### Gowers/Grothendieck/Gromov

Gowers/Grothendieck/Gromov

#### theory building

theory building

Exploring MM space

[ Maximize branchial expansion ]

#### problem solving

problem solving

Finding specific geodesics

[ Maximize branchial contraction ]

### Branch equivalence

Branch equivalence

[ For a one-way MW system, BE is a “metamodel” ]

In[]:=

ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},{"A"},5,"StatesGraph"]

Out[]=

In[]:=

ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},"CanonicalKnuthBendixCompletion"]

Out[]=

{}

If this was not causal invariant, then branch equivalence actually says something.

In[]:=

ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},"A",5,"KnuthBendixCompletion"]

Out[]=

{AABBABAB,ABABAABB,AABBABBA,ABBAAABB,AABBABBBBB,ABBBBBAABB,ABABABBA,ABBAABAB,ABABABBBBB,ABBBBBABAB,ABBAABBBBB,ABBBBBABBA}

[[ looks only at unresolved items ]]

In[]:=

Table[ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},"A",t,"KnuthBendixCompletion"],{t,4}]

Out[]=

{{},{},{AAABBB,ABBBAA},{AABABA,ABAAAB,AABABBBB,ABBBBAAB,ABAABBBB,ABBBBABA}}

In[]:=

Flatten[%]

Out[]=

{AAABBB,ABBBAA,AABABA,ABAAAB,AABABBBB,ABBBBAAB,ABAABBBB,ABBBBABA}

In[]:=

ResourceFunction["MultiwaySystem"][Join[{"A""AB","BB""A"},%270],{"A"},5,"StatesGraph",GraphLayout"LayeredDigraphEmbedding"]

Out[]=

## Causal invariance

Causal invariance

Theorem proving is trivial; no wrong turns

[It could take a while to converge]

## Models

Models

In[]:=

ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},{"A"},5,"StatesGraph"]

Out[]=

Is ABBBB provable from AA? Answer: not generically.

Multiplication table: maximal set of relations between words

Less extreme model: one that adds some random relation not provable in the original system

In[]:=

ResourceFunction["MultiwaySystem"][{"A""AB","BB""A","AA""ABBB"},{"A"},5,"StatesGraph",GraphLayout"LayeredDigraphEmbedding"]

Out[]=

Need to foliate the second object with the first:

In[]:=

{ResourceFunction["MultiwaySystem"][{"A""AB","BB""A","AA""ABBB"},{"A"},5,"StatesGraph",GraphLayout"LayeredDigraphEmbedding"],ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},{"A"},5,"StatesGraph"]}

Out[]=

,

In[]:=

ResourceFunction["MultiwaySystem"][{"A""AB","BB""A","AA""ABBB","ABBB""AA"},{"A"},5,"StatesGraph",GraphLayout"LayeredDigraphEmbedding"]

#### With a multiplication table, you are adding a relation between every element and A, or B.

With a multiplication table, you are adding a relation between every element and A, or B.

### Construct foliations

Construct foliations

## Holonomy

Holonomy

Curvature is reflected in the presence of unresolved branch pairs....

[ Delay in resolution of branch pair reflects curvature ]

Curvature leads to the difficulty of theorem proving

Causal disconnection prevents theorems from being proved [across event horizon]

## Causal Graph

Causal Graph

[ Using prior lemmas to prove subsequent lemmas ]

I.e. ABABB is used to prove ABBAA [ insofar as they are all coming from the “big bang” ]

#### Different case: every node is a proposition [cf FindEquationalProof]

Different case: every node is a proposition [cf FindEquationalProof]

So then every single event is a complete proof.....

But to prove a particular thing may take many steps, with many nodes as lemmas...

#### Difficulty of finding a proof depends on the number of causal edges that come in ...

Difficulty of finding a proof depends on the number of causal edges that come in ...

Causal edge density leads to branch pairs [ Einstein equations ]

Roughly: the more prior results you “know”, the more wrong turns you can make

[ Causal edges cause the contraction of corollary balls ]

[ If the curvature grows, the branch pair density increases ... does this lead the corollary ball to compress to nothing? ]

[i.e. do a given set of initial conditions all converge to the same thing? ]

[i.e. do a given set of initial conditions all converge to the same thing? ]

#### [Completion approach: structure of space changes]

[Completion approach: structure of space changes]

#### [ Is the infinite future of math like the universe: a bunch of black holes? ]

[ Is the infinite future of math like the universe: a bunch of black holes? ]

# Older Notes

Older Notes

#### I could choose to “contextualize” my math experiments by adding a bunch of data to each end of each string....

I could choose to “contextualize” my math experiments by adding a bunch of data to each end of each string....

#### Theorems in math A turns into B: physics A evolves to B

Theorems in math A turns into B: physics A evolves to B

#### Each “theorem path” is like a transition amplitude for QM

Each “theorem path” is like a transition amplitude for QM

#### What is the analog in math of coarse graining in physics?

What is the analog in math of coarse graining in physics?

Nearby theorems: corollaries....

Some theorems are shortly provable from others; start with a bundle of “nearby terms”

Powerful theorem: big backbone in the graph

Powerful theorem: big backbone in the graph

#### In actually doing math, you don’t go on a long path; you use theorems you already have, and go on a fairly short path.

In actually doing math, you don’t go on a long path; you use theorems you already have, and go on a fairly short path.

#### The refactoring of axioms : what is the relation to abstraction?

The refactoring of axioms : what is the relation to abstraction?

#### Speciation in math: event horizons in the MW graph might correspond to the breaking off of different fields of math...

Speciation in math: event horizons in the MW graph might correspond to the breaking off of different fields of math...

Note: vertex list is different in this case:

If we are tracing all possible paths, which lemma is most worth introducing to shorten the average path?