Thoughts Guiding Systems Corp.,

Ottawa, Canada.

e-mail:karasik@sympatico.ca

Word "dual" is sometimes applied to one object (e.g. dual citizenship) and sometimes to two objects (e.g. dual theorems). In the first case something is dual if it possesses two like (or, conversely, opposite) properties or features, or consists of two like (or, conversely, opposite) objects. (E.g. the physical matter is dual because it has properties of both wave and particle.) In the second case "dual" means that objects are inverted in a sense.

These invertions can be internal and external. External invertions are invertions with respect to how objects interact with (or act upon) other objects, or with respect to relationships with other objects.

Canceling each other is an example of the external invertion in the way the dual objects act upon other objects. In this regard numbers X and -X are dual because they cancel each other when "act upon" (i.e. added to) other numbers: Y + X + (-X) = Y. They are said to be dual with respect to addition. Similarly numbers X and 1/X are dual with respect to multiplication: Y • X • 1/X = Y.

Besides external invertions there are internal ones. They are invertions not in the way the objects act/interact but in respect to their structures. For example, expressions ((a•b) + (c•d)) and ((a + b)•(c + d)) are dual because their structures are inverted.

Now we are in a position to give the following definitions:

__Definition 1:__Let A → B or B ← A denote interaction/action/combination of object A with/upon/with object B and (A → B) and (B ← A) respectively denote the result of such an interaction/action/combination. Then two objects A and A^{*}are dual (with respect to interacting with, or acting upon, or combining with object B) if either (A → B ← A^{*}) = B, or (A^{*}→ (A → B)) = B, or any similar equality holds, which is obtained from the previous two by transposing A and A^{*}, → and ←, etc.

__Definition 2:__Let a set S be given that for any A and B from the set (A → B) = (A ← B). Then ← is identical to → and both can be denoted by one symbol ⊗ Then A and A^{*}are dual with respect to operation ⊗ over this set if for any X from the set A^{*}⊗ A ⊗ X = X.

__Definition 3:__Objects A and A^{*}are called structurally dual if at least two elements of their respective structures are transposed.

All mathematical theories of duality deal with dualities of the latter kind where structures of some objects or expressions are transposed. For example, logical expressions ¬(a ∧ b) and (¬a ∨ ¬b) are dual because everywhere negation is replaced by not negation and vise versa, and ∧ is replaced by ∨ and vise versa. The following two statements are also dual:

- every two straight lines in the plane have at most one common point;
- every two points in the plane have at most one common straight line.

Each and every mathematical theory of duality boils down to a theorem stating that dualities of a certain kind are equivalent in some respect. For example, in geometry the duality theorem states that propositions about points and straight lines where "points" and "straight lines" are transposed are both either true or false. In logics the duality theorem states that any two logical expressions (like ¬(a ∧ b) and (¬a ∨ ¬b)) where ∧ and ∨ are transposed and the unary operations ¬ and "" are also transposed are both either true or false, e.g. ¬(a ∧ b) = (¬a ∨ ¬b).

The author, however, discovered a more intriguing fact that not only most branches of mathematics have such duality theorems stating equivalence of dualities of a kind but that ANY theorem in mathematics can be represented as a statement about equivalence of some dualities. This means that mathematics is nothing else than a science about equal dualities.