### Logic for Children - Workshop at UniLog 2018 (Vichy)

Official pages (at uni-log.org):
http://www.uni-log.org/start6.html
http://www.uni-log.org/wk6-CHI.html
http://www.uni-log.org/registration6.html
The workshop will happen during the "congress" (jun 21-26), not during the "school" (jun 16-20).

This page is uglier than the official one but we can update it more easily.

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.

2. A second description of what the workshop is about (used in a call for help):

Hi list,

we - me and Fernando Lucatelli - are trying to organize a workshop called "Logic for Children", that will happen in the UniLog 2018 in Vichy, France, in june 21-26...

The "children" in "logic for children" means "people without mathematical maturity", which in its turn means people who:

• have trouble with very abstract definitions,
• prefer starting from particular cases (and then generalize),
• handle diagrams better than algebraic notations,
• like to use diagrams and analogies (as in [BP2006]).

If we say that categorical definitions are "for adults" - because they may be very abstract - and that particular cases, diagrams, and analogies are "for children", then our intent with this workshop becomes easy to state. "Children" are willing to use "tools for children" to do mathematics, even if they will have to translate everything to a language "for adults" to make their results dependable and publishable, and even if the bridge between their tools "for children" and "for adults" is somewhat defective, i.e., if the translation only works on simple cases...

We are interested in that bridge between maths "for adults" and "for children" in several areas. Maths "for children" are hard to publish, even informally as notes (see this thread in the Categories mailing list), so often techniques are rediscovered over and over, but kept restricted to the "oral culture" of the area.

Our main intents with this workshop are:

• to discuss (over coffe breaks!) the techniques of the "bridge" that we currently use in seemingly ad-hoc ways,
• to systematize and "mechanize" these techniques to make them quicker to apply,
• to find ways to publish those techniques - in journals or elsewhere,
• to connect people in several areas working in related ideas, and to create repositories of online resources.

It was based on these slides - the video is here.
The UniLog youtube channel is here.

4. Presentations:

5. Keynote speakers:
• Ralf Krömer: Category theory and its foundations: the role of diagrams and other "intuitive" material.
Abstract and slides of his talk.
• Bob Coecke: Quantum Theory for Kids.
Abstract and slides of his talk.
• (Mateja Jamnik almost became a keynote, but Krömer and Coecke confirmed first.)

6. Other people / related work / resources:

7. Connections with λ-calculus, type theories and proof assistants:
• Gross, Chlipala and Spivak's Experience Implementing a Performant Category-Theory Library in Coq (PDF) - in page 4 they explain that in their (first) implementation a category C is a record with eight fields, like this: C = (COb, CHom, Co, C1, CAssoc, CLeftId, CRightId, CTruncated).
• In sections 12 and 19 of IDARCT (PDF) there's a sketch of an idea for implementing skeletons of constructions and proofs: in the "real world" a category is a 7-uple and in the "syntactical world" it is a 4-uple; a projection from the "real world" to the "syntactical world" drops the last 3 fields; when we have to handle both worlds at once the shorter structure is called a "protocategory", and we also have protofunctors, proto-NTs, proto-adjunctions, proto-fibrations, and so on; skeletons are these proto-things from the syntactical world; the easy direction of the "bridge" between the real and the syntactical world just drops some fields, and the hard direction infers or reconstructs them.
• I never worked out the details of proto-categories, proto-fibrations, proto-CCCs and so on beyond the level of detail of these seminar notes from 2010. =(
• We (Ochs/Lucatelli) are trying to get in touch with people who work with proof assistants - it would be great to have some of them in the workshop!

8. Our next step: getting more submissions

Note: there is still a tiny chance that the workshop won't happen "officially" and that people would have to present their work either on other workshops at the UniLog or at the main conference, and meet informally... officially each workshop needs at least 10 presenters that are neither organizers nor keynotes, so we (Ochs/Lucatelli) are going to try a trick that Jean-Yves Beziau suggested to us, which is to send individual e-mails to people that can be interested in participating (he said that sending "about 100 invitations" usually works)...

If you have people to suggest please send us their names, and, if possible, their e-mails... we work mostly on Category Theory, and we think that this workshop could of interest to people working on, e.g., Education, Diagrammatic reasoning, Visualization of algorithms - and we don't know many people in these areas yet...

9. If you would like to help us, here are some trivial ways:

• you can send us pointers to related work,
• you can send us pointers to people that we should get in touch with,

And here are some less trivial ways:

• you can send us folklore ideas (without pointers),
• you can attend the workshop,
• you can submit something to the workshop. =)

10. On funding:

Some people have asked me if I can obtain some kind of funding to support their trip to the UniLog. I've tried to ask around, but I didn't even get any useful hints... The conference has a page listing its sponsors, and that's all I know at this moment. Hints welcome!!! =\

11. My contacts:
`eduardoochs@gmail.com`