I thought it might be useful to post more references and reorganise
into more focused questions. I appologise for the more playful first
post. It's sometimes hard for me to put aside creativity for more
formality.

There are Heyting algebras in a lot of fields. They are one of the
more useful logical forms for our models in these fields.

In the study of cognitive research we have models of neural
interaction, and many experiments match well to the models. There is
a school of thought that says concepts are attractors in the neural
models dynamics. This allows us to pose dynamical propositions in the
neural models. The propositional structure would then likely be
Heyting in such a model.

"Neural organisation: structure, function, and dynamics" by Arbib,
Erdi, and Szentogathai
"The algebraic structure of sets of regions" by Stell and Worboys
"A canonical model of the region-connection calculus" by Jochen Renz
"A relation-algebraic approach to the region-connection calculus" by
Duntsch, Wang, and McCloskey

Interestingly, there are also very similar things found in the logical
study of the way we reason about the world.

"How logic emerges from the dynamics of information" by Peter
Gardenfors
"Bimodal logics for reasoning about continuous dynamics" by Davoren
and Gore
"Categorical and Kripke semantics for constructive S4 modal logic" by
Alechina, Mendler, de Paiva, and Ritter
"The algebra of topology" by Tarski and McKinsey
"Sentential calculus and topology" by Tarski

And this fits in with a particular history of research in the logics

http://plato.stanford.edu/entries/logic-modal/
http://plato.stanford.edu/entries/model-theory/
http://en.wikipedia.org/wiki/Formal_language
http://en.wikipedia.org/wiki/Algorithmic_information_theory

which builds the context for the way we reason about the world in
terms that become computable. Its at the point where the models are
able to process the information in the manner of a state machine with
a discrete, directed collection of states (often represented by time)
that languages are formalised. The models become dynamic.

This as well is interesting since the well-known Curry-Howard
isomorphism describes a natural relationship between lambda calculi
and Heyting algebras.

"Lectures on the Curry-Howard isomorphism" by Morten Heine B. Sorensen
and Pawel Urzyczyn

Language and the dynamics of conversations can be modeled by these
automatons.

"Representing communicative intentions in collaborative conversational
agents" by Matthew Stone
"Speech and language processing" by Jurafsky and Martin
"Denotational semantics for agent communication languages" by Frank
Guerin and Jeremy Pitt
"From continuous dynamics to symbols" by Herbert Jaeger
"Attractors in the development of communication" by E. D. de Jong
"Autonomous formation of concepts and communication" by E. de Jong

And when you look at the computational ability of our models of brain
activity

"On the computational power of neural nets" by Siegelmann and Sontag

we see an intimate connection forming. This is not meant to affirm a
logical linguistic programme, only to show that there are Heyting
algebras associated to the fields around such a programme from several
different directions, and the theory of computation in particular has
strong foundations in structures that are associated with these
algebras. There are other logics also associated with
turing-completeness

"The Turing-completeness of multimodal categorial grammars" by Bob
Carpenter

which shows that the logical programme of linguistics can arive to
equivalence to lambda calculi from different directions. There are
many interesting relationships between Heyting algebras and other
logics known, such as

"Combining possibilities and negations" by Greg Restall
"A semantical study of orthologics" by Miyazaki Yutaka
"Algebras and frames for modal logics"

And so on...

Now, Heyting algebras are all over the place in mathematics. Besides
the above (which also mentions the classic result of topologies
defining Heyting algebras), there is the general theory of topoi.

"Topoi: the categorial analysis of logic" by Robert Goldblatt
http://en.wikipedia.org/wiki/Topos
http://math.ucr.edu/home/baez/topos.html

which shows that the study of mappings between mathematical objects
often occurs in categories that are topoi. All topoi have a natural
Heyting algebra structure.

Many models of our world contain reasoning inside topoi, on objects
that are members topoi and the transformations between them. In
particular, we have evolutionary theories on graphs and digraphs
(which include trees and directed trees)

"A guided tour in the topos of graphs" by Sebastiano Vigna

where we find both topological logics and some proposed connections
with fuzzy logics, both in modeling and the inverse problem

"A general study on genetic fuzzy systems" by Oscar Cordon
"A formulation of fuzzy automata and its application as a model of
learning systems" by Wee and Fu

and we have again Heyting algebras found for the logic of fuzzy sets.

"Lattice of fuzzy sets" by Takashi Mitsuishi and Grzegorz Bancerek
"A natural interpretation of fuzzy sets and fuzzy relations" by Mamoru
Shimoda

So there's seems to be something about abstraction, computational,
evolutionary logics that tend to draw Heyting models. Something
connectional and topological.

And still, we have the wonderful Fotini Markopoulou-Kalamara and her
programme of causal sets and their Heyting structure.

"The internal description of a causal set: What the universe look like
from the inside" by Fotini Markopoulou
"An insider's guide to quantum causal histories" by Fotini Markopoulou
"Dynamics of causal sets" by David Rideout
"Causal sets and frame-valued set theory" by John L. Bell
"Quantum spacetime as a qunatum causal set" by Ioannis Raptis
"Topos theory and consistent histories: the internal logic of the set
of all consistent sets" by C. J. Isham

Which ties itself up nicely with, to me, one of the neatest Heyting
algebra of all, that of the operator projection calculus on a quantum
Hilbert space.

"Operational quantum logic: an overview" by Coecke, Moor, and Wilce
"Quantum logic in intuitionistic perception" by Bob Coecke
"Constructive mathematics and quantum physics" by Douglas Bridges and
Karl Svozil

and of course we now have ties with quantum computing.

"Locality, weak or strong anticipation and quantum computing II:
constructivism with category theory" by Heather and Rossiter

So those are the models I see when I think about Heyting algebras. It
seems to me to have something intimate to do with process. And these
are all quite interesting models, about learning and our symbol
systems, about calculation, evolution, the quantum.

And these particular algebras mentioned are the non-Boolean kind.

Some mathematicians may also be familiar with Goedel algebras (also
Heyting).

http://math.chapman.edu/cgi-bin/structures.pl?Goedel_algebras

So, for the mathematicians, I think the thing that ties it all
together is the theory of realisability. Its a well established field
that, in many ways, works at describing these various relationships
mentioned.

"Realizability: an historical essay" by Jaap van Oosten

Thats pretty much why I wanted to discuss Heyting algebras. Their
beautiful things, and I thought, from my exposure to its ubiquity in
all of these fields, maybe there might be others who had seen at least
a piece of it in their own fields. I thought maybe all the of all the
people out their from the various directions, maybe one or two from
each field would be hanging around the newsgroups. Maybe a few others
not active working on it, but maybe had seen some of the work. This
bibliography is nothing. I didn't want to waste all my time tossing
everything I could together, but just what this presents, and their
bibliographies and new articles referring to them, the numbers of
people working is large. But there is also Carlos Leguizamon and his
programme on the algbraic-relational theory in biological systems.

"Review: the algebraic-relational theory and its applications" by Alba
Zeretsky

with their pseudo-Booleans found in antigen-antibody reactions and
other biological modelings.

Their are many beautiful models with more exotic structures, but
Heyting algebras seem to have this special universality principle that
somehow seems connected to the Church-Turing thesis. So when I
mentioned education in my first post, I thought these algebras would
be the least controversial to put prior to Boolean. It pretty much
fits right there, with fewer defining relations, full computational
ability, potential cognitive primacy. Its useful in all sorts of
evolutionary models. But in some ways, its also a step to a better
education about models in general, and our ability to calculate in
agreemant with our observations.

Because new models come out all the time. Its important to understand
their logical structure to reason correctly within them. And I
thought maybe this might be a good time to start thinking about
teaching some of these connections I mentioned. Nearly all my friends
went through logic classes as prerequisites in their undergraduate
studies, and these never explained anything outside of quantified
classical logic (often without much algebra). And it seems to me that
with just a little restructuring, such a course could include
constructive logic as it builds the axiom set. All of the various
research in many different directions easily provides a large example
set. Interesting new cases come up all the time.

"An information-based theory of conditionals" by Wayne Wobcke

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galathaea: prankster, fablist, magician, liar