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
particularization and 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. Projections lose
dimensions and may collapse things that were originally different;
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 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 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 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.
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.
Call for papers
Topics of interest to the workshop include, but are not limited to:
* Ways to visualize logics or other algebraic structures
* (The many roles of) diagrams in Category Theory
* Categorical semantics (e.g. topos theory, linear logic, type theory)
* Translations between digrammatical languages and formal languages
Contributed talks should not exceed a duration of 30 minutes including
discussion. To submit a contribution, please send a one-page abstract
by October 5, 2017 to: eduardoochs@gmail.com