Instead of being too intuitive, they're 2-intuitive

I now know less about tensors than I ever did, but I feel much better about myself.

Quote of the day: "One man's trash is another man's intuition."

Quaternions: 0 intuition of the mathematical object itself, but here I am rotating spacecraft in my 3d game engine like a boss

mathematical ragebaiting

Consider this intuitive definition: spicy matrices.

Dude reframing it as being too intuitive is so brilliant, I got it immediately

Realest video i've ever seen in my life, 12 and a half minutes of spitting facts

6:37 bro took shots at his own vid 😭

1:57 Calling a 2-category a "monoidal monoidoidoid" is pretty funny. To quote a blog post, "This is possibly the worst way to introduce 2-categories, and has the added benefit of being incredibly silly to say out loud. 10/10. No notes. I’m absolutely adding this to my vocabulary. But what does it mean? Well first things first, we want to parenthesize this correctly. A 2-category is a (monoidal monoidoid)oid. ..."

The joke is that -oid is often used for a particular type of generalization: "In general if X is a gadget that can be seen as a one-object category, then an X-oid will be that same kind of gadget, but allowing multiple objects. For instance, a group can be seen as a one object category where every arrow is invertible. So a groupoid is any category where every arrow is invertible."

Of course the -oid suffix is not at all exclusive to this construction, the term "monoid" predates category theory. But a monoid can be viewed as a single-object category (with the monoid being the set of endofunctors of that object), so a "monoidoid" is a silly way to just say "category". And a monoidal category can be viewed as a single-object 2-category (its "delooping"), hence a 2-category could, if you're being very silly, be called a monoidal categoroid or monoidal monoidoidoid.

I think one of the most confusing aspects is people refusing to make the distnction between tensors and tensor fields. People who do physics or differential geometry call both just tensors.

I personally hate the "tensor is an array of numbers" since that doesn't tell me what a tensor IS, just how it can be written.

0:18 I almost died laughing at that one

Honestly, all this does is make me want a resource that dives into all three views and explores the connections between them. "In most cases these connections are not needed" just makes me want it even more.

My favorite is the option "A tensor is a scalar-valued multilinear map from some fixed number of vectors and covectors" because at least I already know what all those words mean, so it forms a fully-defined concept without further research

A tensor is that thing I plug numbers into to get my principal stresses, idk why everyone makes it seem so complicated

A very simple notion of "tensor" is this.  A "tensor" is something that may be represented in a vector space that does not depend on which basis you are using.  For instance, a particular vector, v, is a tensor because it is invariant as we transform from one basis to another, while the coordinates of vector v are not tensors because the coordinates of vector v change from one basis to another.  As a "real world example", look at some object in front of you, like your coffee-cup.  The coffee-cup never changes, but as you move your head around, viewing the coffee-cup from different locations, its shape changes as perceived by your eyes.  From some locations, the handle sticks out to one side; from another location the handle is not visible.  Yet, in spite of all these perceived changes in the shape projected onto the retina of your eye, the coffee-cup itself never changes.  Thus, the coffee-cup is a tensor, while the perceived shape of that coffee-cup is not a tensor.  The coffee in your coffee-cup may be thought of as consisting of time-varying tensor fields, but this is a topic for another discussion.  This is why all great scientists drink coffee.

There is a famous quote attributed to Albert Einstein, who contributed to the theory of tensors when he came up with his theory of Relativity.

"Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore."
— Often attributed to Einstein in discussions about his work (quoted in P. A. Schilpp, 1949)

I think what he meant by this is that mathematicians took his simple, elegant ideas and "formalized" them, transforming them into an intractable mess of complicated notation and abstract theory.  The mathematician Levi-Civita actually corrected some mistakes in Einstein's tensor manipulation in a famous paper.  Thus, abstract tensor theory was even confusing to the brilliant Albert Einstein, a giant in Physics.

My big problem of "edutainment" is it tricks you into thinking you learned something but then when you try to use what you learn you realize you don't really know what your doing. Edutainment is fun but nothing beat an actual education and we'll build curriculum from a professor, and it might actually harm your learning

Teaching tensors to bioinformaticians (former biologists) right now. My definition was "tensor is an element of a tensor product", and the motivation was the space of random variables obtained from two different random variable spaces by joining their probability measures independently (which is basically the same reason why they appear in quantum mechanics)

2:05 I nearly cried