:: towards a constructive education :: (news server friendly)

From: galathaea (galathaea_at_excite.com)
Date: 02/06/04 

Date: 5 Feb 2004 16:04:39 -0800

I posted this earlier this week, but discovered that many news servers
(and Google) would not carry it, due to the number of newsgroups
posted to, and this also prevented some who could access it from
replying. I feel this is an important topic to discuss in an
interdisciplinary style, and I still believe that there is much
benefit to all newsgroups and their respective professions which I
posted to, but have broken the linking up in order to expose this to a
wider audience of news servers. I appologise if you received both
postings on your news server, and I hope that you do not consider it
spam. I have read all charters and FAQs I could find and the only
thing that is troubling is the double post, as the content applies to
all fields in a quite straightforward way.

-=-=-=-=-=-=-=-=--=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-

While you are reading this, your retinal rods and cones are
interpreting images projected onto them. Each letter, like the

W

that begins this post is given a neural focus, an "attention", as
rhythmic waves refresh the focus regularly. The transmission through
the optic ganglions and along the optic fibres of the raw inforation
of the

W

goes through a processing which has been well studied in the
literature. One successful modeling approach is that of the
connectionists or found in the neural net literature. The oculomotor
loop and the importance of saccades is established. We have many
models on how neural nets detect edges, identify topological
components in their optic information, determine orientation. The
retinotopic map is formed.

These models are useful to the extent that there is actual technology
built with these models, and they make good money today. They agree
with our experiments extremely well. We can usefully pattern match
the shape

W

with a fair degree of accuracy.

The logic of propositions in the models on shape theory, the theory of
the topological, metrical, and orientation-based classification
calculus, the dynamical logic of the attractor spaces in neural net
models, the region-connection calculus, all of these logics are
Heyting.

Of course, human pattern matching involves much additional information
processing. When the

W

is blurry, when the information is vague, we may look to the
surrounding

*hile

and call on recollections of known forms of full words. We've learned
to attach meanings to utterances and visual shapes, and while you are
reading this you are building structures of meaning to the words which
can assist in the recognition, determining patterns of abstractions
connected in a mental system of relationship which is still poorly
understood. But every year more information is gathered. The
schematic decomposition of the process is being studied, and regularly
receives more insight towards the architectonic basis of the function
and dynamics of thought.

And, we can formalise the game of language. Formal languages have a
well studied mathematical foundation, and the reasoning in a semantic
system can be formalised in a model theory. When we look to the many
sources of "reasoning about", from necessities and modalities to
counterfactual possible worlds, satisfaction (semantic), operational
resolution, etc. we find many different logics. But they are all
Heyting, and this substructure common to all semantic theory allows
for a rich functor structure of equivalences and dualities amongst
these logics. There are common dictionaries to translate between
various related semantics.

And there appears to be a very important reason this Heyting
substructure is found so predominantly in the formal languages and
foundations of mathematics. Formal reasoning has a structure of a
state and its transitions, it is in many ways the analysis of an
automaton whose transition graph is specified by the logic of the
reasoning. Proof theory is very intimately related to the theory of
computation and the lambda calculus. The lambda calculus has a well
known Heyting structure, and we have the Church-Turing thesis over our
shoulders hinting that this may very well be the most primary
structure found in all algorithmics.

When we look to topics like Martin-Lof type theory to explain the
evolution of the objects of or conceptions and the structure of
describing our world, again we have its Heyting structure well known
and studied.

Constructivism is very much the study of process and information, of
the logic of propositions on information structures. Processing
information allows us to channel it to useful decision making. It
appears to be prior epistemically to the abstraction of a "reality".

And we use this structure in our theories about the world around us.
In general, the semantics of evolving collections of objects and
causality is known to be Heyting. Applications of Heyting semantics
have been detailed for antigen-antibody interactions or the deviation
of a cancer process, but one of the most fascinating applications for
me has always been the analysis of quantum propositions and the
evolution of quantum systems. In fact, the work of von Neumann and
Birkhoff is one of the most stark expressions of this generality,
where it was shown that contrary to much early physics, the logic of
the universe may very well be that of a Heyting algebra that is not
Boolean.

