Very often a mathematician considers his colleague from a different domain with disdain—what kind of a perverse joy can this guy find in his unmotivated and plainly boring subject? I’ve tried to learn the hidden beauty in various things, but still for many areas the source of interest is for me a complete mystery.
My theory is that too often people project their human weaknesses/properties onto their mathematical activity.
There are obvious examples on the surface: for instance, the idea of a classification of some objects is an incarnation of collector instincts, the search for maximal values is another form of greed, computability/decidability comes from the desire of a total control.
Fascination with iterations is similar to the hypnotism of rhythmic music. Of course, the classification of some kinds of objects could be very useful in the analysis of more complicated structures, or it could just be memorized in simple cases.
The knowledge of the exact maximum or an upper bound of some quantity depending on parameters gives an idea about the range of its possible values. A theoretical computability can be in fact practical for computer experiments.
Still, for me the motivation is mostly the desire to understand the hidden machinery in a striking concrete example, around which one can build formalisms.
If one tries to go further towards the “dehumanization”of mathematics, a natural next step would be to consider the real numbers (which emanate from the basic properties of the physical world) as just another complicated non-algebraically closed field. In some sense it is true; the complex numbers are much more beautiful.
But in another sense the real numbers are truly fundamental as they incarnate the idea of a bound, of a control of abstract algebraic structures. In a deep sense we are all geometers.