Quality and Quantity in the Book of Numbers

Bamidbar, fourth of the Five Books of Moses, is correctly translated as “In the wilderness” or “In the desert.” Yet, like Deuteronomy, the English name “Numbers” has Jewish roots and reflects the nature of the work. Numbers, Sefer HaP’kudim, famously begins with a census of the Hebrew tribes and proceeds with some of the most wondrous and mysterious stories in all the Chumash. Despite its numbers, quantity is not the (sole) focus of the book of Bamidbar. Rather, the book and its stories instruct us in the sublimation of quantities, the divine quality of numbers, and how they figure into the proper worship of G-d.

That numbers matter is hardly a given. For much of history the wise have considered a focus on numbers in human affairs1 to be a concession to the coarse and unintelligent, even to the animal. Is the number of children you have more important to you than their personalities or their individual souls? If your neighbor has three kids and you have two, does that mean their kids are better?

Thousands of years ago, the only people convinced the truth of reality is deeply mathematical were mystical cultists of Pythagoras, idolators who believed salvation came through alignment with mathematical harmonies. Not until the enlightenment and the last few centuries was their belief resurrected and pursued with remarkable result by scientists. Perhaps in some sense, one fifth of the Torah, ancient and extremely non-idolatrous, sat quietly for millennia, full of stories allegedly about numbers, waiting to speak to a quantitative age…

Now, the Torah doesn’t teach that mere numbers confer importance or truth—after all, it is the book of “indeed, you are the smallest of peoples.” On the other hand, we do find in Jewish law that numbers do lend a certain reality to things. Things that are counted cannot be nullified. A minyan for prayer consists of ten Jewish men, regardless of righteousness; nine holy tzaddikim cannot replace it.2 So while quantity itself has no inherent value, a quantity of qualities can itself lend new qualities3; ten Jews become a quorum, and the tenth brings in the Shechina, presence of the Ultimate Quality. Quantity is important, but only as it reflects quality.

Quantity and quality are in fact deeply intertwined. Learn if from Parshas Nasso, the most repetitive portion in the entire Torah, a procession of quantities, each the same, and each, Rashi explains, reflecting the years of Adam’s life. The offerings count the same for each tribe, and each is based on the same essential reasoning, yet there are still twelve different intentions and so a twelvefold repetition of the words4. It is the quantity of Adam’s years that allow the multifaceted interpretation that mere quality would deny, but each of the twelve facets is imbued with the quality of each tribe and the unity of their general mission. Quantity is important, but only as it reflects quality.

We learn it again in Parshas Beha’aloscha, wherein the Torah does not flow with the qualitative passage of time. The story of Pesach Sheini actually occurred before the first two portions of the book. Just as the people rejected the rule of time and asked to celebrate Passover out of its time, so is the Torah revealed to transcend such quantifiable concerns.5 On the contrary, the way the Torah is grounded in time, in the successive days of the week or months of the year, and the consecration of certain periods, is shown to be a qualitative concern, the Torah’s choice born of G-d’s will and its divine nature rather than its subjugation to quantity. By adhering to Torah, the flow of measurable time is elevated. Quantity is important, but only as it reflects quality.

In Shlach and Korach, we see what happens if the balance of quantity and quality is disturbed. First the spies reduce quantity to quality, arguing that quantity is not important, that the desert and the holy land are not two instance of one thing (e.g. two creations, sharing the nature of all creations of subservience to G-d’s will). They rather viewed the wilderness and the land as two particular things, one that allowed for G-d’s miracles and one that didn’t.6

Korach sees the mistake of the spies and seizes on the opposite extreme, reducing quality to quantity. He insists Moshe and Aharon are one, at essence, belonging the same group as Korach, mere holier Levites. He errs in refusing to see the irreducible qualities of his cousins, that in fact they are infinitely greater, as king and high priest, than he, a gap no addition can cross and no generative divine algorithm can iterate across.

As if to emphasize the point, the portion of Chukas launches immediately into a discussion of the red heifer and the Torah’s laws of ritual purity and impurity. The laws are the most purely qualitative in the entire Torah7, rebelling against the mind’s tendency to homogenize through quantification and comparison. Touch a corpse with one finger or your whole arm, your body is just as ritually impure. Try to divide the purity from impurity in the heifer which purifies and corrupts simultaneously. The suprarational chok decree expands Korach’s lesson to all of Judaism. Never can the Torah be called a mere means to some complex or composite end.

