Physics of Tzimtzum I — The Quantum Leap

Introduction

“In the beginning G‑d created the heavens and the earth.” (Genesis 1:1)

“In the beginning G‑d created the heavens and the earth,” the Torah says. However, what was before the “beginning”? It is like asking, What was before the Big Bang? In physics, until relatively recently, such questions were discouraged. The prevailing wisdom was that time and space had been created by the Big Bang, and there was no “before” before the Big Bang. Mishnah discourages such thinking, too. The sages point out that the first letter of the Torah, the letter bet, is open on the left and closed on the right:[1]

The text of the Torah and the history of the world proceed from that opening on the left. The closed right side of the letter bet visually walls off the beginning of the text. The sages interpret this as a message: What comes after this letter is open for us to learn, but what came before is closed to us.[2] We should not inquire into what “was” before the “beginning.”[3] However, this injunction never stopped Kabbalists from peering back behind the letter bet, beyond the beginning.

Modern cosmology no longer discourages this question either, and the field is abuzz with various proposals as to what might have existed before the Big Bang. One proposal, for example, imagines another universe that collapsed into a giant black hole, on the other side of which we find our present universe emerging with the Big Bang that is a mirror reflection of the collapse of the “mother” universe (to which we might still be connected by an Einstein-Rosen bridge, otherwise known as a wormhole).[4]

Medieval Kabbalists entertained such bold ideas centuries before modern cosmologists. What they described is one of the most mystical and profound doctrines in Kabbalah—the doctrine of Tzimtzum (primordial contraction).

In the next several posts, please G‑d, we will take a deep dive into the concept of Tzimtzum and analyze it in terms of physics and mathematics. We will uncover uncanny parallels not only with modern cosmology, but also with the quantum paradigm, the quantum-mechanical collapse of the wave function, and the procedure of renormalization employed in quantum field theory. As we shall see, the doctrine of Tzimtzum may have foreshadowed some of the most radical concepts in modern physics. So, buckle up and join me for a trip beyond the event horizon.[5]

The Problem

From the earliest times, Jewish sages, philosophers, and mystics were faced with the paradox of creation, which can be expressed in several ways:

1. How can an Infinite Being[6] give rise to finite worlds, when no continuous causal chain can lead from infinity[7] to finitude?[8]

2. How can the monistic Deity, who is absolutely one[9] in every sense of the word (the one and only,[10] the First Cause,[11] the unique existence,[12] solitary Being,[13] simple in His oneness[14]), create multifarious creations?[15]

3. How can creation[16] by G‑d, who is, by definition, absolute perfection,[17] result in an imperfect world?

4. How can any creature feel its independence, when there is no place devoid of G‑d’s presence?[18]

5. How can G‑d, who is utterly transcendent,[19] be involved in the affairs of this lowest of the worlds?[20]

6. How can we, mortal creatures, have a relationship with the infinite and transcendent G‑d?[21]

7. How can hidden G‑d be revealed when there is nothing outside of G‑d?[22]

These are different ways of asking the same question—how can we bridge between the infinite,[23] perfect,[24] and transcendent Creator and the finite, imperfect world, in which His providence is manifest?[25]

The paradox arises from the apparent logical incompatibility of the notions of G‑d and creation. Indeed, we can ask a question: Is creation a part of Infinite G‑d? There are only two possible answers—“yes” or “no”—but either answer gets us into trouble. If creation is part of G‑d, then, when created, it was an addition to G‑d. However, the argument goes, nothing could be added to infinity.[26] If we, on the other hand, assume that Creation is not part of G‑d, that would limit G‑d, which also contradicts our assumption of G‑d’s infinity.[27] Thus the notion of Infinite G‑d, who is the Creator, gets us caught on the horns of a dilemma.[28]

Medieval Jewish philosophers attempted to answer this paradox using the doctrine of emanations. This concept was originated by the Greco-Roman Neoplatonic philosopher Plotinus,[29] who described it in his Enneads. This concept was developed to bridge the gap between the perfect and infinite G‑d and the imperfect and finite creation, through a series of continuous, hierarchically descending radiations (“emanations”) from their source in the G‑dhead, through intermediate stages, to this physical universe. This metaphysical theory is known as emanationism. It implies a continuous process of creation, by gradual diminution and degradation of the Divine Light seen as the creative force emanating from the G‑dhead. Whereas original emanationism, as developed by Plotinus and his school, is at odds with creationism, which teaches creatio ex nihilo, Jewish Neoplatonic philosophers (such as ibn Gabirol in his Fons Vitae) reconceptualized emanationism as originating in G‑d’s will, so as to make it compatible with the Torah.

