This article concerns the ongoing (Scottish) neo-logicist program in the philosophy of mathematics. The enterprise began life, in something close to its present form, with Crispin Wright’s seminal [1983]. It was bolstered when Bob Hale [1987] joined the fray on Wright’s behalf and it continues through many extensions, objections, and replies to objections (see Hale and Wright, 2001). The overall plan is to develop branches of established mathematics using abstraction principles in the form:  

formula
where a and b are variables of a given type (typically first-order, ranging over individual objects, or second-order, ranging over concepts or properties), Σ is a higher-order operator denoting a function from items of the given type to objects in the range of the first-order variables, and E is an equivalence relation over items of the given type....

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