Yet, in Balak, Bilaam saddles his talking donkey8 to ride off to curse the Israelites, attempting to pervert9 the very notion of the suprarational chok, to take advantage of the parity of qualitative reason.10 Bilaam tries to show that worship totally beyond reason can allow evil to arise, that the suprarationality of the Hebrews’ worship could in turn collapse to the irrational, for there is no standard for comparison, no ratiocinative quantitative reason that can divide evil from good in the realm of the suprarational.11 Answers G-d that Avraham beat Bilaam to the punch, that the suprarational is, itself, mysteriously and immutably Good. His curses are transformed into blessings. We thus find that quantity is important, but only as it reflects quality, and quality is important only as it reflects the will of its Creator.

This mysterious transcendence beyond quality, to the One Who Lends Quality To All Qualities—a Cause of Causes, if you will—is reflected in the total dedication of Pinchas beyond the dictates of the Torah (and Moshe) itself.12 Pinchas learned not only from the repentance of the second Pesach, which pointed beyond the quantifiable, but from the failure of Bilaam, which pointed beyond the qualitative. At a moment when wisdom provided no answer, he was able to find one; his actions reflected the suprarational will of G-d.

It is first in Pinchas that the quantitative fully conveys the qualitative and is united with it.13 After all the preceding portions, Pinchas sees the root of quality lies above quality. Why, then, should he ever intend to act according to quantity or quality alone? On the contrary, quality, and the quantity to which it must speak, can only be guided by faith and total surrender of one’s will to G-d. Only when quality is appraised in terms of its source, rather than in the context of speaking to quality, is that quality then able to speak to quantity without becoming corrupted.

It is thus by living in a way of Pinchas that quality and quantity are properly balanced and united. Numbers depend on souls which depend on G-d. Only then is the mission of the Book of Numbers fulfilled: To elevate numbers and reveal their holiness. The Book of Numbers shows us, the descendants of the Hebrews who fill its stories, how to live a G-dly life in an ever-more measured world.14

1That numbers are objects of wisdom and in their relation involve eternal truths is hardly a modern idea. However, particularly through the Aristotelian influence, the primacy of the numerical in perceiving the natures of things was relegated to the outskirts of western thought for centuries.

2A law derived from the book of Numbers, specifically from the incident with the spies.

3See Likkutei Sichos, vol. II, p. 293ff.

4See Likkutei Sichos, vol. VIII, p. 41ff.

5See Likkutei Sichos, vol. XXIII, p. 62.

6See Likkutei Sichos, vol. IV, p. 1041ff.

7See Likkutei Sichos, vol. XIII, p.68ff.

8G-d warning Bilaam through a talking donkey should have tipped the sorcerer off. When the Rogatchover Gaon wants to characterize Korach’s heresy, he compares it to thinking a human being is nothing but a donkey plus some further, ‘humanizing’ traits, that a donkey’s eating and human eating are more-or-less the same. For Bilaam’s beast to suddenly leap across this divide is almost like G-d saying, “I choose which incomparable creations are actually comparable around here. Korach thought he was like Aharon, when he and Aharon were as distant as a man to a donkey. You think that by accessing the suprarational, you rise to the level of the Hebrews, but you are more different from the Hebrews than a donkey is from a man…” See footnote 10 below.

9See Likkutei Sichos, vol. XXVIII, p.157ff.

10 Unlike the spies, who focus too much on quality and ended up denying G-d’s power over certain natures, Bilaam focuses on quality to, in a sense, broaden G-d’s power, to claim the same connection to G-d for evil as for good. Quality seems to divide from quantity at two extremes, then: (a) to become estranged qualities (b) to become One. The spies abandon quantity in order to divide; Korach abandons quality in order to unite; Bilaam abandons quantity in order to unite. Of course, the unity Bilaam seeks, in irrationality, is the loneliest of all unities, in which true communication between individuals is probably impossible. The spies’ division at least maintained G-d as a principle, and where there are principles there is common ground. Bilaam unites under a G-d so dissociated that evil and good are equal, and so divides; the spies divide the wilderness from the land under a G-d who is Good, and so unite. Thus, not every medium (e.g. rationality) separates and not every immediacy (e.g. a decree of pure will) unites.

11Think, again, of the red heifer, in which pure and impure, qualities, are rationally inseparable.

12See Likkutei Sichos, vol. XVIII, p. 318ff.

13See Likkutei Sichos, vol. XXXIII, p. 164ff.

