Mathematics

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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 [...]

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 [...]

Space – Between Future and Past

Meditations on the Maaseh Merkavah – I We do science by studying nature. We study physics in a lab, peering into space, or working out mathematical models with pencil and paper to see if they fit experimental data. However, great mystics of the past were able to see how this world operates by gazing into spiritual worlds above. On Shavuot, the holiday when we celebrate the revelation on Mount Sinai, we read the prophecy of Ezekiel (the First Vision of Ezekiel) called Maaseh Merkavah (or Ma'aseh Merkabah) – Work of the Chariot. Masters of Kabbalah have taught us that understanding the Work of the Chariot gives the initiated an understanding of the works of nature. This Shavuot, studying and thinking about Maaseh Merkavah, I came to understand some profound insights about the physics [...]

Ten Sephirot and Special Relativity

The Special Theory of Relativity, published by Albert Einstein in 1905, was a game changer. It ushered the era of new physics dubbed by some as the “Jewish physics.” And Jewish physics it was. At the core of Special Relativity is the notion of the relativity of motion – all motion is not absolute, as Newton thought, but is relative to something else, to a particular point of view. In physics, we define a point of view as a frame of reference. If you are standing in a moving train, leaving the railroad platform, in your frame of reference (which is always at rest) the train is at rest relative to your frame of reference. The railroad platform, on the other hand, is moving with respect to your frame of reference. For someone [...]

The Puzzle of Pi

March 14 is celebrated by nerds around the world as the Pi Day. When written in digits, 3/14 represents first three digits of the number traditionally represented by the Greek letter “π” (pronounced pi) – 3.14. Pi Day is celebrated by eating pie and discussing the significance of π. Some folks who have too much time on their hands compete at memorizing decimal digits representing the value of pi. The record presently stands at 67,890 digits! While eating pie is optional, discussing the significance of π is truly mandatory. So, what is π and why is to so important? π is arguably the most famous mathematical constant, expressing the ratio between the circumference of a circle and its diameter. If you take a circle with a diameter of 1, the circumference of this [...]

Vectors, Spins, and Gender

Vectors, Spins, and Gender Speak to the children of Israel, and have them take for Me an offering; from every person whose heart inspires him to generosity, you shall take My offering. (Ex. 25:2) Every physics professor teaching physics 101 to freshmen spends the first lecture teaching vector algebra. Why vectors are so important? Because most quantities in mechanics (a branch of physics dealing with motion and forces) are vectors. A vector is a mathematical object that has a magnitude and a direction. For example, velocity is a vector – it shows not only how fast the object is moving, but also in what direction it is moving (it is different from “speed” which is an average velocity and is just a number). Similarly, acceleration is a vector, force is a vector, angular [...]

Fractal Tu B’Shevat

When thou shalt besiege a city a long time, in making war against it to take it, thou shalt not destroy the trees thereof by wielding an axe against them; for thou mayest eat of them, but thou shalt not cut them down; for is the tree of the field man, that it should be besieged of thee? (Deut. 20:19) On Tu B’Shevat we celebrate the New Year of the trees. There is much to be said about the legal, ecological, homiletical, and mystical meaning of this day. And, indeed, much has been written about this minor holiday from these perspectives. In this post, however, I’d like to focus on the mathematical and biological aspects of this holiday that, to my knowledge, had not been discussed until now. When the Torah rhetorically asks, [...]

Jacob Meets Esau and his 400 men

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.” (Gen. 32:7) Why does the Torah take pains to tells us the exact number of men accompanying Esau? Servants of Esau are of no particular significance. It is apparent that, by specifying the number of men in Esau's entourage, the Torah is trying to tell us something about Esau. It seems to me that Jacob understood the significance of the number 400. This number is a sum of the powers of 7 from 0 to 3: 70+71+72+73=400. This four powers of 7 hint at the lunar month that has, approximately, 28 days (4×7). The significance of this will become apparent later. 400 is a repdigit in base [...]

The Meaning of Life as Taught by Bayesian Angels

And he [Jacob] dreamed, and behold! a ladder set up on the ground and its top reached to heaven; and behold, angels of God were ascending and descending upon it. (Gen. 28:12) Rabbi Shneur Zalman of Liadi (a.k.a Baal HaTanya, or the Alter Rebbe) once remarked in Yiddish, “lieben mit di tzait” – live with the time, meaning one should live with Torah portion of the week.  Last week I was taught a lesson on how the current Torah portion affects my thinking, whether I know it or not. Last week I shared with my wife, who is a biophysicist, that while thinking about system biology, I realized that all organisms – from a single cell to multicellular organisms – are Bayesian systems. I knew that the brain is a Bayesian engine, but [...]

Tisha B’Av on Shabbat – A Relativistic Perspective

Yesterday was the 9th day of the month of Av or, in Hebrew, Tisha B’Av. Usually, Tisha B’Av is marked by mourning and fasting. Yesterday, however, we ate festive meals, drank wine and were prohibited from fasting or displaying any signs of mourning. Because yesterday was Shabbat. Shabbat pushes off the observances of Tisha B’Av by a day. Indeed, today, Sunday, we fast and mourn the destruction of the First and the Second Holy Temple  – Bet Hamikdash – in Jerusalem, we remember the Holocaust and many other tragedies that befell the Jewish people. Why couldn’t we observe Tisha B’Av on Shabbat? After all, that was the day when on the 9th of Av, both Temples were destroyed! The simple explanation, of course, is that on Shabbat there is no mourning. On Shabbat, [...]

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