Philosophical Ramblings on Some Mathematical Dichotomies and the Concept of Scale

This article looks at the issues relating to and arising out of self similarity of scale with links to "fractal" geometry and more generally investigates the possibility of finding the limit of a dynamical process given an initial generator/set of generators as also the inverse problem of finding the initial image that results in a particular attractor given the process. The latter could also be related to finding the point at which the image vanishes (including the possibility of oscillatory phenomenon). This problem is viewed through the "Operator paradigm" and links to recursive reflective phenomenon such as consciousness are examined. The above is also linked to the "Satisfiability problem" in computer science with links to Godel's theorem in mathematical logic. At a more constructive level, methods of constructing more general shapes out of the basic shapes such as "Seiperenski's Triangle" through basic operations (including diffraction) are examined with the possibility of lifting the results obtained to functions defined on the above shapes. Some paradoxes relating to scale which emerge in this process such as Zeno's paradox and other related paradoxes are examined. The article concludes by examining the possibilities of applying the outlined techniques to mathematical logic/meta-mathematics itself underlining the constructive nature of mathematical proof (with possible links to the area of optimization, especially discrete and combinatorial optimization).