Tom Etter's Relation Arithmetic Glossary

as interpreted by James Bowery with LLM assist so don't blame Tom

pre-alpha so this is far from "minimal" let alone correct

Glossary Definitions

Hover over bold terms to see their definitions.

Discriminator: A *discriminator* $x$ is a structural device that fixes a notion of **Relative Identity** and thereby a **World**$(x)$: for any $y,z$, the statement $x(y=z)$ is decided solely by how $y,z$ appear in the **Structure Tables** and **Count Tables** that $x$ uses. In GA, a discriminator is implemented as a chosen family of **Selections** and **Linkings** (projection operators) acting on multivectors.
World of a Discriminator: The *world* of a **Discriminator** $x$ is the class of **Things** that $x$ can distinguish from itself by some pattern in its **Structure Tables**: $$ {World}(x) = \{ t : x(t=x)\}, $$ where $x(t=x)$ is **Relative Identity**. In GA, ${World}(x)$ is the set of multivectors not identified with the ``state'' of $x$ under the projections determined by $x$.
Thing: A *thing* $t$ is any entity that lies in the **World** of at least one **Discriminator**: there exist $x,y$ with $x(t=y)$ under **Relative Identity**. Thus, ``thinghood'' is defined by participation in some pattern of discrimination. In GA, a thing is any multivector that survives non-trivial **Selection** for some discriminator.
Relative Identity: *Relative identity* is the identity statement read ``$y$ and $z$ are the same *for* discriminator $x$''. Formally, $x(y=z)$ holds iff, in all **Structure Tables** and **Count Tables** used by $x$, the rows and columns containing $y$ and $z$ are interchangeable under the same **Linkings** and **Selections**. In GA, $x(y=z)$ means that the projection operators implementing $x$ send $y$ and $z$ to the same multivector.
Quine Identity: Given a single binary relation $R$ whose extension is encoded in a **Structure Table**, the *Quine identity* $Q_R$ is the **Relative Identity** induced solely by the way elements appear in the rows and columns of that table. Concretely, $Q_R(x=y)$ holds iff $x$ and $y$ occur in exactly the same positions within the **Structure Table** (same incident rows and columns) and remain interchangeable under all **Selections** and **Linkings** generated from $R$. In GA, $Q_R$ identifies terms whose associated $R$-operators act identically on all multivectors.
Intrinsic Identity (of a predicate / system): The *intrinsic identity* of a primitive predicate (or of an axiom system) is the finest **Relative Identity** obtained by closing the relevant Quine identities $Q_R$ under all **Linkings**, **Selections** and **Structural Evolution Operators** that are allowed by the system. Two **Things** are intrinsically identical if every such structurally permitted transformation preserves $Q_R(x=y)$ for all predicates $R$ in the language. In GA, intrinsic identity is the equivalence of terms generated by all definable operators in the chosen model.
Structure Table: A *structure table* is a **Count Table** whose range has been canonically indexed so that only structural relations between rows and columns matter. It is the basic carrier of structure on which **Selections**, **Linkings**, **Hidings**, and **Structural Evolution Operators** act. In GA, a structure table corresponds to a fixed choice of basis for multivectors representing the same structural pattern.
Count Table: A *count table* is an extension represented as rows, columns, and integer counts (possibly **Negative Counts**), so that multiple identical relationships are combined into a single row with a total count. Count tables support **Empty Pairs**, **Interference**, and the normalization to **Density Count Matrices**. In GA, a count table is a multivector whose coefficients are these integer counts.
Density Count Matrix: A *density count matrix* is a count matrix derived from a **Count Table** by appropriate normalization (typically by total non-empty count), so that it can be used to assign ``structural probabilities'' to link values. It is the object that evolves under **Structural Evolution Operators** and is constrained by **Structural Unitarity** and **Past--Future Symmetry**. In GA, it is the density operator built from multivectors whose coefficients come from count tables.
Negative Count: A *negative count* is an entry $-n$ in a **Count Table**, understood only as part of a pattern of **Empty Pairs** and **Interference**. Negative counts cannot appear in isolation in a **Proper** structure, but only in combinations that can be cancelled or transformed by **Structural Evolution Operators**. In GA, a negative count is a negative coefficient on a blade in the multivector.
