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Wolfram Physics Project: Working Session Tuesday, Aug. 4, 2020 [Empirical Physical Metamathematics]

Wolfram・154 minutes read

The text explores the computational nature of physics and mathematics, delving into the commonality between the two fields and the challenges they face. It discusses the importance of lemmas, multi-way systems, and causal invariance in understanding complex concepts and the potential impact of accumulated lemmas on the structure of mathematics.

Insights

  • Mathematics and physics theories are computational, forming networks of proofs from axioms, revealing a shared computational nature.
  • Multi-way systems in mathematics mirror the underlying structures in physics, particularly quantum mechanics, showcasing a deep connection between the two disciplines.
  • The concept of accumulated lemmas in mathematics impacts theorem proving and the structure of mathematical space, akin to spontaneous symmetry breaking, altering paths and introducing curvature.
  • Causal invariance underpins the infiniti groupoid, enabling effective random searches in mathematics and ensuring results will eventually be found, despite computational limitations.

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Recent questions

  • What is the relationship between physics and mathematics?

    The discussion delves into the computational nature of physics and mathematics, highlighting how both fields build networks of proofs from axioms. By exploring the commonality between the two disciplines, the physicalization of matter mathematics is introduced, importing physics knowledge into mathematics to create a cohesive understanding. The network of theorems in mathematics mirrors multi-way systems underlying physics, especially in quantum mechanics, showcasing the interconnectedness between the computational aspects of both physics and mathematics.

  • How do multi-way systems represent proofs in mathematics?

    Multi-way systems involve mathematical rules and rewrite rules, generating a well-defined expression that represents proofs in mathematics. By growing multi-way systems from axioms, the symmetry and structure of the multi-way system representing mathematics are explored. Implementing relations and inverses in a multi-way group allows for the generation of elements and a deeper understanding of group theory. The concept of evaluators traversing through multi-way graphs to effectively perform proofs is also discussed, shedding light on the intricate process of proof representation in mathematics.

  • What is the significance of lemmas in theorem proving?

    Lemmas in mathematics play a crucial role in theorem proving by representing unexpected relationships between coordinates, reflecting curvature in metamathematical space. The introduction of lemmas is likened to spontaneous symmetry breaking, altering the structure of mathematical space and impacting the deflection of geodesics. Accumulated lemmas can influence theorem proving and the overall structure of mathematics, potentially affecting the curvature of mathematical space. The presence of lemmas aids in representing equivalent branches and enhancing the depth of mathematical proofs.

  • How does mathematics approach computational irreducibility?

    Mathematics navigates computational irreducibility by relying on certain axioms that lead to the development of complex concepts like relativity and quantum mechanics. The discussion emphasizes how mathematics describes itself through quantization, utilizing various axiom systems to explore computational irreducibility and boundedness. Constructive proofs in mathematics involve creating models to derive proofs, showcasing the adaptability of mathematics in approaching computational challenges. The success of mathematics in overcoming computational limitations and causal invariance is attributed to the strategic reuse of lemmas and the inherent nature of mathematical coordination.

  • What are the implications of causal invariance in mathematics?

    Causal invariance in mathematics ensures that random searches will eventually yield results, aiding in the understanding of complex concepts within the field. The infiniti groupoid is believed to embody causal invariance, providing a framework for exploring intricate mathematical relationships. The relationship between accumulating theorems and proofs in mathematics is a key focus, highlighting how causal invariance influences the progression of mathematical concepts. The discussion also touches on the challenges mathematics may face, such as undecidability and meta-mathematical obstacles, showcasing the importance of causal invariance in navigating complex mathematical landscapes.

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Summary

00:00

"Physics and Mathematics: Computational Nature Revealed"

  • Discussion with Taliban about the physicalization of matter mathematics and the computational nature of physics and mathematics.
  • Mathematics and physics theories are computational, building networks of proofs from axioms.
  • Exploring the commonality between the computational nature of physics and mathematics.
  • Importing physics knowledge into mathematics to create a physicalization of matter mathematics.
  • The network of theorems in mathematics resembles multi-way systems underlying physics, especially quantum mechanics.
  • Growing multi-way systems from axioms to represent proofs in mathematics.
  • Demonstrating a simple example with commutativity axioms and proofs in a graph layout.
  • Exploring the symmetry and structure of the multi-way system representing mathematics.
  • Implementing relations and inverses in a multi-way group to generate elements and understand group theory.
  • Contrasting operator systems with multi-group systems and discussing the differences in structure and transformations.

