In the first installment on Tzimtzum (see “Physics of Tzimtzum I — The Quantum Leap”), we gave a general overview of the mystical doctrine of Tzimtzum—the cornerstone of the Lurianic Kabbalah. It is time to get into the details.

The first phrase that describes the process of Tzimtzum states:

Ein Sof “contracted” (tzimtzum) Himself in the point at the center, in the very center of Ohr Ein Sof. (Etz ChaimHeichal A“K, 2)

This sentence raises several difficult questions:

1.   What could it possibly mean that the Infinite (Ein Sof) “contracted” (tzimtzum) Himself? In Hebrew, the word tzimtzum comes from the root tzom, which means “to diminish” or to “fast,” i.e., to “diminish” oneself.[1] It can also mean “to be precise,” i.e., to remove ambiguity.[2] The repetition of this root is a grammatical form of “doubling down” or emphasizing the meaning in the extreme.[3] Thus, the traditional translation of tzimtzum is “contraction” or “contracted.” However, the word “contracted” means “became smaller in size or volume.”[4] Needless to say, this meaning cannot apply to Infinite God, who has no body, form, size, or volume whatsoever. Such reading would clearly contradict the core principles of monotheism, which postulate that God is infinite and incorporeal. Moreover, “size” or “volume” are geometric concepts that exist in a metric space (i.e., space where the distance between two points is defined). No such space exists at this stage, as space is yet to be created. As we see, the expression “Ein Sof contracted Himself” cannot be understood literally. Even the remote possibility of such literal interpretation is so abhorrent to monotheistic beliefs that many authors translate tzimtzum as “concealed” instead of “contracted.” However, far from the literal translation of the word tzimtzum, this rendition merely sweeps the question under the rug. So, what does this expression, “Ein Sof contracted Himself,” mean?

2.   The next few words—Ein Sof “contracted” (tzimtzum) Himself in the point at the center”—presents another problem. What is a “center” in this context? The word “center” has several meanings in physics and mathematics. In physics, we use the expression “center of mass.”[5] We also use the expression “center of gravity.”[6] However, any matter to which these concepts may apply is yet to be created. So, these meanings of the word “center” are of little help to us. In mathematics, a “center” means a point that defines a circle on a plane or a sphere in space (or a hypersphere in a multi-dimensional space) as a set of points equidistant from the center. Also, the word “center” implies the presence of spherical symmetry.[7] However, physical space has not been created yet. So, what does the “center” mean in relation to Ein Sof? Similarly, the point mentioned in the expression “in the point at the center” is a geometric concept. What could it mean in the context of Tzimtzum before the creation of physical space?

3.   Lastly, the narrative starts with Ein Sof contracting Himself in the point at the center but then clarifies that this center is in the very center of Ohr Ein Sof. Why the sudden change from Ein Sof to Ohr Ein Sof?

As we shall soon see, the answer to all of these questions lies in the notion of conceptual space (or, in the language of mathematics, abstract space). We will conclude that that all geometrical terms used above (“point,” “center,” “contracting”) must be interpreted in the context of conceptual space.

In this section, we shall analyze the end-product of Tzimtzum—the makom hapanui (the “empty space” or “vacated space”) or chalal (the “vacuum”) to understand what happened during Tzimtzum. In the end, we will draw an analogy between the first phase of Tzimtzum, “contracting,” with the quantum-mechanical collapse of the wave function.

This is the first of two sections dedicated to makom hapanui. In the first, we will discuss resolving contradictions, and in the second, removing infinities. But first, we need to understand what this space created by Tzimtzum is.

 

Conceptual Space

Let us start with the second question—What is the nature of the point in the center, and what is the meaning of the center itself?

The biggest mistake one can make thinking about Tzimtzum is to imagine that the “empty space” created by Tzimtzum was our three-dimensional physical space. The physical space is not to be created until many stages later, two universes down the path of the ontological order of creation called Seder Hishtalshelut, i.e., the “chain-like order.”[8]

At the same time, the geometric concepts of point and center mentioned in the Etz Chaim, unmistakably point to space. If this space is not our familiar physical space, then what kind of space is it?

It is neither the khôra[i] (Greek for “space”) of Plato[9] or is it topos[10] (Greek for “place”) of Aristotle.[11] It is a conceptual space.

A conceptual space is a geometric structure that represents a number of quality dimensions, which denote basic features by which concepts and objects can be compared. In the traditional conceptual space,[12] points denote objects, and regions denote concepts. Conceptual space may include physical space as its sub-space. Since before the Creation, there were no objects, we shall consider a conceptual space where points denote ideas rather than objects. Our conceptual space is the space of ideas.

A simple example of a conceptual space is a color space. We can construct a three-dimensional space based on the RGB (red-green-blue) color model wherein the X-axis (dimension) is color red, the Y-axis (dimension) is color green, and the Y-axis (dimension) is color blue. Since any color could be represented by a triplet of numbers representing respective values for the red, green, and blue color, we can model each as a point in a three-dimensional space wherein three coördinates represent the respective RGB values. Another example is a face space (not be confused with phase space used in physics)—a multi-dimensional space where human faces are stored as points in space based on their invariant features.

