Tensor Fields and Simplicial Complexes

Listening to Freya Holmér last night I started to get glimmers of an idea I had long ago about how to represent vector spaces in computational processes using this recursive abstract type: 

abstype 'a point = POINT
  of {getx :  'a vector,
      diff :  'a point -> 'a point,
      move :  'a point -> 'a point,
      scale : 'a -> 'a point,
      proj :  'a point -> 'a}
with
   fun new i (op +) (op -) (op * ) dot =
   let fun self x = POINT
      {getx = x,
       move = fn (POINT pr) => 
         (self (x + (#getx pr))),
       diff = fn (POINT pr) =>
         self (x - (#getx pr)),
       scale = fn i =>
         (self (x * i)),
       proj = fn (POINT pr) => 
         dot(x, (#getx pr))}
       in self i
       end
   fun getx (POINT pr) = #getx pr
   fun diff (POINT pr) = #diff pr
   fun move (POINT pr) = #move pr
   fun scale (POINT pr) = #scale pr
   fun proj (POINT pr) = #proj pr
end

Then we can use points to instantiate vectors of 'a points starting with the unit type for a 0-simplex. 

So the idea is to start with simplicial complexes rather than fixed vectors. In a sense, that is what vector spaces already are: we never have actual data in any physical space which define  what the basis vectors of the space really are, they are just orthonormal vectors in some unspecified units, and in a Tensor field we don't even know what orthogonal really means either. So the angle and length scales are a matter of convention, not of "physically evidential" fact. Then on the basis of just a simplicial complex we can define a metric tensor field. That would be much more like space as we experience it, I think. 

See Gabriele Carcassi on The Correspondence Between Quantum and Classical Mechanics

They are running an Assumptions of Physics Summer School, according to this post. See Assumptions of Physics - Summer School 2026 - June 24-26.

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Norman Wildberger thinks Algebraic Calculus is the way forward.

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See About Logic with Seunghyun Song and Jordi Fairhurst for more on vector spaces. 

Toby made a great video about preorders today:

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I guess that means I am supposed to learn Lean now:

My comment:

In Eilenberg's 1976 book "Automata, Languages and Machines" he defines Relations, Monoids and Categories and then writes (on the last page I have access to) "An example of a category is the category R of relations. The objects are sets and the morphisms in R(X, Y) are all the relations f: X → Y, with composition being composition of relations." There is a copy of volume II on the Internet Archive but you aren't allowed to look inside,... It's a three-volume work intended to lay a rigorous mathematical foundation for a chunk of computer science. I'm sure you know about it. 

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