Jessica Carter
"The role of diagrams in mathematics"
The topic of visualisation and the role of diagrams in mathematics has recently become popular in the philosophy of mathematics. Previously the view was that diagrams should play no role in mathematical reasoning. This view is contested in a number of different ways. Historically one may trace the view to the developments during the 19th century leading to modern mathematics, where mathematicians realised that diagrams could be misleading, and so gave up on
the geometric foundation. Some scholars, like Azzouni, have argued that diagrammatic reasoning can in fact be rigorous, others, in particular Giaquinto, have pointed to the many different roles (for understanding, in reasoning and discovery) that diagrams or visualisation play.
It is thus clear - whether or not reasoning using diagrams is rigorous - that visualisation has played and still plays a role in mathematics. In the talk I will talk about the role of diagrams for discovery in mathematics, and give some examples of other roles they play. In addition I intend to speculate about the reason for why representing in general is fruitful.
the geometric foundation. Some scholars, like Azzouni, have argued that diagrammatic reasoning can in fact be rigorous, others, in particular Giaquinto, have pointed to the many different roles (for understanding, in reasoning and discovery) that diagrams or visualisation play.
It is thus clear - whether or not reasoning using diagrams is rigorous - that visualisation has played and still plays a role in mathematics. In the talk I will talk about the role of diagrams for discovery in mathematics, and give some examples of other roles they play. In addition I intend to speculate about the reason for why representing in general is fruitful.
| ruc20maj2014.pdf |
Melvin Fitting
"Forcing and Intensional Set Theory"
Forcing is a fundamental tool in set theory. I am most decidedly not a settheorist, but I am interested in the tool. One pedagogically helpful way to look at it uses modal logic. (It was used extensively in a book I wrote with Raymond Smullyan.) It is simple and intuitive (at least relatively so). But also, it suggests that what is going on is a kind of intensional set theory. I will discuss the basic ideas, and show how intensional constructs play a central role.
| fittingforcingslides.pdf |
Tinne Hoff Kjeldsen
"Reflections on mathematics through a historical lens: what can history of mathematics tell us about mathematics besides who did what when?"
In this talk I will discuss what a historical lens can contribute to our understanding of mathematics. Through examples from history, most notably “Minkowski’s inventive art” as Hilbert called it, we will touch upon questions like: when do mathematical thoughts inspire to further research, when do they lead to new creations. Based on these discussions, we will raise the questions: how do we develop a critique of mathematics and can history contribute to that?
| tinne-talk-mathematical-reflections_ruc-20-maj-2014-webpage.pdf |
Øystein Linnebo
"Dynamic abstraction"
Abstract: Traditional Fregean and neo-Fregean approaches to abstraction operate with a fixed domain. I show how to make progress on several fronts by proceeding in a more dynamic manner, which allows the domain to "grow".
| dynamic-slides.pdf |
Stig Andur Pederson
"Hilbert’s non-formalistic view of mathematics"
Usually, Hilbert is considered as the creator and the main advocate of the formalistic conception of mathematics. This is correct when we define formalism in a narrow sense as Hilbert’s finistic proof theoretical programme. But if we look at his view of the role and methods of practical mathematics we get a very different view - a view that is far from being formalistic.
| hilbertnonformal.pdf |
Stewart Shapiro
"Potential infinity and intuitionistic logic"
Abstract: I will give a model for the Aristotelian notion of potential infinity (taking geometry to be the case at hand). It will use Øystein Linnebo's formulation of the "potential" nature of the iterative hierarchy. One item of note is the extent to which the framework can be adapted to an intuitionistic background.
| more_aristotelian_for_intuitionists_slides.pdf |