Conceptual space employed in the doctrine of Tzimtzum is a space where ideas are represented by the points in this space, i.e., each point is “occupied” by an idea according to its qualities. Distance between points is determined by the similarity of the ideas these points represent. Two ideas that are equivalent to each other occupy the same point (having the same coördinates) in the conceptual space. In other words, ideas A and B occupy the same point in the conceptual space if and only if A = B. Two ideas that are opposite, A and not-A, are infinitely far apart. The more similar two ideas are to each other, the closer they are in the conceptual space. The distance between two ideas is the measure of the conceptual similarity between the two.[13]

For example, kindness is an idea. One can define a dimension of “kindness.” Two equally kind people (or two acts of equal kindness) would occupy the same point on the kindness line. The one who is a more kind would be on the right from the person (or act) chosen as the reference point, and one who is less kind would on the left. We could add another dimension, honesty, to the dimension of kindness. Two equally honest people would occupy the same point on the honesty line. The more honest one would be on the right from the person chosen as the reference point, and the less honest one would on the left. These two dimensions taken together would constitute a plane—a conceptual two-dimensional space. We can add to it a third dimension, say, humility, to obtain a three-dimensional conceptual space, etc. [14]

Instinctively, we use the notion of the conceptual space in our everyday speech without realizing it. When we characterize a politician as one who is on the right off-center (or left off-center) we imply that in a conceptual political space (this space is one-dimensional—a line) this politician is on the right off-center (say, the center of a particular party) on that line. However, there may be a need for more granularity. In this case, the one-dimensional line may be replaced by a two-dimensional plane where one axis represents one’s attitude to fiscal issues and a perpendicular axis represents one’s attitude on social issues. A person may characterize herself as a social liberal and fiscal conservative occupying a dot, whose (X, Y) coördinates represent particular values on the respective axis.

The first to introduce the conceptual space was one of the oldest books of Kabbalah, Sefer Yetzirah (the Book of Formation):[15]

A depth of beginning, a depth of end;

A depth of good, a depth of evil;

A depth of above, a depth of below;

A depth of east, a depth of west;

A depth of north, a depth of south.[16]

Each line in this passage defines one dimension in a five-dimensional conceptual space by describing the opposite extremes (“depths”) of each direction. For example, “A depth of beginning, a depth of end” defines the dimension of time extending infinitely into past and future (from the reference point of the observer who is in the present). Similarly, “a depth of good, a depth of evil” defines a spiritual or “moral” dimension. The other lines describe our physical three-dimensional space. In this picture, our four-dimensional physical spacetime continuum is embedded in five-dimensional conceptual space, in which an extra dimension is a spiritual or moral dimension:

Dimension – ∞ + ∞ Meaning
Time A depth of beginning (past) A depth of end (future) Physical time
Moral dimension A depth of good A depth of evil Spiritual dimension
Up-Down direction A depth of above A depth of below Three-dimensional physical space
East-West direction A depth of east A depth of west
North-South direction A depth of north A depth of south

 

As Rabbi Aryeh Kaplan stresses in his introduction to The Bahir, the sages of Kabbalah were well aware that the space created by the Tzimtzum was not a physical but a conceptual space.[17]

Now we can explain the passage from Etz Chaim in the context of the conceptual space and answer the second question. In this space, a point is an idea, and the point at the center is the central idea—the idea of God. We have nothing to say about God per se, as He is utterly unknowable. Everything we say about God is not really about God per se, but about our idea about God, i.e., about godliness (Elokut).[18] In Kabbalah, this is called God’s Name. Certain ideas we hold about God are reflected by different names.[19] Divine names, their meaning, manifestations, as well as their use in meditative practices are major topics of study in Kabbalah. In this sense, Kabbalah is an epistemic doctrine that is concerned with our ideas or knowledge of God, rather than with God per se, who is unknowable.

We note an interesting parallel between Kabbalah and quantum physics insofar as they both are epistemic theories[20]. Quantum mechanics, at least in the traditional Copenhagen interpretation, does not describe the physical objects per se, but only their wavefunction, which represents our knowledge about the physical objects. From this point of view, quantum mechanics is also an epistemic theory.

It becomes now clear, why the narrative had to turn from Ein Sof to Ohr Ein Sof. Ein Sof, in His ultimate simplicity,[21] has nothing that could even remotely be identified as a “center.”[22] Ohr Ein Sof, however, has a natural center. In the metaphor of light, radiated light (ohr) implies a luminary (ma’or) as the source of the radiation at the center. Similarly, the emanation (for which light serves as a metaphor) is, by definition, the radiation from the source.[23] Thus, the center of emanation—Ohr Ein Sof—is the luminary (ma’or)—Ein Sof, as the source of the emanation.[24]

However, this makes our first question even more difficult. According to our interpretation of the center as the idea of Ein Sof, the first part of the narrative can now be restated as Ein Sof[25] contracts Himself into a point that is an idea of Ein Sof. What does it mean? The meaning of this will become apparent soon enough.

