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Wolfram Physics Project: Working Session Tuesday, July 28, 2020 [Metamathematics | Part 3]

Wolfram・157 minutes read

Meta mathematics explores the connection between mathematics and physics using multi-way systems, showing equivalence between different representations through paths in the system. Different structures and axioms impact the computational essence of mathematics and physics, with constructs like the axiom of choice influencing proof redundancy and event horizons.

Insights

  • Meta mathematics explores the connection between mathematics and physics through multi-way systems, revealing equivalence between representations of rules in both fields.
  • Foliations of the graph in multi-way systems correspond to underlying system models, showcasing relationships between elements and enabling the creation of mappings between equivalent states.
  • Homotopies between paths in multi-way systems induce critical pair completions, impacting the global effects within the system.
  • The univalence axiom in physics plays a crucial role in defining total orders and vibrations within branched structures, allowing paths to be treated as a single entity and morphisms to be localized into isomorphisms.
  • Constructive and non-constructive proofs, influenced by axioms like the axiom of choice, shape the mathematical space, with constructive mathematics focusing on computable proofs and non-constructive axioms hindering finite completion procedures.

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Recent questions

  • What is the relationship between mathematics and physics?

    Meta mathematics explores the connection between mathematics and physics, focusing on multi-way systems as rules for both. By representing the rules of physics and mathematical axiom systems through multi-way systems, equivalence between different representations can be shown. The progression of equalities between representations is derived by following paths in the multi-way system, indicating the deep interplay between mathematics and physics.

  • How are mathematical theories represented in multi-way systems?

    Mathematical theories are represented by replacement rules in multi-way systems, implementing the axioms of the theory. The multi-way system explores all possible proofs, with paths representing individual proofs and their relationships in the overall system. By adding all possible rules progressively, akin to a completion limit, the system delves into the intricate details of mathematical theories and their interconnections.

  • What is the significance of the univalence axiom in physics?

    The univalence axiom plays a crucial role in physics, particularly in relation to foliations and vibrations in multi-way systems. This axiom allows treating paths as a single entity, which is essential in defining total orders and vibrations in branched structures. By inducing equivalence between foliations, vibrations, and completions, the univalence axiom provides a foundational framework for understanding complex relationships within the system.

  • How does the axiom of choice impact mathematical knowledge?

    The axiom of choice introduces mathematical knowledge and induces proof redundancy, leading to the creation of equivalence vectors in mathematical space. This axiom boosts the growth rate of mathematical knowledge, resulting in more proof redundancy. However, the axiom of choice is considered too powerful, generating excessive mathematical knowledge and promoting non-constructivism in mathematical reasoning.

  • What is the role of constructivism in mathematics?

    Constructive mathematics focuses on proofs that can be witnessed by computable models, emphasizing the importance of constructive proofs with computable constraints. Realizability, related to intuitionistic logic, allows extracting more information from constructive proofs. Constructivism contrasts with non-constructive axioms like the axiom of choice, which hinder finite completion procedures and generate branch pairs in mathematical space.

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Summary

00:00

"Meta mathematics: Exploring connections in physics"

  • Meta mathematics explores the connection between mathematics and physics, focusing on multi-way systems as rules for both.
  • Multi-way systems can represent the rules of physics and mathematical axiom systems, showing equivalence between different representations.
  • The progression of equalities between representations is derived by following paths in the multi-way system.
  • Foliations of the graph correspond to models of the underlying system, indicating relationships between different elements.
  • The groupoid structure of the system allows implications to be turned into equalities, while topos structures enable localizing morphisms into isomorphisms.
  • The rule limit is reached by adding all possible rules progressively, akin to a completion limit.
  • Homotopies between paths induce critical pair completions, leading to global effects on the system.
  • Vibrations of the multi-way system inherit a topos structure, allowing morphisms to be turned into isomorphisms locally.
  • Mathematical theories are represented by replacement rules, implementing the axioms of the theory.
  • The multi-way system explores all possible proofs, with paths representing individual proofs and their relationships in the overall system.

19:40

Inheritance and Equivalence in Foliations and Vibrations

  • The discussion involves the inheritance of the topos structure from the groupoid structure.
  • Actualizing a foliation involves adding completions to create mappings between equivalent states.
  • By adding relations, branches that were independent can behave as one branch.
  • Equivalence relations exist for mutual independence and total order induced by the model.
  • The univalent axiom states equivalence between foliations, vibrations, and completions.
  • A foliation induces a total order on the multi-way graph.
  • Vibrations can limit branches to consider a subset as an individual fiber.
  • Completion rules determine equivalence between paths and identity relations.
  • An AI perceiving multiple branches can't construct a causal invariant representation.
  • Multi-threaded AI in branchial space contrasts with single-threaded experience in physical space.

