In the previous post “Physics of Tzimtzum I—The Quantum Leap”, we gave a general overview of the mystical doctrine of *tzimtzum*—the cornerstone of Lurianic Kabbalah. It is time to get into the details.

The first phrase that describes the process of *tzimtzum* in Etz Chaim states:

Ein Sof“contracted” (tzimtzem) Himself in the point at the center, in the very center ofOhr Ein Sof.

This sentence raises several difficult questions:

First, what could it possibly mean that the Infinite (*Ein Sof*) “contracted” (*tzimtzem*) Himself? In Hebrew, the word *tzimtzum* comes from the root TZM, which means “to diminish” or “to fast,” that is, to “diminish” oneself.^{[1]} It can also mean “to be precise,” that is, to remove ambiguity.^{[2]} The repetition of the root TZM is a grammatical form of doubling down, an extreme emphasis of the basic meaning.^{[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 G‑d, who has no body, form, size, or volume whatsoever. Such a reading would clearly contradict the core principles of monotheism, which postulate that G‑d is infinite and incorporeal. Moreover, “size” or “volume” are geometric concepts that exist in a metric space (that is, space where the distance between two points is defined). No such space exists at this stage, because space is yet to be created. Thus, the expression “*Ein Sof* contracted Himself” cannot be understood literally. Even the remote possibility of such a literal interpretation is so abhorrent to monotheistic belief that many authors translate *tzimtzum* as “concealed” instead of “contracted.” However, by deviating from the literal meaning of the word *tzimtzum*, this rendition merely sweeps the question under the rug. So, what does this expression, “*Ein Sof* contracted Himself,” mean?

The next few words—*Ein Sof* “contracted” (*tzimtzem*) Himself *at the point at the center*”—presents another problem. What, in this context, is the “center”? 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, at this point in our narrative, any matter to which these concepts may apply is yet to be created. So, the physical connotations of the word “center” offer little help to us. In mathematics, a “center” is a point that defines a circle on a plane or a sphere in space (or a hypersphere in a multidimensional space) as a set of points equidistant from the center. Furthermore, 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 “*at the point* at the center” is a geometric concept. What could it mean in the context of *tzimtzum* before the creation of physical space?

Third, and last, the narrative starts with *Ein Sof* contracting Himself at 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 all the geometric 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 when thinking about *tzimtzum *is to imagine that the “empty space” created by *tzimtzum* was our three-dimensional physical space. Physical space was not created until many stages later, two universes down the path of the ontological order of creation called *Seder Hishtalshelut*, or “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*^{[9]} (Greek for “space”) of Plato^{[10]} nor is it the *topos*^{[11]} (Greek for “place”) of Aristotle.^{[12]} 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,^{[13]} points denote objects, and regions denote concepts. Conceptual space may include physical space as its sub-space. Since no objects existed before the creation, we shall consider a conceptual space as a 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 red, the Y-axis (dimension) is green, and the Z-axis (dimension) is blue. Since any color could be represented by a trio of numbers representing respective values for the red, green, and blue colors, we can model each as a point in a three-dimensional space wherein three coordinates represent the respective RGB values. Another example is a face space (not be confused with phase space used in physics)—a multidimensional 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 points in this space, that is, each point is “occupied” by an idea according to its qualities. Distance between points is determined by the similarity of the ideas that these points represent. Two ideas that are equivalent to each other occupy the same point (having the same coordinates) in conceptual space. In other words, ideas A and B occupy the same point in 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 conceptual space. The distance between two ideas is the measure of the conceptual similarity between the two.^{[14]}

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. However, if the people are not equally kind, the person who is more kind would be on the right of a person (or act) chosen as the reference point, and the person 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. But if the people are not equally honest, 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.^{[15]} And so on.

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

The first text to introduce conceptual space was one of the oldest books of Kabbalah, *Sefer Yetzirah* (The Book of Formation):[16]

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.^{[17]}

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, shown in the following table.

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 |

Medieval Jewish philosophers also understood conceptual space well. Thus, Maimonides, for example, writes:

below the level of the other” does not refer to height in a spatial sense as [one might say], he is sitting higher than his colleague, [but rather in regard to spiritual level]. For example, when speaking about two sages, one of whom is greater than the other, we say, “one is above the level of the other.” Similarly, a cause is referred to “above” the effect [it produces].

