Hans Halvorson

I argue against the claim -- advocated by Albert Einstein, Bernard Williams, and Ted Sider, among others -- that a description is objective only if it says how the world is in itself. Instead, I argue for the claim -- inspired by comments of Niels Bohr -- that a family of descriptions is objective only if they co-vary with their respective descriptive contexts. Moreover, I claim that "there is a shared objective reality" simply means that it is possible to satisfy this kind of covariance requirement.

Major figures of twentieth-century philosophy were enthralled by the revolution in formal logic, and many of their arguments are based on novel mathematical discoveries. Hilary Putnam claimed that the Löwenheim-Skølem theorem refutes the existence of an objective, observer-independent world; Bas van Fraassen claimed that arguments against empiricism in philosophy of science are ineffective against a semantic approach to scientific theories; W. V. O. Quine claimed that the distinction between analytic and synthetic truths is trivialized by the fact that any theory can be reduced to one in which all truths are analytic. This book dissects these and other arguments through in-depth investigation of the mathematical facts undergirding them. It presents a systematic, mathematically rigorous account of the key notions arising from such debates, including theory, equivalence, translation, reduction, and model. The result is a far-reaching reconceptualization of the role of formal methods in answering philosophical questions.

The Univalent Foundations of mathematics take the point of view that all of mathematics can be encoded in terms of spatial notions like "point" and "path". We will argue that this new point of view has important implications for philosophy, and especially for those parts of analytic philosophy that take set theory and first-order logic as their benchmark of rigor. To do so, we will explore the connection between foundations and philosophy, outline what is distinctive about the logic of the Univalent Foundations, and then describe new philosophical theses one can express in terms of this new logic.