24:26

Exploring Multi-Way Systems in Mathematics

  • Multi-way systems involve mathematical rules and rewrite rules, generating a well-defined expression.
  • Evaluators traverse through multi-way graphs, effectively performing proofs.
  • The concept of arbitrary evaluation chains and the study of multi-way systems are not adequately explored.
  • Symbolic systems and combinators are related to multi-way generalizations.
  • The study of semi-groups in multi-way systems is crucial, with semi-groups being a key element.
  • The Cayley graph complements the representation of multi-way systems, identifying equivalent elements.
  • Applying group relations generates equivalent words under specific group rules.
  • Mathematics faces challenges similar to physics, with undecidable propositions and computational irreducibility.
  • The analogy between the universe's history as a multi-way graph and mathematics as a giant multi-way graph is drawn.
  • General principles in mathematics may apply universally, beyond the sampled theorems, akin to laws in physics.

43:25

"Generators alter counts, change system dynamics"

  • Introducing generators on the left and right would change the system, potentially quantifying how many times a rule is applied.
  • Different rules could lead to quantization, affecting the counts of 'a' and 'b'.
  • Generators adding new 'a's and 'b's would alter the counts, creating a non-trivial coordinate scheme.
  • The system models axiom operations, but new generators may disrupt the scheme.
  • The current system lacks generators, maintaining the initial count of 'a's and 'b's.
  • Adding a completion step could shorten certain proofs.
  • Homotopy or homology could be introduced, affecting path lengths.
  • Adding a lemma could create shortcuts in the system.
  • By adding rules consistent with the existing ones, paths can be altered.
  • The goal is to derive relativistic and generalistic results from the system, exploring model independence and causal graphs.

01:03:24

"Group Theory: Relations, Models, and Equivalence"

  • Adding relations to the free group creates a model of the free group.
  • Different groups can have multiple calligraphs based on selected generators.
  • Certain truths are universal for all groups, while others are specific to individual groups.
  • Demonstrating general group theory results can be done by applying them to specific group models.
  • Models in bulk are akin to reference frames in multi-way graphs.
  • Reference frames create equivalence classes of states, defining simultaneity.
  • Foliation coordinates in multi-way graphs represent natural selection or preferences.
  • The mutation graph contains all possibilities, but foliation limits occurrences.
  • Finite groups have a finite number of equivalence classes under relations.
  • Homotopies in theorem proofs involve adding relations to transform paths.

01:21:41

Linear maps, groupoids, and metamathematical relationships.

  • Composing linear maps from rn to rp results in a linear map from rn to rp.
  • Initial object in this context is the empty vector space, while the final object is not clearly defined.
  • Products in this scenario are akin to potential products in an obvious manner.
  • In multi-way systems, a sequence of morphisms forms a one category, with transformations between proofs resembling two categories.
  • The rulial multi-way graph, derived from mappings of proofs and mappings of mappings of proofs, is considered an infinity groupoid.
  • An infinity groupoid encompasses categories and higher categories, evolving through mappings of morphisms.
  • The infinity groupoid includes all possible rules, leading to isomorphisms and identity morphisms within the groupoid.
  • A groupoid differs from an ordinary group by having morphisms that map between objects, not just back to a single object.
  • The infinity groupoid serves as a renormalization group fixed point, embodying all possible axiom systems in mathematics.
  • Lemmas in mathematics represent unexpected relationships between coordinates, reflecting curvature in metamathematical space.

01:40:49

Exploring Graph Theory in Mathematics

  • The significance of graph theory in mathematics is explored, focusing on proof complexity and theorem sizes.
  • Terminating a wavefront of proof complexity early can yield elegant or powerful theorems.
  • A toy domain rich enough to generate various theorems is needed for practical applications.
  • The theorem network of Euclid is examined, revealing its structure and potential for analysis.
  • The theorem network of Euclid is available for exploration and analysis in the Wolfram Language.
  • The proof of the verified Platonic solids is identified as the longest to arise in Euclid's elements.
  • Clusters in the theorem network may correspond to different books in Euclid's work.
  • Common notions and postulates in Euclid's work are analyzed for dependencies and relationships.
  • Geodesic distances between points in the theorem network are used for coordinate mapping.
  • Different forms of coordination, such as distance from vertices or application of relations, are considered for modeling groups and algebraic systems.