Kabbalah also adopts a paradigm of emanations in the context of the Jewish mystical doctrine of sefirot (“lights” or “numbers”), initially introduced in the Sefer Yetzirah (the “Book of Formation”),[30] Sefer HaBahir (the “Book of Illumination”),[31] and the Zoharic literature.[32]

However, emanationism does not solve the problem of the finite world emanating from the infinite G‑d because, as mentioned above, no continual causal chain can lead from infinity to finitude. Lurianic Kabbalah modifies emanationism by postulating Tzimtzum[33]—a discontinuous jump—a “quantum leap” from infinity to finitude by infinite contraction or concealment of the Divine Emanation.

The Quantum Leap

The answer designed to address the paradox of the finite world emerging from the infinite Creator was first explicitly formulated by Rabbi Isaac Luria Ashkenazi, known as the Ari.[34]  Although the books of Bahir and Zohar hint at Tzimtzum,[35] and it is first mentioned explicitly in the medieval Kabbalah book, Sefer HaIyun,[36] the Ari was first to explicate this doctrine of Tzimtzum—the infinite contraction, constriction, diminution, and concealment—perhaps something akin to what modern cosmologists would call the Big Crunch.[37]  This allows us to bridge the gap between the infinity and unity of G‑d, on the one hand, and the finitude and multiplicity of creation.

Most Kabbalists agree that Tzimtzum does not refer to the G‑dhead—i.e., the Essence (Atzmut or Ein Sof[39])—but rather to G‑d’s Emanation (Ohr Ein Sof[40]). Many Kabbalists and all Chasidic masters stress that the infinite contraction called Tzimtzum is not to be understood literally. As Rabbi Shneur Zalman of Liadi writes in Tanya, Tzimtzum occurred

…to conceal from created beings the activating force within them, enabling them to exist as tangible entities, instead of being utterly nullified within their source.” (Tanya, Sha’ar HaYichud veHaEmunah, ch. 4)

Rabbi Shneur Zalman interprets Tzimtzum as concealment of the Light of the Infinite (Ohr Ein Sof)—i.e., concealment of the Divine Emanation.

Furthermore, even the concealment of the Emanation (Ohr Ein Sof) is only an illusion, as it happens only from our point of view; from G‑d’s point of view, there is no concealment whatsoever.[41]

Let us read the description of Tzimtzum given in the name of the Ari by Rabbi Chaim Vital,[42] his principal student.

Prior to Creation, there was only the infinite Ohr Ein Sof (“Light of the Infinite”) filling all existence. When it arose in G‑d’s Will to create worlds and emanate the emanated…He contracted (“tzimtzum”) Himself in the point at the center, in the very center of His light. He restricted that light, distancing it to the sides surrounding the central point, so that there remained a void, a hollow empty space, away from the central point…

After this tzimtzum… He drew down from the Or Ein Sof a single straight line [of light] from His light surrounding [the void] from above to below [into the void], and it chained down descending into that void…. In the space of that void He emanated, created, formed and made all the worlds.” (Etz Chaim, Heichal A”K, anaf 2)[43]

The first sentence of this description lays out the scene “before”[44] the Tzimtzum. There was G‑d, referred to in Kabbalistic terminology as “the Infinite” or “Ein Sof” (i.e., “without end,” or “without limit”), which refers to G‑d “before” emanation. G‑d was[45] alone.

The Zohar describes Ein Sof before the emanation as follows:

Before He gave any shape to the world, before He produced any form, He was alone, without form and without resemblance to anything else.” (Zohar II, section “Bo,” 42b)

There was also the potential of G‑d’s emanation, called the Light of the Infinite or Ohr Ein Sof. There was nothing else. To be sure, the Light of the Infinite mentioned above was not physical—this is just an allegory used in Kabbalah for Divine Emanation. Just as light radiates from its source, so too emanation emanates from G‑d.[46] (As we shall see later, light turns out to be a good metaphor for the Divine emanation on deeper levels, too.) At this stage, there was no “space” for the creation of the world; there was no concept of physical space or time at all.[47]

When we read about the initial contraction at the center and the subsequent expulsion of the Divine Light, Ohr Ein Sof, from the center to the periphery to create space, one cannot miss the uncanny similarity to the cosmological models of the Big Bounce (the contraction followed by the expansion of the universe) or the contraction of the “mother” universe into the center forming a black hole, followed by the Big Bang expansion giving birth to a “daughter” universe—our universe—as we discussed at the start of this essay.[48]

In the first approximation, the lesson of Tzimtzum is that the creation was a “negative” process. Whereas we create by building up, G‑d created spiritual and physical universes by concealing (or removing) His light. To help understand this notion, Chasidic thought often uses a metaphor of speech. We speak by constricting our breath using our throat, palate, tongue, teeth, and lips. Similarly, G‑d created the world by constricting His light.

