Tensors Are Too Intuitive

TITLE: Tensors are TOO intuitive CHANNEL: sudgylacmoe DATE: 2026-04-24 URL: https://youtu.be/NOb8dCMP0Ys ---TRANSCRIPT--- So you want to know what a tensor is. You decide to Google, what is a tensor? And the ever so helpful AI overview tells you that it’s a multi-dimensional array of numbers that generalizes things like scalars, vectors, and matrices. All right.

Then you go to Wikipedia, the most helpful resource for learning mathematics. It says that tensors are algebraic objects that describe multilinear relationships. Huh? What does that have to do with multi-dimensional arrays of numbers?

Upon looking further, you get even more confused. A tensor is something that transforms like a tensor? And this comment doesn’t help, either. What the heck is a covector? Looking into covectors, you start seeing things like dual spaces and linear forms. How are they related to tensors?

Then you see people start talking about covariant and contravariant tensors, where it seems like the only difference is whether we write a subscript or a superscript. Confused, you finally decide to ask people yourself. And then the top answer is, I’m not going to tell you what a tensor is.

Okay, fine. Looking around more on Wikipedia, you notice that there are at least five different pages about tensors. Why are there different pages if they’re all talking about the same thing? I mean, products are an algebraic idea, so tensor algebra and tensor product are basically the same thing, right? And these pages have weird arrow thingies that are called diagrams for some reason. And why in the world are these called universal properties?

You then hear about these things called symmetric monoidal categories, which are somehow related to tensors, so maybe you should figure them out, too. You then spend the next 10 years going down the category theory rabbit hole. By then, you have lost your job, your home, and are about to die of starvation. But at least you now know what a monoidal monoid away is. And yet you still don’t know 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. Instead, I will show you why tensors are so difficult to figure out. In the process, I hope I can equip you to find better resources on tensors for your context.

So why is it hard to understand tensors intuitively? My answer to this question may seem a bit backwards. It’s not that tensors are not intuitive. Instead, it’s that tensors are too intuitive.

To explain what I mean by that, we need to spend some time talking about the role of intuition in mathematics. Intuition has an interesting place in math. I think most professional mathematicians would agree that it’s important to have good intuition when doing math. It helps us see the bigger picture and can guide us when trying to figure things out.

However, intuition has many issues. The first issue is that it can be wrong. People having incorrect intuition is why there are endless debates over things like point nine repeating equaling one. Second, and more importantly, intuition is subjective. What is intuitive for one person is not necessarily intuitive for another.

I feel like a good example of this is category theory. I have seen people that swear by category theory, and if something can be phrased in terms of it, they feel like they understand that thing much better. But for others, category theory is just a bunch of nonsense, and they prefer describing things in very different ways. I have rarely seen discussions between these two types of people go well. I’m not trying to say that either way is better than the other. It’s just that some people’s brains are wired for category theory, while others’ brains are not.

There is a solution to these issues that mathematicians discovered long ago, formality. By presenting mathematical ideas in a formal way, there can be no disagreement over what is being talked about. While formal math can still be wrong, it is much easier to identify where the error is.

In my opinion, good math is a mix of intuition and formality. We use intuition to internally think of mathematics, but when the time comes to share our results with others, we should present with much more formality. While sharing our intuition can sometimes be helpful, it is more important to share the formal math than the intuition behind it.

Because of the importance of formality, mathematicians possibly started focusing on it too much. If you read a random math paper, chances are you will have zero idea what is going on. They almost never give any intuition. Because the public work of mathematicians became this formal, educational material started to follow suit. As a result, math developed a reputation for being overly formal and unintuitive. And to be fair, the educational material had become exactly that, overly formal and unintuitive.

In recent times, there has been a trend, especially in online content, to describe math in much more intuitive ways. While this trend has been helpful, and I would even say necessary for the future of math, I think some of us have gone too far.

The recent intuition-only content has led to several issues that I have seen among people online. The first issue is that people now think that they must have intuition for a mathematical idea before they can use it. While yes, intuition is important, there are times where we just have to roll up our sleeves and use something we don’t have good intuition for. I feel like tensors are actually a great example of this. Depending on your context, you don’t have to understand what tensors are. You just have to know how to work with them. You may not have any intuition behind what is going on, but as long as you know what rules they follow, you can get your job done.

Another problem with the recent intuition-only content is that people start to think that they understand a concept when in reality they only have a surface level intuitive understanding of the idea. To make matters worse, surface level intuition is often wrong. I have seen too many conversations online where somebody acts like they know what something is when it’s obvious that they actually don’t. I think that this is largely due to content that has good production quality but doesn’t actually have good explanations.

A third issue is that just having intuition doesn’t equip you to do actual mathematical work. Let me give a personal example of this. When looking up intuition for adjoint functors from category theory, a common description is that adjunctions describe a weak form of equivalence between categories. Okay, great. Now we have some intuition for adjoints. However, this description did not equip me to actually know how and when to use adjoints. Recently, I found myself wanting to prove the fact that given two disjoint sets A and B, the vector space with a basis given by the union of those sets is the same as the direct sum of the vector spaces with bases given by each set individually. I had an inkling that there was some way to describe this idea categorically, and it turns out it follows from one of the fundamental properties that adjoint functors satisfy. I had no idea that adjunctions would help me here since I had never taken the time to figure out how adjoint functors actually work.

The fourth issue is that, as I said before, what is intuitive for one person may not be intuitive for another. I have seen videos that claim to be telling me an intuitive way of thinking about something, and it did not feel intuitive to me at all. I’m sure it was intuitive for that person, but for me, it just fell flat. One man’s trash is another man’s intuition.

So, what does all of this have to do with tensors? The issue with tensors is that they are possibly the mathematical concept that have the most distinct intuitive ways to describe them. Over the years, I have compiled this list of different ways to think about tensors. While I primarily see it as a funny thing to use whenever somebody asks what a tensor is, it does make a good point. There are way too many different ways to intuitively think of tensors. Given modern educational trends, people feel like they have to focus on intuition when describing tensors. Thus, we get educational content that focuses on all of these different ways of describing them. Given how much educational material exists nowadays, we tend to shop around by looking at a variety of content, but in this case, we get people describing tensors in completely different ways. This confuses everyone that is trying to learn about tensors.

So, in light of all of this, how do we actually learn about tensors? Personally, I think the answer to this question depends on the person. As I said before, intuition varies from person to person, and with how many different ways there are of thinking about tensors, I can’t possibly know which one will work for you. However, I think that we can narrow it down a little by knowing why you want to learn about tensors.

I would say that there are three basic ways of thinking about tensors, which I have named the computational view, the functional view, and the abstract view. The computational view is the one that focuses on multi-dimensional arrays of numbers, along with things like index manipulations and coordinate transformations. The functional view is the one that focuses on the connection between tensors and multilinear maps, and tends to emphasize things like covectors. Finally, the abstract view is the one that focuses on the algebraic and categorical ideas behind tensors, emphasizing things like tensor products and universal properties. There are connections between these three views, but in most cases these connections are not needed. Depending on your circumstance, you usually only need to focus on one of these approaches.

If you are an engineer, you probably want to focus on the computational view. While I personally think that components are overused, I have to admit that when you are actually trying to get the job done, you will need components eventually. If you want to try to avoid the components, as I would suggest, you could also look into the functional view of tensors. I can’t think of any reason for an engineer to try to figure out the abstract view outside of curiosity.

If you are a physicist, I personally would suggest learning the functional view. However, I have to admit that many physicists in the past have used the computational view, so you might have to learn that too if you want to understand what other physicists have said. Furthermore, I don’t know too much about this, but I have seen the abstract view of tensors being used in quantum mechanics, so you might have to figure that one out, too. But you probably don’t need to go all in on figuring it out.

If you are a mathematician, you shouldn’t need to figure out the computational view at all. Depending on your situation, you probably want to focus on either the functional view or the abstract view. And if you do want to figure out the abstract view of tensors, you really should not shy away from universal properties. They’re not that bad once you get to know them.

There is one more group of people that might want to learn about tensors, hobbyists. After all, if you are watching this video, there’s a decent chance you are one of them. In this case, if your goal is to just understand tensors themselves, I would suggest picking one view and only focusing on that, at least for a while. You can go to the other views later if you are curious about them.

So, how do you learn how to use tensors? Just figure out how and why you want to use them, find material that focuses on the view that helps you, and then disregard what anybody else says about tensors. And remember, if you still can’t get intuition for them, that’s fine. There’s no shame in the tried and true method of using tensors without understanding them, like countless people have done in the past.