Before I left, I had mentioned wanting to write up the material I
presented in my MathFest talk for possible submission to a
high-profile math journal, probably the American Mathematical Monthly
or the AMS Notices.
I showed my latest existing "draft" (really just notes) to my dad, and
he had some helpful comments: (1) to be clear about the benefits of
doing things the way I am talking about doing them; (2) that the paper
I showed him really did concern the sort of "technological" ideas I
have been talking about; (3) that I should try to organize the points
thematically.
I have attempted to respond to these comments in the current draft:
given that sometimes these are difficult points to address
definitively, e.g. because they need research, this is only an
incremental improvement. Still, it is readable, and it would benefit
from your comments.
Note that despite the mention of "hyperreal dictionary" in the title,
and the adoption of its goals as the main motivations for the paper, I
have not talked about it by name explicitly in the body of the paper.
However, it should be somewhat clear that the current body of
technological ideas -- Arxana -- informs much of the paper. The
precise wording can change, and I welcome your suggestions.
*
PLANETMATH.ORG AND THE HYPERREAL DICTIONARY PROJECT
META-MOTIVATION
I would like to write an article that will really challenge people's
views on non-"free as in freeedom" publishing in math -- and win!
MOTIVATIONS
* To put mathematical knowledge on the computer as an aid for human
learners.
* To get the computer to help solve "human-style" math problems.
BENEFITS
* PlanetMath is already valuable as a "live" online mathematics
community, and can become increasingly so.
* Representing knowledge on the computer facilitates uses of that
knowledge that aren't there otherwise.
* The free/open model provides universal access to knowledge and
"process", which is as advantageous to producers/purveyors of
knowledge, as it is to learners.
* The free/open model can reduce costs borne by mathematical
societies, while helping math societies live their ideals.
* In addition to these direct benefits, if the the mathematics
community handles this effort well, it can become a societal leader
in the knowledge representation and commons development domains.
AN ONLINE MATH COMMUNITY
PlanetMath is not currently a whole lot finer-grained than the
standard ink-and-paper representations of the material it collects.
However, it seems to have done a pretty good job from the point of
view of "being a mathematical reference work". From the point of view
of sheer information content, it does more than most print
compilations -- although of course quantity is not a very useful
measure of quality. (Things like: internal organization, accuracy,
etc., are at least as important.)
As a result of its size, PlanetMath ends up appearing in a lot of web
searches for math terms, and is visited by tens of thousands of users
each day. This certainly indicates that PlanetMath is "useful".
However, this does not by itself indicate that PlanetMath is providing
a particularly valuable service. Anyone can download and re-host
PlanetMath's content; and, again, if size alone was taken as the
measure of quality, Wikipedia would be a better bet.
The primary service that PlanetMath delivers, which makes it more
valuable than a mere repository of data, is that it provides access to
a mathematical *community*. This community provides and grooms the
all of the content, asks and answers questions about mathematics, and
provides one point of organization for the future of online
mathematics.
Now, frankly I think that the community interactions on PlanetMath are
not anywhere near as useful (or as valuable for that matter) as they
would be after a significant "tuning up" of the PlanetMath
infrastructure PlanetMath. This project is "in the works".
MATHEMATICAL KNOWLEDGE REPRESENTATION
Putting texts on the computer is not really the same as representing
knowledge on the computer. As to what "knowledge" is, that is a good
philosophical question [!], but in any event, text is such a coarse
representation that it would make more sense to call online texts
"information".
Representing *mathematical* knowledge on the computer seems like a
very good preliminary test-case for the general problem, because the
computer is already predisposed towards managing logically-organized
materials (this view has been shared by Turing, McCarthy, and other
computer scientists who think about math).
Note both parallels and differences with Google. ``Access'' to
knowledge should not mean that you merely obtain access to some block
of information that coarsely represents the knowledge in question.
Google definitely integrates "knowledge" into search results or the
maps that enable you to find the location and phone number of a
pizzeria in your neighborhood. But this is a long ways from helping
you find the answer to a specific math question.
ACCESS TO KNOWLEDGE
PlanetMath, or something very much like it, will be needed if the
mathematics community as a whole is going to organize and provide
access to its knowledge in such a way as to go beyond the
techno-social status quo.
Yes, things are changing (thanks to tools like Google, ArXiv, blogs,
Wikipedia, etc.), but they are not changing in a particularly coherent
way. And I would argue that they are not yet changing enough.
[This would be a good place to provide some research results.] Access
to knowledge in many parts of the world is still pretty terrible,
despite increasing access to the internet. Within our own country,
compulsory education has overwhelmingly failed to instill significant
knowledge of mathematics in graduating students (much less in the
population of drop-outs).