Another metaphor relates to light. In the days when movies were still analog (for those of us who still remember), they were recorded onto celluloid film, creating a negative, and then projected onto a screen. The negative images printed onto the film blocked the light of the projector (bright, pure, white light), thereby creating various images on the screen. Thus, showing a film in a cinema was a “negative process”—the process of restricting much of the light, while allowing a small portion to shine through. Similarly, the role of Tzimtzum is to restrict most of the Divine light, while permitting a minute amount to pass through—just enough to create multifarious creations.

According to most Kabbalists (and all Chasidic Masters), the infinite contraction (Tzimtzum) did not apply to G‑d Himself but only to the Divine Emanation—the Light of the Infinite (Ohr Ein Sof)—and only by way of (figurative, not literal) concealment.

The Lubavitcher Rebbe, Rabbi Menachem M. Schneerson, enumerates four different interpretations of Tzimtzum:[49]

The fourth interpretation of Tzimtzum—based on the teachings of Rabbi Shneur Zalman of Liadi, that Tzimtzum took place only in the Light of the Infinite (Ohr Ein Sof) and that it is not meant literally—has become almost universally accepted. In other words, Tzimtzum is merely an illusion.[54]

The quintessential property of Tzimtzum is its discontinuous nature. In mathematics, we have continuous functions that do not have any abrupt changes in their value and discontinuous functions that have abrupt changes in value. On a graph, a continuous function is represented by an unbroken curve over its entire domain, whereas a discontinuous function is represented by a broken curve.[55]

A continuous cubic function

A discontinuous function f(x)=1/x           it is undefined for x=0

The doctrine of emanationism involves a continuous series of emanations connecting the infinite source in the G‑dhead to the finite creation, through a continuously diminishing radiation (presumably describable by a continuous function). The doctrine of Tzimtzum breaks with this continuity and postulates a discontinuous process—an infinite contraction or concealment. This discontinuity necessitates a quantum leap[56] from Infinite G‑d to the finite and multifarious creations.

Kabbalah and Chasidic philosophy teach that the primordial Tzimtzum was followed by myriad subsequent tzimtzumim,[57] which contract G‑d’s light so as to allow for new spiritual realms whose quantity and quality of spiritual light are immeasurably diminished when compared to their predecessors.[58] Thus, a continuous flow of emanations in the philosophy of emanationism is replaced in Lurianic Kabbalah with a discrete quantum process of consecutive contractions and concealments (tzimtzumim) that diminish and occlude the Divine emanation in quantum leaps. It appears that this “quantization” of the Divine emanation expressed in the series of discrete contractions is parallel to—and, indeed, might be the primordial source of—the quantum nature of physical reality.

To understand Tzimtzum better, we need to look at what results from it and understand what has been accomplished. Only then can we walk our way back through the process, to understand how it was done and why it was necessary.[59] We will explore this, please G‑d, in the next installment on Tzimtzum.

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Endnotes

[1] Hebrew is, of course, read from right to left.

[2] Mishnah states, “Whoever reflects on four things it were better for him that he had not come into the world: “what is above? what is beneath? what is before? and what is after?” (Hagiga 2:1) Rabbi Jonah taught in the name of Rabbi Levi that the world was created with a letter bet, because just as the letter bet is closed at the sides but open in front, so one is not permitted to investigate what is above and what is below, what is before and what is behind. Bar Kappara also interpreted Deuteronomy 4:32 to say, “ask not of the days past, which were before you, since the day that G‑d created man upon the earth,” cautioning students not to speculate on what was before the day of creation. Midrash Genesis Rabba 1:10. (See in English Midrash Rabbah: Genesis. Translated by H. Freedman and Maurice Simon, volume 1, page 9. London: Soncino Press, 1939. See also Talmud, Chagigah 16a.

[3] Other Abrahamic traditions have also discouraged peering behind the creation. For example, when asked what G‑d did before the creation, Augustine of Hippo retorted, “G‑d was busy preparing hell for those who pry into mysteries!”

