So I would disagree that the machine learning terminology was coined without regard to the mathematical concept. I would argue that multi-dimensional arrays are

So I would disagree that the machine learning terminology was coined without regard to the mathematical concept. I would argue that multi-dimensional arrays are very naturally thought of as concrete realizations of tensors, just like how floating point numbers are concrete realizations of real numbers. It's true that you can take the "free vector space" on any set under the sun, but I disagree with the statement "every object can be viewed as a vector." Rather, you can construct a vector space and associate to each object an element of that vector space, but the object itself is not a vector. As opposed to an array of floats, which really itself looks like and is fruitful to think about as a vector. Edit: Oh, also regarding the heterogeneous type stuff. I think most would agree that calling a bag of ints, floats, bools, and strings a "tensor" would not make much sense.