02:01:25

Coordinate Functions for Duplicates and Relationships

  • Use the position index function to find duplicates and obtain a list of their occurrences.
  • The function provides coordinates and indices where duplicates occur.
  • The function can be applied to access the entire set of coordinates.
  • Each key represents a coordinate, and the values indicate the indices where the coordinate occurs.
  • Utilize key value map to access keys and values, repeating keys with a slight y offset.
  • Consider using gather or gather by to group coordinates without needing to remember the second coordinate.
  • Implement a function like munge to modify coordinates, adjusting y values.
  • Suggest creating a function to embed modified coordinates into vertex coordinates efficiently.
  • Develop 2D and 3D versions of the coordinate functions for different cases.
  • Analyze the coordinated structure to understand relationships and potential theorems based on the coordination.

02:24:44

"Quantization and Graphs: Paths and Relations"

  • In quantization, if two things are far apart, they must also be distant in the original graph.
  • The original graph's key feature is that everything is equal, with paths connecting everything.
  • Adding generators to the graph creates directed paths and generates all possibilities.
  • By appending elements to strings, directed paths become the primary focus.
  • The multi-way semi-group includes relations like a b goes to b a and a b a goes to a b.
  • A method to differentiate between two types of arrows in the graph is needed.
  • A post-processing function can be used to change the edge shape based on string length.
  • Weighting rules differently can help prevent generators from dominating in the model.
  • The layered graph structure reveals causal invariance when adding elements on both sides.
  • Simplifying the code by using only string generators and relations can enhance efficiency.

02:49:12

"Visualizing Graphs with Metadata and Lemmas"

  • The process involves coloring and creating a layered graph plot.
  • Suggestions are made to use lines instead of arrows for clarity.
  • The idea of capturing metadata for paths produced in a multi-wave system is discussed.
  • Coordinating the graph structure is considered, focusing on string length and color.
  • The concept of using coordinates based on the number of rule applications is explored.
  • The presence of lemmas in theorem proving is explained as a way to represent equivalent branches.
  • The analogy of spontaneous symmetry breaking is used to understand lemma introduction.
  • The relationship between accumulated lemmas and deflection of geodesics in mathematical space is pondered.
  • The potential impact of accumulated lemmas on theorem proving and the structure of mathematics is considered.
  • The discussion delves into the implications of accumulated lemmas on the curvature of mathematical space.

03:12:17

Mathematics: Curvature, Theorems, and Axioms

  • Curvature in mathematics can be intrinsic or a reflection of coordination.
  • Powerful theorems can act as gravitational lenses, allowing access to more distant theorems.
  • Good theorems aid in effective random searches in mathematics.
  • The complete universe of mathematics poses questions about conceptual thinking.
  • Mathematics lacks a counterpart to photons in physics for sensitivity to distant events.
  • Proof expansion has a speed limit, akin to the speed of light.
  • Adding lemmas in mathematics must align with the original axioms to be valid.
  • Constructive proofs in mathematics involve creating models to derive proofs.
  • Mathematics can describe itself through quantization, seen in various axiom systems.
  • Computational irreducibility and computational boundedness influence how mathematics is approached.

03:33:40

"Mathematics, Physics, and Causal Invariance"

  • Mathematics relies on certain axioms that lead to the development of concepts like relativity and quantum mechanics.
  • Reusing lemmas in mathematics allows for progress without reinventing the wheel each time.
  • Mathematics may demand unique coordination or lemmas in different areas, hindering progress without intuition.
  • Physics and mathematics both navigate computational irreducibility, with physics finding ways to work around it.
  • The success of mathematics and physics is due to our computational limitations and causal invariance.
  • The infiniti groupoid is believed to be inherently causal invariant, aiding in understanding complex concepts.
  • Causal invariance ensures that random searches in mathematics will eventually yield results.
  • Mathematics may face challenges like undecidability, making it hard to complete certain tasks.
  • The relationship between accumulating theorems and proofs in mathematics is a key question.
  • Mathematics may hit a wall where it becomes less doable, posing a meta-mathematical challenge.

03:52:37

Exploring extraterrestrial intelligence and metamathematics limitations

  • The discussion delves into the concept of extraterrestrial intelligence and the limitations of human understanding, particularly in mathematics and metamathematics.
  • There is a mention of irreducible lemons and the analogy to maximum entropy, highlighting the complexity of entropy and the absence of ensembles in metamathematics.
  • The conversation shifts to David Deutsch's constructor theories in physics, emphasizing skepticism towards probabilities and the exploration of metamathematics akin to exploring the physical universe, aiming to describe experiences in a similar intuitive manner.
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