The entire so-called STEM domain is shortly going to be given a kick
in the pants by Washington at significant cost -- but is this going to
be efficient? If it isn't efficient, is it going to be effective?
My tentative answer is "no". Pouring money into STEM will have
serious limits if STEM is not rooted in fertile soil. Fertility in
this case can be obtained in one way alone: switching over to a
free/open model. [I like the metaphor of "organic" or "bio-dynamic"
growing versus "intensive farming". I don't know how much work this
metaphor can be expected to do -- but the point is that the free/open
set-up is like a compost heap, digesting materials that are put into
it and making them useful. By contrast, just pouring money into
existing systems is the way to make "stove-piped" and relatively
barren results.]
It is vitally important that "access to knowledge" not be just
uni-directional. Even if people were putting papers and course
materials online (which some are and many are not), foregoing the
majority of the downstream user's additional efforts towards making
useful these materials useful is the proverbial "terrible waste".
ECONOMIC REALITIES AND PHILOSOPHICAL IDEALS
Bring up the idea of giving mathematical knowledge away to math
societies or publishers and they will typically tell you "no way!".
Income to these societies continues to derive largely from journal
revenues.
And yet, if you bring up the ideal that "mathematical knowledge should
be available to everyone", there will be wide agreement.
How can we get past this impasse between this reality and this
widely-held ideal? I think the only honest course is to work towards
our ideals.
In a simplistic model, we could imagine driving the costs borne by
mathematical societies continually down, while replacing many society
functions with free online equivalents. This process will surely have
its limits: but then, once we have found these limits, we could reset
membership fees to cover the newly "slimmed down" demands. Surely
this approach is too simple, but it should be the beginning of a
solution.
POLICY RECOMMENDATIONS
1. Understand the demand(s) for public-good-like resources coming out
of mathematics. Not just the demand for a stream of new knowledge,
but also increasingly accessible platforms for learning and
communication. Note that demand for public goods is not always
expressed by "willingness to pay". The constituencies who can be
helped with these resources are often the most destitute, least
educated, least organized groups that the mathematical community can
*imagine* serving. Sometimes demand from third parties
(e.g. philanthropists) will help cover the costs of serving these
groups.
2. Be prepared to do interdisciplinary work. Disparate groups working
in psychology, communication, education, linguistics, etc., won't do
as well on the problem of "how to make mathematical knowledge
accessible" as they would with as an organized team. Mathematicians
should often take a leadership role on this team, not least because
they will be the primary direct consumers of its product. (This is
not unlike other cases of building repositories of indigenous cultural
knowledge.)
3. Develop or use tools for managing growing, large-scale, and often
interdisciplinary knowledge stores. We shouldn't have to abandon old
or familiar ways of doing things (e.g. writing papers in LaTeX), but
rather, we should put together new organizational strategies that
overlay and combine old methods and results. Presumably these
("scholiumific" or "semantic") methods will be useful not just for
archiving or knowledge-assimilation purposes, but also for many live
aspects of communication and practice.
4. Take into account as many possible uses as possible! What sorts of
things might people want to try? Since it isn't possible to plan for
everything, plan especially for unexpected novel weirdness. Do not
just give permission for innovation (e.g. via suitable legal terms),
but actively support it wherever possible. At the same time, plan for
ways of evaluating different behaviors and integrating them, when
useful, into other activities.
5. Run this project in a "business-like" fashion. Some of the
relevant work will be volunteers. Other parts of the effort will
require further motivation, sometimes in the form of monetary payment.
Some parts of the effort will be carried out by largely-independent
groups (e.g. other disciplinary or interdisciplinary knowledge
representation or commons-development projects), who we will still
need to coordinate with. Be able to evaluate various possible
approaches and combinations of approaches.
CONCLUSION
Why are we talking about representing mathematical knowledge? Why not
just talk about representing general knowledge?
Indeed, many of the tools will be the same, although not all of them
will. As mentioned, in order to make a project like this work well,
significant *interdisciplinary* knowledge and *societal* knowledge
will have to be leveraged; so, this project necessitates the
development of more general tools.
As such, it begins to solve more general problems. How should we do
science in the information age? The solutions offered here hit the
main points -- science, at least academic science, should, like
mathematics, be as free and open as possible. [The economic issues
with "science" are even more tricky than they are in math, since so
much research is done by for-profit companies; and even research done
on university campuses often gets fed into the for-profit machine.]
But new questions are raised, for example: How to attribute `credit'
for the kind of incremental (small) improvement, change, or
contribution that free/open model, together with a suitable support
infrastructure, facilitates?
I hope that this paper will help launch a discussion of these and
other important questions for our time.
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