The meeting will run from friday evening 27/iii until sunday
afternoon 29/iii in Meeting Room 4 in the Centre for Mathematical
(Most) people from outside Cambridge will be accommodated in Churchill College. Please present yourself at the Porters' Lodge to be given a packet containing instructions and things like keys-to-your-room. Two UEA people (needing twin rooms) will be put up in significantly more spacious guest rooms in Clare Hall. Again, go to the Porters' Lodge.
The home team consists of Zachiri Mackenzie (who is cunning enough to not have a mobile), Vu Dang (who isn't: 07703735217) and tf (ditto: 07887701562)
Lecture One: The basic Fraissé theory, with examples. Definition of the age of a structure, amalgamation class and homogeneous structure, back-and-forth method. Fraissé's Theorem. As basic examples, linear orders, coloured linear orders, the random graph, the generic partial order.
Lecture Two: The above theory will be illustrated with regard to work currently in progress (with Jenkinson and Seidel, former students). This concerns the classification of the countable homogeneous multipartite graphs. The work follows some of the features of an earlier paper (Torrezão and Truss) where the countable homogenous coloured partial orders were classified. We start with the bipartite case, where the classification is known (Goldstern, Grossberg, Kojman), where there are 5 types, empty, complete, perfect matching and its complement, and generic. Then we consider the tripartite and quadripartite cases, and discuss how the overall classification is handled. There are some interesting and quite complicated features.
Lecture Three: Whereas in Lecture 2 we examined structures where the standard Fraissé theory applies, there have been a number of modifications using essentially the same ideas. The most famous methods are due to Hrushovski, but in this lecture I shall mention a construction of a certain digraph by David Evans, and extensions of this by myself and Daniela Amato. The key idea is once more that we are seeking to construct, or possibly describe, certain (countably) infinite structures, in this case digraphs, which have some reasonable amount of symmetry, and we do this by means of a series of approximations. In the case of homogeneous structures, the approximations are actually finite, but in the modifications they may only be finite in some weaker sense (finitely generated for instance). An important feature is that there must only be countably many structures which are allowed approximations, to ensure that the construction terminates in countably many steps.
A global problem in the history of logic is why progress between Aristotle and the nineteenth century was so painfully slow. Among various likely reasons, one is that several ideas we take for granted today were in conflict with basic and often unspoken principles of traditional logic. I trace this for three ideas.
Lecture One: Relational logic, which was in conflict with the principle of Top-Level Processing. Evidence: Ibn Sina 'Qiyas', Ockham, Leibniz, Frege 'Begriffsschrift'.
Lecture Two: Discharge of assumptions, which was in conflict with the principle of Local Formalising. Evidence: Ibn Sina 'Qiyas', Port-Royal Logic, Frege 'Grundlagen der Geometrie', Lukasiewicz.
Lecture Three: Type-theoretic semantics, which was in conflict with the Aristotle-Porphyry theory of ideas. Evidence: Ammonius, Ibn Sina 'Ibara', Wallis, Frege 'Grundlagen der Arithmetik' and 'Grundgesetze'.
I hope to be able to hand out translations of the relevant essays of
Ibn Sina) . Meanwhile here is the link to
the lecture notes and here is the link to
the relevant texts.