 

Makom hapanui—the Empty Space

To understand the meaning of the word “contracted,” we need to examine the result of this action. As a result of Tzimtzum, makom hapanui (a “vacant place”), or chalal (a “void”—the primordial vacuum) was created. Makom hapanui[26] is not a physical space; it is an abstract space. Because of the importance of this concept to Tzimtzum, we need to understand what is an abstract space.

In mathematics, space is defined as a set (i.e., a collection of elements) with a certain structure (topological[27] and geometrical[28]). If the elements of a set are points of the physical space, then such space, indeed, is a physical space. Abstract space may include the physical space as its sub-space. However, the elements of the set do not have to be points of the physical space. They could be moments in time,[29] people, cars, states of a physical system, or concepts (leading to the conceptual space discussed above). We may still choose to call these elements “points” so long as we remember that these are not points of the physical space.

In mathematics, such space is called an abstract space (or simply space, as all spaces in mathematics are abstract). In Kabbalah, space is a gevul, i.e., “limitation,” as it limits the shape, size, and position of the object in physical space. In mathematics, abstract space is also a limitation, because space is a set limited by its topological and geometrical structure. Conceptual space is a particular case of a more general abstract space.

In physics, we also use abstract spaces. The simplest example is a configuration space used in classical mechanics. The configuration space of a system of particles allows representing the position of a system of particles as a single point in a multi-dimensional space. For a single particle, configuration space is the same as the regular three-dimensional Euclidean space, where each point has coördinates (x, y, z). The configuration space of a system of two particles is six-dimensional.[30] The configuration space of a system of n particles has 3n dimensions—three dimensions for each particle.[ii]

Another important example of abstract space is a phase space used in the dynamical system theory and analytical mechanics. Phase space is a space where points represent all possible states of the system, with each point uniquely representing one particular state of the system.[iii]

A notable example of abstract space is a Hilbert space, which is a generalized version of the phase space used in quantum mechanics and other areas of physics. Hilbert space is a generalization of the Euclidian space (our ordinary three-dimensional space) on any number of dimensions—finite of infinite. In quantum mechanics, all possible states of a quantum-mechanical system are represented as vectors in an infinite-dimensional Hilbert space.[iv] These are all examples of abstract spaces used in physics.

As stated above, the “space” or “void” created by the Tzimtzum is not a physical space[31] but an abstract space. Needless to say, anything that existed before the Creation is beyond our understanding. However, as a rough approximation, perhaps we can think of the space created by Tzimtzummakom hapanui—as a phase space of sorts or a Hilbert space. Perhaps, it can be viewed as a potential phase space of the universe to be created, in which every possible state of the future universe is represented by a unique point.[32] However, since the universe is not created yet, and each potential state in the future is in a superposition of all possible states, this requires a phase space of infinite dimensions. Consequently, this space may be viewed as a Hilbert space. Armed with this understanding, we can now proceed to analyze the process of the Tzimtzum that led to the creation of this space.[33]

 

Sweeping Contradictions under the Rug or Collapse of the Wave Function

So, what does Tzimtzum represent, and why is it necessary? Before we propose an answer to these questions based on the conceptual space model, let us first get some intuition about it using the classic Chasidic metaphor for Tzimtzum—the teacher and a student.

In the teacher-student (in Hebrew, Rav and talmid) metaphor, the teacher is presumed to not only to possess infinitely greater knowledge that his pupil, but to possess a holistic view of the entire Torah (or whatever subject the teacher is teaching). The latter is particularly important to our discussion, because Torah (both written, Torah SheBiKhtav, and oral, Torah SheBe’alPe) is full of what seem to be dialetheia, i.e., contradictory statements. Only a person who mastered the entire Torah (both written and oral) can see the wholistic picture were all pieces of the puzzle fit perfectly together. Such sage can embrace the dialetheism[34] of the Torah. Moreover, as F. Scott Fitzgerald wisely noted, “The test of a first-rate intelligence is the ability to hold two opposed ideas in the mind at the same time, and still retain the ability to function…”[35] While this is true of the teacher in our metaphor, this cannot be presumed about the student, who will experience acute cognitive dissonance upon entertaining two opposite ideas. Thus, the teacher must contract (tzimtzum) his understanding and vacate his mind off the holistic picture of the subject he knows so thoroughly, to be able to descend to the level of his student to teach him one idea at a time. As we see in this metaphor, hiding contradictions is an essential aspect of Tzimtzum. Armed with this intuitive understanding, we can now proceed to examine this matter more rigorously.

As we just concluded, the Tzimtzum resulted in the creation of abstract space—a potential phase space—where each possible state of the future universe would be represented by a unique point. As was mentioned above, in any conceptual space, two contradictory ideas, such as statement A and its negation, non-A, would be represented by two points infinitely far apart from each other. However, God is the ultimate self-referential and self-contradictory construct. God is the master of all possibilities—there is nothing impossible for God. He can be in a state A and a state non-A at the same time,[36] as it were. God is infinite and perfect, and therefore He lacks nothing. It follows that absolute infinity and perfection requires that He does not lack finitude. In other words, as all-powerful[37], Omnipotent, He possesses the powers of infinity and of finitude.[38] A contradiction!

God can be in any state and the opposite state. He can be in all states, as nothing is impossible for God. The Sages termed this self-contradictory aspect of God as nimna hanimna’ot (literally “restricting [all] restrictions,”) i.e., the “paradox of paradoxes”.[39] This enigmatic state of nimna hanimna’ot, of being in contradictory states A and not-A, is reminiscent of what is called in quantum mechanics “superposition of states.” Indeed, God, as Infinite Being, can be and is in a superposition of all states, as it were.[40] God is the ultimate paradox. This paradoxical nature of Ein Sof extends to the Divine emanation, Ohr Ein Sof, as God manifests His powers of infinitude and finitude together.[41]

So long as God’s emanation filled all of the existence,[42] there could not be any conceptual space where two contradictory ideas may not occupy the same point but are infinitely far from each other. Thus, before the conceptual space could be created, God had to conceal the self-contradictory nature of His Light—Ohr Ein Sof (which reflects the self-contradictory nature of Ein Sof), i.e., He had to separate His ko’ah hagvul (powers of finitude) from ko’ah bli gvul (powers of infinitude) to hide the paradoxical aspect of nimna hanimna’ot.[43] Without such concealment, no conceptual space, no abstract space, let alone Creation, would be possible. The necessity to restrict the contradictions in order to create a conceptual space makes Tzimtzum necessary.[44] In a manner of speech, Tzimtzum is God’s way of sweeping the contradictions under the proverbial rug.[45]

It is somewhat akin to the collapse of the wave function[46] in quantum mechanics.[47] Before the collapse, a particle may be in a superposition of two contradictory states. For example, an electron can be in a superposition of spin-up |↑〉 and spin-down |↓〉 states (which is analogous to a classical particle rotating clockwise and counterclockwise at the same time). A popular example of this paradoxical situation is the Schrödinger cat, which is in a state of superposition of being alive and dead at the same time. The collapse of the wave function resolves the ambiguity and the contradiction, resulting in a definitive state (spin-up or spin-down, or being alive or dead, as the case may be). The collapse of the wave function limits or contracts the plurality of possible states to a single actual state. Similarly, if before Tzimtzum, godliness, i.e., the idea of God, i.e., godliness, was in a state of superposition of all possible states, Tzimtzum may be viewed allegorically as the collapse of the “wave function” of godliness, as it were, resulting in resolving contradictory states.[48] Moreover, as it was earlier proposed, it may be preferable to translate the word Tzimtzum as “collapse” rather than “contraction.”[49]

Just as the wave function collapse is relative to the observer,[50] so too is Tzimtzum. In the Wigner’s Friend thought experiment, from the point of view of the friend in a  laboratory, who performed the experiment and observed the system in question, the wave function is collapsed; whereas for Wigner, who stepped outside the laboratory during the experiment, the system is still in the state of superposition. This experiment demonstrates the relative nature of the wave function collapse. The sages of the Kabbala, as well as Chasidic masters, stress that the Tzimtzum occurred only relative to us (sc., it so appears from our frame of reference), whereas, for God, no Tzimtzum ever took place. Based on this understanding of Tzimtzum, they call it “illusory.”[51] Calling Tzimtzum “relative,” as we do here, means the same thing—Tzimtzum only happened from our perspective—but it allows us to employ fruitful metaphors of relativism in physics and further strengthens the analogy with the wave function collapse.

As it happens, the quantum-mechanical wave function also defines Hilbert space. So, the conclusion, at which we previously arrived, that the created void—makom hapanui—the abstract space created by Tzimtzum is also Hilbert space may not be coincidental, after all.[52]

 

Conclusion.

At the beginning of this section, we asked three questions regarding the passage from Eitz Chaim quoted hereinabove. The first question was, What could it possibly mean that God (Ein Sof) “contracted” (tzimtzum) Himself? As explained above, this question was answered by interpreting “contraction” as hiding contradictions to make room for conceptual space where two contradictory ideas cannot be in the same place. We also proposed to translate the word tzimtzum as “collapse” and analogized Tzimtzum to the quantum-mechanical collapse of the wave function. It is apparent from our interpretation that Tzimtzum took place in godliness, not in God, and not literally but only relative to us.

The second question we asked was, What is the meaning of the “point at the center of Ohr Ein Sof”? We proposed that (i) a point is an idea; (ii) the center of Ohr Ein Sof (ohr-illumination) is Ein Sof (ma’or-luminary); and (iii) the point at the center is the idea of God, i.e., Divine name, or godliness.

