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[]=
,
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?