Tensors are TOO intuitive — shantum's notes

ELI5/TLDR

Tensors are notoriously hard to learn, but not for the reason you’d think. The problem isn’t that they’re abstract — it’s that there are too many different ways to picture them, and every teacher picks a different one. If you read five explanations, you’ll get five mental images that don’t obviously match up. The fix is to figure out why you want tensors in the first place, pick the matching view, and ignore the rest.

The Full Story

The setup — a familiar nightmare

The video opens with a routine that anyone who has tried to self-teach a hard math topic will recognize. You Google “what is a tensor” and the AI tells you it’s a multi-dimensional array of numbers. Then Wikipedia tells you it’s an algebraic object describing multilinear relationships. Then somebody on a forum tells you it’s “something that transforms like a tensor,” which is the mathematical equivalent of being told a chair is a thing you sit on chairs with. You chase covectors, then dual spaces, then covariant versus contravariant indices, then universal properties, and ten years later you’ve learned category theory but still cannot tell anyone what a tensor is.

In this video, I will not be explaining what a tensor is. At this point, any explanation I could give would probably just confuse you even further.

That’s the thesis. The video isn’t a tensor explainer. It’s an explainer about why tensor explainers keep failing.

The diagnosis — too intuitive, not too abstract

The author’s claim is contrarian. The usual complaint about math is that it’s unintuitive. His claim is the opposite — tensors are too intuitive. As in, there are too many distinct intuitive pictures available, all of them legitimate, and they don’t obviously line up with each other. Pick any other math object — a derivative, a group, a probability distribution — and you can usually find one or two ways to picture it that most teachers agree on. Tensors don’t have that. They have a dozen.

To explain why this is a problem, he detours into the role of intuition in math itself.

Intuition has two problems

First, intuition can be wrong. He uses the classic example of people insisting 0.999… cannot equal 1, because their gut feeling about infinitely repeating decimals is leading them astray.

Second, and more dangerous — intuition is subjective. What clicks for one person is noise for another. His example is category theory. Some people find it the master key to mathematics; others find it pointless abstraction. Neither side is wrong. Their brains are just wired differently.

One man’s trash is another man’s intuition.

The historical fix mathematicians invented for this is formality. Strip out the gut feel, write everything in symbols and definitions that can’t be misread, and disagreements become identifiable bugs rather than philosophical brawls. The cost is that formal math papers are unreadable to outsiders, and for a long time educational material copied that style. Math got a reputation for being cold and impossible.

The pendulum swung — and overshot

The recent online wave — think YouTube channels with smooth animations and warm narration — pushed back against the formality-only style. The author thinks this was good and necessary, but says some of us went too far in the other direction. He flags four specific failure modes that pure-intuition content creates.

One — people now think they need intuition before they can use a tool. Sometimes you just have to learn the rules and crank the handle. You can compute with tensors for years without a satisfying mental picture, and that’s fine.

Two — people mistake surface intuition for understanding. A polished animation gives you the warm feeling of getting it without the underlying machinery. The feeling and the understanding come apart.

Three — intuition alone doesn’t let you do real work. He gives a personal example. He’d read that adjoint functors in category theory describe “a weak form of equivalence between categories.” Nice and intuitive. Useless when he actually needed to prove something — a theorem about basis-unions of vector spaces — that turned out to follow directly from a property of adjoints. He couldn’t see the connection because he had only the slogan, not the machinery.

Four — what feels intuitive to the explainer may not feel intuitive to you. Same point as before, now applied to the explainers themselves.

Why tensors specifically get hammered by this

Other math concepts have one or two standard intuitions. Tensors have many. The author has actually been collecting them — a list of distinct intuitive descriptions used by different communities. Because modern educators feel obligated to lead with intuition, and because there are so many intuitions to choose from, learners shopping across YouTube channels get a different mental model from each one. The pictures don’t reconcile. The learner feels stupid. They are not stupid. They are watching ten people draw the same elephant from ten different angles, with no one telling them it’s the same elephant.

The three families

The author groups all the tensor intuitions into three buckets:

Computational view — tensors as multi-dimensional arrays of numbers. You manipulate indices, you transform coordinates. This is the world of subscripts and superscripts and Einstein summation.
Functional view — tensors as multilinear maps. Things that eat vectors and covectors and spit out numbers, linearly in each slot. This is where dual spaces and covectors live.
Abstract view — tensors as algebraic objects defined by a universal property. This is the world of tensor products and category theory. The definition doesn’t tell you what a tensor is, it tells you what role it plays.

These three are connected in deep ways, but you usually don’t need the connections. You need the one that matches your work.

Match the view to the job

His prescription is practical. Engineers should learn the computational view, optionally the functional view, almost never the abstract view. Physicists should learn the functional view, can pick up the computational view to read older papers, and should peek at the abstract view if they’re doing quantum mechanics. Mathematicians can skip the computational view and choose between functional and abstract depending on their subfield. Hobbyists should pick one and stick with it for a while.

There’s no shame in the tried and true method of using tensors without understanding them, like countless people have done in the past.

The closing move is permission to be a competent user without being an enlightened one. Pick a lane, learn the rules, do the work.

Key Takeaways

The reason tensors are confusing isn’t that they’re abstract — it’s that there are many legitimate intuitive descriptions of them, and they don’t obviously match.
Intuition in math has two failure modes — it can be flatly wrong, and it’s irreducibly subjective. What clicks for one brain is noise for another.
Formality is math’s escape hatch from the subjectivity of intuition. Symbols don’t disagree about what they mean.
Polished intuitive content can produce a feeling of understanding that comes apart from real understanding the moment you try to do work.
Three canonical “views” of tensors — computational (arrays + indices), functional (multilinear maps), abstract (universal properties + category theory).
Match the view to your job. Engineers — computational. Physicists — functional, possibly all three. Mathematicians — functional or abstract. Hobbyists — pick one.
It is legitimate to use a mathematical tool well without ever having satisfying intuition for it. Mechanical competence is not a consolation prize.
Adjoint functors anecdote — a slogan-level intuition (“weak equivalence between categories”) was useless for actually applying the concept to a real proof. Slogans aren’t tools.

Claude’s Take

This is a short video doing one job well — reframing a frustration most learners blame on themselves. If you’ve ever felt dumb because three tensor explanations refused to add up, the reframe is genuinely useful. The author isn’t selling you a new explanation. He’s saying the reason you’re stuck is structural, not personal.

The strongest part is the diagnosis of intuition-only content. There’s a real failure mode where a smoothly animated video gives you the dopamine of insight without the load-bearing machinery underneath, and you don’t notice the gap until you try to compute or prove something and discover you have nothing to grab onto. The adjoint-functors story is a clean instance of this — a slogan that satisfied the curiosity itch but did zero work when work was needed.

The weakest part is that the three-view taxonomy, while useful, is itself a piece of intuition presented without much justification. Why exactly these three families and not four or two? The video doesn’t say. It works as orientation, not as theory. That’s probably fine for a 12-minute video, but it would have been more honest to flag.

Score 8/10. Doesn’t teach you tensors — explicitly refuses to — but does something more valuable for someone in the early-confused stage. Tells you the maze has multiple correct exits and that picking one is allowed. Useful general advice for any abstract topic, not just this one.