14Perhaps we could say that the unique relationship of the Jew to G-d at a totally suprarational level that in turn permeates down even through the level of basest quantity is reflected in Parshas Matos-Maasei as well. See Likkutei Sichos, vol. XXIII, p. 214ff.

Reckoning With Kabbalah

If G-d is any, He is One. Not the one of counting or one of many, but the One and Only, alone. He has no parts, no aspects, no faculties, no cause. His one is truly infinite. He is not the unit for addition; there is no room for a second; nothing additional can exist.

G-d then creates another, a universe. It could be a universe consisting of an atom or full of fish or our own frustrating cosmos, but no matter which it is, it is Two. The creation is separate, composite, and inherently bifurcated, since however lofty it may be, it always has at least two aspects – itself and its relationship to its Creator. This is the split of all splits, not a big bang but a big crack. Imagine! Something composite, something dependent, something that could exist but doesn’t have to exist.

Two radically reverses our understanding of one. The original one and only, of course, contained no potential for a second. Now that there is two, we must say it came from one. There is now a one-implicit-in-the-two, in addition to the original one-and-only.

The two can be iterated or multiplied to produce as many things as one wants. The first atom can become the one-implicit-in-the-second atom, and the second atom may in turns become the implicit cause to yet a third. In this way, the one-implicit-in-two, cause of causes, and the two, the template for all effects, are the two needles with which all things are sewn.

If the counting ended with two, however, then every further creation would only be a further two – a two-and-a-half, a two-and-a-quarter, an infinity of twos. A world of twos is a sad place indeed. It would call out to us, each two aspiring to be our one, our cause, almighty. They all say, “Trust me, for mine is the way, to escape the pain of two and become one.”

There is no such thing, of course, as our forefather Avraham realized. A caused being is ever a two, and one is beyond us. Avraham was as close to the one as ever walked this earth, but he resigned himself to life in a world of twos.

Then the One gave us three, the Torah of Truth, one and two united. Three contains two – it has a source and it speaks to the world. But three also contains one – it is the mind of G-d, infinite and inscrutable. Three takes one and the other and unites them. When two argues with two, each saying they are superior, they may each turn to three. Three exists beyond their struggle and division. It is the light at their feet that helps them find each other, for it speaks to them from an unassailable place beyond them.

Every two has a place in three; the trick is finding the right place. Generally, the more a two recognizes it is not a one, the more it may coexist with one through the three.

Four is a thing. One, two, and three are the Creator, the creation, and their unity, but even two, close as it is to (at least the implicit) one, is an awareness of being created. “I am after one,” it says when it’s thinking clearly. In fact, “thinking clearly” can be defined as one, two, and three in their right places.

Four is not thinking at all, clearly or otherwise. Four counts not the world as opposed to G-d, but each thing simply as it is. Three helps one and two past their duality, and four is the result. Four is stable being, a thing in itself. It does not need to consider its source; three sorted that out. And three naturally leads to the static realm of four. Four is so much the nature of each thing as it is, that even ten reduces to the tetragrammaton.

But if four is being, five is its limit. Five says that a four only reaches so far, that things have their edges. Five is a number that counts non-being then, the partner of four just as two is the partner of one. Even though four does not think and therefore has no room for others inherent in its nature, there is still a point beyond which it does not extend and a reality to which it is not relevant. That reality, following naturally from four, is five.

Four and five are very different from one and two, however. One and two exist in obvious mathematical dichotomy in a way that four and five do not; one is the number beyond numbers, whereas two begins to actually count limited things. Four and five are not obviously so different. That the difference is concealed, even though four and five are opposites, is itself indicative of the simple, “non-declarative” nature of these higher numbers. They are exactly the non-descript pair for counting “just a thing” and “where the thing ends.”

Now, five, as four’s limit, is still more than four, the way the negative space at the page’s margins is bigger than the image sketched at the center. That negation is more than affirmation is a consequence of the ancient Jewish belief in counting from one instead of from zero.[i] If zero is the first number, if nothing is the ultimate reality, then taking away from four would result in a smaller number. The negative space would be cut out from the page, rather than extend as its margin. Since we start from one, and one is the greatest and the most perfect, when we delineate the limits of four, the end of each thing, we do it by adding one. When we see “not four,” it says to us, “but maybe one.” Negation, the not-four, thus naturally yields five.

Six contains both four and five in one unity. Whereas four is a thing and five it its end, six is the elegant balance between the two.