[4] We know a lot about what happened to the universe starting 10^-13 seconds after the Big Bang. We know a little about what happened between 10^-44 and 10^-13 seconds after the Big Bang, but we know absolutely nothing about what happened before Planck time—10^-44. We know that the theory that must govern physics at the time scale of 10^-13 seconds is quantum gravity, but this theory has not been developed yet. We also know that all modern physics falls apart at the time scale of 10^-44 seconds after the Big Bang, but again, that has never stopped theoretical physicists from speculating on what was before the Big Bang. One such speculative conjecture is that the steady-state universe that preceded the Big Bang was infinitely hot and infinitely dense, and a random quantum fluctuation caused the Big Bang. Another equally speculative proposal suggests that a mirror universe exists on the other side of the Big Bang, in which entropy increases toward the past (instead of toward the future, as in our universe), and consequently time flows in the opposite direction. Still others hypothesize the existence of another universe that collapses into a giant black hole, on the other side of which we find our present universe emerging with the Big Bang, that is a mirror reflection of the collapse of the “mother” universe (to which we might still be connected by an Einstein-Rosen bridge, i.e., a wormhole). There are yet more exotic proposals. For example, string cosmology leads to the “ekpyrotic” universe, which comes from the Greek word for the conflagration. In this theory, the Big Bang is not the beginning, but rather an event sparked by what happened in the previous universe. Then there is the popular Big Bounce proposal, in which the Big Bang is the point in time when contraction of our universe switched to expansion—this so-called cyclic cosmology is also motivated by string theory as well as other theoretical models. The stoics of ancient Greece already entertained cyclic cosmology (albeit without the fancy math of string theory), appropriately calling it ekpyrosis—“out of fire.” Yet another proposal suggests that our baby universe was torn off from the mother universe as the result of a quantum fluctuation. In still another model, a catastrophic collision of two universes floating in higher dimensions led to the Big Bang. Needless to say, all these proposals are pure speculation that is outside the boundaries of normative scientific inquiry—we have neither the theoretical framework to rigorously analyze them nor any experiments to confirm or falsify them.

[5] A trip beyond the event horizon usually involves falling into a black hole, whence there is no return. Fear not! I use this term here only figuratively and guarantee your safe return.

[6] The absolute infinity of G‑d, a quintessential feature of monotheism, originated with Judaism. Greeks did not believe in an infinite G‑d. Plato’s Demiurge was limited, and so was Aristotle’s Nous. Generally, the G‑d of philosophers is not infinite, because it is limited by human logic. Judaism introduced to the world a novel idea of G‑d, who was eternal, omnipresent, omnipotent, omniscient, unconstrained by human logic, and unlimited in every way. Rav Saadia Gaon (882–942) emphasized the infinity of G‑d in his Book of Beliefs and Opinions (Emunot VeDeot).  See also Maimonides, Guide for the Perplexed, 2.1, where he advances the cosmological argument as proof that an Infinite Being must exist. The belief in the eternity of G‑d is the fourth of his Thirteen Principles of Faith. The belief in the omniscience of G‑d is the tenth principle.

[7] The word “finite” can have a meaning different from the colloquial meaning of “limited” or as an antonym of “infinite.” In theology, “finite” can mean “not being able to account for its own existence and able to serve as the basis for a cosmological argument purporting to prove the existence of the Infinite G‑d.” The argument goes as follows: “If a person contemplates the finite nature of the world, i.e., the contingency of things and their inability to account for their own existence, one is liable to come to an understanding that there must be an infinite G‑d, who accounts for the existence of all entities and sustains the complexity of the world. See, for example, Thomas McPherson, Finite and Infinite, Mind, Volume 66, Number 263, July 1957, pp. 379–384, and E. L. Mascall, Existence and Analogy (London, 1949), p. 85 quoted here.

[8] As Rabbi Schneur Zalman of Liadi writes in Tanya, “The effect is encompassed by the cause, in relation to which it is essentially non-existent… Thus, even numerous contractions will not avail to there being matter…” (Tanya, Igeret HaKodesh, Kehot Publication Society, ch. 20).

