Switch to: References

Citations of:

Hilbert's program then and now

In Dale Jacquette (ed.), Philosophy of Logic. Amsterdam: North Holland. pp. 411–447 (2007)

Add citations

You must login to add citations.
  1. Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic.Corey Edward Mulvihill - 2015 - Dissertation, University of Waterloo
    Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Proof Theory.Jeremy Avigad - unknown
    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 set-theoretic language and infinitary methods that served to downplay or suppress computational content. This shift in emphasis away (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Takeuti's Well-Ordering Proof: Finitistically Fine?Eamon Darnell & Aaron Thomas-Bolduc - 2018 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics The CSHPM 2017 Annual Meeting in Toronto, Ontario. Birkhäuser Basel.
    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 well-ordering of ordinal notations in Cantor normal form. The paper begins with a historically informed discussion of (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Anti-Foundational Categorical Structuralism.McDonald Darren - unknown
    The aim of this dissertation is to outline and defend the view here dubbed “anti-foundational 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 (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • On Three Arguments Against Categorical Structuralism.Makmiller Pedroso - 2009 - Synthese 170 (1):21 - 31.
    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 set-theoretical foundations. Against this tendency, criticisms have been made that category theory depends on set-theoretical notions and, because of this, category theory fails to show that set-theoretical foundations are dispensable. The goal of this paper is to show that these (...)
    Download  
     
    Export citation  
     
    Bookmark   4 citations