The Torah dictates the balance of all things, what they are and what they aren’t, and ensures that neither being nor non-being transgresses on the territory of the other. Six is that aspect of all things by which they are neither entirely limited to being what they are not entirely defined by what they are not. G-d, in His mercy, has implanted in each creation the ability to transcend this dichotomy. Four can be even where it is negated; its negation is shown to be part of its definition. The nothingness of five prevails even where something abides, for it, too, is an expression of something, and a deep one, the quintessence.

If the thing and its negation are united in one higher unity containing both, are not the numbers exhausted? What else is there to count?

In truth, there exists in each creation not only its source and interplay with G-d almighty, nor its self-contained being and limitations alone. There is also the result of the balance of being and negation, the way a thing affects others beyond its borders, the way every creation is, in its own way, a father to other creations.

This is not the same way that two can endlessly cause further twos. Three through six have rendered the cause of new creations no longer a single number, but a complex balance of unified contradictions. This is what allows a creation to give of itself without losing itself, to continue even as it ends, to carry on in its children, not as a solitudinous clone but as the soul to the next generation.

Seven, the count of nature, to make the world a certain way. Where one is the source, and four is the self, seven is the remnant, the bullet point remaining after the lecture. One was simple by being beyond complexity; seven is simple because complexity has come and gone. Rather than the interaction of something and nothing, self and other, unity and diversity, seven simply tells the world “this is how I am, and this is how you should be.”

Eight, the supernatural, is the negation of mundane seven. Eight counts secrets and illusions. It says, “Do not take the story of seven at face value.” It is not the stark apartness of two, nor the radical denial of five, but a humble villager telling seven, “You, too, are a number.” In eight we see non-being in the world, that the evaluation of self + world can equal only world. When seven gives a gift to G-d or humanity, eight thanks them for gifts already received.

Nine is the unity unifying all unities. Three weds G-d to universe, six weds being to non-being. Nine shows the other to both exist and not exist, to stand in relation to us and not affect us at all. Nine says that we can become the other and remain ourselves, can separate from each other yet remain untied. Nine, in other words, says that we can have children and our solitude can be shared.

In a way, there is no ten. One through nine, in combination, reckon where everything comes from, what everything is, and where everything is going. They accommodate the negations of source, self, and other. They count the unities and paradoxes of G-d and world, being and non-being, other and none other. There is nothing about anything, including nothingness itself, that is not described in one through nine.

Ten is really the nine in a different place. Just as the nine derive from the One, so does ten derive from the nine. Ten takes the prismatic web of their interaction and communicates it. Ten is the nine in transit and therefore in context, the complete whole repeating, source, self, and effect. Ten reflects the decimal nature of reality. After nine, the One is carried to a new column, and the digits count up again. Ten reveals that the nine are not self-contained, even counting the implicit one. Ten shows that the nine are a message, a signal in the night.

Ten is us, who hear the numbers, add them up, and turn our eyes toward the heavens and our own hearts, looking, always looking, for more.

[i] Incidentally, if you count to ten starting from zero, it yields eleven numbers. Eleven is a rough number in kabbalah, and better avoided.


Originally posted on Hevria.

A Piece of the Torah’s Pi

Here’s a semi-obscure controversy from the Hebrew bible that you may be aware of: The Torah (in two different locations,  I Kings 7:23 and II Chronicles 4:2) gives the incorrect value for π. For those who have forgotten since math class, Pi or π is the ratio of a circle’s circumference to its diameter. It has the same value for every circle, and though it is impossible to calculate its precise value it approximately equals 3.14159.

Here is one of the offending verses, from I Kings: “And he made the molten sea, ten cubits from brim to brim; it (was) round all about, and the height thereof (was) five cubits; and a line of thirty cubits did compass it round about.” And even if you have indeed repressed every memory of math class you probably see that thirty divided by ten is exactly three, and that three does not equal 3.14159.

The Torah using 3 as π’s value is sub-par whether or not you are religious. A secular observer would (and several have) note that other civilizations at the time of the events of I Kings had already approximated Pi much more closely, had at least realized that this fundamental constant of the universe exceeds the round number of three. And a religious observer obviously has trouble squaring how the eternal truths of mathematics can contradict a verse that clearly says the Molten Sea was thirty, ten, and round.