[9] The belief in one G‑d is the sine qua non of monotheism. Jews are enjoined to declare their credo of monotheism twice daily by proclaiming, “Hear O Israel:  the Eternal our G‑d, the Eternal is one” (Deuteronomy 6:4).  In his Mishnah Torah, Maimonides states, “This G‑d is One, not two or more than two, but One whose unity is different from all other unities that there are. He is not one as a genus, which contains many species, is one. Nor is He one as a body, containing parts and dimensions, is one. But His is a unity that which there is no other anywhere” (Yad, Yesode Ha-Torah 1:7). It is the second of the Thirteen Principles of Faith of Maimonides: “The belief in G‑d’s absolute and unparalleled unity.” The belief in one G‑d that pervades all existence, while transcending time and space, is called monistic panentheism. The conception of G‑d in Kabbalah, particularly as it is interpreted by Chabad Chasidic philosophy, falls under this rubric.

[10] In the sense that there are no other gods, as the Prophets said, “Is there a G‑d beside Me?” (Isaiah 44:8); “…And thou knowest no G‑d but Me, and beside Me there is no savior” (Hosea 13:4). Thus, Hannah prayed, “There is none holy as the Eternal, for there is none beside Thee; neither is there any rock like our G‑d” (I Samuel 2:2). The Psalmist also declared, “For who is G‑d, save the Eternal? And who is a Rock, except our G‑d?” (Psalms, 18:32).

[11] “Before Me there was no G‑d formed, neither shall any be after Me” (Isaiah 43:10). “I am the first, and I am the last, and beside Me there is no G‑d” (Isaiah 44:6).

[12] As the Prophet said, “To whom then will ye liken Me, that I should be equal? saith the Holy One” (Isaiah 40:5). Similarly, the Psalmist wrote, “For who in the skies can be compared unto the Eternal…” (Psalms 89:7). For another approach to this uniqueness, see Maimonides, Mishnah Torah, Hilchot Yesodei Hatorah 1:1., where Maimonides explains that G‑d’s existence, unlike any other, is a necessary existence. See also Rabbi Menachem Mendel Schneerson, Toras Menachem: Hadranim al HaRambam V’Shas (Kehot Publication Society, 1992).

[13] Nothing can be said to truly exist besides G‑d, as the existence of the spiritual and physical universes are contingent on the will of G‑d and His continuously maintaining their existence. Thus, the Torah states, “Know this day, and lay it to thy heart, that the Eternal, He is G‑d in heaven above and upon the earth beneath; there is none else” (Deuteronomy 4:39). See also Rabbi Schneur Zalman of Liadi, Tanya, Sha’ar Yichud VeHaEmunah, Chapter 1, et seq.

[14] That is, incorporeal. As the Torah states, “Take ye therefore good heed unto yourselves—for ye saw no manner of form on the day that the Eternal spoke unto you in Horeb out of the midst of the fire (Deuteronomy 4:16). See also Maimonides, who wrote, “There is one simple essence in which there is no complexity or multiplicity of notions, but one notion only; so that from whatever angle you regard it and from whatever point of view you consider it, you will find that it is one, not divided in any way and by any cause into two notions …” Guide 1.51. Moreover, the belief in G‑d’s non-corporeality is the third of Maimonides’ Thirteen Principles of Faith.

[15] As the Psalmist wrote, “How manifold are Thy works, O Eternal!  In wisdom hast Thou made them all; the earth is full of Thy creatures” (Psalms 104:24). How can this multiplicity of creatures—and even more so, the duality of good and evil—come from absolute oneness?

[16] Creation de novo and ex nihilo are novel ideas introduced by the Torah. While Plato believed that the world was created de novo, he maintained it was created from a preexisting material substrate. Aristotle thought that the world existed eternally and, although dependent on a deity, was co-eternal with it. Creation from noting—creatio ex nihilo—is Judaism’s innovation. As it says, “He stretcheth out the north over the empty space, and hangeth the earth over nothing” (Job 26:7).

[17] As Rabbi Meir ibn Gabbay (1480–1540) stated, “The Ein Sof is the ultimate perfection” (Avodat HaKodesh, part I, ch. 8). The belief that G‑d is perfect in the first of the Thirteen Principles of Faith of Maimonides: “Belief in the existence of the Creator, who is perfect in every manner of existence and is the Primary Cause of all that exists” [emphasis added].

[18] As it says, “Holy, holy, holy, is the Eternal of hosts; the whole earth is full of His glory” (Isaiah 6:3).

[19] Judaism holds that G‑d utterly transcends all worlds and human logic. Philosophers, particularly those outside the Abrahamic faith traditions, generally do not accept this belief and limit G‑d to what is logical. Leibnitz, for example, wrote that G‑d was only able to create a world that was logically possible. (According to some, Spinoza may have been the only major philosopher who believed in a transcendental G‑d.)

