And G‑d opened the mouth of the ass, and she said to Bilam: “What have I done to you, that you have struck me these three times?”

And Bilam said to the ass: “Because you have mocked me; I would there were a sword in my hand, for now I would kill you.”

And the ass said to Bilam: “Am not I your ass, upon which you have ridden all your life to this day? Was I ever wont to do so to you?” And he said, “No.”

Num. 22:28-30

Despite the simple dialogue between Balaam (Bilam) and his ass in the Torah portion of Balak, this ass was no ordinary ass. This ass saw an angel where a prophet as great as her owner, Balaam, did not. It stands to reason that Balaam’s ass had a greater prophetic vision that Balaam himself. Perhaps she can teach us a lesson or two. Let us listen to what she has to say…

In his philosophical treatise, Guide of the Perplexed (Moreh Nevuchim), Maimonides (Rabbi Moshe ben Maimon, known by the acronym of his name the Rambam), states that such Biblical stories as talking donkeys (Num. 22:28-30) or talking snakes – “The serpent said to the woman, ‘You surely will not die!’” (Gen. 3:4-5) – must be interpreted allegorically, not literally. Donkeys and snakes don’t speak, which precludes a literal interpretation of these verses. Rather, according to the Guide, these were visions. the Maharal (Rabbi Judah Loew ben Bezalel, famous for his Golem), in Gur Aryeh disagrees sharply with Maimonides, insisting on the literal interpretation of the story of the talking ass. Surprisingly, he agrees with Maimonides that the primordial serpent did not speak with Eve (Havah). This is very strange… If the Maharal was a literalist, he should have applied this approach across the board. Why did he agree with Maimonides that the serpent did not really speak?

The classical explanation of a miracle is that G‑d created the world to run according to the laws of nature but, from time to time, He chooses to break these laws. Today, however, thanks to quantum physics, we can offer another interpretation. In quantum mechanics, many things that seem impossible to us (such as going through a wall without breaking it) are possible, albeit rare. In fact, an electron can go through a wall—it is called the quantum tunneling effect. One can say that a miracle does not necessarily have to involve breaking the laws of nature. The “miraculous” outcome may very well be permitted by the laws of physics but have such a low probability that we never see it. I am not trying to explain biblical miracles as natural phenomena, as some apologists have tried to do; and I am not denying G‑d’s involvement in securing the desired outcome. What I am suggesting is that to make a “miracle,” G‑d does not always have to break the laws of nature created by Him. He merely needs to make sure that an extremely unlikely outcome comes to pass at the right moment and at the right place.

Take for example kriyat yam suf – the splitting of the sea. The verse says, “And the Lord drove the Sea back by a strong East wind all night… (Ex. 14:21)” If G‑d just wanted to break the laws of Nature, He surely didn’t need the wind to help Him do that. It seems, however, that the involvement of the wind indicates that it was a natural phenomenon albeit an extremely unlikely one. In other words, over the course of thousands of years, it is possible that there could occur such a confluence of meteorological conditions that would cause the waters of a sea to split. However, the likelihood of such an occurrence is extremely small, particularly since this event would only be of benefit, if it occurred at precisely the right moment. The fact the waters split exactly at the place and time where and when Jewish people started crossing the Sea was the result of Divine intervention celebrated as a miracle.

In quantum-mechanical terms, the wavefunction that describes a distribution of the probabilities of various outcomes contains all possibilities, some more probable, others less so. To be more precise, the square amplitude of the wavefunction is the probability of finding a particular outcome. If we repeat a scenario many times, the more probable outcome will appear more often; the less probable will be rarely found. With an infinite number of trials, the number (frequency) of occurrences of a particular outcome will be equal to its probability of occurrence.

However, the collapse of the wavefunction reduces the plurality of possibilities to one actuality. The realization of a particular outcome as a result of the collapse of the wavefunction is completely random. It takes Divine intervention to make sure that the wavefunction collapses the “right” way resulting in the desired outcome. This is perceived by us as a miracle.

To get a bit more technical, a quantum system before a measurement is in a state of superposition of all possible outcomes. The measurement causes the collapse of the wavefunction, destroying the superposition and realizing only a single possibility, which becomes the reality. For example, a photon can be in state A, which is horizontal polarization, or state B, which is vertical polarization. Before we measure the polarization, the photon is in the unique state C that is a superposition of states A and B. Roughly speaking, C = A + B.

In another example, an electron can have spin up (state A) or spin down (state B). Before measurement, the electron is in a state of superposition of A and B with the direction of the spin undetermined. We can, however, calculate the probabilities of finding the electron with spin up or down once we measure it. In fact, if we repeat the measurement many times, the frequency of finding the electron with spin up will correspond to the predicted probability. But each individual measurement will give a random result, either up only or down only.

The Schrödinger cat is another example, used many times on this blog. The unfortunate feline is stuck in the blurred state of superposition of being dead and alive. Let’s look at this more carefully. There are four possible “classical” states that can be assigned to the cat:

  1. Alive
  2. Dead
  3. Both
  4. Neither

These four states exhausts the universe of possibilities. The formal logical analysis of the Schrödinger cat experiment reveals that the cat is neither dead, nor alive, nor both, nor neither, but is in a fifth state called the state of superposition. This state is thought to be unique to quantum mechanics and unknown in classical physics. Schrödinger was so taken aback by such “absurd” possibility that he, along with Einstein, held it against the quantum theory he helped formulate.