Now, I appologise for the large crossposting, but I believe this post
to be topical to all newsgroups in my list. I earnestly feel that
there is a common thread which needs to be discussed amongst all of
these communities. I am concerned about the education of constructive
theory and Heyting semantics prior to the study of classical logic and
Boole (and Aristotle when you lack quantifiers, etc.). And when I say
Heyting here, prior, and to come, I also intend co-Heyting, for
obviously there are semantics mentioned who have more reasonable
interpretations in the dual algebras, but that is of course a
contravariant functor away.

For some reason, I feel that this sounds like a radical idea, even
though I also get the impression that Heyting algebras are not the
controversy they were when associated more strongly with Brouwer and
intuitionism. There are obviously many communities using them. They
appear to be intimately related with conceptualisation and the thought
process.

Yet, going through the college system, I found it rare for
non-specialists to be aware of the more general Heyting structures
found in their respective fields, though they were often well prepared
in classical logic. And I often found that this presented a much
_less_ useful tool in which to evaluate the structures of their
fields. The application of Boolean logics to quantum systems is one
of the major sources of "strangeness" under which quantum mechanics is
commonly described, and understanding how to properly make
propositions about quantum systems can clear up much confusion here.
In linguistics and formal foundations, there is regular rediscovery of
basic results on the difficulty of identity and the ambiguity of
negation which are well described by constructive semantics. From the
theory of Cohen forcing to slaving principles and thermodynamic
process, from topoi to decoherence-function-consistent histories, our
fields have many Heyting structures lying just an analysis away.

I am not of the opinion that the Boolean has _no_ use, nor would I
even advocate giving up the Axiom of Choice as a useful tool. But I
do believe that prior to exposing a student to these particular models
of mathematical reasoning, the more constructive foundations should be
explained more thoroughly. Because I do find that alot of
misconceptions about constructivism get propagated at times. In
particular, on these newsgroups I find that questions concerning
issues particularly involved with the general theory around
applications of Heyting algebras get a much diminished audience to
that of more classical analyses of reasoning. Sometimes there is even
derision.

John Baez once stated that,

"Intuitionism proceeds by a method known as Winnowing the Audience.
Essentially, one can avoid the use of formal proofs by making the
proofs so long, tedious, and bewildering that only those in agreement
bother to read them."

Obviously, I have placed this without context, and recent comments by
Baez in support of some of Lawvere's programme may comment to possible
changes in position, but the statement is certainly indicative of a
trend. Many pop-foundations (pop-philosophy, pop-mathematics, etc.)
books still mention the various constructive schools of thought as
minor players, sometimes with actual dismissal and often without much
detail. Now, this may only point to a failing of pop books, and
although I think most professionals deride the pop books of their
professions, there is certainly a community attitude that such books
commonly intend to convey.

Yet, I also find that there is quite a large group of communities
using many different faces of Heyting structures. And I find that
mathematicians as a whole do seem to appreciate constructive proofs
over nonconstructive ones where they can get them. And on some of
these newsgroups I even find most constructive expositions are well
received at times. I just find that those with a good understanding
of the field are either self-trained or studied directly under a
contributer to the field, and there is often lamentation that the
field is more widely applicable. In fact, the field surely seems more
widely applicable than the study of classical Boolean logic, it being
only one form of Heyting structure and not applicable to many of the
structures mentioned or, for example, the well known theory of
decidability founded by Godel or the theory of Kleene truth.