[20] As the Psalmist wrote, “Who is like unto the Eternal our G‑d, that is enthroned on high, that looketh down low upon heaven and upon the earth? Who raiseth up the poor out of the dust and lifteth up the needy out of the dunghill…” (Psalms 113:5-7).

[21] Rabbi Sholom Dovber of Lubavitch (the Rebbe Rasha”b), Sefer Hma’amorim, 5672, ch. 18–19.

[22] Rabbi Azriel of Gerona. (See Gershom Scholem, Kabbalah, Dorset Press, 1974: New York, p. 90.)

[23] The theological notion of G‑d’s infinity is different from mathematical or philosophical notions of infinity. Metaphysics differentiates between potential infinity (e.g., the infinity of natural numbers), actual infinity (e.g., the infinity of real numbers), and transcendental (as exceeding human understanding) infinity. By G‑d’s infinity, theology understands transcendental infinity and, more broadly, the lack of any limitations on G‑d’s abilities. Therefore, in Kabbalah, G‑d is called Ein Sof, i.e., Infinite (lit. “without end”). While some metaphysicians and theologians identify G‑d with transcendental infinity, this is not the view of Judaism, in whose view G‑d transcends any definition and is unlimited even by infinity. Thus, G‑d possesses powers of infinity and finitude, which is to say that G‑d is limited neither by finitude nor by infinity. As the fifteenth-century Kabbalist Rabbi Meir ibn Gabbai wrote, “Just as He has power in the realm of the infinite, so too He has power in the realm of the finite. For should you attribute infinite power to Him but not attribute finite power to Him, you are diminishing His perfection” (Avodat Hakodesh, part 1, chapter 8.)

[24] Although many medieval Jewish philosophers who greatly influenced the development of Jewish thought in general and Kabbalah, in particular, were Neoplatonists, here is where they break rank with Plato. Plato considered G‑d to be perfect. However, like most Greek philosophers, Plato viewed infinity as shapeless, boundless, indefinite, chaotic, and unintelligible—and, therefore, imperfect. Consequently, Plato did not consider G‑d to be infinite. Jewish philosophers, on the other hand—even those who otherwise followed Plato—all believed in the infinity of G‑d, whose abilities cannot be limited in any way. Many of them (such as ibn Gabirol) followed Plotinus, a Neoplatonist who believed that G‑d was infinite.

[25] “Who is like unto the Eternal our G‑d, that is enthroned on high, that looketh down low upon heaven and upon the earth? Who raiseth up the poor out of the dust and lifteth up the needy out of the dunghill” (Psalms 113:5–7). See also the first and the tenth of the Thirteen Principles of Faith of Maimonides, cited above.

[26] For example, any number a added to infinity results in infinity: ∞ + a = ∞ (the mathematical symbol ∞ denotes infinity). Ergo, nothing can be added to infinity.

[27] See Yitzchak Block, “Creation: The Argument from Finitude,” B’Ohr HaTorah, 11 (1999), p. 12.

[28] I tend to think that there are few logical problems with constructing this paradox. First, any question whether or not something is a part of G‑d is illogical, because G‑d has no parts by definition—Infinite G‑d is incorporeal. Asking if something is a part of G‑d is setting up a logical contradiction ab initio. Second, there is no logical prohibition to add to infinity. In fact, in mathematics, an expression such as ∞ + a = ∞ is perfectly legitimate. Although infinity cannot be increased by adding a number to it (it is still infinity), the operation is permitted and is meaningful in the number theory or analysis. Thus, it is incorrect to say that adding something to infinity is a contradiction—far from it. Third, G‑d’s infinity cannot be understood in quantitative terms—G‑d has no measure, because He has no body, and G‑d has no parts because He is incorporeal. The infinity of G‑d is, first and foremost, His transcendence over our understanding and His ability to contain paradoxes. Thus any paradoxical construct attributable to G‑d is no surprise—G‑d is a self-referential and self-contradictory construct by definition.

[29] Plotinus (c. 204/5–270) was a major Hellenistic philosopher who lived in Egypt (which was at the time part of the Roman Empire). He was a Neoplatonic philosopher whose philosophy is described in Enneads. Plotinus was the first Hellenistic philosopher to ascribe infinity to G‑d—a point of departure from Platonic tradition. Plotinus had a significant influence on medieval Jewish philosophers, particularly Solomon ibn Gabirol (Avicebron) and, to a lesser extent, Maimonides.

[30] Sefer Yetzirah, I. See in English, Aryeh Kaplan, Sefer Yetzirah: The Book of Formation. (Samuel Wiser, 1990), chapter 1, pp. 19 ff.