We proposed that the nature of this first phase of Tzimtzum is to restrict contradictions—nimna haminma’ot—by sweeping contradictions under the rug.[53]

At this point it is important to make a disclaimer that our analogies and structural parallels are not part of traditional teachings of Kabbalah or Chasidic philosophy—the thoughts expressed herein are nothing more than speculation of this author. Kabbala is a tradition that is passed down from a teacher to a student; it is received wisdom.[54] This is not something one can dream up, or derive through mathematical or metaphysical speculations. Therefore, it is very important to distinguish the authentic wisdom of Kabbalah from speculative mossing on this subject. This writing is nothing more than a humble attempt to understand highly mystical, abstract, and abstruse concepts of Tzimtzum couched in terms of modern science and philosophy. Therefore, the reader is cautioned to take these writings with a healthy dose of skepticism and a grain of salt.

In the next section, we shall address the second phase of Tzimtzum, which we will explain as sweeping infinities under the rug and analogize it to the procedure of renormalization in quantum field theory.

—————————–

Footnotes:

[1] See Babylonian Talmud, tr. Baba Kama, 29b; See also Etymological Dictionary of Biblical Hebrew: Based on the Commentaries of Samson Raphael Hirsch (Philipp Feldheim, 2000).

[2] As in echshon itzamtzu—to remove doubt.

[3] Compare with such words as damdam—very red (dam mean “blood” or “red” in Hebrew).

[4] The Merriam-Webster Dictionary.

[5] The center of mass is the mean location of a distribution of mass in space, i.e., the unique point where the weighted relative position of the distributed mass sums up to zero. This is a point of balance.

[6] The center of gravity of a body is the point around which the torque due to gravity forces vanishes.

[7] Spherical symmetry means the invariance under spherical rotation. In other words, nothing changes under rotation around the center. Indeed, a circle turned by any degree remains the same circle indistinguishable from the original.

[8] The concepts of space and time first appear in their proto-physical (“spiritual”) form in the Malchut of Atzilut—the first world of the Universe of Tikun, created after the collapse of the universe of Tohu. The physical space and time only appear in the last (lowest) world of the universe of Tikun, the world of Asiyah.

[9] Plato, Timaeus, 51a. “Everything that exists must be in some place [topos] and occupy some space [khôra], and that what is not somewhere on earth or heaven is nothing” (Timaeus 52d). Thus, Plato’s khôra is much closer to the notion of physical space than the concept of makom hapanui.

[10] Aristotle understands space as an extendable limit defined as “The limit of the surrounding body, at which it is in contact with that which is surrounded [i.e., the thing]”, Physics, 212a 5–6.

[11] Aristotle, Physics, Book IV, Delta.

[12] The mathematical theory of conceptual space was developed by Björn Peter Gärdenfors in 2000 (see P. Gärdenfors, “Conceptual spaces: the geometry of thought,” MIT Press, 2000) However, many authors including Rabbi Aryeh Kaplan (See A. Kaplan, The Inner Space and his introduction to Bahir.), wrote about this concepts long before him.

[13] In this space, “up” means “closer to God,” i.e., more God-like. “Down” means further away from God or less God-like. This is why, colloquially, people say that angels are above—because they are closer to God, i.e., more aware of God and less conscious of their own self. Some angels are higher than others for this very reason—the higher angels are even closer to God, i.e., even more aware of God and even less conscious of their own self than the lower angels. See Aryeh Kaplan, Inner Space, (Moznaim Publishing, Jerusalem, 1990), p.

[14] This author, for instance, was awarded five patents on the architecture of a social network based on a conceptual space wherein the distance is computed as the social distance between the members of the network as measured by the similarity of their age, gender, character traits, profession, education, hobbies, interests, social and/or political affiliations, religion/spirituality and other characteristics. Some of these dimensions could be continuous, such as age, others, such as gender, are discrete. This is just an example of how conceptual spaces may be used in computer science.

[15] The tradition attributes the authorship of this book to patriarch Abraham. The second-century sage, tana Rabbi Akiva is credited with redacting the book and transmitting it to us.

[16] Sefer Yetzirah 1:5; English translate by Rabbi Aryeh Kaplan, Sefer Yetzirah, (Samuel Weiser, 1990) p. 44.

[17] “If God filled every perfection, man would have no reason to exist. God therefore constricted His infinite perfection, allowing for a ‘place’ for man’s free will and accomplishment.” Aryeh Kaplan, The Bahir (Samuel Weiser, Inc. 1979), Introduction, p. xxiii, citing Shomer Emunim 2:43 and KaLaCh Pitchei Chokhmah 24.

[18] Godliness is neither God nor a creation, but is a reflection and manifestation of God that retains God’s character, as it where. In the Chasidic philosophy of Chabad, godliness is identified with Ohr Ein Sof—the Light of the Infinite.

