Need (a) theorem statement in the language of the axioms (b) theorem proof in the language of the axioms
We say: what gives the theorem size is the “uncertainty” in its underlying axioms
E.g. in metamath:
pythag, pythi are both statements of the Pythagorean theorem
Same metamath library, but different outcome statements...
In full entailment cone ... the Pythagorean theorem is a region in branchial space
Example of “theorem cloud”
1. At the axiom level
Case 1. Same axioms; different statement
Case 2. Different axioms
Throw in many axioms to the original big bang .... Now any particular version of a theorem might pick out different ones
2. At the eme level
E.g. there are many ways to make an integer
Size of a theorem in emes
Take the theorem statement and write it in the language of the axioms ... and then even lower level
What does an observer coarse grain?
“Nearby” atoms of space might have had different histories ;; their path from the big bang might have different [cf two proof paths]
0. The observer doesn’t care about names of variables
1. The observer doesn’t care about the proof
If you conflate pythag and pythi then you are also conflating a zillion other intermediate results...
Path counting gives the number of ways to reach a theorem
In the entailment cone, how frequently is the theorem reached?
With expression rewriting styles of graph...
Destructive interference: you can’t believe both branches at the same time .... because you’d get shredded
Observer has a certain “branchial range” : how far are they sensing in branchial space?
Between the slits : the passage of the photon through each slit is too far away in branchial space
The initial condition contains not just the axioms we want, but all other axioms as well ....
So ... if we try to conflate things that are too far apart, we might be gulping on random other axioms ... which means we’re concluding things that we consider garbage....
If the proofs get too far apart, they will go into zones that have contribution from axioms you didn’t want to be considering
If things stay sufficiently close, the computational irreducibility from the environment won’t affect you ... and you can see things without “noise”
If the observer adds too many completions, they’ll get explosion...
With only 4, 6 things should be OK and we should only get to the entailment of the first rule...
With 4, 3 we can now get anywhere.....
Consistency + Consistency ?
Can two consistent axiom systems get combined to yield inconsistency?
“Two slits are worse than one”
Not only do you have to run a single thread and see what it does; you have to run all the threads to find out e.g. they do the same thing
Interpretation for math
Two proofs can get far enough apart in metamathematical space that an observer shouldn’t believe they come from the same axiom system
Because if they do believe it they wind up believing in some “intermediate cases” that correspond to different (and “wrong”) axiom systems
Two branches: one geometry, one algebra
At the beginning and end of the proof, we have a comparatively fast-running translator between the geometry and the algebra In the middle, they’re a big mess, and it’s nontrivial to do the translation [to do the translation, you can’t do it at a high level; you have break everything into atoms] If they do say they’re equivalent, they’re making a leap of faith ... because they can’t verify it at that moment [ In the leap of faith, you have no constructive path ] “Jumping conjectures” (e.g. Langlands, Grothendieck, etc.) [Assume there is a conjecture ; but it’s bad ] [ Ramanujan-related stuff ;;; e.g. PrimePi[n] > ....
E.g. Pythagorean claim: all numbers are ratios of integers E.g. one branch is about numbers represented symbolically; one branch is about p/q numbers We are really about rationals ... but we have general statements