[31] Sefer HaBahir, 124. See in English, Aryeh Kaplan, The Bahir (Samuel Wiser, 1979), pp. xix–xx, 47, 104.

[32] Tikkuney Zohar, Patach Eliyahu. (See English translation, for example, in Siddur Tehilat Hashem with English Translation, Kabbalat Shabbat.)  Sefirot are a major theme of the Zohar and appear throughout the text.

[33] Etz Chayim II.

[34] Rabbi Isaac Luria Ashkenazi (1534–572), a.k.a. Ha’Ari (the Lion), the Ar”i HaKadosh (holy Ari), or Ariza”l (Ar”i of blessed memory), was a master Kabbalist and the founder of the Safed school of Jewish mysticism known as Lurianic Kabbalah. He is considered the ultimate and undisputed authority on Kabbalah, and all later Kabbalists and Chasidic masters based their teachings on Lurianic Kabbalah.

[35] See the introduction to The Bahir by Aryeh Kaplan loc cit., p. xxiii.

[36] In a preface to a commentary on the 32 Paths of Wisdom. See Gershom Scholem, Kabbalah (New York: Dorset Press, 1974), p. 129.

[37] The Big Crunch is the Big Bang in reverse—i.e., the process of contraction of the universe back into an infinitesimally small point. According to cosmological models based on Einstein’s general theory of relativity, depending on the average density of matter in the universe, the process of expansion may reverse itself into contraction, leading to the Big Crunch. Current observations confirming accelerated expansion of the universe contradict the prediction of the Big Crunch. To be sure, this metaphor is not to be taken literally with respect to Tzimtzum, which did not take place in a physical universe and is ultimately an illusionary contraction of the Divine Light.

[38]

[39] Ein Sof means “Infinite” (lit. “without end,” i.e., “boundless”). This Kabbalistic appellation for G‑d before the emanation signifies the level above any names ascribed to G‑d. As the Zohar states, “Who then can comprehend how He was before the Creation? Hence it is forbidden to lend Him any form or similitude, or even to call Him by His sacred name, or to indicate Him by a single letter or a single point” (Zohar II, section “Bo”, 42b). Most classical Kabbalah sources identify the G‑dhead with Ein Sof. In Chasidic philosophy, however, it is not the ultimate essence of G‑d, which is referred to as Atzmut, i.e., the “Essence.”

[40] Ohr Ein Sof (literally, the “Light of the Infinite”)—the primordial Infinite Light signifying the original infinite divine emanation.

[41] See Rabbi Schneur Zalman of Liadi, Tanya, ch. 36.

[42] Rabbi Chaim (or Hayyim) ben Joseph Vital (1542–1620) was one of the most prominent Kabbalists of the Safed school of Kabbalah and the foremost student of Rabbi Isaac Luria, known as the Ar”i. The Ari did not commit any of his teachings to writing. Thus, we are only able to study them as recorded by his students. He transcribed and transmitted his master’s teachings in many volumes, chief among them Etz Chaim, which became the foundational text of Lurianic Kabbala, second only to the Zohar.

[43] English translation by Rabbi Moshe Miller, First Constriction, Introduction to the Ari’s concept of ‘tzimtzum,’ https://www.chabad.org/kabbalah/article_cdo/aid/380815/jewish/First-Constriction.htm (accessed on 8/16/2020).

[44] Since Tzimtzum occurred before the creation of time, the word “before” must not be understood in a temporal sense but rather in a conceptual sense. For example, in the set of integers, the number 3 comes before the number 4. The term “before” can be applied to a pair of elements in any ordered set, to indicate that one element precedes the other element. Similarly, in any hierarchical structure, one level of the hierarchy can be said to precede another level, which is the case here, as we are talking about the level above Tzimtzum.

[45] The word “was,” as the word “before,” is not to be understood in a temporal sense, but rather in a conceptual sense, signifying the level “before” (i.e., above) Tzimtzum. On that level, G‑d is still alone, as this level is above time.

[46] In this sense, Ein Sof is the Emanator and the source of the emanation—Ohr Ein Sof. Using the metaphor of light, Ein Sof is referred to as the Ma’or (“Luminary”) and Ohr Ein Sof as the Ohr (the “Light”).

