Workshop on Intensionality in Mathematics

Titie: A Theory of Fregean Abstract Objects

Abstract:In this paper, I am going to present a theory of Fregean abstract objects. The theory deploys a comprehension principle for concepts, and a comprehension principle for objects. On top of it, a principle of plural comprehension is added. These principles are tweaked to the effect that their interaction is consistent, and they also prove to be strong enough to derive as theorems several Fregean abstraction principles in their full mathematical strength.

Titie: Soundness, reflection, and intensionality

Abstract:It is commonly held that acceptance of the axioms of an arithmetical theory T (such as Peano arithmetic) obligate us to accept various formal expressions of T's soundness -- e.g. Con(T), or the uniform or global reflection principles for T. In this talk, I will discuss the nature of this commitment in light of several phenomena which are intensional in character: the paradoxes of Montague (1963) and Thomason (1980), Feferman's (1960) observations about the formulation of consistency statements, and Kreisel & Levy's (1968) observations about the relationship between reflection principles, mathematical induction, and the internal provability of cut elimination.

Titie: On logicality, invariance, and definability

Abstract: The traditional account of what a logical consequence is says that A follows logical from T if for every (re-)interpretation of the non-logical expressions in T and A; if all the sentences in T are true then so is A. This definition rests on the fact that we know how to distinguish between logical and non-logical expressions, this is the problem of identifying the logical constants. I will focus on a model theoretic approach to solve this problem: An operator is a logical constant if it is invariant under the most general transformations. Apart from giving some background I will present recent results (jointly with Denis Bonnay) on Galois correspondences between invariance and definability: The dual character of invariance under transformations and definability by some operations has been used in classical work by for example Galois and Klein. In this talk I will study this duality from a logical viewpoint and generalize results from Krasner and McGee into a full Galois correspondence of invariance under permutations and definability in L_{\infty\infty}. I will also present a similar correspondence related to definability in L_{\infty\infty}^-, the logic without equality.

Titie: Mathematical intensions and intensionality in mathematics

Abstract: There are a number of different research programs associated with the topic of intensionality in mathematics, as will be seen in the variety of talks in this workshop. This talk considers some of the common issues as well as some differences in approaches. In common there is a philosophical and mathematical history, as well as an interest in the epistemology of mathematics. There are significant differences, however, between the way the epistemology of mathematics is approached. I will argue that some of this variation can be associated with the decision to focus on different puzzles, or asymmetries, produced by the distinction between an intension and "its" extension. I conclude by suggesting that we still lack a satisfying account of the basic relation between mathematical intensions and extensions, that is, one that explains which intensions produce definite extensions and why. And for this, intuition is still an appealing "prop".

Titie: Models for absolute provability

Abstract:In this presentation I will discuss models for the notion of absolute provability. Kripke models have been the standard tools for modelling the notion of necessity. And they have also been used fairly successfully to generate models for the concept of absolute provability. Yet I will tentatively argue that branching time models are more suitable for modeling absolute provability.

Titie: Consistency statements in semi-euclidean systems

Abstract: In logical systems where the axioms and rules of inference are allowed to change over time - like in Jeroslow's Experimental Logics - a reasonable definition of theoremhood makes it possible for the statement of the system's consistency to be among its theorems. The talk discusses examples of this, the intension of such statements and knowability of consistency.

Titie: Modality in Mathematics

Abstract: The talk will be concerned with the role that modality can play in a broadly classical mathematics and in particular with the question of how this modality can be interpreted.

Titie: The intensional side of algebraic-topological representation theorems

Abstract: Stone representation theorems and the like are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use will be reviewed and it will be shown how, through the methods of analytic proof theory, they can be circumvented in favour of direct and constructive arguments.

Titie: The First Few Numbers: How Children Learn Them and Why It Matters

Abstract: Cognitive and developmental scientists since Piaget have been interested in how humans acquire number concepts. In this talk, I'll briefly review a decade of research on how children acquire the first few numbers, and explain why this topic is both interesting and important. Then I will present data on the socio-economic gap in young children's number knowledge in Orange County, California; and describe two intervention studies in progress in my lab, both designed to help kids learn numbers. I'll wrap up by discussing how the results of these studies may contribute to our understanding of the origins of number concepts.

Titie: Mathematics & Logic: Between Theory & Reality

Abstract: In this talk I will discuss the relation between the objects (objectual grounding, source of truth) of mathematics and logic and the form their theories take. In particular, I will examine what in reality, if anything, mathematics and logic are about (or grounded in) and how this is related to the structure of their theories. This will lead to the discovery of certain gaps between theory and objects (objectual grounds) in the two disciplines. In the case of logic, the gap has to do with the linguistic nature of our logical theory and the non-linguistic reality grounding it. In the case of mathematics it has to do with the order (level) of mathematical theories and the order (level) of their objects. I will offer a solution to these gaps and show how it enables us to tackle well-known problems in the philosophies of mathematics and logic.

Titie: On the logic of up to Isomorphism

Title: Making sense of maths: a formalist theory of mathematical intension

Abstract: This paper proposes a solution to the problem of the intensional content of mathematical assertions in terms of their proof theoretic properties. It is proposed that an extensional relation symbol, such as =, has extensional content given by its standard mathematical interpretation, as well as inten- sional content constituted by a set of sound formal mathematical theories. It is claimed that we can use this to give a compositional theory of the intension of a mathematical assertion that can serve, among other things, as a semantics of mathematical propositional attitude ascriptions.

Title: Peano numerals as buck-stoppers

Abstract: I will examine three claims made by Ackerman (1978) and Kripke (1992). First, they claim that not any arithmetical terms is eligible for universal instan- tiation and existential generalisation in doxastic or epistemic contexts. Second, Ackerman claims that Peano numerals are eligible for universal instantiation and existential generalisation in doxastic or epistemic contexts. Kripke’s position is a bit more subtle. Third, they claim that the successor relation and the smaller- than must be effectively calculable. These three claims will be examined from the framework of modal-epistemic arithmetic, i.e. arithmetic extended with cer- tain modal, epistemic and modal-epistemic principles. I will present theorems that give support to the claims made by Ackerman and Kripke.