21–27 of 27 posts
Blog post from Google : https://ai.googleblog.com/2019/06/introducing-tensornetwork-...
and an official website about the diagrams : https://tensornetwork.org/
Would going through this course (if one can call it that) help me learn linear algebra, or is more of a "Here is a different angle for those who get it" type of thing?
We have established that there is a homomorphism called θ from B, the PROP of diagrams, to Mat, the PROP of matrices. I've never seen such a complex subject as linear algebra presented with such clarity and simplicity.
Are you being sarcastic?
I looked this page, the whole site and even slept on the questions involved. My experience with being student, doing a bit of math tutoring and doing a bit of math teaching is that linear algebra is one of the simplest "real" math courses a student encounters. Matrices are elaborate but they're just repeatedly doing pretty simple - and the geometric intuition of matrices as translation and rotation of linear spaces, is both useful and one of the easier intuitions to get, among "basic" higher math intuitions.
Now, this page comes along and overall message, "Wow, linear algebra is hard to understand, here's simpler way to approach it" and then [stream of stuff no undergraduate possibly get and that I struggle with having an MA in math] (also a structure that's not at all related to the concrete geometric interpretation of linear algebra). And, after scanning the thing structure, I at least have come away with what I think the motivation involved is.
The thing that's "complicated" and "unintuitive" to our author is that the development of linear begin with operations on individual vectors, describing the mechanics in detail. It then jumps to the qualities of sets of vectors and how one subspace can be orthogonal to another subspace.
So, basically, it takes informal set theory as implicit scaffolding to the entire theory. And if you are a mathematician (or one kind of mathematician), set theory is a grunging, undesirable thing. The mechanics of computation is also grunging. I'm tempted to say everything is grunging until you reach a supreme level of abstraction. But the computation + set thing definitely is constructing the structure "bottom". So what the author is doing is constructing things top down. Combining algebraic operations as strings that's constructing the equivalent of subspaces at the same it's describing what addition "is". Everything gets constructed from one "piece".
I feel like I can see the appeal of this but I can't actually see it as producing a thing useful to me, at the levels of abstraction I like to work with.
It also makes think how set theory is a bit the object-orientation of mathematic. It's a messy, do-anything, constructive exercise. I can understand an urge of someone who like their basis to be enlightening, to base their structures on things of great apparent elegance. Perhaps I'm pessimistic enough to believe that one is always to going to face "grunge".