Empty Pair: An *empty pair* is a pair of rows in a **Count Table** with counts $n$ and $-n$ that together form a net-zero contribution to all **Selections**, **Linkings**, and **Shape**. A finite union of empty pairs that yields net zero counts on all rows is an *empty part*. Adding an empty part to any structure does not change its **Relative Identities**, **Worlds**, or **Shape**. In GA, an empty part is a multivector that is identically zero, built from terms $B + (-B)$.
Interference: *Interference* is the phenomenon that occurs when **Negative Counts** and positive counts in a **Count Table** combine into **Empty Pairs**, thereby changing the pattern of non-empty rows under **Selections** and **Linkings** without altering **Shape**. In GA, interference is just addition of multivectors in which some blades cancel.
Shape: The *shape* of a **Structure Table** is the class of all tables that can be obtained from it by permutations of rows, columns, and value indices that preserve the pattern of **Count Tables**, **Empty Parts**, **Relative Identities**, and **Relation-numbers**. Thus, shape is the global structural invariant of a table under all allowed relabelings. In GA, shape is the orbit of a multivector under the automorphism group that preserves all structural operations. (In Principia Mathematica this was *incorrectly* called **Relation Number** which is, in geometric analogy, congruence rather than mere similarity of shapes.).
Relation-number: The *relation-number* of a part of a **Structure Table** is the number of distinct occurrences of that part in the same **Shape** when we apply all permutations that preserve **Relative Identity** and the **Count Table** structure. It measures structured recurrence of parts within a shape. In GA, a relation-number is the size of the orbit of a sub-multivector under the automorphisms that fix the ambient shape.
Selection: A *selection* is an operation that takes a **Structure Table** and designates some of its rows and columns as *foreground*, leaving the rest as *background*. Selections generate the frames of a **Structural Time** sequence and determine which parts of a structure a **Discriminator** actually uses to fix **Relative Identity**. In GA, selections are projectors onto subspaces spanned by chosen blades.
Linking: *Linking* is the operation that imposes equality conditions between columns (or rows) of one or more **Structure Tables**, by deleting all rows that violate these equalities and then re-indexing the resulting **Count Table**. Linking thus creates joint structures and defines the ``present'' interface between past and future in **Structural Time**. In GA, linking is projection onto the subspace where linked indices coincide.
Hiding: *Hiding* is the operation that removes columns or rows that have been used in **Linking** but are no longer of interest to a given **Discriminator**. Hiding preserves **Relative Identity** on the remaining variables while marginalizing out the hidden ones, within the same **Shape**. In GA, hiding corresponds to partial trace or projection that forgets certain indices.
Structural Time: *Structural time* is the ordering of **Selections**, **Linkings** and **Hidings** applied to **Structure Tables** and **Density Count Matrices**, rather than an external parameter. A *dynamic sequence* is a finite or infinite chain of such operations, possibly governed by a **Structural Evolution Operator**. In GA, structural time is encoded by the composition order of projectors and linear operators on multivectors.
Structural Evolution Operator: A *structural evolution operator* $T$ is a map that sends **Structure Tables** and their corresponding **Density Count Matrices** to new ones in a way that respects **Shape**, **Relative Identity**, and the **Empty Part** bookkeeping. When $T$ is invertible and satisfies **Structural Unitarity**, it generates reversible **Structural Time**. In GA, $T$ is a linear operator (often unitary) acting on the space of multivectors and their density operators.
Structural Unitarity: A *structurally unitary* evolution is one in which a **Structural Evolution Operator** $T$ preserves the total ``count-mass'' and the normalization of **Density Count Matrices**, i.e.\ it preserves all structural probabilities. Structural unitarity thus ties the reversible part of **Structural Time** to invariants of **Shape** and **Relative Identity**. In GA, structural unitarity is realized by norm-preserving linear operators (unitary or orthogonal) on the state space.
Past--Future Symmetry: A system has *past--future symmetry* when its **Structural Time** evolution admits a **Structural Evolution Operator** $T$ and an inverse $T^{-1}$ such that the corresponding **Density Count Matrices** and **Relative Identities** are invariant under reversing the order of applications of $T$ (modulo **Empty Parts**). In this domain, quantum-like **Interference** and **Structural Unitarity** dominate. In GA, this is realized when the dynamics is given by a unitary family $T(t)$ and its adjoint $T(t)^{-1}$, with symmetry under time reversal.

Discriminator

A *discriminator* $x$ is a structural device that fixes a notion of **Relative Identity** and thereby a **World**$(x)$: for any $y,z$, the statement $x(y=z)$ is decided solely by how $y,z$ appear in the **Structure Tables** and **Count Tables** that $x$ uses. In GA, a discriminator is implemented as a chosen family of **Selections** and **Linkings** (projection operators) acting on multivectors.

World of a Discriminator

The *world* of a **Discriminator** $x$ is the class of **Things** that $x$ can distinguish from itself by some pattern in its **Structure Tables**: $$ {World}(x) = \{ t : x(t=x)\}, $$ where $x(t=x)$ is **Relative Identity**. In GA, ${World}(x)$ is the set of multivectors not identified with the ``state'' of $x$ under the projections determined by $x$.

Thing

A *thing* $t$ is any entity that lies in the **World** of at least one **Discriminator**: there exist $x,y$ with $x(t=y)$ under **Relative Identity**. Thus, ``thinghood'' is defined by participation in some pattern of discrimination. In GA, a thing is any multivector that survives non-trivial **Selection** for some discriminator.

Relative Identity

*Relative identity* is the identity statement read ``$y$ and $z$ are the same *for* discriminator $x$''. Formally, $x(y=z)$ holds iff, in all **Structure Tables** and **Count Tables** used by $x$, the rows and columns containing $y$ and $z$ are interchangeable under the same **Linkings** and **Selections**. In GA, $x(y=z)$ means that the projection operators implementing $x$ send $y$ and $z$ to the same multivector.

Quine Identity

Given a single binary relation $R$ whose extension is encoded in a **Structure Table**, the *Quine identity* $Q_R$ is the **Relative Identity** induced solely by the way elements appear in the rows and columns of that table. Concretely, $Q_R(x=y)$ holds iff $x$ and $y$ occur in exactly the same positions within the **Structure Table** (same incident rows and columns) and remain interchangeable under all **Selections** and **Linkings** generated from $R$. In GA, $Q_R$ identifies terms whose associated $R$-operators act identically on all multivectors.

Intrinsic Identity (of a predicate / system)

The *intrinsic identity* of a primitive predicate (or of an axiom system) is the finest **Relative Identity** obtained by closing the relevant Quine identities $Q_R$ under all **Linkings**, **Selections** and **Structural Evolution Operators** that are allowed by the system. Two **Things** are intrinsically identical if every such structurally permitted transformation preserves $Q_R(x=y)$ for all predicates $R$ in the language. In GA, intrinsic identity is the equivalence of terms generated by all definable operators in the chosen model.

Structure Table

A *structure table* is a **Count Table** whose range has been canonically indexed so that only structural relations between rows and columns matter. It is the basic carrier of structure on which **Selections**, **Linkings**, **Hidings**, and **Structural Evolution Operators** act. In GA, a structure table corresponds to a fixed choice of basis for multivectors representing the same structural pattern.

Count Table

A *count table* is an extension represented as rows, columns, and integer counts (possibly **Negative Counts**), so that multiple identical relationships are combined into a single row with a total count. Count tables support **Empty Pairs**, **Interference**, and the normalization to **Density Count Matrices**. In GA, a count table is a multivector whose coefficients are these integer counts.

Density Count Matrix

A *density count matrix* is a count matrix derived from a **Count Table** by appropriate normalization (typically by total non-empty count), so that it can be used to assign ``structural probabilities'' to link values. It is the object that evolves under **Structural Evolution Operators** and is constrained by **Structural Unitarity** and **Past--Future Symmetry**. In GA, it is the density operator built from multivectors whose coefficients come from count tables.

Negative Count

A *negative count* is an entry $-n$ in a **Count Table**, understood only as part of a pattern of **Empty Pairs** and **Interference**. Negative counts cannot appear in isolation in a **Proper** structure, but only in combinations that can be cancelled or transformed by **Structural Evolution Operators**. In GA, a negative count is a negative coefficient on a blade in the multivector.

