Organization : Andrew Arana, David Rabouin, Ivahn Smadja, Sean Walsh The sessions are held at Université Paris Diderot - Site Rive Gauche, Bâtiment Condorcet, 4 rue Elsa Morante, 75013 Paris.

### October 19th, 2009

Common session IHPST - Paris Diderot University and “Chaire d’excellence” ANR Senior *Ideals of Proof* - Michael Detlefsen

Location : Room 454A-Klee, 14h-17h

**Harvey Friedman** (Ohio State University)

*Concept Calculus*

**Abstract** :

We have discovered an unexpectedly close connection between the logic of mathematical concepts and the logic of informal concepts from common sense thinking. Our results indicate that they are, in a certain precise sense, equivalent.

We call this development the Concept Calculus. We begin by a discussion of the two commonsense concepts of "better than" and "much better than", and the Zermelo and Zermelo Frankel axioms for set theory.

We then discuss a number of commonsense contexts involving informal linearly ordered time. This leads to a subject that we call Abstract Cosmology, whereas principles such as "anything that can happen will" are used. Connections between "The Exploding Universe" and large cardinals from set theory are established.

It is conjectured that Concept Calculus fruitfully applies to all (combinations of) commonsense concepts, thereby creating the largest mathematical/philosophical enterprise ever.

### February 15th, 2010

Session organized by Andrew Arana (Kansas State University)

Location : Room 454A-Klee, 14h-16h45

14h-15h15

**Andrew Gelman** (Department of Statistics and Department of Political Science, Columbia University, New York, USA,
Visiting Sciences Po, Paris, 2009-2010)

*Philosophy and the practice of Bayesian statistics in the social sciences*

15h30 - 16h45

**Isabelle Drouet** (IHPST)

*Causal inference from statistical data. What is new with Bayesian networks ?*

**Abstracts** :

*Philosophy and the practice of Bayesian statistics in the social sciences*

I present my own perspective on the philosophy of Bayesian statistics, based on my experiences doing applied statistics in the social sciences and elsewhere. My motivation for this project is dissatisfaction with what I perceive as the standard view of the philosophical foundations of Bayesian statistics, a view in which Bayesian inference is inductive and scientific learning proceeds via the computation of the posterior probability of hypotheses. In contrast, I view Bayesian inference as deductive and as part of a larger Bayesian data-analytic process, different parts of which I believe can be usefully understood in light of the philosophical frameworks of Popper, Kuhn, and Lakatos. The practical implication of my philosophy is to push Bayesian data analysis toward a continual creative-destruction process of model building, inference, and model-checking rather than to aim for an overarching framework of scientific learning via posterior probabilities of hypotheses. This work is joint with Cosma Shalizi.

*Causal inference from statistical data. What is new with Bayesian networks ?*

Bayesian networks appeared at the beginning of the 1980s as formal tools for the representation and treatment of uncertainty in artificial intelligence. Since then, they have come to be used in different sciences, and in more and more diverse contexts. In particular, several propositions were formulated at the end of the 1990s and at the beginning of the 2000s to the effect of using them for causal inference from statistical data. These propositions were accompanied by the claim that Bayesian nets solve a major difficulty here, by allowing to induce causal knowledge from statistical data. My talk aims at assessing this claim. More precisely, I will offer a comparison of the inferences that are allowed by Bayesian nets with the more traditional, hypothetico-deductive, inferences that take statistical data as their premisses and causal statements as their conclusions.

