I’m reading Shapiro’s Thinking About Mathematics, and there’s a quote by Godel which I would like to fully understand, both his intended meaning and how it’s viewed in the wider context of mathematics, truth, and philosophy.
First the quote from page 10:
"In traditional philosophical terms, Poincare rejected the actual infinite, insisting that the only sensible alternative is the potentially infinite. There is no static set of, say, all real numbers, determined prior to the mathematical activity. From this perspective, impredicative definitions are viciously circular. One cannot construct an object by using a collection that already contains it. (Shap)
Enter the opposition. Godel (1944) made an explicit defense of impredicative definition, based on his philosophical views concerning the existence of mathematical objects: (Shap)
...the vicious circle...applies only if the entities are constructed by ourselves. In this case, there must clearly exist a definition...which does not refer to a totality to which the object defined belongs, because the construction of a thing can certainly not be based on a totality of things to which the thing to be constructed belongs. If, however, it is a question of objects that exist independently of our constructions, there is nothing in the least absurd in the existence of totalities containing members, which can be described (i.e. uniquely characterized) only be reference to this totality...Classes and concepts may...be conceived...as real objects...existing independently of us and our definitions and constructions. It seems to be that the assumption of such objects is quite legitimate as the assumption of physical bodies and there is quite as much reason to believe there existence. " (Godel)
Here's what makes sense to me:
Nominalists do define impredicative mathematical objects without the assumption of their realness in Godel's sense.
These nominalists can do so because their impredicative objects are not the same mathematical objects or semantics Godel is talking about.
Godel has some idea of a correspondence theory of truth or meaning. Nominalism is closer to pragmatism, which denies correspondence of truth and meaning.
Both the nominalist and the Godelian realist share the same mathematical language (the same set theory), but the have entirely different meanings, metaphysics, and notions of truth about that language.
We can say in the specific sense of truth and meaning Godel has in mind, realism is necessary for impredicative definitions to not be absurd or viciously circular.
Are these 5 points correct or fair statements? Am I missing any that should be added that seem relevant? Is this a good exercise within philosophy of mathematics-can I hang my hat on these statements?