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Incommensuration Theorem and its implications

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From: jimruttshow8596

The Incommensuration Theorem (ICT) is a theorem arising from Forrest Landry’s Immanent Metaphysics [00:00:48]. It is described as a statement about the nature of the relationship between what Landry calls symmetry and continuity [00:02:00].

Foundational Concepts

The ICT is built upon several core philosophical and epistemic concepts:

Symmetry and Continuity

  • Symmetry is defined as a form of constancy or sameness [00:02:23]. Examples include:

    • A shape that appears the same when flipped [00:02:28].
    • The law of conservation of matter, where the amount of matter remains constant over time [00:02:40].
    • It is formally defined in terms of the intrinsics of comparison as a “sameness of content and a difference of context” [00:57:58]. It represents a way of encoding knowledge, particularly regular observations and causal laws, applying across different contexts [00:56:54].

  • Continuity relates to connectedness, where one can move between places through infinitesimally small steps without hard edges or boundaries [00:02:55]. This concept is similar to continuous functions in mathematics [00:03:26].

    • It is formally defined in terms of the intrinsics of comparison as a “sameness of content where there is a sameness of context” [01:00:36].

Domains

A domain is a concept that extends the notion of a set [00:04:06].

  • While a set is like a “bag with things in it” (elements or points) [00:04:09], a domain includes not just content and context, but also the relationships between the elements as a first-class object [00:04:48].
  • A key characteristic of a domain, in this context, is that it cannot be included in other domains [00:07:07]. This clarifies that a domain extends to what is the largest enclosing set that is not itself enclosed, akin to the concept of a “universe” [00:08:06]. When a domain, such as “baseball,” is referred to within a larger context like “team sports,” it is treated as an element or identity within the larger domain, not as a domain itself [00:14:38].

Three Necessary Concepts for a Domain (Universe Example)

For a comprehensive understanding of a domain, such as the universe, exactly three necessary and sufficient concepts are proposed:

  1. Creation: How things came to be, like the Big Bang in physics [00:16:18].
  2. Existence: The nature of matter and material things [00:16:22].
  3. Interaction: The relationships and forces between existing things [00:16:24].

Landry distinguishes this definition of “domain” from conventional usage by conceptualizing the domain (e.g., the universe) as a concept container for concepts, rather than a physical container for material things. This allows for thinking about the closure of concepts in a different way [00:23:01].

The Concept of Comparison

All epistemic processes (how we know what we know) are contingent upon observation [00:31:11]. Every observation is a type of measurement, and every measurement is a type of comparison [00:31:31]. Comparison, therefore, is fundamental to knowing anything [01:04:12].

Comparison involves six “intrinsics” that are distinct, inseparable, and non-interchangeable [00:30:24]:

  1. Subjective: The perceiver [00:29:27].
  2. Objective: The perceived [00:29:31].
  3. Content: The “figure” or what is being perceived [00:35:25].
  4. Context: The “ground” or environment in which perception happens, including the observer’s state of mind [00:36:09].
  5. Sameness: The notion of non-changingness [00:32:11].
  6. Difference: The notion that a state has changed (e.g., from not knowing to knowing) [00:32:17].

These intrinsics are inseparable; for instance, there can be no content without context, and no sameness without difference [00:54:03].

Defining Symmetry, Asymmetry, Continuity, and Discontinuity

The six intrinsics allow for precise, absolute definitions of these four concepts, which are consistent with their usage across mathematics and physics [01:04:48]:

  • Symmetry: Sameness of content in different contexts [00:58:00].
  • Asymmetry: Different content in different contexts.
  • Continuity: Sameness of content in the same context [01:00:36].
  • Discontinuity: Different content in the same context [01:03:12].

The Incommensuration Theorem

The ICT states that perfect symmetry and perfect continuity cannot coexist [01:11:54].

This arises from combining the concepts based on their intrinsic definitions. When attempting to perfectly combine symmetry and continuity, the balance of sameness and difference (an intrinsic requirement for comparison) is mismatched, leading to a conceptual contradiction [01:12:41].

Therefore:

  • If one desires perfect symmetry, one must allow for perfect discontinuity [01:12:03]. This represents the “digital world” or discrete universe, where particles have distinct identities but share properties [01:13:13]. This is a third-person, objective view of knowledge [01:17:06].
  • If one desires perfect continuity, one must allow for perfect asymmetry [01:12:07]. This represents the first-person, subjective experience of consciousness, characterized by a continuous flow and an inherent asymmetry of time (e.g., we cannot access the future) [01:19:00]. This is the epistemic process itself [01:16:51].

