It was yesterday that my friend Juanmarqs commented my post and pointed me about the origin of writing tensor as an indexed-object. Before that time I always think that it is clear why we write tensor as the way it is now. Since by definition tensor is the mapping from some dual vectors and some vectors (where “some” can means one or zero) to a real number, then the tensor itself is the quantity that has the “combination” of each basis dual vectors and basis vectors as its basis. Moreover, since this mapping is generally not symmetric, then if we interchange the order of those dual vectors and vectors, we will get a different number, which forces us to conclude that the combination of basis in a tensor depends on the order of dual vector spaces and vector spaces which are mapped. Using this way of thinking, the representation of tensor as the indexed-object (to be clear, the indexed-object is indeed its component, not the tensor itself) like we usually encounter is very natural.

After that my other friend recommended me about the Penrose’s diagram representation for tensors (lucky for him to have read Penrose’s The Road to Reality), in which he constructed the rule to write tensor as the electronics component-like diagrams and how to use them for tensor operation. Certainly this is remarkable, although accompanied by a little bit of pain if the number of tensors operated are full-stock. Naturally, then, I am curious about this radical idea. What if the universe is not described with equation-like system, but some other logical one which is internally consistent?

Of course we don’t have to find that because we are quite comfortable with our equation regime. But finding the best representation system of our universe in each part “locally” perhaps can help to understand it better, although it will give us a topology-like homework in the end: Why does this part use this system and that one use another? Well, just go ahead and see all possibilities.