The Kuzari Principle - A Rigorous Formulation

In the Jewish year 5757 (1996-97 C.E.), I decided to become a Torah-observant Jew after I read Rabbi Dovid Gottlieb's book, Living Up to the Truth, (which at that time was entitled The Search for Truth) which presents a powerful logical argument for the belief that the Torah was given to the Jewish people by G-d.

While I was reading Rabbi Gottlieb's book, I was a mathematics graduate student at Johns Hopkins University in Baltimore, Maryland. Having been trained as a mathematician (and also as a magician), I understood that it is very easy to be fooled into believing that a logical argument is valid even if it is fallacious; slight of hand not only exists in magic but in logic as well. This is one of the reasons why, in general, mathematicians are notoriously difficult to convince of the validity of any logical argument that is not presented in the format of a rigorous mathematical proof. In a rigorous mathematical proof, every step must follow logically from the previous step; the rules of logic that mathematicians accept are so well-defined that it is possible to program a computer to recognize whether a given mathematical proof is valid or not, assuming that the proof is presented in a format acceptable to the computer program.

Because of my mathematics training and because Rabbi Gottlieb's book, Living Up to the Truth, was not written in a style of a typical mathematical proof, I was initially unconvinced by Rabbi Gottlieb's arguments. It took me a few months to digest what Rabbi Gottlieb was really saying in his book and to research the claims that he was making. After a few months, I finally believed Rabbi Gottlieb's argument when I was able to translate it into a rigorous mathematical proof. In this note, I shall present this proof. Now before I go claiming that every mathematician and scientist will believe that the Torah was given by G-d after reading this proof, it is necessary for me to say that I have no such expectation of this happening whatsoever for the following three reasons:

1. The proof uses Rabbi Gottlieb's Kuzari Principle, which is induced from empirical observations. Because induction is never 100% certain, some people may choose not to believe the Kuzari Principle and therefore not believe the proof.

2. The arguments in the proof involve the Orthodox-Jewish interpretation of the Torah; the reader may not know enough about the Orthodox-Jewish interpretation of the Torah to be able to ascertain whether the arguments in the proof are reasonable.

3. We human beings are notoriously difficult to convince of any fact that we do not want to believe. It is unnatural for us human beings to base our beliefs on logic alone; we are primarily emotional beings. It takes a great deal of effort for a human being to accept as truth facts that he or she does not want to believe. The Torah is difficult for many Jews to accept. Many of its claims are not politically correct, and it also places a great deal of responsibility on Jews.

Let us consider the following claims of the Jewish religion:

A) At least 600,000 Israelites gathered at the bottom of Mount Sinai over 3,300 years ago.

B) All of the Israelites heard G-d speak to them at Mount Sinai, and they then asked Moses to be His prophet.

C) Moses received the entire Torah from G-d and taught the Torah to all of the Israelites standing at Mount Sinai.

D) The Israelites transmitted the Torah and also the history of the transmission process of the Torah from generation to generation in an unbroken chain of generations for over 3,300 years until today, with at least one hundred thousand Israelites in each generation of the chain.

The following historical writings plus rabbinical commentaries and oral history have been traditionally used by rabbis to support the claims A, B, C, and D:

A) Biblical passages,

Exodus 12:37 - The Children of Israel journeyed from Ramses to Succoth, about six hundred thousand men on foot, aside from children.

Exodus 19:17 - Moses brought the people forth from the camp toward G-d, and they stood at the bottom of the mountain.

Deuteronomy 4:11 - So you approached and stood at the foot of the mountain, and the mountain was burning with fire up to the heart of heaven, darkness, cloud, and thick cloud.

B) Biblical passages,

Exodus 19:9 - Hashem said to Moses, Behold! I come to you in the thickness of the cloud, so that the people will hear as I speak to you, and they will also believe in you forever.

Deuteronomy 4:10 - the day that you stood before Hashem, your G-d, at Horeb, when Hashem said to me, "Gather the people to Me and I shall let them hear My words, so that they shall learn to fear Me all the days that they live on the earth, and they shall teach their children."

Deuteronomy 4:12-13 - Hashem spoke to you from the midst of the fire; you were hearing the sound of words, but you were not seeing a form, only a sound. He told you of His covenant that He commanded you to observe, the Ten Declarations, and He inscribed them on two stone Tablets.