The theorem implies that these two fundamental types of epistemic knowledge are “incommensurate” with one another [01:17:20].

Implications

First-Person vs. Third-Person Perspectives

The ICT provides tools to understand the relationship between first-person (subjective, experiential) and third-person (objective, abstract) perspectives [01:18:24].

  • The first-person experience aligns with continuity and asymmetry: a felt immediacy of continuity (e.g., self-identity over time) and the asymmetry of time (the subjective flow of time where the future is inaccessible) [01:19:10].
  • The third-person abstract body of knowledge, such as mathematics, aligns with symmetry and discontinuity: equations and laws represent discrete ideas (discontinuity) and generalizations (symmetry), but lack subjective flow or surprise [01:21:07].

Access Control Limits in Physics

The concept of “access control limits” relates to what can be known or accessed through signaling [00:42:37].

  • Newtonian Mechanics: Implies a deterministic “clockwork universe” where, in principle, all information (past and future) is accessible [00:47:26]. It suggests no inherent access control limits [00:47:57].
  • General Relativity: Describes fundamental access control limits, such as the inability to receive causal signals from the “absolute elsewhere” outside one’s light cone, or to know what’s inside a black hole due to the event horizon [00:45:00].
  • Quantum Mechanics: Also shows inherent access control limits, as seen in the Planck constant and Heisenberg Uncertainty Principle, where simultaneous knowledge of certain pairs of properties (e.g., position and momentum) is impossible [00:46:21].

The irreversibility of perception and the arrow of time are tied to the inherent asymmetry in the measurement process [01:33:28]. This “one-wayness” of causation is fundamentally connected to the first-person perspective [00:49:52].

Connection to Bell’s Theorem and Quantum Interpretations

Bell’s Theorem, which deals with locality and realism in quantum mechanics, is seen as a special case of the ICT [01:25:20].

  • The notion of “lawfulness” (how laws apply everywhere) is isomorphic to symmetry [01:25:28].
  • The notion of “locality” (connectedness) is strictly isomorphic to continuity [01:25:40].

The ICT implies a fundamental choice:

  • One can emphasize symmetry (lawfulness) and accept discontinuity (non-locality, i.e., instantaneous effects).
  • One can emphasize continuity (locality) and accept asymmetry (non-lawfulness or lack of fundamental randomness).

This framework provides a way to sort through quantum interpretations based on their assertions about subjectivity, hard randomness, and the arrow of time [01:33:04]:

  • The Copenhagen interpretation asserts the existence of hard randomness, subjectivity, and an arrow of time, aligning with the first-person perspective of continuity and asymmetry [01:35:09].
  • The Many-Worlds interpretation (MWI) typically assumes no hard randomness, no subjectivity, and no arrow of time, representing a perfected third-person view [01:34:55].
    • However, if knowledge must be grounded in the scientific method (epistemic process), MWI faces a challenge. The inherent discontinuities (branching universes) in MWI, when viewed through the lens of completeness, imply that there are necessarily other universes that are inherently unmeasurable and unknowable, making MWI an incomplete theory [01:44:00].
  • Superdeterminism, which posits a clockwork universe with no fundamental randomness or free will, would allow for causality and locality [01:28:41].

The ICT, through its fundamental abstract necessity, provides a framework to potentially rule out quantum interpretations like Many-Worlds that attempt to describe a “complete” reality but fail to account for the necessary “discontinuities” or “incompleteness” inherent in knowledge itself [01:44:26].

Connection to Gödel’s Incompleteness Theorems

Gödel’s Incompleteness Theorems in mathematics are presented as a projection of the ICT into the domain of mathematics itself [01:38:09].

  • Consistency (in Gödel’s theorem) is strictly isomorphic to symmetry [01:38:31]. Consistency means a statement is not both true and false, representing sameness of content across different theoretical contexts.
  • Completeness (in Gödel’s theorem) is strictly isomorphic to continuity [01:40:08]. Completeness implies that from a given set of axioms, all provable theorems can be reached, effectively subsuming the totality of all true mathematical knowledge into a single continuous field [01:39:27].

Gödel’s theorem states that within any consistent formal system, there will always be true statements that cannot be proven within that system (incompleteness) [01:37:25]. This aligns perfectly with the ICT’s assertion that you cannot have both perfect symmetry (consistency) and perfect continuity (completeness) simultaneously [01:40:47]. If consistency is preferred (as it usually is in mathematics), then incompleteness (discontinuity) must be accepted [01:42:09]. The ICT offers a simpler, more fundamental proof of this concept by operating on the intrinsics of epistemic process rather than requiring complex self-referential number systems [01:40:53].

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