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How big is the past entailment cone for physics & math?
How big is the past entailment cone for physics & math?
How big is our conflation zone?
How big is our conflation zone?
Physics
Physics
What are human-scale coordinates for space?
In[]:=
933-(63)//N
Out[]=
261.
Space: >> 10^-21 m ; everyday ~ 1 micron [ smaller than a pixel ]
In 1 μm^3, 10^260 atoms of space
Branchial:
Mathematics
Mathematics
How big is the Pythagorean theorem?
How big is the Pythagorean theorem?
Accumulative derivation:
What does the Pythagorean theorem compile to?
(In terms of the primitives in which the axioms are stated)
In[]:=
ResourceFunction["MetamathImport"]["http://us.metamath.org/mpeuni/pythi.html"]
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Symbols,Statements
How big is the “most difficult theorem in Metamath”?
How big is the “most difficult theorem in Metamath”?
Top 100 : https://www.cs.ru.nl/~freek/100/
Top 100 : https://www.cs.ru.nl/~freek/100/
Pythagorean theorem
Pythagorean theorem
In[]:=
mm=ResourceFunction["MetamathImport"][URL["https://raw.githubusercontent.com/metamath/set.mm/develop/set.mm"]];
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tree=MetaMathProof[mm]["pythag","Tree"]
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TreeGraph[%795]
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Graph
,AspectRatio
1
2
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TreeGraph[%]
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Graph[EdgeList[%796],AspectRatio1/2]
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In[]:=
MetaMathProof[mm]["pythag","Graph"]
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tree[]
In[]:=
VertexCount[%800]
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1422
In[]:=
TreeExtract[%795,TreeLevel[{0,1}]]
Out[]=
,
,
,
,
,
,
,
,
In[]:=
treeTake[tree_,n_Integer?Positive]:=Tree[TreeData@tree,treeTake[#,n-1]&/@TreeChildren@tree]treeTake[tree_,_]:=TreeData[tree]
In[]:=
treeTake[%795,1]
Out[]=
In[]:=
treeTake[%795,2]
Out[]=
Analogy: statement of pythag; proof of pythag [=past entailment cone back to the axiomatic big bang]
Analogy: statement of pythag; proof of pythag [=past entailment cone back to the axiomatic big bang]
Statement of theorem:
Statement of theorem:
Entailment cone leading to pythag:
Entailment cone leading to pythag:
Red nodes are inputs for theorem.....
Euclid
Euclid
Axioms of geometry
Axioms of geometry
Need
(a) theorem statement in the language of the axioms
(b) theorem proof in the language of the axioms
Need
(a) theorem statement in the language of the axioms
(b) theorem proof in the language of the axioms
(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
We say: what gives the theorem size is the “uncertainty” in its underlying axioms
E.g. in metamath:
E.g. in metamath:
pythag, pythi are both statements of the Pythagorean theorem
Same metamath library, but different outcome statements...
Same metamath library, but different outcome statements...
In full entailment cone ... the Pythagorean theorem is a region in branchial space
Example of “theorem cloud”
Example of “theorem cloud”
1. At the axiom level
1. At the axiom level
Case 1. Same axioms; different statement
Case 1. Same axioms; different statement
Case 2. Different axioms
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
2. At the eme level
E.g. there are many ways to make an integer
Size of a theorem in emes
Size of a theorem in emes
Take the theorem statement and write it in the language of the axioms ... and then even lower level
Take the theorem statement and write it in the language of the axioms ... and then even lower level
What does an observer coarse grain?
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
0. The observer doesn’t care about names of variables
1. The observer doesn’t care about the proof
1. The observer doesn’t care about the proof
“Destructive interference”
“Destructive interference”
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
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...
With expression rewriting styles of graph...
Destructive interference: you can’t believe both branches at the same time .... because you’d get shredded
Destructive interference: you can’t believe both branches at the same time .... because you’d get shredded
Physics case
Physics case
Observer has a certain “branchial range” : how far are they sensing in branchial space?
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
Between the slits : the passage of the photon through each slit is too far away in branchial space
Math Interference
Math Interference
The initial condition contains not just the axioms we want, but all other axioms as well ....
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....
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 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...
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 ?
Consistency + Consistency ?
Can two consistent axiom systems get combined to yield inconsistency?
Double Slit
Double Slit
“Two slits are worse than one”
“Two slits are worse than one”
Multicomputational Irreducibility
Multicomputational Irreducibility
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
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
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] > ....
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] > ....
Most famous: Hilbert’s decision problem
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
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