Arithmetic, Structures and the Rise of Modern Logic
A Colloquium in Honour of
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15 June 2013, Lecture Room, Philosophy Faculty, Infirmary ( Radcliffe Humanities Building)
To register please follow this link. Registration is free. Lunch will be provided. Attendees may also wish to attend the post-colloquium dinner, which will be held at St Cross and very reasonably priced.
Directions can be found here.
Please click on the titles to see the abstracts.
| 10:00-11:00 | Richard Pettigrew · Why I'm not (that kind of) a structuralist |
| 11:00-12:00 | Alex Paseau · Degrees of adequate formalization |
| 12:15-13:15 | Marcus Giaquinto · Universes of sets |
| 13:15-14:15 | lunch |
| 14:30-15:30 | Paolo Mancosu · On the relationship between plane and solid geometry |
| 15:45-16:45 | Alex Wilkie · A walk on the tame side |
| 17:00-18:00 | Dan Isaacson · Valedictory lecture: Philosophy of mathematics in Oxford |
Organizers: Harvey Brown, Volker Halbach and Alex Paseau
For questions please write to Bryn Harris.
Abstracts
Richard Pettigrew · Why I'm not (that kind of) a structuralist
Structure Realists -- such as Dedekind, Resnik, Shapiro, and others -- hold that, to every isomorphism class of systems, there corresponds a unique pure structure. And they hold that these structures comprise the subject matter of mathematics. I consider four objections to this conception of mathematics: I argue that, while none is decisive on its own, together they provide compelling reason to reject this position.
Alex Paseau · Degrees of adequate formalization
Formalisations of a given natural-language sentence are routinely compared with one another. For instance, Fa&Gb is thought a better first-order formalisation of ‘Fido is a dog but Bruin is a bear’ than FavGb; and we usually regard the formalisation of ‘It is possible that it rains’ in propositional modal logic as superior to its formalisation(s) in a non-modal logic. Can we model such comparative judgments? In as far as the existing literature considers this question, it tends to adopt an all-or-nothing approach: either a formalisation is adequate or it is not. My talk will explore two approaches that go beyond the all-or-nothing tendency. The first tries to exploit the idea that some pairs of natural-language sentences can be closer than others, in some appropriate sense. The second seeks to model how well a particular formalisation captures a sentence’s inferential role.
Marcus Giaquinto · Universes of sets
Is there just one universe of (pure) sets? Or many? If many, are they orderly, or wild? And is the Continuum Hypothesis decided? I plan to respond to Daniel Isaacson's discussion of these and related questions in the final sections of his paper "The reality of mathematics and the case of set theory".
Paolo Mancosu · On the relationship between plane and solid geometry
In the first part I will give a rough sketch of some key episodes in mathematical practice that relate to the interaction between plane
and solid geometry. In the second part, I will look at a late nineteenth century debate (on “fusionism”) in which for the
first time methodological and foundational issues related to aspects of the mathematical practice covered in the first part of the paper came to the fore. I conclude this part of the
talk by remarking that only through a foundational and analytical effort could the issues raised by the debate on “fusionism” be made precise. The third part of the talk focuses on
a specific case study which has been the subject of such an effort, namely the foundational analysis of the plane version of Desargues’ theorem on homological triangles and its
implications for the relationship between plane and solid geometry. Finally, building on the foundational case study analyzed in the third section, in the fourth section
I point the way to the analytic work necessary for exploring various important claims on “purity”, “content”
and other relevant notions.
Alex Wilkie · A walk on the tame side
"That the foundations of topology are inadequate is manifest from the very
beginning, in the form of “false problems” (at least from the point of
view of the topological intuition of shapes) such as the “invariance of
domains”, even if the solution to this problem by Brouwer led him to
introduce new geometrical ideas. Even now, just as in the heroic times
when one anxiously witnessed for the first time curves cheerfully filling
squares and cubes, when one tries to do topological geometry in the
technical context of topological spaces, one is confronted at each step
with spurious difficulties related to wild phenomena." (Alexandre
Grothendieck, Sketch of a Programme, 1984.)
My first aim in this talk is to describe how model theorists have come up with a possible answer to the problem, implicitly posed here by Grothendieck, of creating a "tame" topology. The criteria we use are as follows.
- It should be a framework that is flexible enough to carry out many geometrical and topological constructions on real functions and on subsets of real euclidean spaces.
- But at the same time it should have built in restrictions so that we are a priori guaranteed that pathological phenomena can never arise. In particular, there should be a meaningful notion of dimension for all sets under consideration and any that can be constructed from these by use of the operations allowed under (A).
- One must be able to prove finiteness theorems that are uniform over parameterised collections.
My second aim is to discuss some technical results illustrating the delicate dividing line between theories maintaining a semblance of (A), (B) and (C), and those that are thinly disguised forms of fully blown set theory.
Dan Isaacson · Valedictory lecture: Philosophy of mathematics in Oxford
Philosophy of mathematics has been an established subject in Oxford University since 1946, when Friedrich Waismann was appointed Lecturer in the Philosophy of Science and Mathematics. In 1948 Waismann was made Senior Lecturer in the Philosophy of Mathematics, and in 1950 Reader in the Philosophy of Mathematics, a post he held until 1955. Hao Wang succeeded Waismann as Reader in the Philosophy of Mathematics, from 1956 to 1961, and Michael Dummett held this post in succession to Wang, from 1962 to 1974. I was appointed University Lecturer in the Philosophy of Mathematics in 1975. In this lecture I will pay tribute to my predecessors.