One answer to this quandary is that Jewish law obviously is willing to approximate when demanding of human beings to construct “perfect” circles. That is, the verse is not defining Pi, but explaining that the craftsmen of the Temple vessels measured a circumference of exactly 30 cubits and approximated the diameter, rounding 9.549 upward to ten. The Talmud notes that one is permitted to make approximations in measurement in the direction of stringency. In this case, using the absolute value of Pi would decrease the diameter of the Molten Sea, perhaps making it too small. The approximation is thus called for and perhaps even legally necessary so as not to err on the too-small side of the precise 9.549. Alternatively, the Lubavitcher Rebbe argued that the workmen used the exact 9.549 value, and the Torah was simply describing the circle with approximation, with the maximum precision necessary for practical purposes.

Either way, the sages of the Talmud were clearly aware that Pi does not equal three, as is stated clearly in the ancient Mishanat Ha’Middot, as Maimonides argues in his commentary on the Mishna, and is implied in the Talmud’s complex discussion of the required size of a round Sukkah in Tractate Sukkah 7b-8a.

However, the question remains — it is good and fine that the later sages were aware of more accurate values of Pi, and that the Biblical verses are mere approximations, but ultimately, the Book that is supposed to speak the truth presents a ratio that is misleading, and not misleading merely in practical matters but in a similar eternal truth of our universe, the hard-to-calculate, definitely-not-a-round-number phenomenon that is π.

An elegant and astonishing discovery of Rabbi Max Munk provides an answer of sorts, and to understand it we must first take a moment to speak of traditional Jewish biblical exegesis. The advent of computers has seen the rise of the controversial (and possibly downright-debunked) Bible Codes, an attempt to apply massive computing power to one ancient method of deriving truth from the Tanakh, that of counting spaces between the letters. However, most Orthodox Rabbis would tell you that it was the switch to computers that indicated the project was doomed to fail; like the story of the biblical Pi itself, a demand for ever-more “precision” tends to overlook the fact that the Torah was given to humans to grasp on the scale of human understanding. Besides, the Bible Codes apply only one method of exegesis, and far from the most common or important one.

Rabbi Munk, on the other hand, applies two better-established methods, and applies them not with a broad brush entering anything he can think of into a search box but with surgical precision, to the extent that it seems almost inconceivable that the verse should give rise to the meaning he discovered by accident. But you be the judge. Here is the verse that was translated earlier into English in the original Holy Tongue:

Rabbi Munk’s first method is to observe the parenthetical statement on the last line. It is a note on the three-letter word preceding it (remember, Hebrew read right-to-left). That word, וקו, is quite important to the verse; it is the “and a line” that refers to the 30-cubit circumference of the Molten Sea. The parenthetical statement tells the reader of the verse that even though the word is to be pronounced וקו (“V’kav”) it is written וקוה, (“V’kavah”). Now, the difference in the meaning of these two words is quite slight, the difference between “and a line” and “and its line.” However, the difference between the written form and the pronounced form has exegetical significance; the tradition of the book of Kings says that there are two superimposed realities in the verse, its written form and its pronounced form. One of the ways of dealing with this bifurcation is to view the pronounced version as the “revealed truth” whereas the written version is a deeper or “inner truth” of the verse.

Rabbi Munk’s second method is to apply Gematriah, the classic Jewish numerology in which each letter of the Holy Tongue is assigned a numerical value, to this verse. In the reckoning of the Gematriah, קו, the word that means “line” in the pronunciation of the verse, has a value of 106. The word that means “line” in the verse’s written form, however, is קוה, with a value of 111.

So, to sum up the two methods, the pronounced, revealed truth of the verse for the world has a value of 106. The written, secret, deeper truth under the surface, however, has a value of 111.

Rabbi Munk reasons that if we’re looking at a verse whose revealed meaning is problematic (because its approximation of a circle’s dimensions are so far off), maybe we can fix it by applying the verse’s hidden meaning. That is, we can perform an operation a little like dimensional analysis with the verse’s numbers. One can find the number of inches in three feet by multiplying (3 feet) x (12 inches / 1 foot), with the foot units cancelling out and leaving us with 36 inches. Similarly, our verse has a possible conversion: The revealed value of Pi into the verse’s deeper, truer meaning. The formula for this is:

(3, the revealed value of pi from the verse) x (the deeper truth of the verse, 111 / the “revealed” value of the verse, 106).

This, using only numbers put into the verse when the book of Kings was written, yields 3 x (111/106) = 3 x 1.04716981132 = 3.14150943396. And that is Pi to four decimal places.