Comfortable with shehes
From: F William Lawvere, To: Posina Venkata Rayudu, Date: 30 Aug 2011
The lack of structure expressed by “completely distinguished from each other but with no distinguishing properties” for the elements of an abstract set is of course not an end in itself, but an attempt to insure that when we model mathematical objects as diagrams of maps in the category of sets, that background contaminates as little as possible and all properties are a result of the explicit assumptions in a particular discussion. The ancient Greeks had a similar concept of arithmos, in some sense the minimal structure derived from a given situation to which one could assign a number; in other words, an abstract set is just one hair less abstract than a cardinal number, but that enables them to potentially carry mappings, which cardinals are too abstract in concept (being equivalence classes) to do.
This sharp contradiction becomes very objective whenever we have a topos defined over another one by a
Discrete …. Points …. Codiscrete
where each is left adjoint to the next. The two opposites map the lower more abstract category to opposite ends of the upper more cohesive or variable category, while the middle abstracts from each object in the upper its denuded version.