"A New Perspective on Applying Mathematics"
"Russell's Influence on Carnap's Aufbau.
Synthese.
Abstract: This paper concerns the debate on the
nature of Rudolf Carnap's project in his 1928 book The Logical Structure of the World or
Aufbau. Michael Friedman and Alan Richardson have initiated much of this debate.
They claim that the Aufbau is best understood as a work that is firmly grounded in neo-Kantian philosophy.
They have made these claims in opposition to Quine and Goodman's "received view" of the
Aufbau. The received view sees the Aufbau as an attempt to carry out in
detail Russell's external world program. I argue that both sides of this debate have made errors
in their interpretation of Russell. These errors have led these interpreters to misunderstand
the connection between Russell's project and Carnap's project. Russell in fact exerted a crucial influence on Carnap
in the 1920s. This influence is complicated, however, due to the fact that Russell and
Carnap disagreed on many philosophical issues. I conclude that accounts of the Aufbau that
ignore Russell's influence are incomplete.
"A Structuralist Approach to Applying Mathematics."
PhD Dissertation, University of California, Berkeley.
Abstract:
"What is the relationship between mathematics and the physical world?
Arguments in natural science frequently employ mathematical concepts and theorems, and do so in order to draw conclusions about the physical world. However, this generates a philosophical problem since the subject matter of mathematics seems distinct from the physical world. For mathematics appears to be about objects such as numbers, functions and sets, and these seem to be not part of the physical world. How then can an appeal to such objects be of any use in the natural sciences? My dissertation seeks to give a precise formulation of this problem, and considers the major proposals for solutions, which I find to be insufficient. My own solution addresses the problem in terms of the structural similarities of the mathematical and physical worlds, and in the final chapters I provide a series of case studies that develop and support my 'structuralist' account of the application of mathematics in natural science."