Changes

By way of providing a conceptual-technical framework for organizing that discussion, let me introduce the concept of a ''category of structured individuals'' (COSI).  There may be some cause for some readers to rankle at the very idea of a ''structured individual'', for taking the notion of an individual in its strictest etymology would render it absurd that an atom could have parts, but all we mean by ''individual'' in this context is an individual by dint of some conversational convention currently in play, not an individual on account of its intrinsic indivisibility.  Incidentally, though, it will also be convenient to take in the case of a class or collection of individuals with no pertinent inner structure as a trivial case of a COSI.

By way of providing a conceptual-technical framework for organizing that discussion, let me introduce the concept of a ''category of structured individuals'' (COSI).  There may be some cause for some readers to rankle at the very idea of a ''structured individual'', for taking the notion of an individual in its strictest etymology would render it absurd that an atom could have parts, but all we mean by ''individual'' in this context is an individual by dint of some conversational convention currently in play, not an individual on account of its intrinsic indivisibility.  Incidentally, though, it will also be convenient to take in the case of a class or collection of individuals with no pertinent inner structure as a trivial case of a COSI.

For some reason — that I won't stop to examine right now — I tend to think ''category of structured individuals'' when the individuals in question have a whole lot of structure, but ''collection of structured items'' when the individuals have a minimal amount of internal structure.  For example, any set is a COSI, so any relation in extension is a COSI, but a 1-adic relation is just a set of 1-tuples, that are in some lights indiscernible from their single components, and so its structured individuals have far less structure than the k-tuples of k-adic relations, when k exceeds one.  This spectrum of differentiations among relational models will be useful to bear in mind when the time comes to say what distinguishes relational thinking proper from 1-adic and 2-adic thinking, that constitute its degenerate cases.

For some reason — that I won't stop to examine right now — I tend to think ''category of structured individuals'' when the individuals in question have a whole lot of structure, but ''collection of structured items'' when the individuals have a minimal amount of internal structure.  For example, any set is a COSI, so any relation in extension is a COSI, but a 1-adic relation is just a set of 1-tuples, that are in some lights indiscernible from their single components, and so its structured individuals have far less structure than the k-tuples of k-adic relations, when k exceeds one.  This spectrum of differentiations among relational models will be useful to bear in mind when the time comes to say what distinguishes relational thinking proper from 1-adic and 2-adic thinking, that constitute its degenerate cases.

Still on our way to saying what brands of iconicity are worth buying, at least when it comes to graphical systems of logic, it will useful to introduce one more distinction that affects the types of mappings that can be formed between two COSI's.

Still on our way to saying what brands of iconicity are worth buying, at least when it comes to graphical systems of logic, it will useful to introduce one more distinction that affects the types of mappings that can be formed between two COSI's.

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Let <math>S\!</math> be the set of rooted trees and let <math>S_0\!</math> be the 2-element subset of <math>S\!</math> that consists of a rooted node and a rooted edge. We may express these definitions more briefly as <math>S = \{ \operatorname{rooted~trees} \}</math> and <math>S_0 = \{ \ominus, \vert \}.</math> Simple intuition, or a simple inductive proof, will assure us that any rooted tree can be reduced by means of the axioms of the primary arithmetic either to a root node [[Image:Cactus Node Big Fat.jpg|16px]] or else to a rooted edge [[Image:Cactus Spike Big Fat.jpg|12px]]&nbsp;.

Let <math>S\!</math> be the set of rooted trees and let <math>S_0\!</math> be the 2-element subset of <math>S\!</math> that consists of a rooted node and a rooted edge.

 

| <math>S\!</math>

| <math>=\!</math>

| <math>\{ \text{rooted trees} \}\!</math>

| <math>S_0\!</math>

| <math>=\!</math>

| <math>\{ \ominus, \vert \} = \{</math>[[Image:Cactus Node Big Fat.jpg|16px]], [[Image:Cactus Spike Big Fat.jpg|12px]]<math>\}\!</math>

 

Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Cactus Node Big Fat.jpg|16px]] or else to a rooted edge [[Image:Cactus Spike Big Fat.jpg|12px]]&nbsp;.

For example, consider the reduction that proceeds as follows:

For example, consider the reduction that proceeds as follows: