Minimal model of mathematics
Minimal model of mathematics
Every statement of mathematics is an expression
Every statement of mathematics is an expression
The expression contains atoms like Plus
equal[plus[1,1],2]
State of mathematics = a collection of statements;
new state is derived by applying laws of inference
State of mathematics = a collection of statements;
new state is derived by applying laws of inference
new state is derived by applying laws of inference
To set up a particular math, we throw in certain initial statements [e.g. axioms], or add others [e.g. to make models]
To set up a particular math, we throw in certain initial statements [e.g. axioms], or add others [e.g. to make models]
Some statements may be in the future light cone of certain other axioms
Atoms in the statements are concepts of mathematics (e.g. “1” or “plus”)
Atoms in the statements are concepts of mathematics (e.g. “1” or “plus”)
Physical space is much more uniform than mathematical space
Chaitin’s claim (interpreted by Jose Manuel):
Chaitin’s claim (interpreted by Jose Manuel):
“Randomness is the true foundation of math”
analogous to: the uniformity of physical space is a consequence of microscopic randomness
analogous to: the uniformity of physical space is a consequence of microscopic randomness
Computation universality exists across many axiom systems ...
PCE homogeneity in metamathematical space
PCE homogeneity in metamathematical space
Claim: “math by typical mathematicians is done at the level of tables and chairs, not atoms of space”
Claim: “math by typical mathematicians is done at the level of tables and chairs, not atoms of space”
[Exception: people who make proof assistants]
There is a “mathematical vacuum” which consists of an infinite collection of “bubbling proofs”
There is a “mathematical vacuum” which consists of an infinite collection of “bubbling proofs”
Mathematicians are observers
Mathematicians are observers
E.g. Peano arithmetic is like a reference frame ??? [ induction is like a foliation ??? ]
We use the univalence axiom to simplify
We use the univalence axiom to simplify
[ Univalence potentially allows us to talk about objects independent of their origins ]
I.e. this 4 is the same as the 4 made in any other way
I.e. this 4 is the same as the 4 made in any other way
In math, is it the case that the details of the axioms don’t matter because of a layer of irreducibility?
In math, is it the case that the details of the axioms don’t matter because of a layer of irreducibility?
Particle / antiparticle
Particle / antiparticle
Because human mathematicians are coarse grained wrt rulial space, they do not stick to a single set of multiway rules
Because human mathematicians are coarse grained wrt rulial space, they do not stick to a single set of multiway rules
Can we estimate ρ (maximum rulial speed) for human mathematicians?
Can we estimate ρ (maximum rulial speed) for human mathematicians?
Do different approaches to math “see the same math” , just wrt different reference frames?
Time dilation: if you translate to a different kind of math, it might be easier to prove something
Time dilation: if you translate to a different kind of math, it might be easier to prove something
Maybe some reference frames assume bigger axiom collections, and therefore “prove faster”
Maybe some reference frames assume bigger axiom collections, and therefore “prove faster”
Simple example: my Boolean algebra axiom, which is slow unless you add commutativity
For Boolean algebra we can measure speed of proof
For Boolean algebra we can measure speed of proof
Inertial frame ?~ fixed sets of axioms
Inertial frame ?~ fixed sets of axioms
Acceleration ~ addition of axioms
Too many axioms decidability normal form black hole [ too many axioms termination ]
[ if the axioms lead to contradiction: then there is a one-step proof of everything ]
[ if the axioms lead to contradiction: then there is a one-step proof of everything ]
(E.g. utility of naive set theory : still useful because contradiction is far away)
There is light cone from the axioms [+ assertions] that gives everything one can prove
Proofs are paths in the space
Proofs are paths in the space
More axioms distort the space
Given more axioms you can prove more
Given more axioms you can prove more
Proving more means you can reach further in the space ; light cone reaches further
More statements within the light cone
More statements within the light cone
[[Energy ~ activity in the network ~ density of proof]]
[[Energy ~ activity in the network ~ density of proof]]
Analogy of relativity :
things like duality [metatheorems]
Analogy of relativity :
things like duality [metatheorems]
things like duality [metatheorems]
Equivalence of theories ?
To be spacelike separated, statements have to be independent ignoring the axioms
To be spacelike separated, statements have to be independent ignoring the axioms
Erasing history, theorems are like axioms; they can be in a spacelike slice together if they are independent
Consider a contradiction:
Consider a contradiction:
Two statements whose future evolution terminates in a contradiction
p && ~p False
[The two branches will give two disconnected universes]
Lemmas vs theorems
Lemmas vs theorems
A theorem is “pushing the boundaries”, whereas lemmas just fill in the bulk (?)
Geodesics
Geodesics
In “spacetime”, it is the shortest derivation of a theorem
In “theorem space”, [ at a particular step, erasing history all theorems are independent ]
??? https://www.csee.umbc.edu/~lomonaco/ams2009-talks/Brandt-Paper-Final-Revized-Version.pdf
Continuum limit of programs:
Continuum limit of programs:
SU(n) for n wires