I committed some time to the study of the above readings. No regrets.
Poincaire also had thoughts on "infinity" that are most apropos. I found this paragraph illuminating:
""" Poincare's prohibition of impredicative definitions is also connected with his point of view on infinity. According to Poincare, there are two different schools of thought about infinite sets; he called these schools Cantorian and Pragmatist. Cantorians are realists with respect to mathematical entities; these entities have a reality that is independent of human conceptions. The mathematician discovers them but does not create them. Pragmatists believe that a thing exists only when it is the object of an act of thinking, and infinity is nothing but the possibility of the mind's generating an endless series of finite objects. Practicing mathematicians tend to be realists, not pragmatists or intuitionists. """
As a practicing Wittgensteinian I would say I am not a mathematical realist so much as a pragmatist and intuitionist.
Both Platonism and nominalism share the same view of meaning as a kind of pointing and think "perfect circles", if not evident in nature, must mean something as "objects in the mind's eye" i.e. their meaning must be somehow witnessed in an observational moment or act.
The Platonist (called a "realist" by such as the Catholic Encyclopedia) thinks mathematical objects such as "perfect spheres" and "perfectly flat infinite planes" really exist in some "realm" accessible to the mind as a kind of private witness.
An observer (also Platonic) ostensively defines "the meaning" of these ideas by means of some private act, not unlike a private act of worship people might say "points to" their deity (who might be a perfect circle for all we know, some secret beetle in some private box we never get to peer into).
The nominalist is typically portrayed as sharing the Platonist / realists notion that words (nouns especially, not so sure about prepositions) are "pointers" with their meanings being objects in the shared and/or private vista.
The nominalist just refuses to believe in the Platonic Realm and considers "generalizations" such as the perfect circle or perfect sphere to be an extrapolation, perhaps an averaging, of actual experiences with material objects.
As I shared with the physics teachers recently -- on a list with partially overlapping membership from here -- referring to my own reading program in philo:
""" Ryle's ridicule of the 'private ostensive definition' -- as if we could "see" what we mean by [some mental phenomenon ] rolls into this, as early 1900s Oxford - Cambridge wrestled with its metaphors for mentation, pre-computer (back then, a "computer" was a type of paid employee, so no help there, in terms of providing an analogy, Von Neumann's PC e.g. a 386 running Windows, still in the future).
This training (reading course) removes many temptations to introspect in search of what cognitive terms such as "understanding" might mean, i.e. there's no closing the eyes to study "my understanding process" as if some "process" were in need of witnessing (observer / observed) for the term itself to have meaning. """