Reengineering Mathematics

MOTIVATION

This is a survey of the fields of endeavor relevant to an "activist anthropology of contemporary mathematics". The aim of this project is to reengineer the field of mathematics to make it more like free software, turning the everyday language of mathematics into "source code". The survey is growing in segments; key segments will be developed into survey papers, and these papers will then be welded together into a unified prospectus.

• For historical background on the HDM project, see http://wiki.planetmath.org/AsteroidMeta/HDM.
• The approximate idea here is to develop a toolchain along the lines of scanning->ocr->parsing->ai->database->UI. Some of these components are further along than others.
• Keep in mind that this all takes place in a social context, with pitfalls, dead ends, and red herrings. I've tried to sum up the situation, as I currently understand it, with a picture: The Road to Free Math

FORMAL SELF-DESCRIPTION OF THIS DOCUMENT

The document is formatted so that "things we need" are presented as succinct capitalized section headings, which are then followed by a brief paragraph describing that topic.

• "Good things" related to these topic-areas are then to be added as bulleted lists below the descriptive paragraph.
• If you do not want to read any more "introductory crap", feel free to skip ahead to MATHEMATICS or other sections below the conveniently-placed line separating "research topics" from "philosophical background".
• Any of the "good things" can be implemented at any time and in any order, and it will be good, since they are all "good things".
• However, if there is any "good thing" that is high priority within its topic area, that item can be sorted "up" within the bulleted list. Similarly, if a given topic area contains items with higher global priority than some other topic area, it should be sorted "up" at this bulkier level of granularity. If it isn't clear which of two items has a higher priority, then, just as clearly, it doesn't really matter how they are ordered here. From a subjective point of view, no one will know whether two items that are next to each other have effectively equal priority or whether one is much higher priority than the other. If we notice any "edit wars" having to do with priority, that will be an interesting chance for out-of-channel discussion or debate.
• Items that are logically nested within other items should just be included in the section for the containing idea (whether that container is a "need" or a "good"), so the document is formally only two levels deep, no matter how detailed the individual sub-items are.
• If any item becomes too complicated to be manageable, it should be branched out onto another page, and only a high-level summary retained here.

TIME AND STABILITY

For working on any task, we need time that isn't already spoken for by some other task with higher priority.

• One basic way to "make time" for a given task is to work on that task as a side-effect of working on some other task. In this fashion, Joe has produced a description of how Arxana relates to his creative writing work as part of an application for an artistic commission. [1]
• The other good way to "make time" is to work on the task directly.

SYSTEMATIC UNDERSTANDING

Knowing how various tasks, desires, and needs fit into a "bigger picture" can give us some confidence that we will get things done in a balanced way.

• Working on this Plan helps achieve this goal.

INCOME

This seems to be an almost physical need, since it relates to biological subsistence.

• Ideally, income will be produced as a side-effect of working on things one enjoys doing.
• In particular, some income could conceivably be produced by developing and implementing BUSINESS MODELS that relate to the ideas we talk about in this plan.
• One aspect of having TIME AND STABILITY is knowing with some certainty when the income is going to arrive and how much of it there will be!

COMMUNICATION

In order to coordinate possibilities for income and collaboration, we need to be syncing up with relevant parties.

• Communicating estimates of time allocations will be spent is a particularly relevant thing to communicate about.
• It is ideal for communication to take place in public whenever the communication is not inherently private in nature. For this reason, the Meteorite wiki seems to be a suitable place to continue major communications about reengineering mathematics.
• It would be of significant advantage to various different projects that have a stake in mathematics and the culture of mathematics to be in communication (and, where possible, further collaboration) with one another.

MATHEMATICS

Any of the following sub-topics will should help "mathematical culture" grow. One of the high-level goals of our work is to create an effective online 'learning platform' for mathematics (cf. section on LEARNING PLATFORMS).