^{[18]}

In his commentary on Sefer Yetzirah, Rabbi Aryeh Kaplan writes:

This constriction or hollowing of the Divine Essence did not occure in physical space, but rather, in conceptual space.

^{[19]}

Rabbi Kaplan stresses that the sages of Kabbalah were well aware that the space created by the *tzimtzum* was not a physical but a conceptual space.^{[20]}

Now we can explain the passage from *Etz Chaim* in the context of the conceptual space and answer the second question (that is, What is the nature of the point in the center, and what is the meaning of the center itself?). In this space, a point is an idea, and the point at the center is the central idea—the idea of G‑d. We have nothing to say about G‑d per se, as He is utterly unknowable. Everything we say about G‑d is not really about G‑d per se, but about our *idea* about G‑d, that is, about godliness.^{[21]} In Kabbalah, this concept is called G‑d’s name. Certain ideas we hold about G‑d are reflected in different names.^{[22]} Divine names, their meanings and manifestations, as well as their uses 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 G‑d, rather than with G‑d per se, who is unknowable.

We note an interesting parallel between Kabbalah and quantum physics insofar as they both are epistemic theories.^{[23]} Quantum mechanics, at least in the traditional Copenhagen interpretation, does not describe physical objects per se, but only their wave function, which represents our knowledge about those 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,[24] has nothing that could even remotely be identified as a “center.”^{[25]} *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.^{[26]} Thus, the center of the emanation—*Ohr Ein Sof—*is the luminary (*ma’or*)—*Ein Sof*, the source of the emanation.^{[27]}

However, this makes our first question (that is, What does it mean that the Infinite (*Ein Sof*) “contracted” Himself?) 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*^{[28]} contracts Himself into a point that is an idea of *Ein Sof.* What does it mean? The meaning will become apparent soon enough.

*Makom Panui***—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*, *chalal* (a “void”—the primordial vacuum) or *makom panui* (a “vacant place”) was created. *Makom hapanui*^{[29]} is not a physical space; it is an *abstract space*. Because of the importance of this concept to *tzimtzum*, we need to understand what an abstract space is.

In mathematics, space is defined as a set (that is, a collection of elements) with a certain structure (topological^{[30]} and geometrical^{[31]}). If the elements of a set are points in physical space, then such space, indeed, is a physical space. Abstract space may include physical space as its subspace. However, the elements of the set need not be points in physical space. They could be moments in time,^{[32]} 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 in 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*, that is, a “limitation,” in that it limits the shape, size, and position of objects 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 us to represent the position of a system of particles as a single point in a multidimensional space. For a single particle, configuration space is the same as the regular three-dimensional Euclidean space, where each point has coordinates (*x*, *y*, *z*). The configuration space of a system of two particles is six-dimensional.^{[33]} The configuration space of a system of *n* particles has *3n* dimensions—three dimensions for each particle.^{[34]}

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.^{[35]}

Yet another 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 Euclidean space (our ordinary three-dimensional space) on any number of dimensions—finite or infinite. In quantum mechanics, all possible states of a quantum-mechanical system are represented as vectors in an infinite-dimensional Hilbert space.^{[36]}

As stated above, the “space” or “void” created by the *tzimtzum* is not a physical space^{[37]} but an abstract one. 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 *tzimtzum*—*makom panui*—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.^{[38]} However, since the universe has not yet been created, and each potential state in the future is 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.^{[39]}

**Sweeping Contradictions Under the Rug**

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 not only to possess infinitely greater knowledge that his pupil, but also 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 the Torah (both written, *Torah SheBiKhtav*, and oral, *Torah SheBe’alPe*) is full of what seem to be contradictory statements. Only a person who mastered the entire Torah (both written and oral) can see the holistic picture where all pieces of the puzzle fit perfectly together. Such a sage can embrace the *dialetheism ^{[40]}* 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.”

^{[41]}Although 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 from his mind 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, G‑d is the ultimate self-referential and self-contradictory construct. G‑d is the master of all possibilities—nothing is impossible for G‑d. He can be in state *A* and state *non-A* at the same time.^{[42]} G‑d is infinite and perfect, and therefore He lacks nothing. It follows that absolute infinity and perfection require that He does not lack finitude. In other words, as Omniponten,^{[43]} He possesses the powers of infinity and of finitude.^{[44]} A contradiction!