Deuteronomy 5:4-5 - Face to face did Hashem speak with you on the mountain, from amid the fire. I was standing between Hashem and you at that time, to relate the word of Hashem to you - for you were afraid of the fire and you did not ascend the mountain.

Deuteronomy 5:20-24 - It happened that when you heard the voice from the midst of the darkness and the mountain was burning in fire, that all the heads of your tribes and your elders approached me. You said, Behold! Hashem, our G-d, has shown us His glory and His greatness, and we have heard His voice from the midst of the fire; this day, we saw that G-d will speak to a person and he can live. But now, why should we die when this great fire consumes us? If we continue to hear the voice of the Hashem, our G-d, any longer, we will die! For is there any human that has heard the voice of the Living G-d speaking from the midst of the fire, as we have, and lived? You should approach and hear whatever the Lord, our G-d, will say, and you should speak to us whatever the Hashem, our G-d, will speak to you - then we shall hear and we shall do.

C) Biblical passages,

Deuteronomy 4:5 - See, I have taught you decrees and ordinances, as Hashem, my G-d, has commanded me, to do so in the midst of the Land to which you come, to possess it.

Deuteronomy 4:14 - Hashem commanded me at that time to teach you decrees and ordinances, that you shall perform them in the Land to which you cross, to possess it.

Deuteronomy 5:25-28 - Hashem heard the sound of your words when you spoke to me, and Hashem said to me, "I heard the sound of the words of this people that they have spoken to you; they did well in all that they spoke. Who can assure that this heart should remain theirs, to fear Me and observe all My commandments all the days, so that it should be good for them and for their children forever? Go say to them, 'Return to your tents.' But as for you, stand here with Me and I shall speak to you the entire commandment, and the decrees, and the ordinances that you shall teach them and they shall perform in the Land that I give them, to possess it."

D) All written history including the Tanach (The Five Books of Moses, Prophets, and Writings) and the Babylonian and Jerusalem Talmud.

However, as of the last hundred years, these historical writings plus rabbinical commentaries and oral history have failed to convince many Jews of claims A, B, C, and D. In order to present these Jews with a valid reason to accept claims A, B, C, and D, Rabbi Gottlieb wrote his book Living Up to Truth. And in order to strengthen Rabbi Gottlieb's argument in his book, I shall present a mathematical proof of claims A, B, C, and D:

This proof relies on what Rabbi Gottlieb calls the Kuzari Principle:

Rabbi Gottlieb's Kuzari Principle:
Suppose that at least one hundred thousand people claim to witness a certain event. Then almost certainly, this event must have occurred.

(This is actually a modified version of Rabbi Gottlieb's Kuzari Principle.)

As we can see, the Kuzari Principle is really just plain common sense. It is basic to every legal system in the world that lots of witnesses claiming X implies that X is true. The overwhelming majority of people have no problems accepting this principle. We shall use the Kuzari Principle to prove A, B, C, and D:

"Kuzari" Proof of Claims A, B, C, D:

First, let us modify claim D to a new claim D(N), where N is a positive integer:

D(N) - The Torah and also the history of the transmission process of the Torah have been transmitted from the first generation of Israelites to the Nth generation of Israelites in an unbroken chain of generations, with at least one hundred thousand Israelites in each generation of the chain.

Now we will use the principle of mathematical induction to prove that for any N>0, if generation N, of at least one hundred thousand Israelites, accepts claims A, B, C, and D(N) to be true, then the claims A, B, C, and D(N) must be true:

First, we shall prove true for N=1: Since D(1) is vacuous if claims A, B, and C are true, it suffices to prove that if the first generation, of at least one hundred thousand Israelites, accepts claims A, B, and C, then claims A, B, and C must be true. And this follows directly from the Kuzari Principle, so we have proven true for N=1.

Now, let us assume that if generation N, of at least one hundred thousand Israelites, accepts claims A, B, C, and D(N), then the claims A, B, C, and D(N) must be true. Then we shall prove using this assumption that if generation N+1, of at least one hundred thousand Israelites, claims that A, B, C, and D(N+1) are true, then the claims A, B, C, and D(N+1) must be true:

Suppose that generation N+1, of at least one hundred thousand Israelites, claims that A, B, C, and D(N+1) are true. Then generation N+1 is claiming that they are witnesses to generation N transmitting both the Torah and the history of the transmission process of the Torah to generation N+1, as this is implied by the claim D(N+1). By the Kuzari Principle, this implies that generation N+1, of at least one hundred thousand Israelites, did witness generation N transmit both the Torah and the history of the transmission process of the Torah to generation N+1, which implies that generation N claims that A, B, C, and D(N) are true. By our induction hypothesis, this implies that the claims A, B, C, and D(N) are indeed true. Then since generation N transmitted both the Torah and the history of the transmission process of the Torah to generation N+1, of at least one hundred thousand people, we can conclude that claim D(N+1) is also true, completing our induction step.