[47] This begs the question: How can there be an emanation before the creation of space? If the word “emanation,” as discussed above, is a radiation from the source, that relationship seems to imply that the radiation is outside the source. However, before Tztimtzum (and after, according to Chasidic philosophy), G‑d filled all of existence. There was (and is) no “outside.” This can be understood, perhaps, by comparing emanation with speech and thought. Speech is projected outside the speaker. Thoughts, on the other hand, are hidden to those on the outside. Even if a person thinks in words, as many of us do, this internal speech—which is also an emanation—stays internal and does not require an outside listener. Similarly, the Divine Emanation before Tzimtzum could be allegorized as G‑d’s thoughts—an internal emanation, as it were.

[48] To be sure, this parallel is not to be understood literally. Whereas cosmology studies the physical universe, Kabbalah talks about spiritual universes and abstract (non-physical) concepts. There is nothing whatsoever physical about Tzimtzum. For a mystic attempting to achieve closeness (or deeper understanding of) godliness, it is critical to shed all physical associations. However, I had no intention of taking my readers on a mystical journey—our only quest is to find structural parallels between Torah and Jewish mysticism on the one hand, and modern physics on the other.

[49] Igrot Kodesh, vol. 1, p. 19.

[50] This opinion was apparently held by Rabbi Eliyahu of Vilna, known as the Vilna Gaon or the GR”A. According to some sources, the GR”A believed that the Tzimtzum was to be understood literally. See Allan Nadler, The Faith of the Mithnagdim: Rabbinic Responses to Hasidic Rapture (Johns Hopkins Jewish Studies) JHUP, 1999. See also: Tamar Ross, “Two Interpretations of the Theory of Tzimtzum: Rabbi Hayyim of Volozhin and Rabbi Schneur Zalman of Liadi,” Mehkarei Yerushalayim be-Mahsheveth Yisrael, 2 (1982) pp. 153-69; and Norman Lamm, Torah Lishmah: Torah for the Torah’s Sake: In the Works of Rabbi Hayyim of Volozhin and His Contemporaries (Sources and Studies in Kabbalah, Hasidism,) (New York, 1989), p. 99 n139. Others, however, maintain that it is not so and that the GR”A interpreted Tzimtzum also figuratively, but believed that people not versed in the esoteric tradition of Kabbalah are incapable of grasping these mysteries. See E. J. Schochet, The Hasidic Movement and the Gaon of Vilna. Jason Aronson, Inc. (1993). This debate was at the root of the conflict between Chasidim (particularly, Chabad Chasidim, the followers of Rabbi Schneur Zalman, who was a staunch proponent of the first view—that Tzimtzum is not to be understood literally) and Mitnagdim (the “Opponents”), who followed the Vilna Gaon. The Lubavitcher Rebbe, Rabbi Menachem Mendel Schneerson, asserted that the GR”A’s principal pupil, Rabbi Haim of Volozhin, disagreed with his master. Having been exposed to the teachings of Rabbi Schneur Zalman in Tanya, his explanation of Tzimtzum matches the third interpretation in the matrix above. See Igrot Kodesh, vol. 1, p. 19

[51] Sha’ar Hashamayim; Shomer Emnunim; Rabbi Chayim of Volozhin, Nefesh HaChayim.

[52] Mishnat Chasidim; Reb Yonatan Eibishetz.

[53] The Alter Rebbe-Rabbi Schneur Zalman of Liadi.

[54] See the introduction by Rabbi Aryeh Kaplan to The Bahir, p. xxiii.

[55] More precisely, a function f with variable x is said to be continuous at the point c on the real line, if the limit of f(x), as x approaches the point c, is equal to the value f(c): Iim f(x)=f(c); the function is said to be continuous if it is continuous at every point x. A function is said to be discontinuous at some point when it is not continuous there.

[56] To be sure, this metaphor is not to be taken literally—quantum leaps or, more precisely, quantum jumps, which are abrupt transitions of a quantum system from one quantum state to another (e.g., when an electron jumps from one atomic orbit to another), are finite, whereas Tzimtzum is an infinite leap from infinity to finitude.

[57] Tzimtzumim is the plural form of tzimtzum—i.e., “contractions.”

[58] Ma’amarei Admu”r HaZaken; The Tzemach Tzedek (Rabbi Menachem M. of Lubavitch), Ohr Hatorah, Inyanim; Rebbe Rashab, Hagohot L’Dibur HaMatchil Patach Eliyahu; Rebbe Rasha”b, Sefer HaMa’amorim, 5666.

[59] Needless to say, words like “was” are not to be understood in a temporal sense—Tzimtzum is outside time and, therefore, from our perspective, continuously occuring.