#### Addition facts: [“statements”]

Addition facts: [“statements”]

In[]:=

Flatten[Table[eq[Nest[s,0,i+j],plus[Nest[s,0,i],Nest[s,0,j]]],{i,3},{j,3}]]

Out[]=

{eq[s[s[0]],plus[s[0],s[0]]],eq[s[s[s[0]]],plus[s[0],s[s[0]]]],eq[s[s[s[s[0]]]],plus[s[0],s[s[s[0]]]]],eq[s[s[s[0]]],plus[s[s[0]],s[0]]],eq[s[s[s[s[0]]]],plus[s[s[0]],s[s[0]]]],eq[s[s[s[s[s[0]]]]],plus[s[s[0]],s[s[s[0]]]]],eq[s[s[s[s[0]]]],plus[s[s[s[0]]],s[0]]],eq[s[s[s[s[s[0]]]]],plus[s[s[s[0]]],s[s[0]]]],eq[s[s[s[s[s[s[0]]]]]],plus[s[s[s[0]]],s[s[s[0]]]]]}

#### Metalogic [“laws of inference”]

Metalogic [“laws of inference”]

Derive new statements from old

#### In physics, limited number of relations contain a given atom of space...

In physics, limited number of relations contain a given atom of space...

In math, could always have “equivalence relations”

eq[s1,s2]

#### Can make statements of math into strings using e.g. Polish notation (or combinators)

Can make statements of math into strings using e.g. Polish notation (or combinators)

Derive new math statements just using rules of inference [ a entails b ]

#### Geodesic path: optimal derivation of math result

Geodesic path: optimal derivation of math result

Ultimately everything derives from the “big bang” .. axioms

#### Proof cone: what would no longer be true if that statement were removed

Proof cone: what would no longer be true if that statement were removed

#### Godel’s theorem: inaccessible regions of metamathematical space

Godel’s theorem: inaccessible regions of metamathematical space

Can take an arbitrarily long path to reach a given region...

#### “Physical space” for math expressions: look at structural connectivity for statements of Boolean algebra etc.

“Physical space” for math expressions: look at structural connectivity for statements of Boolean algebra etc.

#### Branchial space: common ancestry for math expressions

Branchial space: common ancestry for math expressions

### Singularity theorems

Singularity theorems

Too much activity all geodesics lead to a single point [ theory is ended ]

All paths terminate in finite time decidable

All paths terminate in finite time decidable

#### Event horizon

Event horizon

These proof paths can’t lead to different parts of metamathematical space

### Reference frames

Reference frames

[Compare with adding additional axioms]

Given a foliation, things in the same slice can’t be proved equivalent

Adding more proof edges (e.g. from additional axioms) more equivalences provable

#### Construct “proof lattice”

Construct “proof lattice”

### Mathematical consciousness

Mathematical consciousness

Follow a single thread of derivation? Find a particular proof: i.e. care what’s true but not “why”

Causal invariance: it doesn’t matter which proof you find...

#### Quantum mathematician

Quantum mathematician

Consider many paths of proof...

Destructive interference? Two statements when glued together with very different proof paths ??

### Motion: ~functors

Motion: ~functors

### Ruliad

Ruliad

All possible rules of inference ; rulial reference frame particular rules (?)

Different inference rules: different views of math

In some, can derive 1+1=7