Hence, we can conclude that for any N>0, if generation N, of at least one hundred thousand people, accepts claims A, B, C, and D(N) to be true, then claims A, B, C, and D(N) must be true. It is well known that as of today, the Torah-observant Jewish community, of at least one hundred thousand people, accepts claims A, B, C, and D. Then since D is really D(N) where N is the number which corresponds to today's generation of Israelites, we can conclude that A, B, C, and D are true. QED

Notice that this proof uses only mathematical logic. And the only way to argue with proof which uses only mathematical logic is to argue with the assumptions that the proof is based upon, in this case:

1. The Kuzari Principle.

2. The fact that Torah-observant Jewish community today, of at least one hundred thousand people, accepts claims A, B, C, and D.

Based on my own research and personal experience, I am comfortable accepting both of these assumptions. However, some readers may still be skeptical of the truth of claims A, B, C, and D. We shall now address some of the more common concerns:

Concern 1: The major events which the Torah describes, e.g., the revelation at Mount Sinai, have never been verified scientifically by physical evidence. A central tenant of modern science is that all conclusions about nature must be supportable by physical evidence to be believable.

Response to 1: The scientific method of only accepting a claim to be true if it is supported by physical evidence is very reliable; however, the scientific method is not the only reliable way to understand our world. For instance, the claim that the Declaration of Independence was signed on exactly July 4, 1776 C.E. cannot be verified by modern science - it is possible, although extremely unlikely, that the people signing it, in their haste, wrote down the wrong date by mistake. Yet, we still believe the witnesses to this event, even though only a small number of people actually saw the event, much smaller than the Kuzari Principle requirement of at least one hundred thousand people.

Concern 2: The Torah contradicts many facts established by modern science. For instance, according to modern science, all of the miracles which are described in the Biblical account of the Exodus from Egypt, including G-d speaking with the Israelites, are impossible; however, modern science does not rule out the possibility that the Israelites suffered from a "mass hallucination" and believed that these miracles happened, even though they did not. Why should I believe the Israelites over modern science?

Response to 2: A mass hallucination has never been witnessed in the history of the world, so it is unlikely that such a mass hallucination occurred in the time of the Exodus or any time after the Exodus. But then again, the miracles which are described in the Biblical account of the Exodus from Egypt have never been known to occur in modern times, so one might think that it is also unlikely that they could have occurred in ancient times. So what happened back then? Clearly, a phenomenon which has never been observed in modern times must have occurred back then - either a mass hallucination or miracles from G-d. Because we have hundreds of thousands of witnesses to one of these possible phenomena, the miracles from G-d, but we do not have any witnesses to the other, a mass hallucination, it makes more sense to believe the phenomenon in which there were hundreds of thousands of witnesses, the miracles from G-d, over the phenomenon in which there were no witnesses, a mass hallucination.

Concern 3: So you are saying that lots of witnesses to a phenomenon which contradicts the known laws of nature are more credible than the known laws of nature? Why?

Response to 3: Let us imagine that we lived in a world in which anytime someone makes a discovery that contradicts the known laws of nature, that person would automatically be disbelieved. For instance, if someone were to discover magnets, then that person would be ridiculed, because everyone knows that objects cannot be moved without touching them. And if someone were to discover fire, then that person would be considered a lunatic, because everyone knows that only the sun is hot. If we were to live our lives dismissing any testimony of others which contradicts what we know, then we would never learn anything. And this is precisely why the testimony of many witnesses is always stronger than even the most powerful circumstantial evidence.

Concern 4: Scientists assert that the universe is billions of years old, not 5,766 years old as Jewish tradition has it. Also, scientists assert that science teaches us that man evolved from lower apes; the Torah claims that G-d created one man named Adam and one woman named Eve from scratch.

