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\begin{document}

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\section*{Brouwer-Heyting-Kolmogorov interpretation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{constructivism_realizability_computability}{}\paragraph*{{Constructivism, Realizability, Computability}}\label{constructivism_realizability_computability}

[[!include constructivism - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Brouwer-Heyting-Kolmogorov} interpretation of [[intuitionistic logic]] is a description of [[proofs]] of [[propositions]] in [[intuitionistic logic]] as [[functions]], often [[computable functions]], where it is also called the \emph{[[realizability]] interpretation}.

This is otherwise known as the paradigm of \emph{[[propositions as types]]} and \emph{[[proofs as programs]]}, and in a precise form as the [[Curry-Howard correspondence]]. See there for more.

The name ``Brouwer-Heyting-Kolmogorov'' is due to Troelstra, and it is a matter of some dispute whether Brouwer's name should be included. Brouwer never explicitly formulated any interpretation of this sort, and remained against all formalism his entire life. Moreover, \hyperlink{EscardoXu}{Escardo-Xu} have shown that Brouwer's famous intuitionistic theorem ``all functions $\mathbb{N}^{\mathbb{N}} \to \mathbb{N}$ are continuous'' is actually inconsistent under a literal version of this interpretation (i.e. without including [[propositional truncation]]). Thus, perhaps it should only be called the ``Heyting-Kolmogorov'' interpretation.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[realizability topos]]


\item [[computational trinitarianism]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/BHK_interpretation}{BHK interpretation}}


\item L. E. J. Brouwer, \emph{Points and Spaces} , CJM \textbf{6} (1954) pp.1-17. (\href{http://cms.math.ca/openaccess/cjm/v6/cjm1954v06.0001-0017.pdf}{pdf})


\item H. Freudenthal , \emph{Zur intuitionistischen Deutung logischer Formeln} , Comp. Math. \textbf{4} (1937) pp.112-116. (\href{http://archive.numdam.org/article/CM_1937__4__112_0.pdf}{pdf})


\item A. Heyting , \emph{Die intuitionistische Grundlegung der Mathematik} , Erkenntnis \textbf{2} (1931) pp.106-115.


\item A. Heyting , \emph{Bemerkungen zu dem Aufsatz von Herrn Freudenthal ``Zur intuitionistischen Deutung logischer Formeln''} , Comp. Math. \textbf{4} (1937) pp.117-118. (\href{http://archive.numdam.org/article/CM_1937__4__117_0.pdf}{pdf})


\item A. Kolmogoroff, \emph{Zur Deutung der intuitionistischen Logik} , Math. Z. \textbf{35} (1932) pp.58-65. (\href{http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002373467}{gdz})


\item G. Kreisel, \emph{Mathematical Logic} , pp.95-195 in Saaty (ed.), \emph{Lectures on Modern Mathematics III} , Wiley New York 1965.


\item E. G. F. D\'i{}ez, \emph{Five observations concerning the intended meaning of the intuitionistic logical constants} , J. Phil. Logic \textbf{29} no. 4 (2000) pp.409--424 . (\href{https://webs.um.es/picazo/miwiki/lib/exe/fetch.php?id=inicio&cache=cache&media=five_observations_2000_jour_of_phil_logic_29_4_pp409_424.pdf}{preprint})


\item [[Jean-Yves Girard]] et al., \emph{Proofs and Types} , CUP 1989.


\item [[Anne Sjerp Troelstra]], \emph{Principles of Intuitionism} , Springer Heidelberg 1969. (\S{}2)


\item [[Anne Sjerp Troelstra]], \emph{Aspects of Constructive Mathematics} , pp.973-1052 in Barwise (ed.), \emph{Handbook of Mathematical Logic} , Elsevier Amsterdam 1977.


\item [[Anne Sjerp Troelstra]], \emph{History of Constructivism in the Twentieth Century} (1991). (\href{https://www.illc.uva.nl/Research/Publications/Reports/ML-1991-05.text.pdf}{preprint})



\end{itemize}
\begin{itemize}%
\item Wouter Pieter Stekelenburg, \emph{Realizability Categories} , (\href{http://arxiv.org/abs/1301.2134}{arXiv:1301.2134}).


\item [[Martin Escardo]] and Chuangjie Xu, \emph{The inconsistency of a Brouwerian continuity principle with the Curry--Howard interpretation} . (\href{http://www.cs.bham.ac.uk/%7Emhe/papers/escardo-xu-inconsistency-continuity.pdf}{pdf})



\end{itemize}
Links to many papers on realizability and related topics may be found \href{http://www.staff.science.uu.nl/~ooste110/realizability.html}{here}.

For a comment see also

\begin{itemize}%
\item [[Robert Harper]], \emph{Extensionality, Intensionality, and Brouwer's Dictum} (\href{http://existentialtype.wordpress.com/2012/08/11/extensionality-intensionality-and-brouwers-dictum/}{web})

\end{itemize}
[[!redirects Brouwer Heyting Kolmogorov interpretation]] [[!redirects Brouwer-Heyting-Kolmogorov interpretation]] [[!redirects Brouwer–Heyting–Kolmogorov interpretation]] [[!redirects BHK interpretation]]

[[!redirects realizability interpretation]] [[!redirects realisability interpretation]]

[[!redirects BHK correspondence]]



\end{document}