The Ways of Abstraction and the Indispensability of Reification

In this talk I present a brief survey of certain possible understandings of the operation of
“mathematical abstraction” and I suggest that - especially when abstraction is considered as
second-order - a certain principle of reification of concepts is indispensable to explicate such
operation. I start by distinguishing four sorts of explications of abstraction: (1) direct
eliminative; (2) conceptual by reification; (3) conceptual by order reduction; (4) by recarving the
content. (1) corresponds to the classical account of abstraction which dates back to Aristotle and
may be partially found in Cantor’s definition of cardinals and in Peano’s school. (2) corresponds
to Frege’s account of abstraction in the Grundlagen. (3) corresponds to Frege’s account in the
Grundgesetze. (4) corresponds to the strategy sketched in §64 of the Grundlagen which Frege
rapidly abandoned. I argue that, when higher-order abstraction principles are concerned, all these
accounts require a principle for reification of concepts which may be preliminary understood as
the introduction of set-like entities. Moreover, I will show that (2) and (3) may be somehow
circular and (1) is less explanatory and may engender some logical difficulties. I draw the
conclusion that given the dependence of abstraction upon a basic set theory, there are little
chances to use this operation to provide a foundational account of arithmetic and real analysis.