Conference on Reconciling Nominalism and Platonism

Department of Philosophy, Columbia University
April 22–23, 2016

FRIDAY APRIL 22 (Philosophy Hall, Room 716)

Achille Varzi (Columbia University), Marco Panza (IHPST)
Welcome and Introduction
John Burgess (Princeton University)
Reconciling Anti-Nominalism and Anti-Platonism in the Philosophy of Mathematics
15:45–16:00 Break
Haim Gaifman (Columbia University)
Reconfiguring the Problem: “Platonism” as Objective, Evidence-transcendent Truth
Sébastien Gandon (Université Blaise Pascal)
Describing What One is Doing. A Philosophy of Action Based View of Mathematical Objectivity

SATURDAY, APRIL 23 (Philosophy Hall, Room 716)

Mirna Džamonja (University of East Anglia and IHPST)
An Unreasonable Effectiveness of ZFC Set Theory at the Singular Cardinals
11:00–11:30 Break
Hartry Field (New York University)
Platonism, Indispensability, Conventionalism
13:00–15:00 Lunch
Justin Clarke-Doane (Columbia University)
The Benacerraf Problem in Broader Perspective
16:30–17:00 Break
Michele Friend (George Washington University)
Is the Pluralist Reconciliation between Nominalism and Platonism too Easy?
18:30 Conclusions

Floyd: Gödel on Russell

Gödel on Russell: Truth, Perception, and an Infinitary Version of the Multiple Relation Theory of Judgment
Juliet Floyd (Boston University)
4:10 pm, May 8, 2014
Faculty House, Columbia University

Beziau : Round Squares are No Contradictions

Round Squares are No Contradictions
Jean-Yves Beziau (Federal University of Rio de Janeiro and UC San Diego)
4:10pm, Friday, April 24, 716 Philosophy Hall, Columbia University

Abstract. When talking about contradictions many people think of a round square as a typical example. We will explain in this talk that this is the result of a confusion between two notions of oppositions: contradiction and contrariety. The distinction goes back to Aristotle but it seems that up to now it has not been firmly implemented in the mind of many rational animals nor in their languages.According to the square of opposition, two propositions are contradictory iff they cannot be true and cannot be false together and they are contrary iff they cannot be true together but can be false together. The propositions “X is a square” and “X is a circle” cannot be true together according to the standard definitions of these geometrical objects, but they can be false together: X can be a triangle, something which is neither a square, nor a circle. A round square is a contrariety, not a contradiction. Aristotle insisted that there were two different kinds of oppositions, from this distinction grew a theory of oppositions that was later on shaped in a diagram by Apuleius and Boethius.It is easy to find examples of contrarieties, but not so of contradictions. Many pairs of famous oppositions are rather contraries: black and white (think of the rainbow), right and left (think of the center), day and night (think of dawn or twilight, happy and sad (think of insensibility), noise and silence (think of music), etc. Examples of “real” contradictions are generally from mathematics: odd and even, curved and straight, one and many, finite and infinite. We can indeed wonder if there are any contradictions in (non-mathematical) reality or if it is just an abstraction of our mind expressed through classical negation according to which p and ¬p is a contradiction.

Bjorndahl: Language Based Games

Language Based Games
Adam Bjorndahl (Carnegie Mellon University)
10:30 AM to 12:30 PM, Friday, March 20, 2015
Room 7395, CUNY Graduate Center

Abstract: We introduce a generalization of classical game theory wherein each player has a fixed “language of preference”: a player can prefer one state of the world to another if and only if they can describe the difference between the two in this language. The expressiveness of the language therefore plays a crucial role in determining the parameters of the game. By choosing appropriately rich languages, this framework can capture classical games as well as various generalizations thereof (e.g., psychological games, reference-dependent preferences, and Bayesian games). On the other hand, coarseness in the language—cases where there are fewer descriptions than there are actual differences to describe—offers insight into some long-standing puzzles of human decision-making.

The Allais paradox, for instance, can be resolved simply and intuitively using a language with coarse beliefs: that is, by assuming that probabilities are represented not on a continuum, but discretely, using finitely-many “levels” of likelihood (e.g., “no chance”, “slight chance”, “unlikely”, “likely”, etc.). Many standard solution concepts from classical game theory can be imported into the language-based framework by taking their epistemic characterizations as definitional. In this way, we obtain natural generalizations of Nash equilibrium, correlated equilibrium, and rationalizability. We show that there are language-based games that admit no Nash equilibria using a simple example where one player wishes to surprise her opponent. By contrast, the existence of rationalizable strategies can be proved under mild conditions. This is joint work with Joe Halpern and Rafael Pass.

Hartmann : Learning Conditionals and the Problem of Old Evidence

Learning Conditionals and the Problem of Old Evidence
Stephan Hartmann (Ludwig Maximilians-Universität München)
4:10 pm, February 13, 2015
Faculty House, Columbia University

Abstract. The following are abstracts of two papers on which this talk is based.

The Problem of Old Evidence has troubled Bayesians ever since Clark Glymour first presented it in 1980. Several solutions have been proposed, but all of them have drawbacks and none of them is considered to be the definite solution. In this article, I propose a new solution which combines several old ideas with a new one. It circumvents the crucial omniscience problem in an elegant way and leads to a considerable confirmation of the hypothesis in question.

Modeling how to learn an indicative conditional has been a major challenge for formal epistemologists. One proposal to meet this challenge is to construct the posterior probability distribution by minimizing the Kullback-Leibler divergence between the posterior probability distribution and the prior probability distribution, taking the learned information as a constraint (expressed as a conditional probability statement) into account. This proposal has been criticized in the literature based on several clever examples. In this article, we revisit four of these examples and show that one obtains intuitively correct results for the posterior probability distribution if the underlying probabilistic models reflect the causal structure of the scenarios in question.

Gruszczyńsk: Methods of constructing points from regions of space

Rafał Gruszczyńsk (Nicolaus Copernicus University, Toruń) will give an informal, non-colloquium talk this Friday, Nov. 21, at 2pm, in the seminar room (Philosophy 716). The title of the talk is “Methods of constructing points from regions of space”. Everybody is invited. The talk should be of special interest to colleagues and students working in logic, ontology, the philosophy of mathematics, and the philosophy of space and time.