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This is especially fascinating when you consider graphs/diagrams are a way to encode math.

• owlbite 3y ago
  The fact that you can represent a graph (the mathematical abstract object) as a diagram is sort of by-the-by here. The most important thing is that graph algorithms and concepts have a strong relation to numerical aspects of the linear algebra and can be used to accelerate computation. (You could of course argue that the act that graphs can be represented as a diagram helps humans come up with such algorithms, but that's basically equivalent to saying that you can represent a matrix as a block of numbers and that helps humans look at it).
  • zmgsabst 3y ago
    I was pointing out the other direction: Diagrams are how you encode categorical models of semantics, which naturally can be represented as graphs. Those graphs can in turn be encoded as matrices. So you have a way to encode semantic foundations as matrices — which you can then use graph algorithms to analyze. Being able to move your semantic models (eg, diagrams) into a computational framework (eg, linear algebra) is neat.
• meindnoch 3y ago
  Umm... What? Please show us the graph that "encodes" the fundamental theorem of algebra.
  • zmgsabst 3y ago
    https://en.wikipedia.org/wiki/Category_theory https://en.wikipedia.org/wiki/Category_theory The fundamental theorem of algebra is a fact about the diagram which relates polynomials via division by monomials. Every polynomial of degree n is n divisions of a monomial away from the empty product. - - - - This is probably easier to see the other direction: Polynomials of degree n are isomorphic to n-products of monomials, and you can build a graph of the assembly where each arrow represents a multiplication by a particular monomial. (Then reverse the arrows, to get my original diagram.)