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(2017/2018) I Semester: Paolo Mancosu (University of California Berkeley): "Abstraction and Infinity"  

 

Prof. Paolo Mancosu (U.C. Berkeley): "Abstraction and Infinity"

 

Neo-logicism is an attempt to revive Frege’s logicist program by claiming that important parts of mathematics, such as second-order arithmetic, can be shown to be analytic. The claim rests on a logico-mathematical theorem and a cluster of philosophical argumentations. The theorem is called Frege’s theorem, namely that second order logic with a single additional axiom, known as N= or Hume’s principle, deductively implies (modulo some appropriate definitions) the ordinary axioms for second order arithmetic. The cluster of philosophical claims is related to the status (logical and epistemic) of N= or Hume’s principle. In the Fregean context the second order systems have variables for concepts and objects (individuals). In addition, there is a functional symbol # that when applied to concepts yield objects as values. The intuitive meaning of # is as an operator that when applied to concepts yields an object corresponding to the cardinal number of the objects falling under the concept (numbers are thus construed as objects). Hume’s principle has the following form:

HP      ("B)("C) [#x:(Bx)=#x:(Cx) iff B ≈ C]

where B ≈ C is short-hand for one of the many equivalent formulas of pure second order logic expressing that “there is a one-one correlation between the objects falling under B and those falling under C”. As remarked, the right-hand side of the equivalence can be stated in the pure terminology of second-order logic. The left hand-side gives a condition of numerical (cardinal) identity for concepts. The concepts B and C have the same (cardinal) number just in case there is a one-one correlation between the objects falling under them.

            In the seminar, I will briefly review Frege’s logicism, introduce the neo-logicist program, and then present two new lines of investigation which are my original contributions to this foundational position. The first consists in rooting the use of abstraction principles in the mathematical practice of the nineteenth century. The second, concerns alternative definitions for assigning numbers to infinite concepts (or sets). I will show that there infinitely many abstraction principles that are alternatives to Hume's principle in the sense that they share with Hume's principle the same epistemic virtues (and, importantly, from each of them one can derive the axioms for second-order arithmetic) but differ from it in the assignments of numbers to infinite concepts. In the seminar, I will present these new investigations and probe their epistemological, ontological, and foundational consequences.

 

Syllabus: 

November 6: Logicism

Readings: Frege, Foundations of Arithmetic, selections

14:30-16:30 Sala "Enzo Paci"

 

November 13: Abstraction Principles and the nature of abstracta from Grassmann to Weyl.

Readings: Mancosu, Abstraction and Infinity, OUP, 2017, chapters 1 and 2

11:00-13:00 Sala "Enzo Paci"

 

November 20: Neo-Logicism

Readings: Wright, Frege’s conception of numbers as objects, 1983, selections

14:30-16:30 Sala "Enzo Paci"

 

November 27: Measuring the size of infinite collections of natural

numbers: Was Cantor’s theory of infinite number inevitable?

Readings: Mancosu, Abstraction and Infinity, OUP, 2017, chapter 3

14:30-16:30 Sala "Enzo Paci"

 

December 4: In good company? On Hume’s Principle and the

assignment of numbers to infinite concepts.

Readings: Abstraction and Infinity, OUP, 2017, chapter 4

14:30-16:30 Sala "Enzo Paci"

 

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