Syntax of category theory

Generalization of the syntax
Morphism; functors [ multimorphisms ; operads (n inputs; 1 output) ]
Without transitive closure: have path dependence

Relationship to type theory

Arrows in type theory are functions
type constructors are rules in WM
Propositions as Types
Types as mw systems

Spatialization of limiting versions [?]

Hierarchy
breaking down universal computational into a geometrical hierarchy

Rulial multiway system

{a->b->c,a->d->c}
In[]:=
Graph[{a->b,b->c,a->d,d->c},VertexLabelsAutomatic]
Out[]=
In[]:=
Graph[{a->b,b->c,a->d,d->c,b->d,d->b},VertexLabelsAutomatic]
Out[]=
Necessarily, we’ve defined a face here...
Given a triangle in any graph, it could be a face. [[But ... many cycles in a graph aren’t naturally faces.]]
1 morphism : joins 2 objects [ maps from 0-morphisms to 0-morphisms ]
2 morphism : maps from 1-morphisms to 1-morphisms
operad : n-ary function mapping from n 0-morphisms to 1 0-morphisms [?]

Space of all symbolic transformation rules

Can be decomposed into transformation for things without , and things with 

Correspondence with type theory

A multiway system is like a type; the rules are the type constructor ; the terms of the type are states of the multiway systems ;; paths are proofs of equality
inhabitation of a type: multiway system has at least one state
multiway systems that are not inhabited:
In[]:=
ResourceFunction["MultiwaySystem"][{"x"->"x"},{},4]
Out[]=
{{},{},{},{},{}}
In[]:=
WolframModel[{{1}->{}},{}]["AllExpressions"]

Set theory

{{},{{}},{}}
Only possible base is {}
ur-element : has base elements other than { }

Analog of consciousness constraint for math:

Think only in terms of n-categories for small n....
Light cones: provability cone
Within the coordinate system defined by the slice
“Faster than proof travel”: have to go outside of a given mathematical framework
e.g. Goodstein needs ZFC

Foundations of mathematics

Axioms are transformations for symbolic expressions
​
two interpretations of mw graphs:
i) expression to expression (expressions are propositions)
ii) nodes are propositions
​
the act of doing mathematics is the act of coarse-graining...
​
a mathematician is a coarse-grainner
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models are labels the mathematical observer is using
​
axiomatic carving of rulial space is the multiway graphs
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A model of a mathematical observer...
​
category theory is like axiomatic systems, GR is like our models of physics

The foliation of an algebraic system & Cayley graphs
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