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{\bf Logics for Children - Unilog 2018, Vichy (proposal)}

\url{http://www.uni-log.org/wk6-logic-for-children.html}

\url{http://www.uni-log.org/workshops6.html?sessions=Workshops}

\medskip

When we explain a theorem to children --- in the strict sense of the
term --- we focus on concrete examples, and we avoid generalizations,
abstract structures and infinite objects.

When we present something to children'', in a wider sense of the
term that means people without mathematical maturity'', or even
people without expertise in a certain area'', we usually do
something similar: we start from a few motivating examples, and then
we generalize.

One of the aims of this workshop is to discuss techniques for {\it
particularization} and {\it generalization}. Particularization is
easy; substituing variables in a general statement is often enough to
do the job. Generalization is much harder, and one way to visualize
how it works is to regard particularization as a projection: a coil
projects a circle-like shadow on the ground, and we can ask for ways
to lift'' pieces of that circle to the coil continously. {\it
Projections} lose dimensions and may collapse things that were
originally different; {\it liftings} try to reconstruct the missing
information in a sensible way. There may be several different liftings
for a certain part of the circle, or none. Finding good
generalizations is somehow like finding good liftings.

The second of our aims is to discuss {\it diagrams}. For example, in
Category Theory statements, definitions and proofs can be often
expressed as diagrams, and if we start with a general diagram and
particularize it we get a second diagram with the same shape as the
first one, and that second diagram can be used as a version for
children'' of the general statement and proof. Diagrams were for a
long time considered second-class entities in CT literature ([2]
discusses some of the reasons), and were omitted; readers who think
very visually would feel that part of the work involved in
understanding CT papers and books would be to reconstruct the
missing'' diagrams from algebraic statements. Particular cases, even
when they were the motivation for the general definition, are also
treated as somewhat second-class --- and this inspires a possible
meaning for what can call Category Theory for Children'': to start
from the diagrams for particular cases, and then lift'' them to the
general case. Note that this can be done outside Category Theory too;
[1] is a good example.

Our third aim is to discuss {\it models}. A standard example is that
every topological space is a Heyting Algebra, and so a model for
Intuitionistic Predicate Logic, and this lets us explain {\sl
visually} some features of IPL. Something similar can be done for
some modal and paraconsistent logics; we believe that the figures for
that should be considered more important, and be more well-known.

\bigskip

References:

[1]: Jamnik, Mateja: Mathematical Reasoning with Diagrams: From
Intuition to Automation. CSLI, 2001.

[2]: Krömer, Ralf: Tool and Object: A History and Philosophy of
Category Theory. Birkhäuser, 2007.

\bigskip

{\bf Call for papers}

Topics of interest to the workshop include but are not limited to:

\begin{itemize}
\item Ways to visualize logics or other algebraic structure

\item (The many roles of) diagrams in Category Theory

\item Translations between digrammatical languages and formal languages

\end{itemize}

\bigskip

Organizers of the workshop:

Eduardo Ochs - Universidade Federal Fluminense (UFF), Rio das Ostras, RJ, Brazil

eduardoochs@gmail.com, \url{http://angg.twu.net/math-b.html}

\medskip

Fernando Lucatelli Nunes - Centro de Matemática da Universidade de Coimbra, Coimbra, Portugal

lucatellinunes@student.uc.pt, \url{http://fernandolucatelli.wordpress.com/}

\bigskip
\bigskip
\bigskip

My (E. Ochs) papers related to diagrams and things for children (note:
these papers motivated me to propose this workshop, but their links
won't appear in the page of the workshop; this is just for the keynote
speakers):

\medskip

Planar Heyting Algebras for Children (2017, submitted):

\url{http://angg.twu.net/math-b.html\#zhas-for-children-2}

\medskip

Internal Diagrams and Archetypal Reasoning in Category Theory

(2013, Logica Universalis):

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

\end{document}

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