Minimal model of mathematics

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

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”)

Physical space is much more uniform than mathematical space

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
Computation universality exists across many axiom systems ...

PCE  homogeneity in metamathematical 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”

Mathematicians are observers

E.g. Peano arithmetic is like a reference frame ??? [ induction is like a foliation ??? ]

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

In math, is it the case that the details of the axioms don’t matter because of a layer of irreducibility?

Particle / antiparticle

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?

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

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

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 ]
(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
More axioms distort the space

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

[[Energy ~ activity in the network ~ density of proof]]

Analogy of relativity :
things like duality [metatheorems]

Equivalence of theories ?

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:

Two statements whose future evolution terminates in a contradiction
p && ~p  False
[The two branches will give two disconnected universes]

Lemmas vs theorems

A theorem is “pushing the boundaries”, whereas lemmas just fill in the bulk (?)

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:

SU(n) for n wires