of coherent phrases and sentences, many of which convey explicitly meaningful information about the Torah and shed light on the Torah itself.
The critics who “discovered” codes in works of English literature did not claim to have used proper scientific protocol. They concluded that since they were able to produce patterns that looked like our codes, this must have been how we got our results.
There are strict mathematical criteria that determine the definition of a code. What is, and is not, a code? Appearing as an ELS is only one of several criteria. The most important criterion is that the configuration of ELSs found must have some well-defined mathematical or structural property that is highly unexpected. In other words, it must be highly unlikely that this so-called “code” occurred by chance. This criterion is crucial. The standard scientific criterion for “highly unexpected” is that the odds against its occurring by chance are greater than, or equal to, 1,000 to 1. This must be determined by use of sophisticated software that executes what is known as randomization tests. Once one applies the criterion to the critics’ “codes,” a mathematician can see immediately that they do not qualify as codes.
JA: Torah Codes are associated with the idea, widely popularized in the late 1990s, that they can be used to make predictions. Are you ever tempted in the course of your research to seek answers about the future?
Gans: Many of the codes that we have observed refer to events that happened long after the Torah was given–in other words, codes about future events. However, we human beings cannot use Torah Codes to make predictions. Trying to do so necessarily involves guesswork, and in unscrupulous hands, or even well-meaning hands, such work easily lends itself to sensationalism. Such experiments, conducted by people who have not been scientifically trained, have not, to the best of my knowledge, been conducted on an a priori basis, and are thus not subjected to the type of controls that I consider necessary in order to determine if a given phenomenon has occurred by chance.
All that human beings can do is to observe codes that describe past events. Any attempt to predict the future based on Torah Codes is a subjective interpretation of words or phrases.
If, for example, on September 10, 2001, we had done a computer search for the word migdalei (“towers”) and had observed its proximity to the word hateumim (“twin”) either we wouldn’t have noticed the proximity, or—if we had noticed—probably would not have made anything of it.
Dr. Rips conducted a search for the words migdalei and hateumim on September 12, 2001, the day after the attacks, and found them in close proximity to each other as well as to other key words associated with the attack on September 11. At this point the context was clear, and he was able to calculate a mathematically precise probability of having found all these words as ELSs in close proximity. The odds against this having happened by chance were greater than 19,000 to 1.
JA: An Orthodox rabbi who is one of your critics declined to be interviewed for an article that would lend credence to Torah Codes. One of the reasons he opposes the codes is that even if they are real, it’s dangerous to publicize them because of all the distortion to which they have been subjected, especially by Christian groups. And he said that if they are ever proven false—even if only in the minds of the Jewish public—belief in Torah would be undermined.
Gans: These issues are valid. The Christian groups, in particular, pose a serious problem. Once the computer programs became publicly available, Christians began conducting their own unverifiable “scientific experiments” to convince the public that Bible Codes prove Christian dogma and theology.
For these reasons, I personally didn’t know whether we should make our research public. In the 1990s, we consulted gedolim for guidance on this question. We were told that the information should be presented to the Jewish public in spite of the potential dangers. Doron Witztum obtained this response from Rabbi Shlomo Zalman Auerbach, Rabbi Shlomo Fisher, Rabbi Boruch Shmuel Deutch and Rabbi Shlomo Wolbe. I received the same guidance from Rabbi Shmuel Kamenetsky and Rabbi Moshe Heinemann.
JA: I wonder if you realize how improbable the whole idea of Torah Codes seems to the average layman.
Gans: I certainly do, because that’s precisely how I felt. If not for the Great Rabbis Experiment, I would not have agreed when, in 1996, Aish HaTorah approached me with a request that I research to determine, as conclusively as possible, if Torah Codes exist. I remember the words of Rabbi Eric Coopersmith: “If this stuff is real, we’re not going to let anyone scare us away. But if it’s not, we’re going to drop it like a hot potato.”
I spent the next two years in intensive study.
JA: If Hashem wants to give messages to mankind, why would He hide them in these incredible crossword puzzles?
Gans: I don’t know.
JA: What do you foresee for the future of Torah Codes research?
Gans: Our research began with ELS simply because it’s the most obvious and most easily deciphered pattern of the various patterns cited in the mesorah. We hope to develop other methodologies that will enable us to research other code patterns listed by Rabbi Cordovero, but we’re still a long way off from that. Our team of mathematicians and computer programmers has been researching ELS codes for the past twenty to twenty-five years, and we know that what we’re seeing is just the tip of the iceberg.
JA: At this point, do you feel that you have absolute proof that the Torah Codes are real?
Gans: There is strong—one can even say overwhelming—evidence that Torah Codes are real, but in this world there is no such thing as having absolute proof of anything associated with belief in Torah or Hashem. It would take away our bechirah [free will]. I would, however, say that the evidence for Torah Codes provides the closest thing to absolute proof of Torah min Hashemayim, short of open miracles such as the Splitting of the Sea or the Giving of the Torah on Mount Sinai.
The question is whether the Torah Codes we observe are random or intentional. Were the existence of Torah Codes to be definitively disproved tomorrow, that would have no bearing whatsoever on the fact that the Torah was written by Hakodesh Baruch Hu. If, on the other hand, the existence of the Codes were to be proven definitively tomorrow, those who doubt Torah’s Divine origin would be confronted by strong evidence of its miraculous nature.
We believe that our evidence for intentional occurrence is valid. At this point, it requires considerable effort to evaluate the evidence that we have versus the claims of the critics. But it can be done, even by people with little mathematical background. All that is required is the conviction that the effort is worth it. Our goal is to produce a mass of evidence that can be evaluated by a reasonable layman who wants to know the truth.
Sarah Shapiro is the author of, most recently The Mother in Our Lives (New York, 2005) and Wish I Were Here (New York, 2006).