## Syntax of category theory

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

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 [?]

Spatialization of limiting versions [?]

Hierarchy

breaking down universal computational into a geometrical hierarchy

#### Rulial multiway system

Rulial multiway system

{a->b->c,a->d->c}

In[]:=

Graph[{a->b,b->c,a->d,d->c},VertexLabelsAutomatic]

Out[]=

In[]:=

Graph[{a->b,b->c,a->d,d->c,b->d,d->b},VertexLabelsAutomatic]

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

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

Space of all symbolic transformation rules

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

## Correspondence with type theory

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

Set theory

Only possible base is {}

ur-element : has base elements other than { }

## Analog of consciousness constraint for math:

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

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

e.g. Goodstein needs ZFC

## Foundations of mathematics

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

models are labels the mathematical observer is using

axiomatic carving of rulial space is the multiway graphs

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

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

models are labels the mathematical observer is using

axiomatic carving of rulial space is the multiway graphs

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