A Conversation Between Jonathan Gorard and Stephen Wolfram (September 1, 2023)
Sep 12, 2023
auto_awesome
Jonathan Gorard joins Stephen Wolfram to discuss ongoing science research. They explore the concept of rulial relativity, the distinction between normalizing and strongly normalizing systems in term rewriting, decidability and primitive recursive functions, non-terminating rewriting sequences, evolutionary perception and course screening, regular patterns and proof theory, observers in computation, randomness and complexity, primitive recursive functions and limited number systems, linear logic and sub-integer set theory, state and event canonicalization in multi-way systems, gravitational radiation and bronchial space, defining fluid velocities, and functoriality with a geometrical interpretation.
Different levels of computational sophistication in the hypo-reulyad can lead to distinct looping behaviors characterized by ordinal numbers.
The abilities of observers in different computational hierarchies can determine the types of equivalences they can identify in the hypo-reulyad.
The presence of geometry in the hypo-reulyad suggests the existence of local statements true at various levels of the rule hierarchy, similar to the polynomial hierarchy in computational complexity theory.
The reformulation of the continuum hypothesis in a computational sense suggests the existence of non-computable elements and a level of non-randomness between algebra and geometry.
Observers and constructors are inverse notions of each other, shedding light on the interplay between computation and perception in exploring computational universes.
Deep dives
The nature of the hypo-reulyad and limitations of observer
In the hypo-reulyad, observers may have limited computational abilities, hindering their ability to determine certain equivalences. They may only be able to perform decidable computations or have bounded recursion levels. This limitation can make the world seem interesting to them, as they are unable to see the underlying patterns or regularities that would indicate a lack of termination. The observer's computational sophistication determines the types of equivalences they can identify, and in the hypo-reulyad, certain non-trivial Turing machine looping behaviors may appear as distinct interesting theorems.
Ordinal numbers and characterization of looping behavior
Looping behaviors in the hypo-reulyad can be characterized using ordinal numbers. Each looping behavior can correspond to a specific ordinal number, such as omega for the loop x=x+1. These ordinal numbers signify the level of looping behavior and classification possible in the hypo-reulyad.
Equivalence, computational sophistication, and P vs NP
Different levels of computational sophistication in the hypo-reulyad can lead to different notions of equivalence. The ability to perform stronger equivalences depends on the observer's computational abilities. The question of P vs NP is relevant, as it asks if the equivalence functions and state evolution functions can be effectively merged or if they exhibit differences in computational complexity. These distinctions may provide insights into the capabilities of observers in different computational hierarchies.
Geometry and evidence of locality in the rule hierarchy
The existence of geometry and the notion of continuum geometry indicates the presence of local statements that can be made at any level in the rule hierarchy. Similar to the polynomial hierarchy in computational complexity theory, there may be translation-independent statements that are true locally within the hierarchy. The concept of geometry itself can be seen as evidence of these local statements and their validity across different levels of the rule hierarchy.
The continuum hypothesis and its computational reformulation
The continuum hypothesis, which asks whether there exists a set with intermediate cardinality between the integers and the reals, can be reformulated in a computational sense. This reformulation suggests that the real numbers may be the smallest set that contains non-computable elements. It also implies that there might be a level of non-randomness that is below the level of pure geometry, an intermediate between algebra and geometry. The concept of forcing, used in set theory, can be analogized to the hypo-rudio civilization, allowing them to construct different models of set theory with restricted computational power. They may experience a version of the continuum hypothesis where they are unable to construct a set of size 11, but are also unable to prove that no such set exists. This computational perspective on the continuum hypothesis sheds light on the limits of observation and construction within different computational frameworks.
The role of observers and constructors
Observers and constructors are two foundational concepts in exploring computational universes. An observer compresses the complexity of the world into a more manageable form, making decisions based on limited information and ultimately perceiving the world through a lens of computation. A constructor, on the other hand, implements decisions and actualizes them in the world, determining what can be built or modified within a given computational framework. Observers and constructors are inverse notions of each other, targeting the same point from different extremes, and they shed light on the interplay between computation and perception.
Maintaining quantum coherence and the speed of light
As observers in spacetime, we exhibit a slower pace relative to the speed of light, allowing us to maintain quantum coherence for longer periods of time. This is in contrast to the theoretical existence of photon consciousness, which would experience rapid entanglement and populate the entire universe quickly. The fact that our quantum coherence is sustained suggests that we only observe a small fraction of reality and are limited in our entanglement with the surrounding microstates. Our localization and limited observational apparatus contribute to this prolonged coherence, allowing us to observe specific subsets of the wider reality.
The Analog of Photons in Branchial Space
In branchial space, the analog of photons is not well defined. While gravitational waves may have an analog in branchial space, there is no clear concept of particles or photons in this realm. However, it is speculated that there might be states in branchial space that exhibit maximal entanglement rates, analogous to photons in physical space. The entanglement cone, which represents the surface of maximal entanglement, could potentially hold the key to understanding the behavior of entangled states in branchial space.
Concepts and Functoriality in Branchial Space
Branchial space offers a unique perspective on the idea of functoriality, where translating a concept from one region to another involves not only preserving but also distorting certain aspects. This can be thought of as a geometrical transformation in which concepts experience defamation while traversing through intermediate regions. The analogy to concepts being packaged like particles in branchial space is intriguing. Furthermore, the potential existence of a geometric structure, analogous to the Fubini-Study metric in projective Hilbert spaces, opens up possibilities for defining curvature and Einstein equations in branchial space. Exploring the relationship between motion, quantization, and defamations in branchial space promises to deepen our understanding of this realm.
Stephen Wolfram plays the role of Salonnière in an on-going series of intellectual explorations with special guests. In this episode, Jonathan Gorard joins Stephen to discuss ongoing science research. Watch all of the conversations here: https://wolfr.am/youtube-sw-conversations
Remember Everything You Learn from Podcasts
Save insights instantly, chat with episodes, and build lasting knowledge - all powered by AI.
