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\begin{document}




\noindent {\bf Notes on a meaning for ``for children''}

\noindent {\bf Eduardo Ochs, 2015}


\bsk



\noindent ...this was taken from the slides of my minicourse in the UniLog 2015
(Istanbul):

\noindent \url{http://angg.twu.net/math-b.html#istanbul}

\bsk


\begin{itemize}
\item Why study Category Theory {\sl now}?

  Public education in Brazil is being dismantled --- maybe we should
  be doing better things than studying very technical \& inaccessible
  subjects with no research grants ---

  {\sl (Here I showed a photo called ``The New Girl From Ipanema'' ---
    a girl walking on the Ipanema beach at night with a gas mask, with
    a huge cloud of tear gas behind her)}

\item Category theory should be more accessible

  Most texts about CT are for specialists in research universities...
  {\sl Category theory should be more accessible.}.

  To whom?...

\begin{itemize}
\item Non-specialists (in research universities)
\item Grad students (in research universities)
\item Undergrads (in research universities)
\item Non-specialists (in conferences - where we have to be quick)
\item Undergrads (e.g., in CompSci - in teaching colleges) - (``Children'')
\end{itemize}

\item What do we mean by "accessible"?

\begin{itemize}
\item Done on very firm grounds: mathematical objects made from
  numbers, sets and tuples; FINITE, SMALL mathematical objects
  whenever possible. Avoid references to non-mathematical things like
  windows, cars and pizzas (like the object-orientation people do);
  avoid reference to Physics; avoid Quantum Mecanics at all costs;
  time is difficult to draw, prefer {\sl static} rather than {\sl
    changing}

\item People have very short attention spans nowadays

\item Self-contained, but not {\sl isolated} or {\sl isolating}; our
  material should make the literature more accessible

\item We learn better by doing. Our material should have lots of space
  for exercises.

\item Most people with whom I interact are not from Maths/CS/etc

\item {\sl Proving} general cases is relatively hard. {\sl Checking}
  and {\sl calculating} is much easier. People can believe that
  something can be generalized after seeing a convincing particular
  case. (Sometimes leave them to look for the right generalization by
  themselves)
\end{itemize}
\end{itemize}


% \msk
 
% Most books on advanced mathematics mention, in their introductions,
% how much ``mathematical maturity'' a reader needs to have in order
% to understand their contents... the term ``mathematical maturity''
% means, among other things, the ability to {\sl work on very abstract
% settings}, to {\sl generalize}, to {\sl particularize}, and to {\sl
% use infinite objects}, besides familiarity with the notation,
% methods, and main concepts in mathematics. A nice term for people
% with very little mathematical maturity is ``children''.
 
% I've tried to write this paper in a way as to makes it as accessible
% as possible to ``children'' like humanities students, philosophers
% with little mathematical background, and freshmen Computer Science
% students. Most of the sections were written after I presented the
% material corresponding to them in a {\sl very} basic introductory
% course on lambda-calculus and logic that I gave in the second half
% of 2016, whose audience was a group of six CompSci students; the
% exercises that they solved in class are not included here.
 
% I've had a handful of opportunities to present parts of this
% material to humanities students --- with them I had to start with
% exercises on expressions, quantifiers, evaluation, functions, sets,
% and lambda-notation, that are not included here.




% I've been using ``for children'' in titles for a while. This is a bit
% of a marketing strategy, of course, but the term ``children'' here has
% a precise, though unusual, meaning: it means ``people with very little
% mathematical maturity'', where I am taking these as the main aspects
% of ``mathematical maturity'': the ability to {\sl work on very
%   abstract settings}, to {\sl generalize}, to {\sl particularize}, and
% to {\sl use infinite objects}.
% 
% Writing things ``for children'' in this sense results in material that
% [is accessible] [exercises, not included here] [visual, easy to check]
% [who I've tested this with]
% 
% \msk
% 
% {\sl A note for ``adults''.} In [Ochs2013] I sketched a method for
% working in a general case and in a particular case (an ``archetypal
% case'') in parallel, and also a way to prove things in the archetypal
% case and then ``lift'' the proofs to the general case. This paper is
% an offspring of that one; I believe that planar Heyting Algebras
% presented here (ZHAs, sec.\ref{ZHAs}) are archetypal Heyting Algebras,
% and when we add ``closure operators'' to ZHAs (as in the seminar notes
% \url{http://angg.twu.net/math-b.html\#zhas-for-children}, pp.13--30;
% they are called ``J-operators'' there) we get something that is
% archetypal for studying toposes and sheaves; that will be the subject
% of a sequel of this paper.
% 
% [Topos theory books are too hard for me] [a bridge between
%   philosophers and toposophers]

% besides familiarity with the notation, methods, and main concepts in
% mathematics.






\end{document}

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