alt Hacker News> You might find it circular reasoning but it is not
Um, yes it is. "A foo is an object that transforms as a foo" is a circular definition because it refers to the thing being defined in the definition. That is what "circular definition" means.
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To 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...
Of course, there are some definitions that reference themselves that are not circular, or alternatively, some circular definitions that are useful. For example, the definition of "ancestors" might be "your parents and your parents' ancestors" (with the implication that if you don't exist, you don't have any ancestors). But I agree that this is not a useful definition for "tensor".