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

This is the previous talk he mentions:

On the physical significance of the existence of a well-ordering on the reals, see basis theorem in nlab:

The original proof of the existence of Hamel bases (after which the concept was named) was for the case of the real numbers regarded as a rational vector space and used (not directly Zorn's lemma but) the well-ordering theorem.

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

See Christoph Benzmüller - Many Logics, One Methodology for some ideas about how automated theorem provers can be used in such an exercise. 

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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. 

A finite-dimensional vector space is isomorphic to its dual, but only if you pick a basis. It's also isomorphic to its double dual, and that isomorphism requires no choices at all. Mathematicians called the second kind "natural" long before anyone could say what that meant precisely. In 1945, Samuel Eilenberg and Saunders Mac Lane invented an entire branch of mathematics to make the distinction rigorous. The framework they built, consisting of categories, functors, and natural transformations, turned out to capture something far more general than its origin. This video builds category theory from the ground up: categories as the minimal axioms for composition, functors as structure-preserving translations between mathematical domains, natural transformations as the maps that commute with everything, universal properties as the principle that what a construction does matters more than what it's made of, and adjunctions as the paired constructions that tie it all together.

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Frederic Schuller's lecture on tensor spaces: 

1:27:54 Vectors and matrices are just witch mathematics! 

1:55:37 On the importance of the order in which things are presented: first without the basis, then with the basis to get the dimension, then back to basis-free representation, then introducing the components, ... If you start out thinking of a vector as a list of numbers then all sorts of confusions arise.

2:00:52 On n-forms and determinants. I am sure there is a way to understand this in terms of simplicial complexes.

Tangent vector spaces from smooth manifolds:

TL;DR: Position vectors are just pointless confusion! 

See the full series of 28 lectures: The Geometric Anatomy of Theoretical Physics. 

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Norman Wildberger's idea is to synthesize smooth functions from power-series representations

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Marketing! See Gabriele Carcassi - A Simple Explanation of Quantum Mechanics and Frederic Schuller on Metric and Topological Spaces.