It seems, at times, that physicists and mathematicians mean different things when they say the word "tensor." From my perspective, when I say tensor, I mean "an element of a tensor product of vector spaces."

For instance, here is a segment about tensors from Zee's book *Einstein Gravity in a Nutshell*:

We already saw in the preceding chapter that a vector is defined by how it transforms: $V^{'i} = R^{ij}V^j$ . Consider a collection of “mathematical entities” $T^{ij}$ with $i , j = 1, 2, . . . , D$ in $D$-dimensional space. If they transform under rotations according to $T^{ij} \to T^{'ij} = R^{ik}R^{jl}T^{kl}$ then we say that $T$ transforms like a tensor.

This does not really make any sense to me. Even for "vectors," and before we get to "tensors," it seems like we'd have to be given a sense of what it means for an object to "transform." How do they divine these transformation rules?

I am not completely formalism bound, but I have no idea how they would infer these transformation rules without a notion of what the object is *first*. For me, if I am given, say, $v \in \mathbb{R}^3$ endowed with whatever basis, I can *derive* that any linear map is given by matrix multiplication as it seems the physicists mean. But, I am having trouble even interpreting their statement.

How do you derive how something "transforms" without having a notion of what it is? If you want to convince me that the moon is made of green cheese, I need to at least have a notion of what the moon is first. The same is true of tensors.

My questions are:

- What exactly are the physicists saying, and can someone translate what they're saying into something more intelligible? How can they get these "transformation rules" without having a notion of what the thing is that they are transforming?
- What is the relationship between what physicists are expressing versus mathematicians?
- How can I talk about this with physicists without being accused of being a stickler for formalism and some kind of plague?