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\section*{philosophy}

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

\hypertarget{philosophy}{}\paragraph*{{Philosophy}}\label{philosophy}

[[!include philosophy - contents]]

\hypertarget{philosophy_2}{}\section*{{Philosophy}}\label{philosophy_2}

\noindent\hyperlink{scope}{Scope}\dotfill \pageref*{scope} \linebreak
\noindent\hyperlink{nonscope}{Nonscope}\dotfill \pageref*{nonscope} \linebreak
\noindent\hyperlink{idea_of_relevance_of_higher_categories}{Idea of relevance of higher categories}\dotfill \pageref*{idea_of_relevance_of_higher_categories} \linebreak
\noindent\hyperlink{metaphysics}{Metaphysics}\dotfill \pageref*{metaphysics} \linebreak
\noindent\hyperlink{enquiry}{Enquiry}\dotfill \pageref*{enquiry} \linebreak
\noindent\hyperlink{philosophical_sentiments}{Philosophical sentiments}\dotfill \pageref*{philosophical_sentiments} \linebreak
\noindent\hyperlink{references_and_links}{References and links}\dotfill \pageref*{references_and_links} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{books}{Books}\dotfill \pageref*{books} \linebreak
\noindent\hyperlink{articles}{Articles}\dotfill \pageref*{articles} \linebreak
\noindent\hyperlink{blog_and_forum_discussions_conferences}{Blog and forum discussions, conferences}\dotfill \pageref*{blog_and_forum_discussions_conferences} \linebreak


\hypertarget{scope}{}\subsection*{{Scope}}\label{scope}

In the philosophical part of $n$Lab we discuss higher category theory and its repercussions in philosophy. More widely, the future entries on philosophy in nLab should also contain [[philosophy of mathematics]] in general, and of [[logic and foundations]] in particular. As it is usual for philosophy and the study of thought, it is usefully carried on via study of historical thinkers and their ideas, hence some idea-related aspects of the [[history of mathematics]] are welcome.

\hypertarget{nonscope}{}\subsection*{{Nonscope}}\label{nonscope}

There are many articles which are not philosophy, but rather essays on general mathematics, and so on, often opinion pieces on what is important and so on. That is not philosophy \emph{per se}, but it may be relevant thoughts and we could link them rather at related pages, like [[opinions on development of mathematics]].

\hypertarget{idea_of_relevance_of_higher_categories}{}\subsection*{{Idea of relevance of higher categories}}\label{idea_of_relevance_of_higher_categories}

Philosophical interest in [[n-category|n-categories]] may be characterised as belonging to one of two kinds.

\begin{itemize}%
\item \textbf{[[metaphysics|Metaphysical]]}: The formation of a new language which may prove to be as important for philosophy as predicate logic was for [[Bertrand Russell]] and the [[analytic philosophy|analytic philosophers]] he inspired.


\item \textbf{Illustrative of mathematics as intellectual enquiry}: Such a reconstitution of the fundamental language of [[mathematics]] reveals much about mathematics as a tradition of enquiry stretching back several millennia, for instance, the continued willingness to reconsider basic concepts.



\end{itemize}
\hypertarget{metaphysics}{}\subsection*{{Metaphysics}}\label{metaphysics}

[[metaphysics]]

\begin{itemize}%
\item [[higher category theory|Higher category theory]] provides a new foundation for mathematics - [[foundations|logical]] and [[foundations and philosophy|philosophical]].


\item [[higher category theory|Higher category theory]] refines the notion of sameness to allow more subtle variants. It advocates the avoidance of [[evil]].


\item There ought to be a [[vertical categorification|categorified]] [[internal logic|logic]], or 2-logic. There are some suggestions that existing work on modal logic is relevant. Blog discussion: \href{http://golem.ph.utexas.edu/category/2007/11/concrete_groups_and_axiomatic_1.html}{I}, \href{https://golem.ph.utexas.edu/category/2008/01/2toposes.html}{II}, \href{http://golem.ph.utexas.edu/category/2008/02/2galois_and_2logic.html}{III}, \href{http://golem.ph.utexas.edu/category/2008/03/worrying_about_2logic.html}{IV}, \href{http://golem.ph.utexas.edu/category/2008/11/the_internal_language_of_a_2to.html}{V}. Mike Shulman's project: \href{http://ncatlab.org/michaelshulman/show/2-categorical+logic}{2-categorical logic}.


\item [[Homotopy type theory]] may be thought of as a vertical categorification of logic to $(\infinity,1)$.


\item [[higher category theory|Higher category theory]] may provide the right tools to take [[physics]] forward. \href{http://math.ucr.edu/home/baez/history.pdf}{A Prehistory of n-Categorical Physics} See also [[physics]].


\item [[higher category theory and physics]], [[geometry of physics]]


\item More speculatively, [[category theory]] may prove useful in [[biology]].



\end{itemize}
\hypertarget{enquiry}{}\subsection*{{Enquiry}}\label{enquiry}

\begin{quote}%
``Mathematical wisdom, if not forgotten, lives as an invariant of all its (re)presentations in a permanently self--renewing discourse.'' (Yuri Manin)


\end{quote}
To [[vertical categorification|categorify]] mathematical constructions properly, one must have understood their essential features. This leads us to consider what it is to get concepts `right'. Which kind of `realism' is suitable for mathematics? Which virtues should a mathematical community possess to further its ends: a knowledge of its history, close attention to instruction and the sharing of knowledge, a willingness to admit to what is currently lacking in its programmes?

\hypertarget{philosophical_sentiments}{}\subsection*{{Philosophical sentiments}}\label{philosophical_sentiments}

\begin{itemize}%
\item [[Platonism]]


\item [[idealism]]

\begin{itemize}%
\item [[objective idealism]]


\item [[subjective idealism]]



\end{itemize}

\item [[positivism]]


\item [[universal exceptionalism]]



\end{itemize}
\hypertarget{references_and_links}{}\subsection*{{References and links}}\label{references_and_links}

\hypertarget{related_entries}{}\subsubsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[foundations and logic]], [[logic]], [[reasoning]]
\item [[category (philosophy)]], [[modality]]
\item [[logos (in philosophy)]], [[Nous]]
\item [[history of mathematics]], [[structuralism]], [[opinions on development of mathematics]]
\item [[metaphysics]]
\item [[hermeticism]], [[mysticism]], [[gnosticism]], [[speculative philosophy]]
\item \emph{[[Science of Logic]]}
\item [[analytic philosophy]]
\item [[philosophy of mathematics]], [[predicative mathematics]]
\item [[philosophy of science]], [[philosophy of physics]]
\item [[Judea Pearl]]

\end{itemize}
\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item wikipedia: \href{http://en.wikipedia.org/wiki/Philosophy_of_mathematics}{philosophy of mathematics}; wikipedia.ru \href{http://ru.wikipedia.org/wiki/%D0%A4%D0%B8%D0%BB%D0%BE%D1%81%D0%BE%D1%84%D0%B8%D1%8F_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B8}{ }
\item \href{http://www.iep.utm.edu}{Internet Encyclopedia of Philosophy}, ``A peer reviewed academic resource''
\item Stanford Encyclopaedia of Philosophy \href{http://plato.stanford.edu}{online}, \href{http://plato.stanford.edu/contents.html}{contents}
\item Stewart Shapiro, \emph{[[The Oxford handbook of philosophy of mathematics and logic]]}, Oxford University Press 2005
\item \href{http://www.openculture.com/free-philosophy-ebooks}{free-philosophy-ebooks} at openculture
\item \href{http://lesswrong.com}{Less Wrong Wiki} ``a community blog devoted to refining the art of human rationality''; including Eliezer Yudkowsky's \href{https://wiki.lesswrong.com/wiki/Highly_Advanced_Epistemology_101_for_Beginners}{Epistemology 101}

\end{itemize}
\hypertarget{books}{}\subsubsection*{{Books}}\label{books}

\begin{itemize}%
\item [[Hegel]], \emph{Wissenschaft der Logik} ( \emph{[[Science of Logic]]} )

\begin{itemize}%
\item [[unity of opposites]]\begin{itemize}%
\item [[becoming]] $\colon$ [[nothing]] $\dashv$ [[being]]

\end{itemize}


\end{itemize}

\item [[Hegel]], \emph{[[Lectures on the History of Philosophy]]}


\item [[Ludwig Wittgenstein]], \emph{[[Tractatus Logico-Philosophicus]]}


\item [[Bertrand Russell]], \emph{[[A History of Western Philosophy]]}


\item [[Albert Lautman]], \emph{Mathematics, ideas and the physical real}, 2011 translation by Simon B. Duffy; English edition of \emph{Les Math\'e{}matiques, les id\'e{}es et le r\'e{}el physique}, Librairie Philosophique, J. VRIN, 2006


\item Michael D. Potter, \emph{Set theory and its philosophy: a critical introduction}, Oxford Univ. Press 2004


\item [[Fernando Zalamea]], \emph{Filosof\'i{}a sint\'e{}tica de las matem\'a{}ticas contempor\'a{}neas}, (Spanish) Obra Selecta. Editorial Universidad Nacional de Colombia, Bogot\'a{}, 2009. 231 pp. \href{http://www.ams.org/mathscinet-getitem?mr=2599170}{MR2599170}, ISBN: 978-958-719-206-3, \href{&lt;http://files.acervopeirceano.webnode.es/200000065-18c1b19bb9/Zalamea-Fil-Sint-Mat-Cont.pdf>}{pdf}. Transl. into English by Zachary Luke Fraser: \emph{Synthetic philosophy of contemporary mathematics}, Sep. 2011. \href{http://www.urbanomic.com/pub_syntheticmath.php}{bookpage}. Some excerpts \href{http://ifile.it/2c3qgz5}{here}.