G‑d can be in any state and the opposite state at the same time. He can be in all states, because nothing is impossible for G‑d. The sages termed this self-contradictory aspect of G‑d as *nimna hanimna’ot *(literally “restricting [all] restrictions”), that is, the “paradox of paradoxes.”^{[45]} 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 the “superposition of states.” Indeed, G‑d, as Infinite Being, can be and *is* in a superposition of all states.^{[46]} G‑d is the ultimate paradox. This paradoxical nature of *Ein Sof* extends to the Divine emanation, *Ohr Ein Sof, *as G‑d manifests His powers of infinitude and finitude together.^{[47]}

So long as G‑d’s emanation filled all of the existence,^{[48]} 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, G‑d had to conceal the self-contradictory nature of His Light—*Ohr Ein Sof* (which reflects the self-contradictory nature of *Ein Sof*), that is, He had to separate His *ko’ach hagvul* (powers of finitude) from His *ko’ach bli gvul* (powers of infinitude) to hide the paradoxical aspect of *nimna hanimna’ot*.^{[49]} 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.^{[50]} In a manner of speech, *tzimtzum* is G‑d’s way of sweeping the contradictions under the proverbial rug.^{[51]}

**The Collapse of the Wave Function**

*Tzimtzum* is thus somewhat akin to the collapse of the wave function[52] in quantum mechanics.^{[53]} Before the collapse, a particle may be in a superposition of two contradictory states. For example, an electron can be in a superposition of both the spin-up |↑〉 and the spin-down |↓〉 states, which, in classical physics, is analogous to a 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, that is, the idea of G‑d, 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.^{[54]} Moreover, as earlier proposed, it may be preferable to translate the word *tzimtzum* as “collapse” rather than as “contraction.”^{[55]}

Just as the wave function collapse is relative to the observer,^{[56]} 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 collapse of the wave function. The sages of the Kabbalah, 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 G‑d, no *tzimtzum* ever took place. Based on this understanding of *tzimtzum*, they call it “illusory.”^{[57]} Calling it “relative,” as we do here, means the same thing—*tzimtzum *happened only from our perspective—but it allows us to employ fruitful metaphors of relativism in physics and further strengthens the analogy between *tzimtzum* and the collapse of the wave function.

As it happens, the quantum-mechanical wave function also defines a Hilbert space. So, the conclusion at which we previously arrived that *makom panui*—the abstract space created by *tzimtzum* is also a Hilbert space may not be coincidental after all.^{[58]}

**Conclusion**

At the beginning of this section, we asked three questions regarding the quoted passage from *Eitz Chaim*. The first question was, What could it possibly mean that G‑d (*Ein Sof*) “contracted” (*tzimtzum*) Himself? As explained above, this question was answered by interpreting “contraction” as hiding contradictions so as to make room for conceptual space where, ordinarily, two contradictory ideas could not share 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, rather than in G‑d Himself, and not literally but relative to us only.

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 G‑d, i.e., the 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.^{[59]}

At this point it is important to offer a disclaimer that nothing in this section is part of traditional teachings of Kabbalah or Chasidic philosophy—the thoughts expressed herein are only the speculation of this author. Kabbalah is a tradition that is passed down from a teacher to a student; it is received wisdom.^{[60]} 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 musings 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 read this essay with a healthy dose of skepticism.

In the next section, we shall address the second phase of *tzimtzum*, which we will explain as sweeping infinities under the rug, a sweeping that we will analogize with the procedure of renormalization in quantum field theory.

### Endnotes:

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

[2] As in the dispute whether *efshar l’tzamtzem*—whether it is possible to remove uncertainty.

[3] Compare with such words as *adamdam*—very red (*dam* means “blood” or “red” in Hebrew), or *yerakrak*—very green.

[4] Definition of “contracted,” *The Merriam-Webster Dictionary* (see www.merriam-webster.com/thesaurus/contract).

[5] The center of mass is the mean location of a distribution of mass in space, that is, the unique point where the weighted relative position of the distributed mass adds 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 gravitational 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 protophysical (“spiritual”) form in the *Malchut* of *Atzilut*—the first world of the universe of *Tikun*, created after the collapse of the universe of *Tohu*. Physical space and time appear only in the last (lowest) world of the universe of *Tikun*, the world of *Asiyah*.

