September 20: Peter Koepke: Felix Hausdorff and the Foundations of Mathematics
16.30--15.30
Abstract:
Felix Hausdorff was connected to the development of the foundations of mathematics in the beginning of the last century in several ways. Hausdorff's philosophy of mathematics can be described as a (non-symbolic) formalism that emphasizes the free but sensible selection of axioms, definitions, and problems. On the other hand Hausdorff did not actively participate in the formation of *axiomatic* set theory or of mathematical logic. There are parallels of Hausdorff's mathematical position and his early anti-metaphysical philosophical book "Das Chaos in kosmischer Auslese" (Chaos in cosmic selection).
To main page

November 21 : John Mumma: Free-rides and overdetermined alternatives in Euclid's diagrams
17:00-18:00
Abstract:
A crucial feature of diagrammatic representation in mathematical argument is, in the terminology of A. Shimojima, the dual phenomena of free-rides and overdetermined alternatives. For any pair of objects represented in a diagram, some relation is depicted. This can be good, if the relation follows from the argument's premises (it is then a free-ride), or bad, if the relation does not (it is then an overdetermined alternative). In this talk I discuss the two different approaches to the phenomena in formalizations of Euclid's diagrammatic arguments. The first, taken in N. Miller's formal system FG, utilizes disjunctive arrays of diagrams. The second, taken in my formal system Eu, provides rules whereby free-rides can be identified in diagrams. I first argue that the second approach provides a more plausible analysis of Euclid, and then consider how, if at all, diagrams by the analysis play a significant role as formal syntactic objects in Euclid's geometry.
To main page

February 29, 2012: Joost Joosten: Well-orders in the Japaridze Algebra
17:00-18:00
Abstract: GLP stands for poly-modal Gödel-Löb's Provability Logic. In this logic there is for each ordinal ß a modal operator [ß]. We shall say some words on how these kind of logics can be employed for ordinal analyses. In particular we shall study certain well-orders that naturally live in the corresponding GLP algebra. These well-orders will provide alternative ordinal notations.
To main page

May 10, 2012: George Metcalfe: Admissibility in Finite Algebras
16:00-17:00
Abstract: The admissibility of a quasiequation in a finite algebra corresponds to the validity of that quasiequation in a finite free algebra, and is hence decidable. However, a naive approach to checking admissibility leads to computationally unfeasible procedures even for small algebras, and tells us little about the properties of admissible quasiequations for the algebra in question. The aim of this talk is to explain, first, a uniform method for obtaining algorithms for checking admissibility in finite algebras, and, second, a strategy using natural dualities for axiomatizing admissibility in these cases.
To main page

June 8, 2012: Giuseppe Greco: Sequent Calculus for the Logic of Public Announcements
16:00-17:00
Abstract: In this talk we present a display-style, cut-free sequent calculus for both the intuitionistic and the classical versions of Public Announcement Logic (PAL). This calculus is sound and complete with respect to both the algebraic and the relational semantics of PAL, and the cut rule is shown to be admissible. This calculus enjoys a weaker form of display property, which is still enough to prove the cut-admissibility. This calculus is modular, which makes it easily generalizable to different Dynamic (Epistemic, Deontic, ...) Logics just by modifying the structural rules.
To main page

June 11, 2012: Michael Beeson: Logic of Ruler and Compass Constructions
16:00-17:00
Abstract: We describe a theory ECG of ``Euclidean constructive geometry''. Things that ECG proves to exist can be constructed with ruler and compass. ECG permits us to make constructive distinctions between different forms of the parallel postulate. We show that Euclid's version, which says that under certain circumstances two lines meet (i.e., a point of intersection exists) is not constructively equivalent to the more modern version, which makes no existence assertion but only says there cannot be two parallels to a given line.
Non-constructivity in geometry corresponds to case distinctions requiring different constructions in each case; constructivity requires continuous dependence on parameters. We give continuous constructions where Euclid and Descartes did not supply them, culminating in geometrical definitions of addition and multiplication that do not depend on case distinctions. This enables us to reduce models of geometry to ordered field theory, as is usual in non-constructive geometry.
To main page

June 22, 2012: Kenneth Manders: Expressive Means and Mathematical Understanding
16:00-17:00
Abstract: What, beyond proof, makes mathematics a powerful form of understanding?
We approach this question by attending to the way mathematics modularizes, by shaping special-purpose contents by the expressions it deploys (and avoids) in special contexts.
To main page

June 22, 2012: Thomas Streicher: Computability in Quantum Theory
16:00-17:00
Weihrauch's TTE (Type Two Effectivity) provides a framework for computability on complete separable metric spaces and domains (as used in computer science) and possibly more. A more abstract account can be given by embedding this into the function realizability topos. Its full subcategory of Sigma-extensional objects corresponds to the world of TTE as shown by M. Schroeder and A. Simpson. It is equivalent to the category of so-called QCB_0 spaces (T_0 quotients of countably based spaces) and all continuous functions between them.
In this talk we will show how the Hilbert lattice, states on it and observables can be subsumed under this framework. We discuss the relation to the more common approach based on functional analysis and raise the question how quantum states should be topologized appropriately.
To main page