Empty Pair

An *empty pair* is a pair of rows in a **Count Table** with counts $n$ and $-n$ that together form a net-zero contribution to all **Selections**, **Linkings**, and **Shape**. A finite union of empty pairs that yields net zero counts on all rows is an *empty part*. Adding an empty part to any structure does not change its **Relative Identities**, **Worlds**, or **Shape**. In GA, an empty part is a multivector that is identically zero, built from terms $B + (-B)$.

Interference

*Interference* is the phenomenon that occurs when **Negative Counts** and positive counts in a **Count Table** combine into **Empty Pairs**, thereby changing the pattern of non-empty rows under **Selections** and **Linkings** without altering **Shape**. In GA, interference is just addition of multivectors in which some blades cancel.

Shape

The *shape* of a **Structure Table** is the class of all tables that can be obtained from it by permutations of rows, columns, and value indices that preserve the pattern of **Count Tables**, **Empty Parts**, **Relative Identities**, and **Relation-numbers**. Thus, shape is the global structural invariant of a table under all allowed relabelings. In GA, shape is the orbit of a multivector under the automorphism group that preserves all structural operations. (In Principia Mathematica this was *incorrectly* called **Relation Number** which is, in geometric analogy, congruence rather than mere similarity of shapes.).

Relation-number

The *relation-number* of a part of a **Structure Table** is the number of distinct occurrences of that part in the same **Shape** when we apply all permutations that preserve **Relative Identity** and the **Count Table** structure. It measures structured recurrence of parts within a shape. In GA, a relation-number is the size of the orbit of a sub-multivector under the automorphisms that fix the ambient shape.

Selection

A *selection* is an operation that takes a **Structure Table** and designates some of its rows and columns as *foreground*, leaving the rest as *background*. Selections generate the frames of a **Structural Time** sequence and determine which parts of a structure a **Discriminator** actually uses to fix **Relative Identity**. In GA, selections are projectors onto subspaces spanned by chosen blades.

Linking

*Linking* is the operation that imposes equality conditions between columns (or rows) of one or more **Structure Tables**, by deleting all rows that violate these equalities and then re-indexing the resulting **Count Table**. Linking thus creates joint structures and defines the ``present'' interface between past and future in **Structural Time**. In GA, linking is projection onto the subspace where linked indices coincide.

Hiding

*Hiding* is the operation that removes columns or rows that have been used in **Linking** but are no longer of interest to a given **Discriminator**. Hiding preserves **Relative Identity** on the remaining variables while marginalizing out the hidden ones, within the same **Shape**. In GA, hiding corresponds to partial trace or projection that forgets certain indices.

Structural Time

*Structural time* is the ordering of **Selections**, **Linkings** and **Hidings** applied to **Structure Tables** and **Density Count Matrices**, rather than an external parameter. A *dynamic sequence* is a finite or infinite chain of such operations, possibly governed by a **Structural Evolution Operator**. In GA, structural time is encoded by the composition order of projectors and linear operators on multivectors.

Structural Evolution Operator

A *structural evolution operator* $T$ is a map that sends **Structure Tables** and their corresponding **Density Count Matrices** to new ones in a way that respects **Shape**, **Relative Identity**, and the **Empty Part** bookkeeping. When $T$ is invertible and satisfies **Structural Unitarity**, it generates reversible **Structural Time**. In GA, $T$ is a linear operator (often unitary) acting on the space of multivectors and their density operators.

Structural Unitarity

A *structurally unitary* evolution is one in which a **Structural Evolution Operator** $T$ preserves the total ``count-mass'' and the normalization of **Density Count Matrices**, i.e.\ it preserves all structural probabilities. Structural unitarity thus ties the reversible part of **Structural Time** to invariants of **Shape** and **Relative Identity**. In GA, structural unitarity is realized by norm-preserving linear operators (unitary or orthogonal) on the state space.

Past--Future Symmetry

A system has *past--future symmetry* when its **Structural Time** evolution admits a **Structural Evolution Operator** $T$ and an inverse $T^{-1}$ such that the corresponding **Density Count Matrices** and **Relative Identities** are invariant under reversing the order of applications of $T$ (modulo **Empty Parts**). In this domain, quantum-like **Interference** and **Structural Unitarity** dominate. In GA, this is realized when the dynamics is given by a unitary family $T(t)$ and its adjoint $T(t)^{-1}$, with symmetry under time reversal.