Tensor Products Aren’t Real
Which is why they’re actually kind of cool.
Halfway through last semester, I was at a point in my mathematical journey where I was familiar with tensor products, but I had no idea why I should care about them. Sure, given two vector spaces V and W, I knew I could make a new vector space V⊗W to which the words “bilinear” and “universal property” somehow applied, but… why would I want to do that? Going to class never really answered this question for me, because the introduction of tensor products was always accompanied by a huge increase in abstraction, which instinctively made me stop paying attention.¹ I resigned myself to a fate of never truly understanding them. Then, I came across a Numberphile video about an application of the tensor product that so perfectly illustrates why I should care about them that from now on it will be my go-to example: the Dehn invariant.
In 1900, David Hilbert (of hotel fame) asked the question,
Given two polyhedra of equal volume, is it always possible to cut up one of the polyhedra with finitely many straight cuts and rearrange the pieces to form the other one?
In the same year, his student Max Dehn answered,
No.²
Dehn’s solution used the observation that there are two ways a cut can affect any given edge of a polyhedron: It can either split the length of the edge or the dihedral angle of the edge. In the first case we have a length invariant l = l₁ + l₂, and the second case we have an angle invariant θ = θ₁+ θ₂. How can we construct an invariant that somehow covers both cases? We use the tensor product! Bilinearity is exactly what we need.
Dehn proved that the sum of l ⊗ θ over all edges of a polyhedron is invariant under cutting and rearranging pieces. Calculations show that two polyhedra of equal volume (a cube and a tetrahedron, for instance) may have different values for this invariant, thus answering Hilbert’s question in the negative. (Do watch the video for a better explanation!)
Dehn’s result was what convinced me that tensor products are worth having around. Unlike the super abstract settings where I had seen tensor products before, problem here is simply about cutting shapes to make other shapes. At the same time, the result is substantial: No matter how hard you try to turn a tetrahedron into a cube, you will fail.
Sometimes a mathematical idea is just innately satisfying, like the Fibonacci thing I mentioned in my previous post. I think I’ve figured out that the tensor product is not one of those things. It seems more like a neat tool for making things bilinear. Looking back at the definition of tensor product, it’s clear that its purpose is purely utilitarian. You smush V and W together, then you literally force them to play nicely with each other by explicitly stating all the rules you want.
Tensor products aren’t real in the sense that they’re not a natural construction (to me at least). Rather, I think of them as being invented in order to help us understand particular problems. Many concepts which people today think of as natural must have felt very invented when they were first introduced. Take negative numbers, for instance, which were invented so that equations like x + 1 = 0 have solutions. Though I have no qualms with negative numbers now, I still remember a time in grade school when it felt really weird to me that the product of two negative numbers is positive. Similarly, the ancient Greeks argued over whether or not the square roots were a thing, but nowadays it’s a function on every calculator. When you’re doing mathematics, you have the freedom to invent new things, without regard to physical constraints, in pursuit of whatever cool result you please.³ This creative freedom is one of my favorite aspects of math, and although I still have much to learn about tensor products, they make a little more sense to me when I put them into this narrative of people inventing new, weird concepts to prove cool results.
1. Anyone who has ever sat next to me in lecture can attest to this.
2. Hilbert asked this question as part of a talk on hard questions that he thought mathematicians should work towards proving in the 20th century, which included things like the continuum hypothesis (proven to be undecidable) and the Riemann hypothesis (unproven to this day). So, it’s a little unfortunate that Dehn solved the question discussed in this post so quickly. It’s even more unfortunate that the question had been solved twenty years before Hilbert even asked.
3. Graph theory was invented so that Euler could prove to the people of Königsberg that their bridges were bad for tourism.
