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