37:34

"Multi-way causal networks and invariance"

  • The discussion revolves around the concept of multi-way causal networks and the ability to mod out in different directions.
  • Multi-way causal graphs can be projected spatially or branchially, leading to different combinatorial structures.
  • Claims are made about causal invariance and multi-way invariance, with the former being considered plausible.
  • Completing causal invariance could potentially alter the perception of time and space in the universe.
  • The discussion delves into inducing multi-way invariants through actions like teleportation.
  • The significance of the univalence axiom in physics is explored, relating to foliations and vibrations in multi-way systems.
  • The univalence axiom allows treating paths as a single entity, crucial in defining total orders and vibrations in branched structures.
  • The conversation transitions to choices of foliations and defining tip foliations by adding relations in groups.
  • A visualization of a free group on one generator is presented, showcasing equivalent words and the unrolled structure before solving the word problem.
  • The concept of undecidable word problems in groups is mentioned, highlighting the existence of arbitrarily long paths in such structures.

53:59

Constructing inertial foliation in spacetime dynamics.

  • The discussion revolves around constructing a foliation in spacetime, focusing on a partial obelionization.
  • Differentiating between a and b, and b and a, is crucial in constructing the foliation.
  • The foliation is defined as an inertial foliation, treating paths as equivalent.
  • Exploring the full foliation involves slicing everything with a specific slicing method.
  • Placing foliation slices in various locations to achieve the inertial foliation is discussed.
  • The need for a unique foliation consistent with the constraints is emphasized.
  • Defining an inertial frame involves a single additional equivalence relation between space-like separated events.
  • The concept of a cosmological rest frame is linked to the cosmic microwave background.
  • The discussion delves into the uniqueness of inertial frames and the constraints involved.
  • The rule graph is highlighted as the maximally non-free structure, crucial in understanding the quantum-classical spectrum.

01:12:33

"Algebraic Structures: Complexity and Beauty"

  • The structure generated by a multi-way system for a model of computation is independent of the underlying model or axioms.
  • Every algebraic structure has a limit point under repeated completion.
  • Algebraic structures based on sets with binary operations form fibers in an appropriate vibration space.
  • The geometry of a structure changes if it does not involve binary operators.
  • The ring version of algebraic structures is anticipated to offer more complexity.
  • Free abelian groups are appreciated for their beauty and structure.
  • The distinction between a group and a semigroup matters in the analysis of structures.
  • The multi-way system involves relations going both ways, defining a semi-semigroup.
  • Different frames correspond to different relations and foliations in the system.
  • The distance in proof space represents the time difference in the cosmological rest frame.

01:32:57

Mathematics: Velocity, Distance, and Time Relations

  • The Pythagorean theorem applies in Hilbert spaces of any dimension.
  • The relation between velocity and distance spanned by a relation is discussed.
  • The comparison of distance spanned by a relation to the speed of light is considered.
  • Time dilation is explored by modding out relations that span larger distances.
  • Higher velocity frames lead to an increase in proof time.
  • Adding lemmas can speed up proofs but increase computational complexity.
  • Mathematics speeds up as more theorems are published.
  • New axiom systems in mathematics correspond to curvature of meta-mathematical space.
  • Each relation adds a new direction for deformation in proof-like hypersurfaces.
  • Pairwise synchronization of events in mathematics is akin to clock synchronization in relativity.

01:52:04

"Relativity, Clocks, and Axioms: Understanding Models"

  • Synchronization in ordinary relativity involves defining time-like successors of events without time tags.
  • Synchronization is challenging without clock information, limiting the ability to create a permanent synchronization.
  • Time-like curves with clocks allow for a discrete approximation to standard relativistic concepts.
  • Space-like hypersurfaces connect identical tick numbers on time-like curves, aiding in understanding relativistic principles.
  • The completion of tick numbers on time-like curves corresponds to the completion of vectors in space-time.
  • Adding vectors in space-time leads to curved space-like hypersurfaces, indicating the presence of curvature.
  • The axiom of choice permits non-constructive proofs, akin to defining light-like paths in proof space.
  • The axiom of choice forces proofs to be non-constructive, similar to how event horizons necessitate light-like paths.
  • Constructivism focuses on models that saturate to a point where they represent something concrete.
  • Morally categoricity asserts that certain axiom systems can define unique models up to isomorphism, impacting the understanding of infinite models in logic.

