IMAGINARY EXPRESSIONS

Beyond imaginary numbers
"Imagination is nature" (Goethe).

"Imagination is reality" (Patrick Harpur).

"Both real and imaginary numbers are equally the products of the human imagination" (Ian Stewart).

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Definition

An imaginary expression is a substitution expression (of the form x=y, where x is not necessarily a name. It can be any expression. In traditional programming languages only variable names are allowed.

The paradigmatic example of imaginary expression is (i2 = −1). This is the famous definition of the imaginary unit i, an entity that we do not know what it is, but we know that its square is −1. The entity i is not √−1 because (i2 = −1) is a substitution, not an equality.

This interpretation as a substitution is most appropriate for two reasons:

1. Because it allows it to be generalized to all kinds of imaginary expressions.
2. Because it allows us to define higher order imaginary expressions.

Depending on the types of x and y expressions, there are imaginary expressions of arithmetic, logical, algebraic, etc. types.

Anyway, from the point of view of MENTAL, there is nothing imaginary. It is just the application of a degree of freedom (the substitution) applied to two expressions.

One of the advantages of imaginary expressions is that they allow you to define new operations without the need to define new operators. For example, the operations of logical addition and logical product of logical quantities can be defined:

⟨( f1*V + f2*V = max(f1 f2) )⟩ // logic sum

⟨( f1*V * f2*V = min(f1 f2) )⟩ // logic product

These imaginary expressions modify the corresponding standard semantics, which would be respectively,

⟨( f1*V + f2*V = (f1+f2)*V )⟩ //sum.

⟨( f1*V * f2*V = (f1*f2)*V )⟩ // product


Examples

1. (ε*ε = 0)
   
   This is the definition of an infinitesimal (ε) specified by an imaginary expression.
   
   
2. ⟨( ( (v+c = c) ← v≤c ← v≥0)⟩
   
   This is the law of invariance of the speed of light in Einstein's theory of special relativity, with v being a velocity and c being the speed of light. So we can say that light has an "imaginary" behavior. For example, 13+c is evaluated as c.
   
   
3. ⟨( (∞+n = ∞) )⟩
⟨( (∞*n = ∞) )⟩
⟨( (∞^n = ∞) )⟩
   
   These are some properties of the arithmetic of infinity.
   
   
4. ⟨( x+x = x )⟩
   
   Specifies that the sum of two equal expressions produces the same expression. For example,
   
   4+4 // ev. 4
abc+abc // ev. abc
   
   
5. ⟨( (1 2 x) = (3 x) )⟩
   
   This can be thought of as the definition of an imaginary function.
   
   (1 2 abc) // ev. (3 abc)
   
   
6. ⟨( r÷0 = ∞) ← r>0 )⟩
⟨( r÷0 = −∞) ← r<0 )⟩
   
   By this definition, every division by zero of a positive real number is infinity, and every division by zero of a negative real number is minus infinity.
   
   1÷0 // ev. ∞
112.34÷0 // ev. ∞
−7÷0 // ev. −∞
   
   
7. ⟨( ( (r1^2 + r2^2 = −1) )⟩ // define the points of an imaginary circle
   
   
8. ⟨( (r1^2 + r2^2 + r3^2 = −1) )⟩ // defines the points of the surface of an imaginary sphere
   
   
9. {1 −1}^2 = −1 // imaginary expression with constants
   
   
10. (a*0 = 23) // the product of 'a' times zero is 23
    
    
11. ⟨( x←y = y )⟩ // imaginary logic example.
a←4 // ev. 4
    
    
12. {a b}>5 // example of declarative imaginary expression

Higher order imaginary expressions.

These are imaginary expressions of imaginary expressions. There are two forms:

1. The imaginary expression is part of another imaginary expression.
2. A subexpression of an imaginary expression is part of another imaginary expression.

Examples:

1. (i*i = −1)
(i^i = i+7)
   According to this definition,
   (i^i + 5) // ev. i+12
   
   
2. (((i^i>i = −1)^2 = −1)
   According to this definition,
   (((i*i = −1)^2 + z) // ev. −1+z