The Limits of Abstraction

The untold history of the multiracial making of the border between Canada and the United States.Often described as the longest undefended border in the world, the Canada-United States border was born in blood, conflict, and uncertainty. At the end of the American Revolution, Britain and the United States imagined a future for each of their nations that stretched across a continent. They signed treaties with one another dividing lands neither country could map, much less control. A century and a half later, they had largely fulfilled those earlier ambitions. Both countries had builtnations that stretched from the Atlantic to the Pacific and had created an expansive international border that restricted movement.The vision that seemed so clear in the minds of diplomats and politicians was never so well-defined on the ground. As A Line of Blood and Dirt argues, both countries built their border across Indigenous lands using hunger, violence, and coercion to displace existing communities and to disrupt their ideas of territory and belonging. Drawing on oral histories, map visualizations, and archival sources, Benjamin Hoy reveals the role Indigenous people played in the development of theinternational boundary, as well as the impact the border had on Indigenous people, European settlers, Chinese migrants, and African Americans. Unable to prevent movement at the border's physical location for over a century, Canada and the United States instead found ways to project fear across international lines.Bringing together the histories of tribes, immigration, economics, and the relationship of neighboring nations, A Line of Blood and Dirt offers a new history of Indigenous peoples and the borderland.What is abstraction? To what extent can it account for the existence and identity of abstract objects? And to what extent can it be used as a foundation for mathematics? Kit Fine provides rigorous and systematic answers to these questions along the lines proposed by Frege, in a book concerned both with the technical development of the subject and with its philosophical underpinnings.Fine proposes an account of what it is for a principle of abstraction to be acceptable, and these acceptable principles are exactly characterized. A formal theory of abstraction is developed and shown to be capable of providing a foundation for both arithmetic and analysis. Fine argues that the usual attempts to see principles of abstraction as forms of stipulative definition have been largely unsuccessful but there may be other, more promising, ways of vindicating the various forms ofcontextual definition. The Limits of Abstraction breaks new ground both technically and philosophically, and will be essential reading for all who work on the philosophy of mathematics.