What is Lie Theory?

Chris Hillman posted an extensive introduction to Lie groups and Lie algebras to sci.physics.relativity back in September 2000. There's also a snapshot of this post and a couple of sequels available via the Wayback Machine.

Research Proposal

I intend to examine and develop the approach to probability proposed by Edwin Jaynes, as described in his extensive published work and particularly in his book "Probability Theory: The Logic of Science", and to demonstrate the natural resolution this approach provides to a number of current problems in the philosophy of probability.

Jaynes' view of probability is a variant of Bayesianism, one in which all probability ascriptions are conditional probabilities, the rules for manipulating probabilities are derived from elementary desiderata (via Cox's Theorem), and the assignment of prior probabilities is the encoding of available evidence (or assumptions, or premises) in numerical form. Thus the "problem of priors" is viewed as a technical problem, to be solved by methods such as Jaynes' "Maximum Entropy" algorithm.

Jaynes worked as a physicist rather than a philosopher, and so was interested in demonstrating that his methods were internally consistent, were powerful enough to solve practical problems, and avoided the conceptual confusions of rival methods (such as frequentist statistics). My first aim is therefore to fill in the gaps left by Jaynes in order to provide a philosophically coherent package, by giving a careful and detailed account of the underlying assumptions and interpretation of probability implicit in his view.

After examining the philosophical requirements and implications associated with a Jaynesian view of probability, I will compare it with a number of alternative accounts in the literature, in particular focusing on the puzzles, problems and apparent paradoxes that are frequently discussed. The advantage of the Jaynesian approach will be demonstrated by its ability to solve (and in some cases simply avoid) many of the common difficulties, while naturally resolving some long-standing debates in an intuitively sensible manner.