Mathematics

Urim and Thummim as a Metaphor for Science

By Alexander Poltorak Introduction In the first installment of this essay, “Collapsing the Wave Function of Urim and Thummim,”[1] we discussed the inherent duality of Urim and Thummim—the duality of human (High Priest) and the oracular device (Urim VeThummim). In this sequel, I want to step back and consider broader parallels between Urim and Thummim and science. The Urim and Thummim, embedded within the High Priest’s breastplate (ḥoshen), constitute the most enigmatic oracular tradition described in the Torah (Exodus 28:30; Numbers 27:21). While the Torah credits their use to Moses and Joshua in determining divine will in moments of crisis (Lev. 8:8; Num. 27:21; 1 Sam. 28:6), it never divulges the precise mechanism by which light (Urim) and perfection (Thummim) conveyed answers. Later Jewish sources—from the tannaitic Sifre Bamidbar (chap. 23) to medieval exegetes like Ibn Ezra—stress two points: first, Urim and [...]

By Alexander Poltorak|2025-05-12T10:00:55-04:00May 11th, 2025|Collapse of the wavefunction, Exodus (Shemot), Information, Parshah, Pentateuch (Chumash), Philosophy, Quantum Theory, Science, Shemot, Tetzaveh, Time, Uncategorized|0 Comments

Tzaraath, Black Holes, and Holographic Principle

By Alexander Poltorak Abstract This essay explores the profound connections between the biblical laws of tzaraath (Leviticus 13-14) and modern concepts of entropy and the holographic principle in physics. Tzaraath, often misunderstood as mere leprosy, is interpreted as a physical manifestation of underlying spiritual and social disorder and discord, particularly linked to destructive speech (lashon hara). The essay argues that the elaborate rituals of diagnosis, isolation, and purification for tzaraath function to contain and reduce this spiritual/social entropy, restoring social order and inner harmony. Parallels are drawn to Jacob Bekenstein’s discovery that black hole entropy is proportional to its surface area, and the subsequent idea that the information content of a spatial volume might be encoded on its boundary—the holographic principle. It is suggested that a spiritual analog of Beckenstein-Hawking entropy expressing the [...]

By Alexander Poltorak|2025-05-04T18:59:15-04:00May 2nd, 2025|Cosmology, Entropy, Information, Leviticus, Metzora, Parshah, Pentateuch (Chumash), Physics, Tazria, Tazria u’Metzora, Thermodynamics, Uncategorized|1 Comment

Shemini,Phase Space,Hamiltonian Mechanics: Kabbalah and Physics of Seven and Eight

By Alexander Poltorak Abstract The Torah portion of Shemini records the climactic eighth day of the Tabernacle’s dedication, the moment when divine fire finally descends. Classical commentators see in the contrast between the first seven days and the eighth the difference between the natural and the supra‑natural. This essay revisits that symbolism from the standpoint of contemporary physics. First, it reassesses the traditional link between seven and the “natural order.” Second, it proposes that the six spatial–momentum coordinates of classical phase space, together with time, provide a more precise physical analogue for the seven midot than the usual six directions of space plus a temporal axis. Finally, it explores an unexpected resonance between the Hamiltonian action integral and the Tetragrammaton, suggesting that the principle of stationary action mirrors the scriptural affirmation, “For I Y‑H‑W‑H do [...]

By Alexander Poltorak|2025-05-04T18:42:44-04:00April 28th, 2025|Freedom, Kabbalah, Leviticus, Malachi, Mathematics, Mathematics, Names of G‑d, Parshah, Pentateuch (Chumash), Physics, Principle of Least Action, Science, Sefirot, Shemini, Space, Time, Uncategorized, Vayikra|0 Comments

Did G‑d Create Mathematics?

By Alexander Poltorak "G‑d is a mathematician"Carl Friedrich Gauss I. Can We Prove that G-d created Axioms of mathematics? 1. Introduction A reader challenged me with a question, “Could you prove G-d created basic propositions (axioms) of mathematics?” It is a profound question that merits a more detailed answer. There is no universally accepted “proof” in the mathematical sense that G‑d authored the axioms (or “basic propositions”) of mathematics. The question of whether G‑d is the ground—that is, the metaphysical foundation—or source of mathematical truths is a longstanding philosophical and theological debate. Here are a few perspectives on this issue, along with reasons why no formal, universally agreed-upon proof exists. 2. What Would a “Proof” Even Look Like? i. Nature of Mathematical Proof In mathematics, proofs demonstrate that a conclusion follows logically from a set of axioms. [...]

