Eric Weinstein: The Square Root is the Psychedelic of Mathematics…
I’ve heard Eric say this a few times, but in his recent podcast with comedian Duncan Trussell, he goes a bit further into explaining this perspective, which, for those of us who are fascinated by the interior mysteries of deep mathematics… turns out to be profound. So here’s my transcription of his brief divergence into ‘The Panic Room in your mind that you didn’t know you had…”
“But what I told him was the most danger — one of the most dangerous secrets in mathematics which is that the square root — the lowly square root — is the Psychedelic of mathematics. It opens up the Panic Room in your mind that you did not know you had.”
“How?”
“Well, if I say what’s the square root of four, you say two. Okay so an integer gets you to an integer. Now I say what’s the square root of two, it goes from an integer to an irrational number — but it’s an algebraic number. Now I say what’s the square root of -2 — suddenly you’re in the complex numbers. So the real numbers are a one-dimensional system but the complex numbers are a two-dimensional — you just broke out of Flatland. Okay well, that’s pretty crazy. What’s the square root of a determinant? Something called the Pfaffian, which you’ve never heard of. What’s the square root of the Pontryagin class? Oh, it’s the Euler class. And now you’re just like: what’s the square root
of vectors and tensors? Oh it’s spinors — and you know objects that require 720 degrees of rotation to come back to normal — and you’re just thinking doesn’t 360 do it for everything?
So the square root is the most powerful Dangerous Drug imaginable.”
My friend Nancy Metrick ( hilarious name onomatopoeia ) asked chat GPT to explain this and received ( and posted ) the following:
Eric Weinstein’s commentary on the square root as a “dangerous drug” in mathematics emphasizes its ability to expand understanding from basic arithmetic to increasingly complex mathematical concepts.
Let’s break down each derivation of the square root that he references:
### 1. **Square Root of 4 → Integer**
- **Explanation**: The square root of 4 is 2, a simple integer. This is the most basic concept of square roots that most people learn early in their education: if you take the square root of a perfect square, you get a whole number. This introduces the idea that the square root can map an integer to another integer, which seems straightforward at first.
### 2. **Square Root of 2 → Irrational Number**
- **Explanation**: The square root of 2 is approximately 1.414, but it is **irrational** — it cannot be expressed as a ratio of two integers. The decimal expansion of √2 goes on forever without repeating. This moves you from the realm of integers to irrational numbers, which are still part of the **real number system** but are not as intuitive. The discovery of irrational numbers historically marked a profound shift in mathematical understanding.
### 3. **Square Root of -2 → Complex Numbers**
- **Explanation**: The square root of -2 introduces the concept of **imaginary numbers**. The square root of any negative number cannot be found within the real number system, so mathematicians extended the number system to include **complex numbers**, where \(i = \sqrt{-1}\). The square root of -2 would be written as \(i\sqrt{2}\). This expands mathematics from the real number line (a one-dimensional system) to the **complex plane**, a two-dimensional system. This transition from real numbers to complex numbers represents a fundamental leap in mathematical thinking, analogous to breaking out of “Flatland” into higher dimensions.
### 4. **Square Root of a Determinant → Pfaffian**
- **Explanation**: The **determinant** is a scalar value that can be calculated from a square matrix and has important implications in linear algebra and geometry. The **Pfaffian** is a mathematical object that arises as the “square root” of the determinant of a skew-symmetric matrix (a matrix where the transpose is equal to its negative). While the determinant provides information about the matrix, such as its invertibility, the Pfaffian is tied to more specialized concepts, such as spinor fields in physics. Most people have never encountered the Pfaffian, illustrating how taking the square root of more complex mathematical structures leads to more exotic objects.
### 5. **Square Root of the Pontryagin Class → Euler Class**
- **Explanation**: **Pontryagin classes** and **Euler classes** are concepts from **topology**, specifically in the study of vector bundles (a generalization of vector spaces that vary smoothly over a base space). The Pontryagin class is a characteristic class that provides information about the curvature of a manifold. The Euler class is related to the orientation and critical points of a vector field. Saying that the Euler class is the square root of the Pontryagin class implies a deep relationship between these topological invariants. This points to the fact that, in certain contexts, taking a square root can lead to a more fundamental or simpler underlying structure, even in abstract areas of mathematics.
### 6. **Square Root of Vectors and Tensors → Spinors**
- **Explanation**: **Vectors** and **tensors** are objects in linear algebra and physics used to describe quantities with direction and magnitude, and systems with multiple components, respectively. Taking the “square root” of these objects leads to **spinors** — more abstract mathematical entities used in quantum mechanics. Spinors are peculiar because they require a **720-degree rotation** to return to their original state, unlike vectors, which only require a 360-degree rotation. This is deeply connected to the geometry of space and how quantum particles behave, revealing that the square root operation can open up entirely new realms of mathematical and physical phenomena, such as spin and particle physics.
### 7. **The Square Root as a Metaphor for Expansion**
- **Psychedelic Effect**: Weinstein uses the metaphor of the square root being a psychedelic or a “dangerous drug” because of its power to take simple concepts and transform them into more complex and mind-expanding ideas. Each application of the square root moves you further from familiar territory, leading to more sophisticated mathematical structures. Just as a psychedelic experience can challenge and expand one’s perception of reality, so too can the concept of the square root expand one’s understanding of mathematical reality.
### Summary:
The square root, while seemingly simple at first, has profound implications across different areas of mathematics. From basic arithmetic to complex numbers, algebraic topology, and quantum mechanics, the square root opens up new dimensions of thought. Weinstein’s playful yet insightful analogy illustrates how mathematical operations like the square root can take you on a journey far beyond elementary math into the depths of abstract and sophisticated theory, mirroring the transformative experiences associated with psychedelic substances.
###7 and beyond is rather tepid. I can do better, by far. Weinstein wants us to attain ‘escape velocity’ from 40 years of failed physics due, primarily, to the ‘cognitive gravity well’ comprised of Ed Witten and String Theory ( the dog that doesn’t hunt ). What he’s hinting at is that interstellar humanity isn’t as far away as we commonly imagine.
Which, actually, is a lot more dangerous … than anything imaginable. Because, at the level of the species, we are an insane, zero-sum game-theoretical nightmare. We’re fairly good at inventing technologies, but abject failures … so far … at establishing intelligent societies.
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