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Philosophy Speakers: Neo-logicist Foundations: Inconsistent Abstraction Principles and Part-Whole

Date & Time:
December 7, 2018 | 3:00 pm - 5:00 pm
1253 SS
Paolo Mancosu, University of California, Berkeley

About the talk

Neologicism emerges in the contemporary debate in philosophy of mathematics with Wright’s book Frege’s 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
(HP) ∀C, D (#(C)=#(D)↔C ≅ D)
(and definitions). This has led to a detailed scrutiny of so-called abstraction principles, of which Basic Law V
(BLV) ∀C, D (ext(C)=ext(D)↔∀x (C(x)↔D(x)))
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. This is joint work with Benjamin Siskind (UC Berkeley).

About the speaker

Paolo Mancosu is Willis S. and Marion Slusser Professor of Philosophy at the University of California, Berkeley. He has made significant contributions to the history and philosophy of mathematics and logic, especially the philosophy of mathematical practice, mathematical explanation, the history of 20th century logic, and neo-logicism. His most recent book, Abstraction and Infinity (Oxford Unversity Press, 2017), concerns the use of abstraction principles in the philosophy of mathematics. He previous books include Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford University Press, 1996), From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s (Oxford University Press, 1998), The Philosophy of Mathematical Practice (Oxford University Press, 2008), The Adventure of Reason. Interplay between Philosophy of Mathematics and Mathematical Logic: 1900–1940 (Oxford University Press, 2010), and Inside the Zhivago Storm. The Editorial Adventures of Pasternak’s Masterpiece (Feltrinelli, 2013).

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