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% (find-angg "LATEX/2015children.tex") % (defun c () (interactive) (find-LATEXsh "lualatex -record 2015children.tex")) % (defun d () (interactive) (find-xpdfpage "~/LATEX/2015children.pdf")) % (defun e () (interactive) (find-LATEX "2015children.tex")) % (defun u () (interactive) (find-latex-upload-links "2015children")) % (find-xpdfpage "~/LATEX/2015children.pdf") % (find-sh0 "cp -v ~/LATEX/2015children.pdf /tmp/") % (find-sh0 "cp -v ~/LATEX/2015children.pdf /tmp/pen/") % file:///home/edrx/LATEX/2015children.pdf % file:///tmp/2015children.pdf % file:///tmp/pen/2015children.pdf % http://angg.twu.net/LATEX/2015children.pdf \documentclass[oneside]{book} \usepackage[colorlinks]{hyperref} % (find-es "tex" "hyperref") %\usepackage[latin1]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{pict2e} \usepackage{color} % (find-LATEX "edrx15.sty" "colors") \usepackage{colorweb} % (find-es "tex" "colorweb") %\usepackage{tikz} % % (find-dn6 "preamble6.lua" "preamble0") %\usepackage{proof} % For derivation trees ("%:" lines) %\input diagxy % For 2D diagrams ("%D" lines) %\xyoption{curve} % For the ".curve=" feature in 2D diagrams % \usepackage{edrx15} % (find-angg "LATEX/edrx15.sty") \input edrxaccents.tex % (find-angg "LATEX/edrxaccents.tex") \input edrxchars.tex % (find-LATEX "edrxchars.tex") \input edrxheadfoot.tex % (find-dn4ex "edrxheadfoot.tex") \input edrxgac2.tex % (find-LATEX "edrxgac2.tex") % \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} % Local Variables: % coding: utf-8-unix % ee-anchor-format: "Â«%sÂ»" % End: