mathematical articles in online encyclopedias

[Paul Taylor <pt08@PaulTaylor.EU>]

It was Vaughan Pratt who first introduced the Wikipedia thread, in
response to someone who said that he hadn't heard of Heyting algebras.
I added my penniworth, having in mind its treatment of Dedekind cuts
and Locally compact spaces.   Recently, however, discussion has
centred on the notion of Topos.

I have changed the Subject line because Wikipedia is not the only
site of its kind.  Anyone thinking of writing for it should perhaps
also consider:
-  citizendium.org - which looks like Wikpedia because it is run by
the latter's co-founder and now unperson;  citizendium has a strict
policy of using real names and qualifications;
- planetmath.org - in which authors "own" the pages that they have
written until they've demonstrably abandoned them;
- mathworld.wolfram.com - beware that this is owned by Wolfram.

It seems to me that toposes are not a good example on which to base
this discussion, being too advanced a topic.   On the other hand,
if you have opinions about what Wikipedia and the other sites should
say about them, then go ahead and write your article, instead of
discussing it here!

But before you write, please bear in mind that these are resources
for the general educated user, not for specialists in the field.
Have a look around for articles that you find informative about
completely different subjects, for example a medical topic or a
place of interest.  It should begin by making sure that the reader
has come to the right place, for example the word "topos" is also
used about poetry.  Then it should tell the lay person why anybody
would spend their time thinking about this thing.   As we all know,
a topos is an elephant.  Its trunk looks like constructive set theory,
its legs look like topological spaces and its tail is a group.

But I really don't think that a specialist in a particular topic
in mathematics should be writing about their speciality.  You need
to see it from a distance.  Wikipedia has policies forbidding
original research.   Encyclopedia articles should provide "general
knowledge" about background topics.

My research programme is a reformulation of general topology, so
I need to talk about this against a background of general knowledge
about traditional point-set topology, locale theory, continuous
lattices, domain theory, constructive analysis and so on.  However,
since I am doing something completely new, I really don't want to
have to give an account of these subjects before I say my own stuff,
so I would like to be able to cite a textbook or other source that
does so.  And I would like that source to be accessible to student
without specialist knowledge, and NOT depend on or be part of some
other partisan presentation.   Chances are that any account of a
topic that is part of a research paper will depend on somebody
else's foundational system.

For example, I would like to refer to an account of locally compact
spaces.  Having been exposed to locale theory and continuous lattices
for 25 years now, I regard it as a matter of "general knowledge"
that the topologies of locally compact spaces are continuous lattices,
and these in turn carry topologies, named after Jimmie Lawson and
Dana Scott.

However, I find NOTHING about this in the Wikipedia article.  That
and more or less every other article there about topological subjects
adheres to the orthodox view in pure mathematics that all self-
respecting topological spaces are Hausdorff.   If I write a new
article about locally compact spaces for Wikipedia then I will find
myself in conflict both with Wikipedia's anonymous cliques and also
with the mathematical establishment.

If you're interested in rings and not non-Hausdorff spaces, then
please substitute the commutative axiom for Hausdorffness in what
I've just said.  If some basic result about rings, fields or modules
can be formulated without assuming commutativity or charactersistic
zero, at no or a small extra cost, then surely it should be so
formulated.   If the more general treatment is more complicated,
but throws light on the subject, the simpler one should be given
first, and an overview of the more general one afterwards.

Another example of this is excluded middle.   Personally, I
foreswore EM about 15 years ago because I was disgusted by some of
the arguments that people were using in domain theory - "bit-picking",
I called it - that didn't form part of any applications or
philosophy.   EM leads to ugly mathematical arguments, and in very
many cases can simply be avoided by stating them more carefully.
In others (for example intuitionistic ordinals and constructive
analysis) there is a more interesting theory when you use the more
delicate tools of constructive reasoning.

Getting back to encyclopedias, remember that they are for teaching,
not research.   Tell students and the general public why the topic
is interesting, and tempt them with some simple point that they
may not have considered.

I'm not claiming to be very good at these things myself, but there
are others who regularly do so on their blogs, as well as in
Wikipedia and the like.   If you don't already know them, you
might like to take a look at the blogs by
- Andrej Bauer  - math.andrej.com
- John Baez et al - golem.ph.utexas.edu/category
- Dan Pioni alias sigfpe - sigfpe.blogspot.com

Paul Taylor