Formal Patterns & Indiscernibility

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Abstractions about formal pattern languages, numbering systems, computational codings, indiscernible elements, structural transformations, and interaction-centered logic.

6 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Cylindrification — Extend a numbering by pairing every original index with an ignored auxiliary coordinate, creating infinitely many computably organized names for each numbered object and converting ordinary reducibility comparisons into one-one form.
• Gower's Distance — Compare mixed-type records by converting each available feature to a bounded type-appropriate similarity or dissimilarity, then taking a weighted pairwise average with missingness and binary-presence rules in the denominator.
• Indiscernibles — Choose elements or tuples whose finite subconfigurations satisfy exactly the same formulas over a declared parameter set whenever their index patterns agree, creating model-theoretic symmetry that supports controlled constructions and automorphisms.
• Ludics — Reconstruct logical propositions and proofs from address-based designs, polarized actions, and successful interaction, defining meaning extensionally through orthogonality rather than presupposed formulas.
• Numbering (Computability Theory) — A surjective coding from natural numbers onto a countable class of mathematical objects, used to transport computability, reducibility, and effective enumeration questions from objects to their indices.
• Pattern Language (Formal Languages) — The formal language generated from one constants-and-variables pattern by consistently replacing each variable with a nonempty terminal string while preserving every constant.