Lawvere wrote in a couple papers that Cantors word “menge” which is usually understood as “set” is actually a cohesive type. And the “kardinale” is the abstraction from this by getting rid of the qualitative aspects of this “whole?” and what you get is a bag of points serving as a universal number was.
What I don’t get is that Cantors description of it seems to be sort of contradictory. Cantor says that he feels that numbers are like the forms of sets. That the number abstracted are actually more United then the whole they are abstracted from.
This seems contradictory to me. Unity seems to be a passage towards a nondual or monistic object. Whatever your conception of it might be. The plurality would imply dicreteness, no? Although cantor never said how points that are homogenous can be distinct. Maybe Nishida’s absolute nothingness concept is an answer. Where he nondual is the ultimate differentia?
I digress. The unity of these numbers seem to me to be the actual separation. His language also expresses an Aristotelian tone where form and matter play a role. Cantor describes the points as the matter and the order underneath as form.
Prime matter however is the potentiality, it is formless and does not exist. Is this nishidas absolute nothingness? If not, then the prime matter seems to be the continuous in Aristotle’s conception and platos ideas are more like discrete distinct forms.
Cantor seems to believe numbers are multitudes of units. These units are unity but the multiplicity of them would imply a separation and discreetness introduced to it, no?
Does Cantor believe quantity comes before quality? Can we not have a system like Hegels where quality is actually first?
I’ve thought that maybe he is referring to prime matter as formless as the “menge” and as the number as the forms. But then how is quality explained? Just differing manifestations or attributes that this number abstracted things may become?
The way he describes menge however is a bunch of distinct thing considered as a whole. The prime matter is not distinctive to anything or it’s both and not at the same time. It is pure potentiality, and Cantors notion only seems to make sense when we consider consciousness as prime matter or the unity “of our consideration”.
If numbers are forms and they seem to be abstracted from the menge which were whole without qualities and then had qualities and then we abstract all qualities away then wouldn’t number he form without matter?
Would not form without matter just equal matter without form?
How can a set be understood as a cohesion in any sense when elements are so important? Oftentimes in set theory we only talk about elements and simply ignore the background which allow them to “register” as something with just a capital letter label. In fact, we start with an empty set and singleton set.
What is this original menge that cantor describes? Is it a continuum with differing qualities and each qualities boundaries are those which tell us that this boundary is a “unit” or element?
Any insights would help. Google translate is not very friendly to the German language as well it seems.