By Alexander Poltorak|2025-05-04T18:39:20-04:00March 17th, 2025|G‑d, Mathematics, Mathematics, Proves of G‑d, Uncategorized|1 Comment

Sarah in a State of Superposition

And it came to pass, when he was come near to enter into Egypt, that he said unto Sarai his wife: “Behold now, I know that thou art a fair woman to look upon. And it will come to pass, when the Egyptians shall see thee, that they will say: This is his wife; and they will kill me, but thee they will keep alive. Say, I pray thee, thou art my sister; that it may be well with me for thy sake, and that my soul may live because of thee.” (Genesis 12:11-13) This passage presents a perplexing scenario. Abraham (then Abram) asks his wife Sarah (then Sarai) to present herself as his sister upon entering Egypt. Classical commentators and the Zohar grapple with Abraham's motives, questioning why he would place Sarah in such a [...]

By Alexander Poltorak|2024-11-10T12:34:00-05:00November 10th, 2024|Bereshit, Collapse of the wavefunction, Fractals, Genesis, Husband and wife, Kabbalah, Lech-Lecha, Pentateuch (Chumash), Physics, Quantum Theory, Sefirot, Song of Songs, Superposition, Superposition, Tanakh, Uncategorized|0 Comments

Gazing at the Shekhinah

Gazing at the Shekhinah

In his commentary on this week’s Torah Portion, Yitro, Rabbi Chayim Vital, writing in the name of his teacher, the Ari-zal, states that Abel was punished for gazing at the Shekhinah—the divine presence.[1] But what relevance does this have to the Torah portion retelling the greatest event in Jewish history (and, indeed, the history of human civilization)—the Sinaitic epiphany—the giving of the Torah? This is the Torah portion, where we read the Ten Commandments. What is the relevance of the sin of Abel to the Ten Commandments? More generally, what is Abel’s connection to this Torah portion? That is easy to understand. The Torah portion Yitro starts with the story of Jethro (Yitro), Moses’s father-in-law, coming to Moses in the Sinai desert with his daughter—the wife of Moses—and her two children. Rabbi Chayim [...]

By Alexander Poltorak|2024-02-05T15:05:18-05:00February 4th, 2024|Bayesian probability, Collapse of the wavefunction, Exodus (Shemot), G‑d, Kabbalah, Neuroscience, Parshah, Pentateuch (Chumash), Reincarnation, Superposition, Superposition, Uncategorized, Wavefunction, Wavefunction of G‑d, Yitro|2 Comments

Principle of Least Action III — History

Principle of Least Action III — History

The spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements. Pierre Louis Maupertuis (1744) Among the more or less general laws, the discovery of which characterize the development of physical science during the last century, the principle of Least Action is at present certainly one which, by its form and comprehensiveness, may be said to have approached most closely to the ideal aim of theoretical inquiry. Its significance, properly understood, extends, not only to mechanical processes, but also to thermal and electrodynamic problems. In all the branches of science to which it applies, it gives, not only an explanation of certain characteristics of phenomena at present encountered, but [...]

By Alexander Poltorak|2025-04-28T14:28:54-04:00September 10th, 2023|G‑d, Kabbalah, Mathematics, Mathematics, Physics, Principle of Least Action, Science, Sefirot, Teleology, Time, Tzimtzum, Uncategorized|0 Comments

Entrainment by the Red Heifer

Entrainment by the Red Heifer

And for the unclean they shall take of the ashes of the burning of the purification from sin, and running water shall be put thereto in a vessel. Numbers 19:17 In my essay “Paradox of the Red Heifer,” I proposed a mechanism by which the ashes of the red heifer remove the impurity of death, where I analogized the procedure with resuscitation using a defibrillator.[1] In this essay, I propose an additional mechanism based on the phenomenon of entrainment, which I will explain below. Entrainment In physics and biology, entrainment refers to the synchronization of two or more rhythmic cycles.[2] This concept can apply to various systems, from physical phenomena to biological rhythms. Here are a few examples: Physics. In mechanical systems, entrainment refers to the process whereby two interacting oscillating systems assume the [...]