• Work on FEM, so we get something concrete to look at.
• Identify and encode a sort of thesaurus of "core mathematical expressions" and grammar of "core mathematical procedures".
• Develop Ray's work on 'mathematical systems'.
• Integrate Joe's earlier math hacking efforts into a new/improved online system. [2]

MATH AND COMPUTERS TODAY

This relates more to the contemporary social and technical aspects of MATHEMATICS than the more explicitly mathematical items discussed there.

• We may want to get other parties who are interested in math and computers in COMMUNICATION with one another by helping found a "math meta-commons". (Cf. [3], [4]) The most obvious place to start is with a sketch of just what is available in this regard. There are easily hundreds if not thousands of projects and many related informal non-projects, and we won't have time to review or list them all -- but if we can at least list the major areas in which work is being done (with a few examples), that would be a good start. It will be important not just to look at applications of math in computers or computers in math, or math to computers or computers to math, but also math through computers (or, I guess, computers through math, although that direction is a little more confusing). In particular, I'm thinking about things like statistics and geography. Finally, we should open our review up to projects that relate on other dimensions (e.g. other "open science" projects, other "content management" projects, other projects that are trying to impact education, etc.).
• Can't we provide links from PlanetMath *into* ArXiv (if not also a module, whether it's something we get put into ArXiv, or perhaps an Adobe plug-in, for getting links running the other direction, i.e. from papers into PlanetMath)?
• We may want to collaborate with the MathJax project ("the next jsMath") as a test site or in some other capacity. I've emailed the project coordinator, but presumably we will need to follow up further.

DATA ACQUISITION AND MANAGEMENT

We have already thought through and seen some decent "proofs of concept" for making electronic editions of public domain works, and managing "deep links" in both public domain and copyrighted texts. Because we have thought through the issues with the public domain end of things pretty well and just need to establish a functional system to do the work, our priority list runs "back-to-front"; in other words, input of brand new data is something we'd do only when we know the various aspects of our backend are going to work well.

• Build facilities for managing a collection of public domain works. Ideally, the data-management facilities will be inter-operable with other data-oriented projects, for example the BKN project [9], or the thesaurus project mentioned in the MATHEMATICS section.
• Run OCR on already-scanned public domain works.
• Scan more public domain works.
• Identify a complete list of public domain works.
• Hand enter or automatically create links to/from any literature that is online, whether public domain or not. (This project is otherwise known as the "cross-index".)

LINGUISTICS

In addition to working with the more traditional methods of linguistics, we will want to work on understanding "large-scale" linguistic issues or ideas, ranging from anaphora, to hypertext (cf. section on HUMAN-COMPUTER INTERACTION).

• Follow up on the use of Relational Grammar as a way of describing mathematical statements.
• Find any computer platforms that work with relational grammar.
• Intersect work with parsers with the work to develop a "thesaurus" of for basic mathematical expressions and procedures.

HUMAN-COMPUTER INTERACTION

For now, the hypertext aspects of a new and presumably useful system for human-computer interaction are fairly well along in the form of the Arxana prototype. Other types of interactive forms may become interesting and will need work, and hypertext itself is still not completely understood. (Some of the "philosophy" of hypertext studies is outlined in [1].)

• Modifications for working with theories needs to be completed.
• Old features related to browsing and editing (e.g. markup and clusions) need to be re-implemented and/or extended.
• New features for search and collaboration need to be added.
• Again it would be nice to have a list of some possible sorts of 'interactions' we would like to carry out (without implementations) so we have a good sense of the directions in which to build.

LEARNING PLATFORMS

I suggest that developing "learning platforms" that will work for human learners is the flip-side of ARTIFICIAL INTELLIGENCE.