[9] “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, in *Plato in Twelve Volumes*, vol. 9, translated by W. R. M. Lamb, Cambridge, MA: Harvard University Press and London, William Heinemann Ltd. 1925. See online at www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0180%3Atext%3DTim.%3Asection%3D51a).

[10] 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 panui* is.

[11] 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.

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

[13] 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 concept long before Gärdenfors did.

[14] In this space, “up” means “closer to G‑d,” that is, more G‑d-like. “Down” means further away from G‑d or less G‑d-like. This is why, colloquially, people say that angels are above—because they are closer to G‑d, that is, more aware of G‑d and less conscious of themselves. Some angels are higher than others for this very reason—the higher angels are even closer to G‑d, that is, even more aware of G‑d and even less conscious of their own selves than the lower angels. See Aryeh Kaplan, *Inner Space: Introduction to Kabbalah, Meditation and Prophecy*, (Moznaim Publishing, Jerusalem, 1990), p. 14, and Aryeh Kaplan, *Sefer Yetzirah: The Book of Creation in Theory and Practice*, (Weiser Books, 1997), p. 14.

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

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

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

[18] Maimonides, *Mishnah Torah*, Book of Knowledge, Laws of Foundations of the Torah 2:6.

[19] Aryeh Kaplan, *Sefer Yetzirah: The Book of Creation in Theory and Practice*, (Weiser Books, 1997), p. 14.

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

[21] G‑dliness is neither G‑d nor a creation, but a reflection and manifestation of G‑d that retains G‑d’s character, as it were. In the Chasidic philosophy of Chabad, godliness is identified with *Ohr Ein Sof*—the Light of the Infinite.

[22] For example, in monotheistic tradition, G‑d is eternal. This notion is reflected in G‑d’s name, the Tetragrammaton—Y‑H‑W‑H (or Y‑H‑V‑H). As the sages teach, this name means *Hayah* (He was), *Hoveh* (He is), and *Yihieh* (He will be). We also hold G‑d to be merciful. This is reflected by the divine name, *A-l*.[22] The name *Elo-him*, on the other hand, reflects our belief that G‑d is just. We also believe that G‑d is the master of the universe. This notion is reflected in His name *Ado-nai*—the Lord. Every divine name represents a particular idea about G‑d.

[23] From the standpoint of the 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.

[24] 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 G‑d.

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

[26] 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.

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

[28] In Kabbalah and Chasidic literature, the 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 term refers to G‑d 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).

[29] *Makom panui* literally means “empty space.” *Makom* means “space,” and *panui* means “empty,” from *lifnot *(to “clear away”) or *pinui* (to “evacuate”).

[30] 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 are no tears or gluing.

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

[32] Hermann Minkowski (1864–1909)—a German-Jewish mathematician and 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.

[33] In the configuration space of a system of two particles, each point has coordinates (*x ^{1}*,

*y*,

^{1}*z*,

^{1}*x*,

^{2}*y*,

^{2}*z*).

^{2}[34] Configuration space is a vector space whose generalized coordinates are the parameters that define the configuration of the system. Configuration space is defined as follows: for a topological space *X*{\displaystyle X}, the *n*^{th} (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 ℝ^{3} is defined by a vector *q* = (*x*, *y*, *z*), the configuration space is a familiar three-dimensional space: *Q *= ℝ^{3}, where Q is configurational space and ℝ^{3} is a three-dimensional Euclidean space. However, for two particles, the configuration space is six-dimensional: *Q *= ℝ^{6} = ℝ^{3}⨉ℝ^{3}, where point coordinates are (*q ^{1}_{x}*,

*q*,

^{1}_{y}*q*,

^{1}_{z}*q*,

^{2}_{x}*q*,

^{2}_{y}*q*,). Generally, in mechanics, the configuration of a system consists of the positions of all particles, subject to kinematical constraints on their movement. 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:

^{2}_{z}*Q = S*, where

^{2}*S*is the surface of the sphere. For a configuration of

^{2}*n*unconnected and non-interacting point particles, the configuration space is

*3n*-dimensional:

*Q*= ℝ

*. 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 railroad) has one degree of freedom, i.e., it moves along the line, 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; and 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 of*

^{n3}*n*particles has

*n*⨯

*3*degrees of freedom, which can be mathematically represented as a

*3n*-dimensional space.