[19] For example, in monotheistic tradition, God is Eternal. This is notion is reflected in God’s name Tetragrammaton—YHWH (or YHVH). As the Sages teach, this Name means Hayah (He was), Hoveh (He is), and Yihieh (He will be). We also hold God to be merciful. This is reflected by the divine name, A-l.[19] The name Elo-him, on the other hand, reflects our belief that God is just. We also believe that God is the Master of the Universe. This notion is reflected in His name Ado-nai—the Lord. Every divine name represents our particular idea about God.

[20] From the point of view of philosophy of science, there are two types of physical theories—ontic and epistemic. Ontic theories describe the evolution of a physical system per se, whereas epistemic theories describe our knowledge of the physical system.

[21] The closest analogy from physics would be such concepts as homogeneity (the same everywhere) and isotropy (the same in every direction). However, such crude metaphors are not to be understood literally, because “everywhere” and “direction” are spatial characteristics that cannot be applied to God.

[22] Moreover, a center defines the set of points equidistant from the center. A set of points is a multiplicity that is not found in God and, therefore, there cannot be a center in God.

[23] This is how emanation was defined from Plotinus to later Neoplatonic philosophers, who developed the doctrine of emanationism, including medieval Jewish philosophers, such as Solomon ibn Gabirol.

[24] Moreover, the switch in the narrative to Ohr Ein Sof may be a hint that Tzimtzum only took place in Ohr Ein Sof, not in Ein Sof Himself.

[25] In Kabbala and Chasidic literature, terms Ein Sof and Ohr Ein Sof are often used interchangeably. When we say Ein Sof, we often imply Ohr Ein Sof, as the only purpose of a luminary is to illuminate. It is important to bear in mind that neither of these terms refer to God Himself. For this reason, most translators translate the passage from Etz Chaim as follows, “Ohr Ein Sof “contracted” itself…” maintaining that the “Ohr” is implied in this context.

[26] Makom hapanui literally means “empty space.” Makom means “space,” and panui means “empty,” from lifnot—to “clear away” or pinui—to “evacuate.”

[27] Topology (from Greek topos—place) studies properties of geometrical figures that remain invariant under continuous deformation, such as stretching, twisting, squeezing, or bending, so long as there is no tears or gluing.

[28] Geometry (from Greek for geos- “earth” and matron, “measurement”) studies properties of space and shape, size and relative position of geometrical figures in space.

[29] Hermann Minkowski (1864–1909)—a German Jewish mathematician, college professor of Albert Einstein, who proposed to unify three-dimensional space with one-dimensional time into a four-dimensional space-time continuum, or, as he called it, “The World.” In his geometrical interpretation of the special theory of relativity, Minkowski treated time as the fourth dimension of a four-dimensional spacetime.

[30] In the configuration space of a system of two particles, each point has coördinates (x1, y1, z1, x2, y2, z2).

[31] It is easy to prove by argumentum e contrario (argument from the contrary). Let us assume that the space created by Tzimtzum is physical space. At this stage, the matter has not been created yet. Thus, this space must be empty and exist independently from material objects yet to be created. (This was the concept of space—an empty container—proposed by Aristotle in his Physics and embraced by Newton in his Principia.) We call it the absolute space. (This notion was vigorously disputed by Leibnitz who believed that space and time are relations between objects and events.) However, the notion of an absolute space was debunked by Albert Einstein in his special theory of relativity. Thus, our assumption that the space created by Tzimtzum is a physical space (and therefor absolute space existing independent of objects in it) is incompatible with modern physics.

[32] Indeed, Tzimtzum is preceded by the Ratzon—the God’s Will to create physical universe. However, God’s Will is not our will. Some people believe that if you will a pink Cadillac in will appear on your driveway. Unfortunately, it does work that way. Our will is the beginning of a long process which may or may not result in achieving our desire. God’s will, on the other hand, is as good as done. If God wills something, it becomes a reality. It follows that if God will that there be physical universe it is enough for the universe to come into existence. And it is—from God’s point of view. From our point of view, however, there are many levels that separate that initial Will and the creation of the physical universe. The first level is the phase space each point of which represents a potential state of the universe. Although, we tend to gravitate towards thinking about the process of Creation in temporal terms—what was before and what came into being later—we need to remember that time was not created yet, so all these level are conceptual (or spiritual) levels that coexist contemporaneously from our time-bound point of view.

[33] Now that we understand that the created void did not represent a physical space, there is no longer any worry that the Tzimtzum would be misunderstood as applying spatial limitations to Infinite God. It was clearly understood by those Kabbalists who maintained that Tzimtzum takes place in Ein Sof. Chasidic philosophy, as well as most contemporary Kabbalists, insist nevertheless that Tzimtzum took place only in Ohr Ein Sof, and only relative to us.

[34] Dialetheism is a philosophical view that embraces dialetheias, i.e., contradictions such as A, and its negation, Law of Non-Contradiction (LNC) is “the most certain of all principles” (1005b24)—firmissimum omnium principiorum.

[35] Francis Scott Key Fitzgerald, “The Crack Up” Esquire Magazine (February 1936).