For me, constructivist theories have always been computational
theories in general. For me, I never found anything "tedious, and
bewildering" and certainly have not noticed that it might be common to
"avoid the use of formal proofs" in intuitionism or other constructive
programmes. But this was because I read early on about intuitionism
and constructivist theories as my notions of logic were developing.
By accidents of choices and self-education, I learned constructive
logics at about the same time as learning classical logics.
Certainly, constructivist math can be more difficult since you lack
some common tools for proof, and I have found myself joking about
difficulty and obscurity many times in my studies when I first entered
certain topics (like algebraic geometry and the analysis of varieties
and schemes, as an example which I still clearly remember making
similar comments), when the pantheon of objects and their
transformation structure was still poorly known by myself, and I find
that quotes like those of Baez above are often indicative of this
early apprehension. So I get concerned about education.

And these are the questions I have for all of our communities out
there:

- is it, with so many applications so fundamentally related in our
fields unified by the common Heyting thread, perhaps time to start
teaching our students more about the theory of the Heyting algebra
structure prior to adding axioms forcing bivalence or existence?
- if not, why not?
- and, somewhat to get a better picture for myself, why do you
believe the more widely applicable Heyting structures and their
semantic analysis is found to be less important to the education of
our students than the lesser applicable Boolean particular case
currently taught?

Obviously, I am of the "yes" opinion to the first question. For the
third question, I do see the historical contigencies that have guided
modern education, but after more than a hundred years of
constructivist thought being advocated, from early rumblings of
Kroenecker and Poincare, through Brouwer, Weyl, Heyting, Tarski, and
the eventual use of the logical structure and its semantic analysis in
numerous foundational fields, and its modern ubiquity, such
explanations still seem to fail for me for modern education.
Obviously, mathematics is an easier enterprise with the additional
tools of excluded middle and transfinite choice, but again I am not an
advocate of _not_ teaching those tools. I am only curious about why
the constructive logics which seem in numerous ways to be prior to
such tools are not taught prior. They certainly can help in the
understanding of the more classical approaches. My hunch for the
third question is that the desire for classical logic is very similar
to the desire to believe in monotheism, that there is this insecurity
in many people concerning the "absolute" and "reality", and questions
of truth and falseness, like questions of good and evil, should be
knowable to some form of completion. I know that is certainly a
controversial opinion in some circles, but I wanted to throw it out
there to give a better view of where I come from and perhaps to
stimulate discussion on the third topic. Paradoxically, I have found
that although I am drawn to multivalent logics and find monotheism
anathema (ironic considering the words origins), I also enjoy studying
the realist interpretations of quantum mechanics, which I suspect are
guided by very similar desires, so perhaps I just like to be
contrary...

In philosophy, epistemics, cognition, linguistics, formal models,
computation, and the possible structure of our world, there are
unifying principles of expressability that I find more and more
useful. It confuses me that I find them still poorly understood in
crowds where, at least to me, it has always seemed they should be more
well known. So I thought I'd try to bring some of the more relevant
communities together and see if I could start some discussion on
broadening education along these important lines, for I feel that such
is urgently needed to prevent a lot of "wheel grinding" and repition
of already known results in the separate fields. I really believe
that such a consolidation of logical education is needed in all of our
fields to place more focus on our respective advances.

-=-=-=--=-=-=--==-=-=-=-=-=-=-=-=-=-=-=-=-=-=

The newsgroups I have broken this up into are divided into 3 groups of
5. They cover the major newgroups discussing issues mentioned, and
whom collectively, I felt, could have the most impact on the
educational concerns I mention. I included two fan newsgroups which I
felt contained overlap with many of the ideas presented as well. For
those interested in all sides of the debate, I wanted to list my
divisions here. I will be participating in all of them.

1
-=-
alt.consciousness
alt.fan.hofstadter
comp.theory
sci.cognitive
sci.logic

2
-=-
alt.philosophy
comp.lang.functional
sci.lang
sci.physics
sci.psychology.theory

3
-=-
alt.fan.noam-chomsky
comp.ai.neural-nets
sci.edu
sci.philosophy.meta
sci.math 

-- 
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar