Completeness, Categoricity, and Dismissal
from MCMP – Philosophy of Science
Michael Stöltzner (South Carolina) gives a talk at the MCMP Colloquium (18 July, 2013) titled "Completeness, Categoricity, and Dismissal". Abstract: In 1939, von Neumann panned Carnap “totally naıve, simplistic views on the issue of ‘completeness’ of the axiomatics of mathematics (‘categoricity’)” and expressed his surprise that philosophers were attracted by them. Taking up recent scholarship on Carnap’s work on axiomatic around 1930 and on von Neumann’s axiomatization of quantum physics, I argue that although von Neumann continued to cherish completeness and categoricity as regulative principles, he became increasingly aware how difficult they were to achieve in quantum logic. Moreover, in von Neumann’s (and Hilbert’s) use, the axiomatic method was never committed to the logical universalism of Carnap’s approach. This episode carries, to my mind, some lessons for the role of mathematics and logic in the context of the axiomatic method, especially if one considers the latter as the core vehicle of mathematical philosophy.
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