[NNexus] Where do you draw the line?

[Deyan Ginev]

Dear PlanetMath and Planetary enthusiasts,

I would like to ask for your help with understanding what ought to be
the domain of NNexus indexing.

NNexus (https://github.com/dginev/nnexus) has traditionally focused on
PlanetMath where this question is easy to answer - every article in
PlanetMath's encyclopedia is intended to be a math concept or topic of
interest.
Thus, the NNexus description is rightfully an auto-linker for "math
concepts".

But, you can already see this is a rather practical, rather than purist,
claim. PlanetMath has not only articles on "concepts", but also
examples, open problems, and possibly other "topical" content.
MathWorld, among others, has pages for different theorems (e.g.
http://mathworld.wolfram.com/FermatsTheorem.html) and theorems are not
"concepts", even though they are mathematical named entities.

The real reason I have been thinking about this is that I have been
trying to write a comparative review between the seven domains I
currently have indexed in NNexus, and have found myself unable to
proceed without a clear answer.

nLab, for example, has a majority of pages titled after, and describing,
mathematicians by name (e.g. http://ncatlab.org/nlab/show/Abraham+Fraenkel).

And, to name the elephant in the room, Wikipedia offers all of these
challenges and more. There we need to decide the limits not only of the
types of objects to index (concepts, names of statements, names of
people, events?, ...) but also where do we draw the border between Math
and "other" scientific disciplines. One example from my current indexing
effort to this extent is " Is econometrics a branch of applied math that
should be in the NNexus index, or is it a subbranch of Economics that
should be excluded?" Same question for Theoretical computer science.

So, with this email I would like to start a brainstorming discussion on
what the precise domain of NNexus indexing ought to be, and why.

I have some first thoughts on the subject, but would like to hear your
opinions first.

Greetings,
Deyan

[Joe Corneli]

Deyan, thanks! I guess we need a little meta-mathematical lexicon. :-)

I've been working on one for my job, where we need something fairly
technical. So far, I just have the "ABCs". I'll append that as it's
not the direct answer for this question, but it might serve as
inspiration.

In this case, I think the use is going to determine what goes in the
index. If we're adding autolinks to nCatLab pages, we might want to
link to their named terms, and since they are a good provider of
biographies, anytime we want biographies, we'll also want this turned
on. This seems a bit like faceted search. It is interesting that
PlanetMath seems to make things easy: every page has a title, and
potentially synonyms, and these are all math terms and should always
be linked.

With Wikipedia, can we extract a similar "PlanetMath-like"
mathematical encyclopedia, just using the Mathematics category? Then,
any other categories (for instance, econometrics, physics) could be
added in a similar purpose-based "faceted" way. I think the default
(which is useful for demos, including an always-on demo on PlanetMath)
should just be to link to terms in the Mathematics category (which
probably just reduces to titles, in the Wikipedia case).

One "trick" I did in my thesis was to keep track of the
[http://example.com marked up words]. In some cases these "marked up
words" might be useful for further analysis, although in many cases
where these links are added by hand, they will be non-mathematical
(and in cases where they are added by NNexus, they will be slightly
redundant, although it's still interesting to know which one of the
multiple "names" for a concept is used). This is something I
discussed briefly with Magdalena when we met.

At a philosophical level, personally I think it's OK for NNexus to
treat any named entity as a concept (although they may not all be math
concepts) -- but as you can see from the lexicon below, this is *not*
how everyone thinks about these things, or even close. Many of the
mathematical "concepts" on PlanetMath would be seen as "domains" in
the lingo of the COINVENT project. Here, "prime number" is a concept,
but "number" is not. Weird, huh?

Anyway, I hope this contributes to the thought process!

Joe

8< 8< 8<

Axiom

For Simon Colton's HR program, an axiom contributes to the definition
of a domain -- the domain might be populated by integers, groups,
integer sequences, graphs, or something else. For HR, the axioms
defining the domain are different from concepts, which are properties
that the elements of the domain may have. For instance, we need to
know what a group is to invent the concept of a cyclic group, and much
as we need to know some certain things about integers in order to
understand the concept of a prime number. We can compare this to the
constructions generated by the "What If Machine": "What if there was a
little cat that was afraid of milk?" presents the hypothetical concept
of a lactophobic feline; the slot-filling terms "cat", "milk" and
"afraid", and the usual relationships between these terms, are taken
as understood.

This is not to say that an axiom or a domain of interest cannot be
invented or discovered by a computer, e.g. we can imagine a program
could invent the definition of "set" or "group" building on some other
lower-level axiomatic material. Somewhat at the other end of the
spectrum, we can imagine a program with a large knowledge base of
"common sense" information which is taken as axiomatic: this is
similar to the What If Machine's use of ConceptNet.

Blend

We tend to understand blend in the category theoretic language of
Joseph Goguen. (Note, we need the "right" definition of concept for
this to be meaningful - work in progress here in the lexicon...)

Concept

One definition for this term is as a function that does some sort of
partition, separating the sheep from the goats, the men from the boys,
or the numbers with k factors from the numbers with j factors when j
is not equal to k; etc. At the most basic level, the concept acts as
a predicate that can be held by some of the members of the domain and
by which members of the partition are recognised (e.g. sheepishness,
manliness, k-divisibility).
A more expanded discussion was presented by Joseph Goguen in the
article "What is a Concept?"

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