[36] To attribute to God any temporal characteristic such as “at the same time” is, of course, an oxymoron, because God utterly transcends time. However, for the luck of better words, we use this expression with the disclaimer that it is not to be understood literally, but nothing more than a manner of speech.

[37] Hence the name, Almighty.

[38] In the words of the Kabbalist Rabbi Meir ibn Gabbai, “Just as He possesses the power of infinity, so does He possess the power of finiteness. For should you say that He possesses the power of infinity but does not possess the power of finiteness, you are detracting from His perfection.” (https://www.chabad.org/parshah/article_cdo/aid/2579/jewish/Tzimtzum.htm); see Maamar Tzion B’mishpat Tipadeh, Sefer Hamaamarim Melukat vol. 3, p. 233.

[39] Shu”t HaRashb”a, I, 413-418. See also Rabbi Hillel Paritcher.

[40] A quantum mechanical system can be in be in superposition of all possible states. God, on the other hand, can be “in superposition” of all state—possible and “impossible,” because nothing is impossible for God. Moreover, we must not forget that this is just a crude allegory—God is not in any state that we can think of, as it says, “I am God and I do not change” (Malachy 3:6). Only our idea of God can be in different states—He can be angry and forgiving at the same time, for example.

[41] The Chabad philosophy teaches that the light (ohr) retains the “nature” or the “character” of the luminary (ma’or)—ohr b’ein ma’or. See, for example Samach Vav.

[42] This is, of course, a tautological statement, because there is no other existence outside of God. However, Torah speaks in the language of man; and we follow the example of the Torah.

[43] The enigmatic state of nimna hanimna’ot would only manifest itself in miracles that violate laws of nature. The well-documented example of such a miracle is found in the description of the measurements of the Bet HaMikdash (the Holy Temple in Jerusalem) recorded in the Talmud. Aron HaBrit—the “Ark of the Covenant”—was the golden box containing the Tablets (Luchot) with the Ten Commandments, placed in the Holy of Holies—the innermost sanctum of the Bet HaMikdash (the “Holy Temple” in Jerusalem). According to the Talmud, the Ark did not occupy any space. The Holy of Holies measured twenty cubits by twenty cubits. The Ark itself measured two-and-a-half cubits. When measured from the South wall to the nearest side of the Ark the distance was  ten cubits. When measured from the side of the Ark to the Northern wall, the distance was also ten cubits. It appears that the Arc occupied no space, although its width was 2.5 cubits. See Talmud, Yoma 21a; Megillah 10b; Bava Batra 99a. As explained by the Sages, this is an example of God’s presence occupying and not occupying physical space— nimna hanimna’ot. See Rabbi Menachem M. Schneerson, Maamar Gadol Yiheyeh Kavod HaBayis HaZeh. See also Sefer HaMa’amarim 5643 p. 100; and loc. cit. 5665 p. 185.

[44] This idea finds its support in the writings of prominent Kabbalists—both students of the Ari—Rabbis Joseph ibn Tabul and Israel Sarug who maintained that, prior to Tzimtzum, the roots of two contradictory tendencies—dinim (judgments) and rachamim (mercy)—were mixed together in Ein Sof. Their separation, i.e., resolution of the contradiction between these two opposite tendencies was a prelude to Tzimtzum. See G. Scholem, Kabbalah, (Dorset Press, 1974), pp. 132-133.

[45] This expression seems particularly appropriate because it reflects the opinion of Rabbi Schneur Zalman that Tzimtzum is to be understood shelo kepshutoi, i.e., not in the literal sense.

[46] The wave function (or wavefunction) is a mathematical description of the quantum space of a physical system. The wave function is not a physical property of a system; it is not an observable. It is an abstract mathematical representation of the system state. According to Born rule, the probability density of finding a particle at a given point is proportional to the square of the magnitude of the particle’s wave function at that point. According to the Bayesian interpretation of quantum mechanics, the wave function represents our level of confidence in our knowledge of the state in which the physical system is.

[47] The interpretation of Tzimtzum as the collapse of the wave function was first proposed by this author in 2014 in his blog post, On Tzimtzum, Sefirot and Cardinal Numbers: “Tzimtzum is the collapse of the universal wavefunction describing the creation… Even the word “tzimtzum,” which means infinite contraction, can be literally translated as a collapse.” See https://www.quantumtorah.com/on-tzimtzum-sefirot-and-cardinal-numbers/ (retrieved on 8/27/20).

[48] Needless to say, God doesn’t have any wave function that could be ascribed to Him. This analogy could be either viewed as a crude metaphor to aid our intuition, or as a structural parallel.

[49] We propose this novel translation not only because it serves our narrative but also because it makes sense grammatically. As mentioned above, the word tzimtzum comes from the root tzom—to diminish, or to fast. However, the repetitious doubling of this root in the word tzim-tzum, is a grammatical device in Hebrew to exaggerate the meaning. Thus, the ultimate diminishing is, indeed, a collapse.

