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# Numerosities, Infinitesimal Probabilities and Infinity

Two Grolog Lectures by Sylvia Wenmackers and Paolo Mancosu

Two Grolog lectures, by Sylvia Wenmackers (Groningen) and Paolo Mancosu (Berkeley)

Sylvia Wenmackers:

#### On numerosities and infinitesimal probabilities

In the first part of this talk, I will introduce numerosity theory, which offers a way of counting infinite sets (developed by Vieri Benci). Normalized numerosities can be employed to assign a uniform probability distribution on an infinite domain in a 'regular' way: assigning non-zero (possibly infinitesimal) probability values to all events other than the empty set. This idea can be generalized, beyond the uniform case, thus leading to the non-Archimedean probability (NAP) theory that I am working on together with Vieri Benci & Leon Horsten (Milan Journal of Mathematics, 2013). In the second part of this talk, I will discuss objections that have been raised against infinitesimal probabilities, most of which also apply to numerosities themselves.

Paolo Mancosu:

#### In good company? On Hume’s principle and the assignment of numbers to infinite concepts

In a recent article (Review of Symbolic Logic 2009), I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign numerosities to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a set A is properly included in B then the numerosity of A is strictly less than the numerosity of B. Numerosities assignments differ from the standard assignment of size provided by Cantor’s cardinality assignments.

In this talk, I generalize some specific worries emerging from the theory of numerosities to a line of thought resulting in what I call a ‘good company’ objection to Hume’s principle. The talk has four main parts. The first takes a historical look at nineteenth-century attributions of equality of numbers in terms of one-one correlations and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part where I show that there are countably infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP and from which we can derive the axioms of second order arithmetic. The third part connects this material to a debate on Finite Hume Principle between Heck and MacBride and states the ‘good company’ objection. Finally, the last part gives a tentative taxonomy of possible neo-logicist responses to the ‘good company’ objection and makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions.

#### Time & Place

Wed, MARCH 27th, 14.00-17.00
Room Gamma, Faculty of Philosophy