Of the infinitesimal, Euclidian points, irrational numbers, imaginary numbers, surreal numbers, and so on: are "Noumena" and "Phenomena" involved, relevant, or even real?

In Kant's philosophy (and in Philosophy in general*) noumenal refers to the realm of things as they are in themselves, independent of human perception and sense. The noumenal realm and noumenal objects are beyond our perception and grasp.  Kant calls this world and these objects the "thing-in-itself." This noumenal realm is contrasted with the phenomenal realm, which is the world as we experience it, moreover as we experience it as (or in) our perceptions--hence the notion that the world is a mental construction, which is a fundamental plank of ideological epistemology in the platform of Continental Philosophy.

A quick glance at the Wikipedia article on the "Infinitesimal" lists a range of mathematical concepts that are commonly used in advanced equations and operations. Among the more intriguing (and amusing) concepts are surreal numbers. Hmm.  Along these lines, there are also "imaginary numbers" and "irrational numbers"--the list of such curiosities goes on and on, as any student of advanced Algebra, Analytic Geometry or Calculus can tell you. What makes these strange numbers (my term) possible is Categorical Theory, a method and (why not?) a field of mathematics with fascinating implications for analytic philosophy and grammatical analysis and description.  Indeed, many of our mathematical concepts and operations are only properly understood (that is to say properly described) when we regard them as grammatical phenomena (and here by "phenomena" I simply mean subjects, objects, activities, propositions, assumptions, utterances, or, of course, things).   Now, when mathematician are manipulating these conceptual numbers, are there in fact noumenal and phenomenal versions of these concepts and numbers?  For example, is there a noumenal imaginary number that is the "imaginary number in itself", that is "the imaginary number as it really is", and as well a distinct "phenomenal" imaginary number that is "the imaginary number or thing as it is perceived"? 

What does infinitesimal mean in noumenal terms? And how could it possibly be different from infinitesimal in a phenomenological sense?  Indeed, could we simply apply Occam's razor (the novacula Occami), and entirely dispense with the noumenal/phenomenal distinction, and talk about infinitesimals in a clear--indeed appropriate--sense, and moreover without the burden of some philosopher's Hog Latin?

 

 

 

 

 

* "Philosophy in general": that is, Philosophy as an academic magisterium consisting of a vocabulary, a curriculum, and as described and exercised through various institutional distinctions and practices.