[50] As many (but not all) interpretations of quantum mechanics hold, including that of this author’s Futurist interpretation of quantum mechanics. See Poltorak A., “The Age of the Universe: Using the Many-Worlds Interpretation of Quantum Mechanics,” B’Or Hatorah (2008) vol. 18, pp. 149-168; and Poltorak, A. “Towards Futurist Interpretation of Quantum Mechanics” (Academia.edu, 2019).

[51] See Rabbi Shneur Zalman of Liadi, Tanya, ch. 36.; loc. cit., Shaar Yihud VeHaEmunah.

[52] Such metaphors are not precise and should always be taken with a grain of salt. At best, this is an imprecise allegory to aid our understanding that requires a poetic license.

[53] The expression, “sweeping contradictions under the rug” may be appropriate here, because, indeed, contradictions are not eliminated—Ein Sof and His emanation, Ohr Ein Sof remain as self-referential and self-contradictory constructs as ever—but only concealed from our sight, i.e., swept under the rug.

[54] The word kabbalah comes from the word mikabel, i.e., to receive. Thus “received wisdom” is the literal translation this word.

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Endnotes:

[i] “So likewise it is right that the substance which is to be fitted to receive frequently over its whole extent the copies of all things intelligible and eternal should itself, of its own nature, be void of all the forms. Wherefore, let us not speak of her that is the Mother and Receptacle of this generated world, which is perceptible by sight and all the senses, by the name of earth or air or fire or water, or any aggregates or constituents thereof: rather, if we describe her as a Kind invisible and unshaped, all-receptive, and in some most perplexing and most baffling way partaking of the intelligible, we shall describe her truly.” (Plato, Timaeus, 51a) Plato in Twelve Volumes, Vol. 9 translated by W.R.M. Lamb. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1925. See online at

[ii] Configuration space is a vector space whose generalized coördinates are the parameters that define the configuration of the system. Configuration space is defined as follows: for a topological space X{\displaystyle X}, the nth (ordered) configuration space of X is the set of n-tuples of pairwise distinct points in X. In other words, configuration space is a configuration manifold, i.e., the set of actual configurations of the system. Whereas we have an intuition for our physical space, in which all physical objects are embedded, we have no intuition for the configurational space, which depends on the system we study. For a single particle, for which the position in the three-dimensional Euclidean space ℝ3 is defined by a vector q = (x, y, z), the configuration space is a familiar three-dimensional space: Q = ℝ3, where Q is configurational space and ℝ3 is a three-dimensional Euclidean space. However, for two particles, the configuration space is six-dimensional: Q = ℝ6 = ℝ3⨉ℝ3, where point coördinates are (q1x, q1y, q1z, q2x, q2y, q2z,). Generally, in mechanics, the configuration of a system consists of the positions of all particles, subject to kinematical constraints on the movement of the particles.  For example, if a particle is connected to a point by a rigid string, it is free to move on a surface of a sphere, whose radius is the length of the string with point to which the particle is attached being the center of the sphere. In this case, the configurational manifold is the sphere: Q = S2, where S2 is the surface of the sphere. For a configuration of n unconnected and non-interacting point particles, the configuration space is 3n-dimensional: Q = ℝn3. It is easy to understand if we keep in mind that each degree of freedom of a physical system is mathematically represented as a dimension in space.  An object that can move only along a line (e.g., a train that can only travel along the rail road) has one degree of freedom, i.e., it moves along the line, which is a one-dimensional space; an object that can move freely on a plane (e.g., a car that can travel not only forward or backward but also right or left) has two degrees of freedom, i.e., it moves in a two-dimensional space; an object that can also move up and down (e.g., a drone that can fly in any direction) has three degrees of freedom, i.e., it moves in a three-dimensional space. Since, generally, a particle can move in three-dimensional space and, therefore, has three degrees of freedom, a system on n particles has n3 degrees of freedom, which can be mathematically represented as a 3n-dimensional space.

[iii] In phase space, every degree of freedom represents two dimensions—one for the value of that degree of freedom, and one for the rate of change on that value over time. Configurational space reflects the positions of all particles in the Euclidean space. However, as we know from Newtonian mechanics, to describe the dynamics of the system, we also need to know the momentum of each particle. Each point of the phase space is a pair of vectors—one for the position of each particle, and one for the momentum.[iii] For an ensemble of n particles, the dimensionality of the configurational space is 6n. In quantum mechanics, the phase space takes on complex values.

[iv] In a mathematic formulation of quantum mechanics developed by John von Neumann, all possible states of a quantum-mechanical system are represented as unit vectors in a complex[iv] infinite-dimensional Hilbert space. The reason for that is this: whereas in classical mechanics, coördinates and momenta of a particle are well defined, in quantum mechanics, they are not—instead, each particle is in a superposition of all possible states, i.e., all possible values for its coördinates and momenta. This requires that the phase space of a quantum-mechanical system be infinite-dimensional Hilbert space.