alt Hacker NewsTo be fair to physicists, the standard physicists' definition isn't "a tensor is a thing that transforms like a tensor", it's "a tensor is a mathematical object that transforms in the following way <....explanation of the specific characteristics that mean that a tensor transforms in a way that's independent of the chosen coordinate system...>".
When people say "a tensor is a thing that transforms like a tensor" they're using a convenient shorthand for the bit that I put in angle brackets above.
My favourite explanation is that "Tensors are the facts of the universe" which comes from Lillian Lieber, and is a reference to the idea that the reality of the tensor (eg the stress in a steel beam or something) is independent of the coordinate system chosen by the observer. The transformation characteristic means that no matter how you choose your coordinates, the bases of the tensor will transform such that it "means" the same thing in your new coordinates as it did in the old ones, which is pretty nifty.
https://www.youtube.com/watch?v=f5liqUk0ZTw&pp=ygURdGVuc29yc...
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Right, but if you fill in the shorthand there’s no reason to think it’s circular; it’s just a normal definition at that point, albeit one without much motivation.
> a convenient shorthand for the bit that I put in angle brackets above.
Yes, but the "convenient shorthand" only makes sense if you already know what a tensor is. That renders the "definition" useless as an explanation or as pedagogy. It's only useful as a social signal to let others know that you understand what a tensor is (or at least you think you do).
> My favourite explanation is that "Tensors are the facts of the universe"
That's not much better. "The earth revolves around the sun" is a fact of the universe, but that doesn't help me understand what a tensor is.
What matters about tensors are the properties that distinguish them from other mathematical objects, and in particular, what distinguishes them from closely related mathematical objects like vectors and arrays. Finding a cogent description of that on the internet is nearly impossible.
> the reality of the tensor ... is independent of the coordinate system chosen by the observer
Now you're getting closer, but this still misses the mark. What is "the reality of a tensor"? Tensors are mathematical objects. They don't have "reality" any more than numbers do.
> no matter how you choose your coordinates, the bases of the tensor will transform such that it "means" the same thing in your new coordinates as it did in the old ones
That is closer still. But I would go with something more like: tensors are a way to represent vectors so that the representation of a given vector is the same no matter what basis (or coordinate system) you choose for your vector space.