Object Theory

Object theory is a theory in philosophy and mathematical logic concerning objects and the statements that can be made about objects.

In some cases "objects" can be concretely thought of as symbols and strings of symbols, here illustrated by a string of four symbols " ←←↑↓←→←↓" as composed from the 4-symbol alphabet { ←, ↑, →, ↓ } . When they are "known only through the relationships of the system, the system is abstract ... what the objects are, in any respect other than how they fit into the structure, is left unspecified." (Kleene 1952:25) A further specification of the objects results in a model or representation of the abstract system, "i.e. a system of objects which satisfy the relationships of the abstract system and have some further status as well" (ibid).

A system, in its general sense, is a collection of objects O = {o₁, o₂, ... o_n, ... } and (a specification of) the relationship r or relationships r₁, r₂, ... r_n between the objects:

Example: Given a simple system = { { ←, ↑, →, ↓ }, ∫ } for a very simple relationship between the objects as signified by the symbol ∫ :

∫→ => ↑, ∫↑ => ←, ∫← => ↓, ∫↓ => →

A model of this system would occur when we assign, for example the familiar natural numbers { 0, 1, 2, 3 }, to the symbols { ←, ↑, →, ↓ }, i.e. in this manner: → = 0, ↑ = 1, ← = 2, ↓ = 3 . Here, the symbol ∫ indicates the "successor function" (often written as an apostrophe ' to distinguish it from +) operating on a collection of only 4 objects, thus 0' = 1, 1' = 2, 2' = 3, 3' = 0.

Or, we might specify that ∫ represents 90-degree counter-clockwise rotations of a simple object → .