## 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