02:10:55

"Complex numbers, event horizons, and theorems"

  • Real numbers have a dense linear order, leading to many possible models.
  • Complex numbers have a smaller number of models and are unique up to isomorphism as a closed algebraic field.
  • Molly categoricity guarantees the uniqueness of complex numbers.
  • The event horizon and the axiom of choice are discussed, with a low probability of them being linked.
  • Dana Scott is not associated with event horizons or physics.
  • Systolic aspects of black hole entropy are mentioned, related to causal graphs and VLSI design.
  • Positive curvature increases proof redundancy, while negative curvature increases theorem diversity.
  • Different types of theorems are discussed, such as difficult, elegant, and deep theorems.
  • The first incompleteness theorem for physics prevents knowing the outcome of the universe's evolution.
  • Constructing a physical girdle sentence shows computational universality prevents computing infinite time features of the universe.

02:32:22

Existence, Physics, and Mathematics: A Complex Relationship

  • Existence cannot be mathematically proven in any axiomatic deduction system.
  • Physical existence is an undecidable predicate even though physics is reduced to mathematics.
  • Axiom of choice plays a role in event horizons and the landscape of mathematics.
  • Constructivism, finitism, and ultrafinitism in mathematics correspond to causality conditions in general relativity.
  • Ultrafinitism is motivated by physics, limiting mathematical objects based on the universe's energy density.
  • Adding axioms increases the mass density of the metamathematical universe, leading to positive curvature.
  • Constructive and non-constructive proofs in theorems determine the presence of event horizons.
  • Theorems with only non-constructive proofs create event horizons in mathematics.
  • Realizability, related to intuitionistic logic, allows extracting more information from constructive proofs.
  • Realizability involves a constructive proof with the additional constraint that the proof is computable.

02:51:08

"Axiom of Choice: Constructive vs Non-Constructive"

  • Constructive mathematics focuses on proofs that can be witnessed by computable models.
  • A non-computable model involves uncountable cardinals and is non-constructive.
  • The axiom of choice introduces mathematical knowledge and induces proof redundancy.
  • Axiom of choice leads to the creation of equivalence vectors in mathematical space.
  • Axiom of choice boosts the growth rate of mathematical knowledge, leading to more proof redundancy.
  • The axiom of choice is deemed too powerful, producing excessive mathematical knowledge and inducing non-constructivism.
  • Constructive proofs are based on foliations, while non-constructive axioms like the axiom of choice hinder finite completion procedures.
  • Axioms inducing non-constructivism generate branch pairs and prevent finite completion procedures.
  • Undecidability of the word problem is linked to the axiom of choice and non-constructive axioms.
  • Axiom of choice allows for infinite jumps forward in the proof graph, creating an infinite hierarchy of jumps.

03:10:42

"Exploring Mathematics, Physics, and Unconventional Proofs"

  • The axiom of choice allows for non-constructive proofs of theorems that could otherwise be proven constructively.
  • Automated theorem provers can generate lemmas, inducing a structured mathematical space.
  • The rural multi-way system emulates the history of mathematics, generating lemmas that shape mathematical space.
  • The limit of mathematics with more constraints converges with the limit of physics, emphasizing computational nature.
  • The limit point of mathematics and physics converges, revealing the computational essence.
  • The rural multi-way system encodes all possible views of the universe, including alien perspectives.
  • The system gives rise to different types of physics and mathematics beyond standard observations.
  • Inter-universal algebra aims to generalize universal algebra, extending beyond current mathematical systems.
  • Different axiomatic systems like universal algebra and set theory may prove different theorems.
  • Univalent foundations and set theory can lead to different conclusions for the same theorem due to their constructive and non-constructive nature.

03:28:11

Alien math, ABC conjecture, IUTech, and more

  • The Omega example illustrates the potential for aliens to possess entirely different axiomatic systems from our mathematics.
  • The ABC conjecture establishes a significant link between addition and multiplication, offering constraints on prime factors in sets of integers that satisfy specific equations.
  • IUTech involves deforming mathematical universes by altering ring structures to compute explicit effects on operations like addition.
  • The Curry-Howard-Lambek correspondence generalizes the connection between proofs, programs, and categories, extending to infinity topoi and homotopies.
  • The Growth-Deep Hypothesis defines topological spaces as infinity groupoids, serving as a fundamental definition in homotopic type theory.
  • Non-constructivism in proof space implies a lack of constructiveness in statements, with hierarchies of non-constructivism posing a separate question.
  • Going faster than non-constructivism would breach consistency with the proof network, potentially contradicting the partial order and veering beyond the light-like path.
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