Logic Colloquium: Neologicist foundations: inconsistent abstraction principles and part-whole

Colloquium | August 31 | 4-5 p.m. | 60 Evans Hall

 Paolo Mancosu, UC Berkeley

 Department of Mathematics

Neologicism emerges in the contemporary debate in philosophy of mathematics with Wright's book Frege's \textit {Conception of Numbers as Objects} (1983). Wright's project was to show the viability of a philosophy of mathematics that could preserve the key tenets of Frege's approach, namely the idea that arithmetical knowledge is analytic. The key result was the detailed reconstruction of how to derive, within second order logic, the basic axioms of second order arithmetic from Hume's Principle \[(\textrm {HP})\quad \forall C,D\,\big (\sharp (C) = \sharp (D)\leftrightarrow C\cong D\big )\] (and definitions). This has led to a detailed scrutiny of so-called abstraction principles, of which Basic Law V \[(\textrm {BLV})\quad \forall C,D \,\big (ext(C) = ext(D) \leftrightarrow \forall x\,(C(x)\leftrightarrow D(x))\big )\] and HP are the two most famous instances. As is well known, Russell proved that BLV is inconsistent. BLV has been the only example of an abstraction principle from (monadic) concepts to objects giving rise to inconsistency, thereby making it appear as a sort of monster in an otherwise regular universe of abstraction principles free from this pathology. We show that BLV is part of a family of inconsistent abstractions. The main result is a theorem to the effect that second-order logic formally refutes the existence of any function $F$ that sends concepts into objects and satisfies a "part-whole" relation. In addition, we study other properties of abstraction principles that lead to formal refutability in second-order logic.