Citations of:
Hilbert's program then and now
In Dale Jacquette (ed.), Philosophy of Logic. Amsterdam: North Holland. pp. 411–447 (2007)
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At the turn of the nineteenth century, mathematics exhibited a style of argumentation that was more explicitly computational than is common today. Over the course of the century, the introduction of abstract algebraic methods helped unify developments in analysis, number theory, geometry, and the theory of equations; and work by mathematicians like Dedekind, Cantor, and Hilbert towards the end of the century introduced settheoretic language and infinitary methods that served to downplay or suppress computational content. This shift in emphasis away (...) 

If it could be shown that one of Gentzen's consistency proofs for pure number theory could be shown to be finitistically acceptable, an important part of Hilbert's program would be vindicated. This paper focuses on whether the transfinite induction on ordinal notations needed for Gentzen's second proof can be finitistically justified. In particular, the focus is on Takeuti's purportedly finitistically acceptable proof of the wellordering of ordinal notations in Cantor normal form. The paper begins with a historically informed discussion of (...) 

The aim of this dissertation is to outline and defend the view here dubbed “antifoundational categorical structuralism”. The program put forth is intended to provide an answer the question “what is mathematics?”. The answer here on offer adopts the structuralist view of mathematics, in that mathematics is taken to be “the science of structure” expressed in the language of category theory, which is argued to accurately capture the notion of a “structural property”. In characterizing mathematical theorems as both conditional and (...) 

Some mathematicians and philosophers contend that set theory plays a foundational role in mathematics. However, the development of category theory during the second half of the twentieth century has encouraged the view that this theory can provide a structuralist alternative to settheoretical foundations. Against this tendency, criticisms have been made that category theory depends on settheoretical notions and, because of this, category theory fails to show that settheoretical foundations are dispensable. The goal of this paper is to show that these (...) 