By Alexander Poltorak|2023-07-06T19:54:37-04:00July 3rd, 2023|Biology, Chemistry, Chukat, Ezekiel, Mathematics, Mathematics, Neuroscience, Numbers, Parshah, Pentateuch (Chumash), Physics, Science, Soul, Spirituality, Uncategorized|1 Comment

The Tree of Knowledge as a Quantum-Mechanical Metaphor

The Tree of Knowledge as a Quantum-Mechanical Metaphor

To Mendel Almost three years ago, in December of 2019, I posted an essay, "The Tree of Knowledge as a Metaphor for Superposition of States and Heisenberg’s Uncertainty Principle." The manuscript of that essay contained an addendum with a short primer on the superposition of states in quantum mechanics and quantum-mechanical treatment of the primordial sin. It expressed the ideas of Maimonides about primordial sin in precise mathematical formulas proving the exact parallel between the fruit of the tree of knowledge of good and evil and a quantum eraser. Realizing that only people familiar with the formalism of quantum mechanics would understand that part of the essay and fearing losing my readers, I decided to omit the addendum entirely. However, the addendum is arguably the most interesting part of the essay, because it [...]

By Alexander Poltorak|2022-10-21T17:10:50-04:00October 20th, 2022|Bereishit, Bereshit, Genesis, Heisenberg Uncertainty Principle, Mathematics, Mathematics, Parshah, Pentateuch (Chumash), Physics, Quantum Eraser, Quantum Theory, Science, Superposition, Superposition, Uncategorized, Uncertainty Principle|1 Comment

Zettaflood

In the six hundredth year of Noah’s life, in the second month, on the seventeenth day of the month, on the same day were all the fountains of the great deep broken up, and the windows of heaven were opened.Genesis 7:11 As the Torah tells us, at the dawn of human history, the flood swept the world. Most people, except for Noah and his family, drowned in that flood. Today, we too are drowning in a flood of a different kind—the informational flood. To put the current flood in perspective, consider this: according to one estimate in 2003, between the beginning of human civilization and the spread of computers, humanity generated cumulatively about 12 exabytes (12x1018 bytes) of data. However, in 2002 alone, we generated 5 exabytes of data. According to another estimate, [...]

By Alexander Poltorak|2021-10-05T16:43:54-04:00October 4th, 2021|Fractals, Genesis, Kabbalah, Mathematics, Noach, Parshah, Pentateuch (Chumash), Uncategorized|1 Comment

Fill the Earth

Fill the Earth

And G‑d created man in His own image, in the image of G‑d created He him; male and female created He them. And G‑d blessed them; and G‑d said unto them: “Be fruitful, and multiply, and replenish the earth, and subdue it . . . .Genesis 1:28 G‑d created man, male and female, and commanded them to be fruitful and multiply, fill the earth, and subdue or conquer it. While the literal meaning of this verse is apparent—fill the earth with your progeny by procreating[1]—it begs a question. Specifically, the phrase “fill the earth” seems superfluous—wasn’t it enough to say “Be fruitful and multiply”? Indeed, the Hebrew word milupim[2] (“to multiply”) is etymologically related to the word mil’u[3] (“to fill”).[4] If humans were to multiply, as commanded by G‑d, they would naturally fill the earth. It seems [...]

By Alexander Poltorak|2021-10-04T15:13:44-04:00October 4th, 2021|Bereishit, Bereshit, Creation, Fractals, Genesis, Kabbalah, Mathematics, Mathematics, Parshah, Pentateuch (Chumash), Space, Uncategorized|0 Comments

Egyptian Exile as a Metaphor for Compactification in String Theory

Egyptian Exile as a Metaphor for Compactification in String Theory

And they made their lives bitter with hard service, in mortar and in brick…Exodus 1:14 On a literal level, the Egyptian exile (Galut Mitzrayim) is a story of the Jewish people’s enslavement in ancient Egypt, where they were forced to do hard labor making mortar and bricks, and building cities for the Pharaoh. On a deeper level, it is axiomatic in Jewish mysticism that suffering is usually a means to purify and rectify a sinner’s soul in this or past incarnation. It is also axiomatic that any sin damages the sinner’s soul and some supernal spiritual levels that obstruct the flow of the divine benefice into the world, impeding the blessing and, therefore, causing suffering to the sinner. Technically, it works as follows. When a sin committed below causes damage above, the resulting [...]

