| I do not believe that it is possible to claim that integer numbers or imaginary numbers do not have a fundamental physical basis. The small integer numbers are not invented by humans, because many other animals can count up to some small number (like 5 or 6). Both are abstractions. That means that they are properties of real physical objects, which are obtained by ignoring all the other properties of those physical objects that are irrelevant in the context of the application. Therefore an abstract property is an equivalence class of physical objects, where all their other properties are ignored, so they are equivalent if they have the same value for the property of interest. Non-negative integer numbers are equivalence classes of collections of physical objects, integer numbers are equivalence classes of pairs of such collections. The imaginary unit is the equivalence class of all rotations by a right angle in the 2-dimensional space. Humans, like many animals, have an innate ability to recognize right angles, like also certain small numbers, so looking around you can perceive as easily all imaginary units like all numbers 3. The complex numbers are the equivalence classes of all geometric transformations of the 2-dimensional space that can be decomposed in rotations and similarities (a subset of the affine transformations). In contrast, the 2-dimensional vectors are the equivalence classes of all translations of the 2-dimensional space (another subset of the affine transformations). All the things that are equivalent from the point of view of an integer number or a complex number, so they are the basis from which such numbers are abstracted, are things that you can see with your own eyes in the physical world (similarity transformations appear in optical projections, e.g. in the shadows of physical objects, and the eyes are based on them). |