`Bayesianism” is used in very different ways according to whether one wants to refer to a view of epistemology, statistics, confirmation theory, decision theory, etc. Modulo that ambiguity, `Objective Bayesianism” is used fairly uniformly to refer to Bayesianism with more-or-less objective (i.e., unique) probabilities.

As to how objective priors are to be achieved varies from author to author. Jaynes advocated the maximum entropy principle and Rosenkrantz followed his usage – I guess that is ”classic” objective Bayesianism. Salmon advocated frequency constraints but still called his view ”objective Bayesianism”. Reference-prior statisticians include an element of convention, but still call it ”objective Bayesianism”.

In my book I characterise objective Bayesianism in terms of stronger constraints on probabilities than are advocated by subjectivists – this fits most of the uses fairly well I think.

Greg and Branden have given you some excellent suggestions. Let me add a few of a different flavor. First, you might enjoy taking a look at something like The Principles of Mathematics by Russell, or perhaps some selections from the Benacerraf and Putnam collection. Second, you might also want to look at something like Steve Awodey’s book on category theory, which provides a very nice introduction to the structural aspects of mathematics, and it does so in a way that should be accessible to many philosophers.