Different approaches to math:

Drive forwards through entailments to a particular statement

Try to build a fabric of many statements that fit together

Create a two-way graph by including both forward & backward entailment cones
The goal of math is to find statements that fit together, and don’t produce white holes
[“obvious white holes” vs. unobvious ones]

Is “inconsistency”/falsity the ability to reach all statements

“Principle of explosion”

If you state a statement, does it mean it’s true?

Constructive math vs. observer math

“4th paradigm metamathematics”

In philosophical logic:

“Statement X” vs. “Statement X is true”
“It is raining” vs. “It is true that it is raining”

Ruliad: all statements are in there....

Which statements can a coherent observer patch together?

If what’s patched contains a white hole, then you’re toast...

Incompleteness

There are statements out there that you cannot reach from the fabric you are building
Axiomatic approach: build your fabric by moving outward from a small number of axioms
Inconsistency: you’re building outwards from the axioms and you hit a white hole
Completeness: your LoI can reach all of a definite set of statements but never hit a white hole

What is negation?

Making fabrics

In[]:=
CombinedFabricTagged[{"A""AAAA","AA""AA","AA""AAA","AA""AAAA","AAA""AAA"},2,VertexLabels->Automatic]
Out[]=
In[]:=
CombinedFabricTagged[{"A""AAAA","AA""AA","AA""AAA","AA""AAAA","AAA""AAA",""<->"A"},2,VertexLabels->Automatic]
Out[]=
In[]:=
CombinedFabricTagged[{""<->"A"},3,VertexLabels->Automatic]
Out[]=
In[]:=
VertexList[%]
Out[]=
{A,AA,,AA,AAA,AAA,AAA,A,AAAA,AAAA,AAAA,AAAA,AAAAA,AAAAA,AA,AAAAA,AAAAA,AAA,AAAAA,AAAAAA,AAAAAA,AAAAAA,AAAAAA,AAAAAA,AAAAAA,AAAAAAA,AAAAAAA,AAAA,AAAAAAA,AAAAAAA,AAAAAAA,AAAAAAA,AAAAAAA,AAAAAAAA,AAAAAAAA,AAAAA,AAAAAAAA,AAAAAAAA,AAAAAAAA,AAAAAAAA,AAAAAAAA,AAAAAAAA}
In[]:=
Map[StringLength,%,{2}]/.TwoWayRule->List
Out[]=
{{0,1},{1,1},{0,0},{0,2},{1,2},{0,3},{2,1},{1,0},{2,2},{0,4},{1,3},{3,1},{0,5},{1,4},{2,0},{2,3},{3,2},{3,0},{4,1},{0,6},{2,4},{1,5},{3,3},{4,2},{5,1},{3,4},{0,7},{4,0},{1,6},{4,3},{5,2},{2,5},{6,1},{4,4},{0,8},{5,0},{5,3},{1,7},{2,6},{6,2},{3,5},{7,1}}
In[]:=
ListPlot[%]
Out[]=
1
2
3
4
5
6
7
2
4
6
8
For a group, this would say that “all word equations are true” [i.e. it is a group which contains only the identity element]

Definition of falsity

Something is false if assuming it makes everything true
Assume 1==2 ... then any number is equal to any other number
If “everything is true” it’s equivalent to “everything is false”

Key question: is there only one kind of falsity?

Is falsity about generating “too many” statements, or is it about generating all statements
Including what turns out to be a false statement: at the beginning, it lets you prove all sorts of useful things ... but it turns it to be “too good to be true”

What is the cardinality of falsity? [How many statements does it reach?]

Truth vs. falsity

Truth: can you get there in your fabric

Falsity: if you get there, you will be able to get everywhere (“explosion”)

Truth becomes a region of statements that have certain coherence and consistency

There might be two different viable entailment fabrics which both avoid white holes, but which contain different stuff [this is a consequence of incompleteness]
[Archipelago of islands of truth : non-finiteness of proofs means it’s not clear the “search party” for other islands will reach something]

Law of excluded middle

[Could take arbitrary time to determine if a statement is true or not?]

How does truth vs. falsity work in multivalued logic?

What happens at the ur level?

You can’t really tell what’s go on : is that clump of molecules actually part of a tornado ?

What happens at the high-level math level?

You imagine that you are always on an island (though you neither know nor care which island)

Equivalences between areas of math

Computation universality implies a very low-level equivalence

E.g. everything is made of (material) atoms [“universality of matter”]
But alchemical transformations are not easy.....
Can you get from one area of math to another without molecular-scale manufacturing?

Motion in physical space

To get from here to there, you might need to be atomized and reconstituted

Can one get beyond “everything is made of atoms” or “everything is computation”