By Alexander Poltorak|2021-01-11T22:54:15-05:00January 11th, 2021|Exodus (Shemot), Kabbalah, Mathematics, Mathematics, Parshah, Pentateuch (Chumash), Physics, Reincarnation, Sefirot, Shemot, Shemot, Spirituality, String Theory, Topology, Uncategorized|0 Comments

Noah’s Ark—Three Layers of Reality

Noah’s Ark—Three Layers of Reality

A light shalt thou make to the ark…with lower, second, and third stories shalt thou make it. (Genesis 6:16) We mentioned in the previous posts that Noah’s ark was a microcosm.[1] As we discussed in the previous post, “Noah’s Ark—Three Layers of Being Human,” most structural parallels related to Noah’s ark are based on its tripartite structure —that is, its having three tiers. In Chasidic thought, the three levels of Noah’s ark correspond to three worlds of BiYA—Beriyah (the World of Creation), Yetzirah (the World of Formation), and Asiyah (the World of Action). In this essay, we will investigate how the tripartite structure of Noah’s ark is reflected in the structure of reality. We might say that the totality of existence comprises three layers—physical, informational, and spiritual. The classification of reality into three [...]

By Alexander Poltorak|2024-06-23T21:53:15-04:00October 28th, 2020|Age of the Universe, Bereshit, Collapse of the wavefunction, Fractals, Genesis, Kabbalah, Mathematics, Mathematics, Noach, Parshah, Pentateuch (Chumash), Physics, Sefirot, Space, Spirituality, String Theory, Superposition, Time, Uncategorized, Wavefunction|1 Comment

Tzimtzum III—Renormalization—Sweeping Infinites Under the Rug

Tzimtzum III—Renormalization—Sweeping Infinites Under the Rug

Sweeping Infinities Under the Rug—or Renormalization Having dealt with internal contradictions in the previous section (see Physics of Tzimtzum I — The Quantum Leap and Physics of Tzimtzum II — Collapse of the Wave Function), we are left with another problem—infinity. Although G‑d concealed His self-contradictory nature by “sweeping paradoxes under the rug” in the process akin to the collapse of the wave function, the Light of the Infinite (Ohr Ein Sof) filled the whole of existence with infinite Divine emanation. This infinite radiation left no room for any finite creation to emerge. What was G‑d to do? Having aced the exam on quantum field theory with flying colors, G‑d employed the favorite trick of theoretical physicists in sweeping infinities under the rug using what is called “renormalization.”[1] Roughly speaking, renormalization solves the [...]

By Alexander Poltorak|2020-10-14T20:39:04-04:00October 14th, 2020|Bereishit, Bereshit, Creation, Deuteronomy, G‑d, Kabbalah, Mathematics, Parshah, Pentateuch (Chumash), Physics, Quantum Field Theory, Quantum Theory, Relativity, Space, Standard Model, Tzimtzum, Uncategorized, Uncertainty Principle, Zohar|1 Comment

Physics of Tzimtzum I — The Quantum Leap

Physics of Tzimtzum I — The Quantum Leap

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

By Alexander Poltorak|2020-08-25T19:56:59-04:00August 17th, 2020|Atomic Theory, Bereishit, Bereshit, Cosmology, Creation, Genesis, G‑d, Kabbalah, Mathematics, Names of G‑d, Parshah, Pentateuch (Chumash), Physics, Quantum Theory, Space, Spirituality, Tzimtzum, Uncategorized, Zohar|5 Comments

Counting Weeks and Days

Counting Weeks and Days

There is a Biblical Commandment to count the days between the Passover and Shavuot, the "Feast of Weeks" (a.k.a. Pentecost). We start counting on the second day of Passover (the first day of the barley harvest in the land of Israel, when the wave-offering of the omer, i.e., “sheaf,” of ripe barley was made in the Jerusalem Temple) and finish on the eve of Shavuot—the day when two loaves of bread made of wheat were offered at the Temple. There are exactly seven weeks (forty-nine days) between these two holidays; we are commanded to count the weeks and the days. These forty-nine days are called days of Sefirat HaOmer (the days of “counting the Omer”) or, simply, days of Sefirah ("counting"). This commandment is given in the following verses of the Torah: And [...]

