October 17, 2018, 6-7:30 PM in 234 Moses Hall

Eric Pacuit (University of Maryland, College Park)

Strategic reasoning: From beliefs about mistakes to mistaken beliefs

A crucial assumption underlying any game-theoretic analysis is that there is common knowledge that all the players are rational. Rationality is understood in the decision-theoretic sense: The players’ choices are optimal according to some choice rule (such as maximizing subjective expected utility). Research on the epistemic foundations of game theory is focused on the game-theoretic implications of different notions of rationality and different assumptions about what the players know and believe about each other. In this talk, I will provide a brief overview of this literature. A key question is what should the players
assume about how their opponents will interpret their moves in the game? Two prominent answers have been discussed in the literature: forward induction and backward induction. According to the former, players rationalize past behavior and use it as a basis for forming beliefs about future moves. According to the latter, players ignore past behavior and reason only about their opponents’ future moves. There are sophisticated mathematical models that formalize the implications of each way of reasoning. In this talk, I will consider a different question that has not received as much attention: How should the players choose between the two styles of reasoning? I will present a formal model of strategic reasoning introduced by Brian Skyrms and show how it needs to be modified to address this question.

February 27, 2019, 6-7:30 PM in 234 Moses Hall

Stewart Shapiro (Ohio State University)

Superplurals, groups, and paradox

There are two views regarding definite plurals like “the students” in

  1. The students gathered around the teacher.

According to singularism, “the students” refers to a certain set-like

object, and (1) will be true if that set-like object has the cumulative property of gathering around the teacher. According to pluralism, “the students” refers to a primitive multiplicity of individuals, as familiar from plural logic, and (1) will be true if that multiplicity has that same cumulative property.

Singularism is the predominant view within linguistic semantics and pluralism is at least the received view among philosophers and logicians who invoke plurals.  The latter is due in large part to the influence of George Boolos.  The primary argument against singularism is that the view is prone to Russell’s paradox. Semanticists are, of course, aware of this threat of paradox, as illustrated by Fred Landman’s “First Amendment” for semantics:

The right to solve Russell’s Paradox some other time shall not be restricted.

The first purpose of this paper is to review the empirical data that supports linguistic singularism.  The issue concerns the so-called “super-plurals”.  The second, and main purpose, of the paper is to discharge the First Amendment and provide a potentialist theory of groups—the semantic value of plural expressions.  On this theory, though it is always possible to form a group from a plurality, it is not necessary that we do so.  The view ties in with both the nature of potential infinity generally and the recursive or generative nature of natural languages.

March 06, 2019, 6-7:30 PM in 470 Stephens

Georg Schiemer (University of Vienna)

How Geometry Became Structural

Joint event with OHST.
Structuralism in the philosophy of mathematics is, roughly put, the view that mathematical theories study abstract structures or the structural properties of their subject fields. The position is strongly rooted in modern mathematical practice. In fact, one can understand structuralism as an attempt to come to terms philosophically with a number of wide-ranging methodological transformations in 19th and early 20th century mathematics, related to the rise of modern geometry, number theory, and abstract algebra. The present talk will focus on the geometrical roots of structuralism. Specifically, we will survey some of the key conceptual changes in geometry between 1860 and 1910 that eventually led to a “structural turn” in the field. This includes (i) the gradual implementation of model-theoretic techniques in geometrical reasoning, for instance, the focus on duality and transfer principles in projective geometry; (ii) the unification of geometrical theories by algebraic methods, specifically, by the use of transformation groups in Felix Klein’s Erlangen Program; and (iii) the successive consolidation of formal axiomatics in work by Hilbert and others.

May 01, 2019, 6-7:30 PM in 234 Moses

Jamie Tappenden (University of Michigan)

Frege, Carl Snell and Romanticism; Fruitful Concepts and the ‘Organic/Mechanical’ Distinction

A surprisingly neglected figure in Frege scholarship is the man Frege describes (with praise that is very rare for Frege) as his “revered teacher”, the Jena physics and mathematics professor Carl Snell.  It turns out that there is more of interest to say about Snell than can fit into one talk, so I’ll restrict attention here to just this aspect of his thought: the role of the concept of “organic”, and a contrast with “mechanical”. Snell turns out to have been a philosophical Romantic, influenced by Schelling and Goethe, and Kant’s/Critique of Judgement/. In Frege’s environment, the “organic/mechanical” contrast, understood in a distinctively Romantic fashion, had reached the status of “accepted, recognized cliché”. More generally, Frege’s environment was more saturated with what we now call ``Continental philosophy” than we might expect.

This context-setting has a payoff for our reading of Frege’s texts: many expressions and turns of phrase in Frege that have been regarded as vague, throwaway metaphors turn out to be literal references to ideas that would have been salient among the people in Frege’s environment that he spent time with day-to-day. In particular, this is true of Frege’s account of “extending knowledge” via “fruitful concepts” and his rejection of the idea that logic and mathematics can be done “mechanically” (as with Jevons’ logic machines, or Fischer’s “aggregative mechanical thought”). When Frege appealed to “organic connection” and speaks of fruitful concepts as containing conclusions “like a plant in its seeds”, he would have expected these phrases to have been understood in a very specific way, as alluding to a recognized contrast between “organic” and “mechanical” connection that was applied by Snell and those close to him not only to distinctions between biological and physical reasoning but also to distinctions of types of reasoning in arithmetic and geometry.