Titres et Abstracts — Titles and Abstracts
- Colin McLarty, Case Western Reserve University
Poincare’s logic in light of his philosophy
Abstract : Philosophers of mathematics have generally understood his philosophy of mathematics to be an outgrowth of his logic. In turn they often understand his logic in considerably modernized terms extrapolated from Russell’s view of it. Jeremy Gray’s recent book, drawing much on materiel assembled by the Archives Henri Poincaré at Nancy, opens the way to a more considered interpretation.
- Christopher Pincock, Philosophy, The Ohio State University
Felix Klein as a Prototype for the Philosophy of Mathematical Practice
Abstract : In this paper I aim to determine the contemporary relevance of the conception of mathematics championed by Felix Klein (1849-1925). Klein is best known among philosophers for his “Erlanger Program” in geometry. However, Klein also devoted a great deal of energy to articulating and defending a philosophical approach to mathematics that went far beyond geometry. Among other things, Kline emphasized the illuminating connections between different areas of mathematics, e.g. the Lectures on the icosahedron and the solution of equations of the fifth degree (1884). A central component of this approach is the interdependence of intuition and logic in the development and clarification of mathematical results. I hope to determine what exactly Klein thought the role of intuition was supposed to be. Is it merely of psychological or pedagogical importance, or does intuition play a role in justifying new mathematical claims ? I will also consider the more radical possibility that intuition provides essential insight into the nature of the subject-matter of mathematics itself. In the end, Klein may offer an instructive prototype for more recent attempts to defend the philosophical significance of non-foundational mathematics.
- Jack Woods, Bilkent University, Ankara
Logical Indefinites
Abstract : The best extant demarcation of logical constants, due to Tarski, classifies logical constants by invariance properties of their denotations. This classification is developed in a framework which presumes that the denotations of all expressions are definite. However, some indefinite expressions, such as Russell’s indefinite description operator eta, Hilbert’s epsilon, and abstraction operators such as ‘the number of’, appropriately interpreted, are logical. I generalize the Tarskian framework in such a way as to allow a reasonable account of the denotations of indefinite expressions. This account gives rise to a principled classification of the denotations of logical and non-logical indefinite expressions. After developing this classification and its application to particular cases in some detail, I show how this generalized framework allows a novel view of the logical status of certain abstraction operators such as ‘the number of’ and correspondingly sheds light on the logical status of abstraction principle’s. I then show how we can define surrogate abstraction operators directly in higher-order languages augmented with an epsilon-operator.
- Thierry Coquand, Gothenburg
Type theory and univalent foundations
At the beginning of type theory, several questions around the axiom of description, axiom of extensionality, impredicativity were formulated by Russell. After recalling how these axioms are represented in Church simple type theory, we explain what happens to these questions in dependent type theory with the axiom of univalence.
- Baptiste Mélès, Clermont II
Computing Tools and Arithmetical Properties
Abstract : Is the arithmetical lexicon relevant to describe computing tools and their practice ? Our basic arithmetical knowledge could lead us to assume that elementary arithmetical operations, such as addition and multiplication, natively and inherently possess all the standard properties we know : associativity, commutativity, distributivity. "Addition" can admittedly be performed on the Chinese abacus — but what are really the properties of addition on the abacus ? We will first show that computer science provides us with finer descriptive concepts than arithmetics, such as data structures, curryfication, denotational and operational semantics. We will then observe that some very elementary arithmetical properties can not even be concretely implemented on computing tools. Concepts of computer science can thus sometimes be used to describe computing tools in a finer way than arithmetics. These concepts could as such deserve to integrate the basic lexicon of historians and philosophers of mathematics.
- Chris Porter, LIAFA Paris VII
Randomness and Accessible Objects in Mathematics
The goal of this talk is to articulate and respond to a specific problem that has been raised for definitions of algorithmic randomness, a problem I call the problem of unruly instances. Roughly, the problem is that for each definition of algorithmic randomness D, there is a collection of objects such that (i) each object in the collection is random according to the definition D, but (ii) there are grounds for holding that each of these objects is not intuitively random. To explain why these unruly instances have been held to be problematic, I will present an account of accessible and inaccessible mathematical objects that is inspired by an account of Borel’s on the distinction between accessible and inaccessible numbers. Further, I will draw on this account of accessible and inaccessible objects to argue that the problem of unruly instances does not undermine the legitimacy of the various definitions of algorithmic randomness.
- Sylvain Cabanacq, SPHERE, Paris VII
The sketches, between logic and geometry : Ehresmann’s theory of species of structures
The sketches, developed by Charles Ehresmann in 1966, can be considered as a categorical form of axiomatization of mathematical structures. Such connection between the formal languages and the sketches can precisely be brought out by the equivalence between sketchability and axiomatizability in an infinitary first order language. But the different kinds of sketches allow to establish, among mathematical structures, a certain measure of complexity, apparently different from the usual logical ones. What is the information provided by the sketch of a given mathematical structure ? In what sense can we say that this complexity is intrinsically geometrical ? In order to make this geometrical content of sketches explicit, this notion will be put into the context of its elaboration : Ehresmann’s idea of an « abstract theory of all the species of possible structures » and his categorical understanding of the connection between local and algebraic concepts.
- Neil Barton, Birkbeck College
Proper Classes and Paraphrases
Abstract : Proper classes are philosophically problematic and formal theories often avoid talk of such entities. However, modern Set Theoretical techniques include the study of large cardinal properties through the use of embeddings between models. Such practice seems to require proper class talk. A natural way of providing a paraphrase is through the use of a first-order definable formula. However, there are results from Set Theory that show this is unsatisfactory. An examination of Category Theory provides an alternative paraphrase, but I argue that it cannot be meshed with mathematical practice in an acceptable manner.
- Monica Solomon, Notre Dame
Surprise in Mathematical Practice
Abstract : Surprising results and the experience of surprise more broadly seem to be ubiquitous in mathematics. To be sure, puzzlement, astonishment or wonder are common reactions in human activities, and especially in learning periods. In the philosophy of mathematical practice there has been very little attention devoted to surprise. Indeed, mathematical practice altogether became the focus for scholarship only recently. While the presence of surprise in mathematics has been noted before, its causes and effects have been sometimes misdescribed and misunderstood.
Surprise is often thought to be a feeling of wonder or astonishment. Other times, surprise is described as a reaction caused by limitations in our epistemic or imaginative states. In these roles, surprise is not important either for the philosophy or for the history of mathematics. On the contrary, following Wittgenstein, I argue that surprise is central to mathematical practice in at least three distinct ways. (1) The cause of surprise shows that for understanding a particular mathematical proof or claim and find them surprising we first need to understand the context of assertion and the history of the training of that individual. (2) Secondly, although it is partly an emotional reaction, surprise has cognitive significance and can act as a mechanism for error-recognition : in this role it facilitates learning and it brings to light some of our unmet expectations. In moments of surprise, we doubt the correctness of the proof, we might look for a gap in the reasoning, and we might start a related investigation. Thus, puzzlement acts as an incentive to mathematical research. Finally, (3) we discover that an interesting feature of a mathematical proof is to dispel surprise. Usually the explanation for this powerful characteristic is to say that the proof increases the mathematical knowledge. In this paper, I emphasize that the roles and the reactions that we attribute to surprise in the course of mathematical practice show that there are different types of mathematical knowledge at work.