[35] 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. For an ensemble of *n* particles, the dimensionality of the configurational space is *6n*. In quantum mechanics, the phase space takes on complex values.

[36] 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 infinite-dimensional Hilbert space. The reason for that is this: whereas in classical mechanics, coordinates 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 coordinates and momenta. This requires the phase space of a quantum-mechanical system to be an infinite-dimensional Hilbert space.

[37] 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, matter has not been created yet. Thus, this space must be empty and exist independent of the 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 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 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, therefore, absolute space, existing independent of objects in it) is incompatible with modern physics.

[38] Indeed, *tzimtzum* is preceded by the *Ratzon*—the will of G‑d to create the physical universe. However, G‑d’s will is not our will. Some people believe that if you will a pink Cadillac it 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 desires. G‑d’s will, on the other hand, is as good as done. If G‑d wills something, it becomes a reality. It follows that if G‑d wills that there be a physical universe, it is enough for the universe to come into existence. And it is—from G‑d’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 toward 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 levels are conceptual (or spiritual) levels that coexist contemporaneously from our timebound point of view.

[39] 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 G‑d. It was clearly understood by those Kabbalists who maintained that *tzimtzum* takes place in *Ein Sof*. Chasidic philosophy, as well as most contemporary Kabbalists, insists nevertheless that *tzimtzum* took place only in *Ohr Ein Sof*, and only relative to us.

[40] *Dialetheism** is a philosophical view that embraces **dialetheias*, i.e., contradictions such as the idea that A and its negation not-A are both true. *Dialetheism has a long tradition in Western philosophy going back to pre-Socratic philosophers. *For example, Heraclitus says: “We step and do not step into the same rivers; we are and we are not” (Fragment 49a, Robinson, 1987, p. 35). Aristotle vigorously opposed *dialetheism, maintaining that *the *Law of Non-Contradiction* (LNC) is “the most certain of all principles” (1005b24)—*firmissimum omnium principiorum.*

[41] F. Scott Fitzgerald, “The Crack Up,” *Esquire *(February 1936).

[42] To attribute to G‑d any temporal characteristic such as “at the same time” is, of course, an oxymoron, because G‑d utterly transcends time. However, for the lack of better words, we use this expression with the disclaimer that it is not to be understood literally, but rather as a manner of speech.

[43] Hence the name, Almighty.

[44] 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.

[45] Responsa of the Rashba (*Shu”t HaRashb”a*), Vol. I, sec. 418; see also Sefer HaChakirah by the Tzemach Tzedek, p. 34b ff.

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

[47] 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*.

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

[49] The enigmatic state of *nimna hanimna’ot* would manifest itself only 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 G‑d’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.

[50] 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, that is, the resolution of the contradiction between these two opposite tendencies, was a prelude to *tzimtzum*. See G. Scholem, *Kabbalah* (Dorset Press, 1974), pp. 132–133.

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

[52] The wave function (or wavefunction) is an abstract mathematical description of the quantum state of a physical system. The wave function is not a physical property of a system; it is not an observable. According to the 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.

[53] 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://quantumtorah.com/on-tzimtzum-sefirot-and-cardinal-numbers/ (retrieved on 8/27/20).

[54] Needless to say, no wave function could be ascribed to G‑d. This analogy could be either viewed as a crude metaphor to aid our intuition or as a structural parallel.

[55] 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 *tzimtzum* (in Hebrew, there are no vowels, so it is a true repetition, *tzm-tzm*) is a grammatical device in Hebrew to exaggerate the meaning. Thus, the ultimate diminishing is, indeed, a collapse rather than a mere diminishment.

[56] As many (but not all) interpretations of quantum mechanics hold, including that of this author’s futurist interpretation of quantum mechanics. See Alexander Poltorak, “The Age of the Universe: Using the Many-Worlds Interpretation of Quantum Mechanics,” *B’Or Hatorah* (2008), vol. 18, pp. 149–168; and Alexander Poltorak, “Towards a Futurist Interpretation of Quantum Mechanics” (2019), available at https://www.academia.edu/40891960/Towards_Futurist_Interpretation_of_Quantum_Mechanics.

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

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

[59] 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 as ever—now only concealed from our sight, i.e., swept under the rug.

[60] The very word *kabbalah* literally means “reception” and is used to denote information received via oral transmission.

Originally published on QuantumTorah.com.

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