\item [[David Corfield]], \emph{Towards a philosophy of real mathematics}, Cambridge University Press, 2003, \href{http://books.google.hr/books?id=s37rhfWs73QC&lpg=PP1&ots=Zobx2bIWFh&dq=david%20corfield%20towards%20a%20philosophy%20of%20real%20mathematics&hl=en&pg=PP1#v=onepage&q&f=false}{gBooks}


\item [[Saunders MacLane]], \emph{Mathematics, form and function}, Springer-Verlag 1986, xi+476 pp. \href{http://www.ams.org/mathscinet-getitem?mr=816347}{MR87g:00041}, \href{http://en.wikipedia.org/wiki/Mathematics,_Form_and_Function}{wikipedia}


\item George Lakoff, Rafael E. N\'u{}\~n{}ez, \emph{Where mathematics comes from}, Basic Books 2000, xviii+493 pp. \href{http://www.ams.org/mathscinet-getitem?mr=1794854}{MR2001i:00013}


\item [[Yuri Manin|Yuri I. Manin]], \emph{Mathematics as Metaphor: Selected Essays of Yuri Manin}, Amer. Math. Soc. 2007


\item [[Ralf Krömer]], \emph{Tool and object: A history and philosophy of category theory}, Birkh\"a{}user 2007


\item [[Jean-Pierre Marquis]], \emph{From a geometrical point of view: a study of the history and philosophy of category theory}, Springer, 2008


\item Ian Hacking, \emph{Why is there philosophy of mathematics at all?}, Cambridge University Press 2014


\item William Bragg Ewald, \emph{From Kant to Hilbert, From Kant to Hilbert: Readings in the Foundations of Mathematics}, 2 vols. (original readings in English translation)


\item [[Roland Omnès]], \emph{[[Converging Realities -- Toward a common philosophy of physics and mathematics]]}, Princeton University Press, 2005



\end{itemize}
\hypertarget{articles}{}\subsubsection*{{Articles}}\label{articles}

\begin{itemize}%
\item Glenn G. Parsons, James Robert Brown, \emph{Platonism, metaphor, and mathematics}, Dialogue \textbf{43} (2004), no. 1, 47--66, \href{http://www.ams.org/mathscinet-getitem?mr=2053877}{MR2004k:00004}
\item John Baldwin, \emph{Model theoretic perspectives on the philosophy of mathematics}, \href{http://homepages.math.uic.edu/~jbaldwin/pub/amstersh.pdf}{pdf}
\item [[Yuri Manin|Yu. I. Manin]], \emph{Mathematical knowledge: internal, social and cultural aspects}, \href{http://arxiv.org/abs/math.HO/0703427}{arXiv:math.HO/0703427}; \emph{Georg Cantor and his heritage}, \href{http://arxiv.org/abs/math.AG/0209244}{arxiv/math.AG/0209244}; \emph{Truth as value and duty: lessons of mathematics}, \href{http://arxiv.org/abs/0805.4057}{arxiv/0805.4057}
\item M. G. Katz, E. Leichtnam, \emph{Commuting and noncommuting infinitesimals}, Amer. Math. Monthly 120 (2013), no. 7, 631-641 \href{http://arxiv.org/abs/1304.0583}{arxiv/1304.0583}
\item M. G. Katz, Thomas Mormann, \emph{Infinitesimals as an issue in neo-Kantian philosophy of science}, \href{http://arxiv.org/abs/1304.1027}{arxiv/1304.1027}
\item [[Mikhail Gromov]], \emph{Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2, 3.} (2013) \href{http://www.ihes.fr/~gromov/PDF/ergologic3.1.pdf}{pdf}; \emph{Structures, Learning and Ergosystems: Chapters 1-4, 6} (2011) \href{http://www.ihes.fr/~gromov/PDF/ergobrain.pdf}{pdf}
\item [[William Lawvere]], \emph{[[Cohesive Toposes and Cantor's ``lauter Einsen'']]}
\item Jeremy Avigad, \emph{Mathematics and language}, \href{http://arxiv.org/abs/1505.07238}{arxiv/1505.07238}

\end{itemize}
Some philosophical aspects of the role of category theory are touched upon in some parts of the introductory paper

\begin{itemize}%
\item [[Jean-Pierre Marquis]], \emph{What is category theory?}, pdf can be found at \href{https://www.academia.edu/16339607/What_is_category_theory}{academia.edu}

\end{itemize}
\hypertarget{blog_and_forum_discussions_conferences}{}\subsubsection*{{Blog and forum discussions, conferences}}\label{blog_and_forum_discussions_conferences}

\begin{itemize}%
\item $n$Caf\'e{} on Mathematical reality: \href{http://golem.ph.utexas.edu/category/2008/10/mathematical_reality.html}{I}, \href{http://golem.ph.utexas.edu/category/2008/11/mathematical_robustness.html}{II}
\item [[GavinWraith|Gavin]] wrote about ``The dangers of category theory'', see $n$Forum \href{http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2472&Focus=21255#Comment_21255}{here}
\item Simplicityonference: Ideals of Practice in Mathematics \& the Arts; videos online at youtube \href{http://s-i-m-p-l-i-c-i-t-y.org/videos.html}{channel}

\end{itemize}
category: philosophy

[[!redirects philosophy]] [[!redirects Philosophy]]



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