• Joe drafted a proposal having to do a 3-year funded project focused on developing learning platforms in both general-purpose and mathematics-specific contexts. (See [5], [6], [7].) The first part of the proposal -- and the first part of the proposed work! -- is a broadly conceived survey of the field of "personalized learning environments".
• As outlined in [8], the flow from "data" to "information" to "knowledge" requires the addition of (a) structure and (b) context. In a context like PlanetMath? that already has a lot of information, we still need to develop context to the information useful for learners. Can we develop standards for what Ray has previously referred to as "quality at the boundary"?
• Aspects of this project may proceed in collaboration with the EduTeX Project.

ARTIFICIAL INTELLIGENCE

One of our long-term goals is to produce a computer program that can answer mathematical questions similar to the questions that humans are able to answer.

• Use Arxana to manage "frames" for the thesaurus project mentioned in the MATHEMATICS section.
• Work on developing a theoretical list of queries for a scholium system that contains mathematical information (no need to worry about implementation right away).
• Study the math/computer science/knowledge management literature on queries (e.g. both 'global' and 'local').

BUSINESS MODELS

For purposes of this discussion, a "business model" needs (a) a consumer (what do they want?); (b) a service or product (i.e. that is to be exchanged for money somehow); (c) a producer (who does the work, how, and with what motivation?).

• In MATHEMATICS, (a) readers and libraries may be interested in purchasing (b) well-edited print versions of the FEM or related works, for sale through Amazon and/or bookstores at around $50 to $100 a copy (c) which we could generate using already-existing code and some new editorial elbow grease, requiring some further investment of time, and perhaps some profit-sharing agreements to motivate participation.
• Once we've gotten some work done in the DATA ACQUISITION AND MANAGEMENT area, we could sell paper books, as in the example above, but they may be even less work to produce.
• In Joe's proposal on LEARNING PLATFORMS [5], each of the "phases" will produce some artifact of non-trivial value to (a) groups of learners, designers, and social science types; (b) who could be charged for; (c) customized versions of the software.

KOINOMICS

Although our productive strategy may not be commons-based per se, aspects of it are; indeed, "scholarship" as a whole seems to be some kind of commons, and, running with this idea, anything that we want to do that we hope will impact an entire field of human endeavor ought to be informed by a fairly solid understanding of the economic features of the landscape. (Here, more properly koinomic -- having to do with the rules of commons and common property, as distinct from the rules of households and private property, although certainly that is involved.)

• Commons-based peer production, "systems", and cybernetics are all relevant -- but no doubt there is a lot more. Coming up with a bibliography of koinomics would be a good first step.

RESEARCH AND DEVELOPMENT PROGRAMMES

Ideally we would have fully-speced out research proposals in each of the major areas. So far, we have a partial proposal on HUMAN-COMPUTER INTERACTION, though it comes at the issue from a somewhat non-scientific angle [1]; one on LEARNING PLATFORMS [5], and at least the beginnings of one on MATH AND COMPUTERS TODAY [3].

• It would be worth doing a literature review and specification of goals, objectives, methodology, and milestones in each of the areas mentioned here, "complete set".
• To the extent that we can integrate these plans, in particular, if we have rather comprehensive knowledge about MATH AND COMPUTERS TODAY, I think we can speak with some authority about the future of this field. But I think we will need research prospectuses if we're going to be taken seriously.
• Converting prospectuses into programmes is a further step, for which we will need to look at BUSINESS MODELS and other viable synergistic models.

REFERENCES

[1] http://metameso.org/~joe/01.pdf

[2] http://metameso.org/~joe/math/

[3] http://wiki.planetmath.org/AsteroidMeta/Surveying_the_Math_Metacommons

[4] http://wiki.planetmath.org/AsteroidMeta/metacommons_manifesto

[5] http://metameso.org/~joe/ple-proposal.pdf

[6] http://kmi.open.ac.uk/studentships/personal-learning-environments.php

[7] http://www.role-project.eu/?page_id=8

[8] Teo, C. B. and R. K. L. Gay (2006). "A knowledge-driven model to personalize e-learning." Journal on Educational Resources in Computing 6(1): 3.

[9] http://www.bibkn.org/drupal/