Midlands Logic Seminar

Week 5: 1 November

Study group: Richard Kaye (Birmingham)
"M-Logic"

M-logic is the associated logic (with an infinitary "M-rule") to the notion(s) of truth for a model M of PA, as discussed last week. Last week we sketched Lachlan's argument that for there to be a truth definition M needs to be recursively saturated. The Krajewski-Kotlarski-Lachlan theorem is a completeness theorem for M-logic: if M is countable and M-logic is consistent then there is a truth definition. (It will turn out that (for countable M) M-logic is consistent if and only if M is recursively saturated.)

Invited speaker: Aaron Sloman (Birmingham)
"From Molecules to Mathematicians: How could evolution produce mathematicians from a cloud of cosmic dust?"

I shall raise some questions about evolution inspired by considering what might have happened if Alan Turing had lived longer and thought of combining his early ideas about discrete Turing machines with his later ideas about chemical morphogenesis involving mixtures of discrete and continuous changes. I'll offer some arguments and speculations about how organisms currently on this planet could have evolved, starting from lifeless chemicals, and identify a long term research programme for filling in the details: the Meta-Morphogenesis project, inspired partly by reading Turing's 1952 paper on chemical morphogenesis. At all stages there are mathematical structures involved in the evolutionary progress and evolution blindly discovers and uses them. In later stages of evolution, individual organisms develop abilities to discover and use mathematical structures and processes, though without realising what they were doing. Later still, humans began to think explicitly about these processes and to discuss their properties. I suggest that sort of history may have led to the collaborative production of Euclid's Elements. At present there's very little we know about the actual history of evolutionary developments. I'll try to show how the meta-morphogenesis project sets out a strategy for trying to fill the gaps, with the hope of answering not only evolutionary questions, but also philosophical questions about the nature of mathematics and how mathematical reasoning and knowledge differs from other kinds. Perhaps that will also help us come up with robots and AI systems that are far more intelligent than the current models are. It may also help us produce much better mathematical education systems based partly on deeper insights into the nature of mathematics and into the nature of biological processes of learning and discovery.

Background to the seminar:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/mathsem.html
including links to various aspects of the Meta-Morphogenesis project.