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




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.



[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.


{\bf Call for papers}

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

\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



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}


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

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


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


Planar Heyting Algebras for Children (2017, submitted):



Internal Diagrams and Archetypal Reasoning in Category Theory

(2013, Logica Universalis):



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