Hey Fabrizio,

thanks for the reference.

However, my motivation is much more mundane — a shorthand notation for contractions (so that one can avoid pesky indices, easy mistakes, and see that some ). It if helps, only for finitely dimensional spaces.

And in this case, there is little to prove, that does not fall in one of the below:
- abstract nonsense (mentally, I am a physicist [*])
- things that go beyond maths itself (e.g. perception /emboddied cognition etc: “Is ‘x’ written in a slightly different shade of gray still the same symbol, or a different symbol?”)

Sure for more advanced operations than contractions, it may be less trivial.

([*] I used Dirac deltas before I studied distributions. And I am fine with that. Also: my approach to teaching is as concrete as possible. To the point that I start linear algebra with 2-dimensional real vectors, certainly not an abstract notion of linear space.

For the same reason, I have a mixed relationship with category theory. Sometimes it is a useful abstraction. The first time I saw a tensor product with as a commutative diagram was illuminating; fortunately, I had a lot of experience with outer/Kronecker product before. :) Otherwise, I have to yet see something from category theory in physics I’ve found useful (for my taste everything that can be represented by a finitely-dimensional matrix does not count).)