### Different approaches to math:

Different approaches to math:

#### Drive forwards through entailments to a particular statement

Drive forwards through entailments to a particular statement

#### Try to build a fabric of many statements that fit together

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]

[“obvious white holes” vs. unobvious ones]

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

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

“Principle of explosion”

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

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

## Constructive math vs. observer math

Constructive math vs. observer math

#### “4th paradigm metamathematics”

“4th paradigm metamathematics”

#### In philosophical logic:

In philosophical logic:

“Statement X” vs. “Statement X is true”

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

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

#### Ruliad: all statements are in there....

Ruliad: all statements are in there....

#### Which statements can a coherent observer patch together?

Which statements can a coherent observer patch together?

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

## Incompleteness

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?

What is negation?

## Making fabrics

Making fabrics

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{A,AA,,AA,AAA,AAA,AAA,A,AAAA,AAAA,AAAA,AAAA,AAAAA,AAAAA,AA,AAAAA,AAAAA,AAA,AAAAA,AAAAAA,AAAAAA,AAAAAA,AAAAAA,AAAAAA,AAAAAA,AAAAAAA,AAAAAAA,AAAA,AAAAAAA,AAAAAAA,AAAAAAA,AAAAAAA,AAAAAAA,AAAAAAAA,AAAAAAAA,AAAAA,AAAAAAAA,AAAAAAAA,AAAAAAAA,AAAAAAAA,AAAAAAAA,AAAAAAAA}

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{{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}}

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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

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?

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?]

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

## Truth vs. falsity

Truth vs. falsity

#### Truth: can you get there in your fabric

Truth: can you get there in your fabric

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

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

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

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?

How does truth vs. falsity work in multivalued logic?

## What happens at the ur level?

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?

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

Equivalences between areas of math

#### Computation universality implies a very low-level equivalence

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?

But alchemical transformations are not easy.....

Can you get from one area of math to another without molecular-scale manufacturing?

#### Motion in physical space

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”

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