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).

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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