### March 22, 2010

Location : room 366A-Klimt, 14h-17h

14h-15h30

**Matthias Schirn** (Ludwig Maximilians Universität, Munich)

*Consistency, Models, and Soundness*

15h30 - 17h00

**Dirk Schlimm** (McGill University)

*Pasch’s Empiricist Foundations of Mathematics*

**Abstracts** :

*Consistency, Models, and Soundness*

In the first part of this talk, I discuss the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. In the second part, I comment on Hilbert’s proof theory (1922-1931). One difficulty that I analyse is the fact that Hilbert’s language of finitist metamathematics fails to supply the conceptual resources for formulating a consistency statement qua unbounded quantification. In the third and final part, I shall take a critical look at Wittgenstein’s views about (in)consistency and consistency proofs in the period 1929-1933. I argue that both his insouciant attitude towards the emergence of a contradiction in a calculus and his outright repudiation of metamathematical consistency proofs are unwarranted. In particular, I argue that Wittgenstein falls short of making a convincing case against Hilbert’s programme. I close with some philosophical remarks on consistency proofs and soundness.

*Pasch’s Empiricist Foundations of Mathematics*

In his groundbreaking "Lectures on Newer Geometry" (1882) Moritz Pasch (1843-1930) gave the first rigorous axiomatization of projective geometry, which paved the way for Hilbert’s famous "Foundations of Geometry" (1899). In these lectures Pasch also presented the two cornerstones of his philosophy of mathematics : a radical version of empiricism, aimed as providing an epistemological basis for mathematics, and a very modern version of deductivism, as the ideal of mathematical rigor. In this talk I will give an introduction to Pasch’s philosophy of mathematics, focusing in particular on his empiricism. Some connections to Frege, Hilbert, and the logical empiricists will also be drawn.

### April 12th, 2010

** Carving mathematics up **

Session organized by Brice Halimi (Université Paris X - Nanterre)

Location : Room 646A-Mondrian, 9h30-17h

9h30-11h

**Sébastien Gandon** (Université Blaise Pascal de Clermont-Ferrand)

*Logicism, universalism and topic-specificity*

11h10-12h40

**Andrew Arana** (Kansas State University)

*Purity and the carving of mathematics*

14h30-16h

**Boris Zilber** (Oxford, Mathematical Institute)

*Quantum space : a model-theoretic point of view*

16h10-17h40

**Paul-André Melliès** (University Paris Diderot -Paris 7, PPS)

*Towards an algebraic presentation of proof theory*

**Abstract** :

The session aims at linking two threads together. The first one bears on logic from a dominantly philosophical point of view, the second one on the connections between logic (more as a branch of mathematics) and ’traditional’ mathematics. The centenary of the publication of Principia Mathematica is a basis for the first thread. Sébastien Gandon (Clermont-Ferrand) and Andrew Arana (Kansas State), will give a talk. Sébastien defends the idea that Russellian logicism can be thought as putting forward a topic-specific view of mathematical fields (in particular of geometry and measure theory), at least much more than one usually makes out. Andrew, on the other hand, will discuss his work on the ideal of purity of proofs in mathematics. The second thread bears on mathematical logic as providing ways of carving mathematics up. Proof-theoretical analysis can indeed be viewed as a sweep of the whole of mathematics, giving rise to original reorganizations, and Paul-André Melliès (Paris 7, PPS) will give a talk about the connections between proof theory and categorical algebra. Finally, Boris Zilber (Oxford, Institute of Mathematics) will set out the concept of categoricity as a general notion to describe mathematical structures, and speak of connections between model theory and geometry. So, to seum up, the general theme will be : Carving Mathematics Up. Is there a natural way of doing it ? This question will be tackled from two different perspectives, philosophy and the Principia on the one hand, and contemporary mathematical logic on the other hand.

### April 13th, 2010

** Proof Complexity **

Session organized by Andrew Arana (Kansas State University)

Location : Room 366A-Klimt, 11h-17h

11h - 12h30

**Alessandra Carbone** (Université Pierre et Marie Curie)

*Logical structures, cyclic graphs and genus of proofs*

14h - 15h30

**Michael Harris** (Université Paris Diderot)

*Complexity as a guide to understanding proofs*

15h30 - 17h

**David Corfield** (University of Kent)

*The complexity of proof and its robustness*

**Tappenden PhilMath Workshop**

May-June 2010 -
Program