By Alexander Poltorak|2024-05-17T18:10:51-04:00April 28th, 2020|Fractals, Kabbalah, Mashiach (Messiah), Omer, Sefirot, Shavuot, Spirituality, Sukkot, Uncategorized, Zohar|2 Comments

Wigner’s Friend Receives a Death Threat

It is related that on the day that Rabbi Yehuda HaNasi died, the Sages decreed a fast, and begged for divine mercy so that he would not die. And they said: Anyone who says that Rabbi Yehuda HaNasi has died will be stabbed with a sword. (Talmud, tr. Ketubot 104a) We find an interesting example of the Wigner’s Friend paradox in the Gemara quoted above. The Gemara discusses the circumstances surrounding the death of Rabbi Yehuda HaNasi (Juda the Prince, knows simply as the Rebbi), one of the greatest sages of Israel, who compiled Oral Torah in the Mishnah. Why did the Sages issued a death threat to “anyone who says that Rabbi has died” promising that he “will be stabbed with a sword”? The medieval Talmudist Rabbi Bezalel Ashkenazi (1520-1592), in [...]

By Alexander Poltorak|2023-03-23T18:17:23-04:00March 19th, 2020|Bayesian probability, Collapse of the wavefunction, Physics, Quantum Theory, Relativity, Superposition, Uncategorized, Wavefunction|2 Comments

The Primacy of the Prime Numbers

And the messengers returned to Jacob, saying: “We came to thy brother Esau, and moreover he cometh to meet thee, and four hundred men with him.” (Genesis 32:7) On this blog, we primarily focus on structural parallels between Torah and Physics and, more specifically, quantum physics. Today I would like to explore an interesting parallel between this Torah portion and the number theory. In my previous essay on the Torah Portion Vayishlach, “Jacob Meets Esau and his 400 men,” I wrote “Esau and his 400 men together were 401 strong. 401 is a prime number that is the sum of seven consecutive prime numbers (43+47+53+59+61+67+71).” In that essay, I focused on the significance of the number seven. Let us now focus on the significance of the prime number. The structural parallel I see [...]

By Alexander Poltorak|2019-12-15T00:54:15-05:00December 13th, 2019|Bereshit, Genesis, Mathematics, Mathematics, Parshah, Pentateuch (Chumash), Randomness, Sefirot, Uncategorized, Vayishlach|3 Comments

Fitting Pieces of the Puzzle Together

Fitting Pieces of the Puzzle Together

Meditations on the Maaseh Merkavah – IV This is the fourth and the final installment in the series of posts related to Ezekiel’s prophesy, Ma’aseh Merkava, “The Making of the Chariot.” For background information, refer to the previous posts, “Space – Between Future and Past,” “Relational Space,” and “Collapse of the Wave Function.” Regular readers of my blog may be wondering about my last post “Futurist Interpretation of Quantum Mechanics,” which was about my new interpretation of quantum mechanics. This is not a physics blog, however; the Quantum Torah blog is about Torah and physics (or, more broadly, Torah and science). What did the last post have to do with Torah? Hopefully, it will all now become clear. Last Shavuot I had a very unusual experience. Sitting in shul listening to the reading [...]

By Alexander Poltorak|2019-06-21T13:21:11-04:00June 21st, 2019|Collapse of the wavefunction, Ezekiel, Freedom, G‑d, Holidays (Yomim Tovim), Kabbalah, Names of G‑d, Physics, Quantum Theory, Relativity, Science, Sefirot, Shavuot, Spirituality, Superposition, Time, Topology, Uncategorized, Wavefunction|2 Comments

Relational Space

Relational Space

Meditations on the Maaseh Merkavah – II This post is a sequel to my previous post, “Space – Between Future and Past.” For background information, the reader is referred to that post. My second Shavuot “epiphany” was that space is relational. Not that I realized that space was relational for the first time – I always thought so. But, for some reason, thinking about the meaning of the four faces of Chayot, the "living creatures" – the angelic beings described in the prophecy of Ezekiel that we read on Shavuot – made it absolutely clear in my mind. Isaac Newton The debate about the nature of space goes back at least to the times of Newton and Leibniz. Isaac Newton believed (as it is apparent from his mechanics and as it [...]

By Alexander Poltorak|2020-10-20T00:26:20-04:00June 17th, 2019|Ezekiel, Kabbalah, Mathematics, Mathematics, Names of G‑d, Physics, Relativity, Sefirot, Shavuot, Space, Time, Uncategorized|8 Comments