Response to 4: The claims that the universe is billions of years old and that man evolved from lower apes cannot be verified directly by experiment or observation; nevertheless, scientists still consider these two claims to be fact because they have found what they believe to be powerful circumstantial evidence supporting them. But while the scientists think they have powerful circumstantial evidence on their side, we have hundreds of thousands of witnesses to the giving of the Torah at Mt. Sinai by G-d on our side. Because hundreds of thousands of witnesses are more credible than even the most powerful circumstantial evidence (see the response to 3) and because these witnesses claimed that the Torah was given by G-d, the Torah account of the origin of the universe and the creation of man is much more reliable than the big bang theory and the theory of evolution.

Concern 5: Still, how does one explain the million-year-old fossils and the appearance of an old universe?

Response to 5: This is not in contradiction with Torah. According to the Torah, Adam and Eve were created as adults. And also, the trees in the Garden of Eden were created in a fully mature state. So we see from these examples that G-d created at least part of the universe to appear as if it has been around longer than it really has. Of course, the appearance of an old universe does not imply that the universe is old.

But this answer does not explain the existence of fossils that appear to be millions of years old, if we are to use modern scientific dating methods. A probable explanation for this is that these fossils are artifacts from before the flood during Noah's time and are less than 5,766 years old. According to the Torah (Genesis 8:22), after the flood, G-d proclaimed that "Continuously, all the days of the earth, seedtime and harvest, cold and heat, summer and winter, day and night, shall not cease", implying that before the flood, nature was different than it is today; therefore, one cannot use modern scientific dating methods, which are based on the assumption that nature has not changed through the years, to accurately predict what really happened before the flood.

Concern 6: How can one guarantee that the transmission process from generation to generation of the Torah was not compromised, like in the famous "Telephone Game"?

Response to 6: The "Telephone Game" is a game which is often used to demonstrate how information can easily become corrupted by indirect communication. In this game, people organize themselves in a line, and a player at the beginning of the line whispers a message as quietly as possible to his or her neighbor. The neighbor then passes on the message to the next player in line to the best of his or her ability. The passing continues in this fashion until it reaches the player at the end of the line, who calls out the message that he or she received. If the game is "successful", the message that the person at the end of the line calls out is different than the original message.

The difference between the "Telephone Game" and the Torah transmission process is the fact that in the telephone game there is only one person at each stage of the message-passing process; in the Torah transmission process, there are at least one hundred thousand people in each generation. Because whispering from one person to another person is not the most reliable method of communication, there is a significant chance that at least one of the people in the line will not relay the message properly to his or her neighbor. And this is enough to alter the message called out by the person at the end of the line.

Now let us consider a modification of the "Telephone Game" in which there are one hundred lines of people instead of only one line of people. The one hundred players at the beginning of each line each pass on the same message to the next player in each of the one hundred lines. Each neighbor in each line then passes on the message to the next player in each line to the best of his or her ability. The passing continues until the message gets to the end of each line. Now let us suppose that the messages called out by the people at the end of each line are not all exactly the same but they still all resemble each other very closely. And let us also suppose that there are parts of the message called out by each person at the end of each line which are all identical. Then would it not be reasonable to assume that these parts of the message were transmitted flawlessly through each of the lines of people? Most people would think so. Of course, there is a chance that these parts of the message were transmitted incorrectly in each of the lines and by coincidence the people in each of the lines made the exact same errors. But the chance of this occurring is so astronomically small that it is reasonable to believe that these parts of the message are identical to the same parts of the original message.

Such a modification of the "Telephone Game" is similar to the Torah transmission process. Everywhere in the world, Torah-observant Jews have received the Torah and the history of the transmission process of the Torah from previous generations. Yes, there are parts of the Torah and the history of the transmission process of the Torah which have varied slightly for different sects of Torah-observant Jews; however, the claims A, B, C, and D have been universally accepted by all sects of Torah-observant Jews. Hence, just as in the modification of the "Telephone Game", it is reasonable to assume that identical messages at the end of each of the one hundred lines implies a perfect transmission of the message, the fact that claims A, B, C, and D are universally held to be true by all sects of Torah-observant Jewry implies a perfect transmission of claims A, B, C, and D, which implies that claims A, B, C, and D are true.

Acknowledgements: I would like to thank Rabbi Gottlieb for his helpful comments.

References:
Rabbi Gottlieb's website - www.dovidgottlieb.com
Wikipedia - http://en.wikipedia.org/wiki/Kuzari