Unbeknownst to physicists and philosophers of science, this state was well-known to Jewish sages hundreds of years before the advent of modern physics. Let us consider the concept of Twilight (bein hashmashot), which is between day and night. There are five opinions as to what bein hashmashot is, each having its own halachic (pertaining to Jewish ritual law) ramifications. I have used Shabbat as an instance of such a ramification. Accoding to various opinions, Twilight (bein hashmashot) is:

  1. Day (according to this opinion, one does not have to start Shabbat until the first stars appear on Friday night);
  2. Night (according to this opinion, one ends Shabbat with sunset on Saturday);
  3. Both (according to this opinion, the period of bein hashmashot has a duration, during which time one if forbidden from work both on the day preceding Shabbat as well as after it, which is the reason why most orthodox Jews start Shabbat with Sunset on Friday night and end it with the nightfall on Saturday);
  4. Neither (according to this opinion, bein hashmashot is an infinitesimally short moment in time separating day from night, which has no duration); and
  5. Neither day, nor night, nor both, nor neither (according to this opinion, bein hashmashot is a unique state, which has its own set of laws).

It is the fifth state that corresponds to the quantum-mechanical state of superposition.

Let’s look at another example—an animal called a “koi,” which is a crossbreed of a goat and a doe. The goat is a domesticated animal, called in Hebrew “beheimah.” Deer, on the other hand, is a wild animal, “chayah” in Hebrew. The question is, what is a koi—is it a domesticated animal, a beheimah, or a wild animal, a chayah. We find a fascinating discussion in the Mishnah (Nazir, 5:7). In the Bible (Num. 6:1-21), we read about a nazirite, or nazarite (Heb., nazir) who takes upon himself a wow to abstain from wine (and any grape products and intoxicating liquors), from cutting his hair and from ritual impurity. The following mishnah discussing a person accepting upon himself the nazirite wow conditional upon the definition of the nature of the koi:

If they saw a koi and one said “I am a nazir if that is a chayah”; one said “I am a nazir if that is not a chayah”; one said, “I am a nazir if that is a beheimah”; one said “I am a nazir if that is not a beheimah”; one said “I am a nazir if that is beheimah and a chayah”; one said “I am a nazir if that is neither a beheimah nor a chayah”; one said “I am a nazir if one of you are a nazir”; one said “I am a nazir if none of you are nazir” and one said “I am a nazir if all of you are nezirim…”

This discussion depends on the nature of the koi. There are several possibilities:

  1. koi is a domesticated animal – beheimah;
  2. koi is a wild animal – chayah;
  3. koi is both a beheimah and a chayah;
  4. koi is neither a beheimah nor a chayah;
  5. koi is neither a beheimah nor a chayah, nor both nor neither but rather a unique type of animal—a crossbreed.

This last possibility corresponds to the quantum-mechanical view that an object—in this case, the koi—is in a state of superposition of two states a beheimah and a chayah. As we see, the state of superposition was known to the Jewish sages at least from the time of the Mishnah preceding quantum mechanics by two millennia!

There is an interesting mishnah in Teachings of the Fathers (Pirkei Avot) that says:

“Ten things were created on the eve of Shabbat, at twilight (bein hashmashot). They are the mouth of the earth; the mouth of the well; the mouth of the donkey; the rainbow; the Manna; the staff; the Shamir; the alphabet; the inscription; and the Tablets.”

The donkey in this mishnah refers to Balaam’s ass who speaks. “And G-d opened the mouth of the ass, and she said…” (Num. 22:28). Although we stated earlier that Maimonides holds in his Guide for the Perplexed, that the ass speaking was just a vision, in his commentary on the Mishnah, the Maimonides states precisely the opposite: that Balaam’s ass did speak and to prove that, he quotes the above Mishnah. But how is this Mishnah a proof that Balaam’s ass actually spoke? On a simple level, the Mishnah seems to be telling us that G‑d created the miraculous speaking mouth of Balaam’s ass when He created the world. I would put a slightly different spin on this explanation. The fact that the Mishnah tells us that the mouth of the donkey was created when the whole world was created tells us that there is nothing inherently miraculous about a speaking donkey—this possibility was created ab initio, from the “get-go”. Moreover, the fact that it was created specifically at twilight—bein hashmashot—hints at the state of superposition, of which bein hashmashot is the primary example in the Torah. Thus the miracle is not that Bilam’s ass spoke, since she was in a perpetual state of superposition of being able to speak and not being able to speak, but that she spoke at the right place and at the right time, just when G‑d wanted to teach Balaam some humility.

Based on this we can understand the nature of the dispute between Maimonides and the Maharal. As a reminder, the Maharal disagrees with Maimonides and maintains that the donkey actually spoke. In the case of the primordial serpent in the Garden of Eden, however, the Maharal agrees with Maimonides that it did not speak. Why?

It seems to me that the answer lies in the timing of the creation of the snake and the donkey. According to the Biblical cosmogony, the snake was created on the third day of creation. The mouth of the donkey was created at the twilight of Sabbath. Accordingly, we can understand the opinion of the Maharal as follows: the donkey, created at bein hashmashot, was made (as bein hashmashot itself) in the state of superposition of being able to speak and not being able to speak. Consequently, it was possible for G‑d to collapse the wavefunction of Bilam’s ass in such a way that the ass spoke, “And G-d opened the mouth of the ass…” The snake, however, was not created at bein hashmashot and was not in a state of superposition of being able to speak and not being able to speak. Snakes never speak! Consequently, it is unreasonable to assume that the serpent actually spoke—it must have been a vision.

These are the lessons of quantum physics and the secrets of Torah taught by the talking ass.