David wrote:

> He says instead to use the universal mapping property to understand the tensor product. But I don't like the idea of abandoning the definition of something in order to understand it.

That's not "abandoning the definition". To this mathematician, at least, the universal mapping property _is_ the definition of the tensor product. It says essentially this: suppose you want to think of bilinear maps out of $M \times N$ as linear maps out of some module. Then the module you want is $M \otimes N$.

If this is too fancy, don't worry: there are lots of different ways to understand tensor products, from geometrical to algebraic, from explicit nuts-and-bolts constructions to nice conceptual characterizations. They're all equivalent, and there's got to be one that's right for you! Everyone has their own favorites.

> Is this a case where it only makes sense to understand things though its morphisms? I hope not, because I like objects as well as arrows.

We all like objects, but the best way to understand them is through their morphisms. To understand what something _is_, nothing beats knowing what it _does_, and what you can _do with it_.

However, if you don't like this philosophy I have no intention of forcing it on you.

> Whereas in the world of vector spaces, tensors have a clearly visualizable representations, things become more subtle when we generalize to modules over a ring.

I wouldn't say that. I _visualize_ them almost the same way. If you're working over a ring with some particular properties, you may want to adapt your visualizations a bit. For example, if your ring has 1 + 1 = 0, now you're in a world where all your vectors "wrap around", obeying $v + v = 0$. But if you're working _in general_, for an _arbitrary_ ring, you might as well visualize things the way you're used to. You just need to not take all the features of your visualization too seriously.

You seem to be reading books that feature highly discouraging quotes about the comprehensibility of tensors. That's too bad! Maybe someone who doesn't understand something shouldn't be writing about it.

> He says instead to use the universal mapping property to understand the tensor product. But I don't like the idea of abandoning the definition of something in order to understand it.

That's not "abandoning the definition". To this mathematician, at least, the universal mapping property _is_ the definition of the tensor product. It says essentially this: suppose you want to think of bilinear maps out of $M \times N$ as linear maps out of some module. Then the module you want is $M \otimes N$.

If this is too fancy, don't worry: there are lots of different ways to understand tensor products, from geometrical to algebraic, from explicit nuts-and-bolts constructions to nice conceptual characterizations. They're all equivalent, and there's got to be one that's right for you! Everyone has their own favorites.

> Is this a case where it only makes sense to understand things though its morphisms? I hope not, because I like objects as well as arrows.

We all like objects, but the best way to understand them is through their morphisms. To understand what something _is_, nothing beats knowing what it _does_, and what you can _do with it_.

However, if you don't like this philosophy I have no intention of forcing it on you.

> Whereas in the world of vector spaces, tensors have a clearly visualizable representations, things become more subtle when we generalize to modules over a ring.

I wouldn't say that. I _visualize_ them almost the same way. If you're working over a ring with some particular properties, you may want to adapt your visualizations a bit. For example, if your ring has 1 + 1 = 0, now you're in a world where all your vectors "wrap around", obeying $v + v = 0$. But if you're working _in general_, for an _arbitrary_ ring, you might as well visualize things the way you're used to. You just need to not take all the features of your visualization too seriously.

You seem to be reading books that feature highly discouraging quotes about the comprehensibility of tensors. That's too bad! Maybe someone who doesn't understand something shouldn't be writing about it.