Hypertext and Metamathematics Preview

[holtzermann17]

Summary

Lifecycle stage: 1 --proto-- 2 ->- [3] --evolving-- 4 ->- 5 --complete-- 6 ->- 7 --mature-- 8

Often distinct from but sometimes overlapping with the ideas in the Computer Math preview, we've been actively looking at and working on issues like these, over the past decade or so:

• Theoretical background work on representing mathematical knowledge as semantic hypergraphs and reasoning with it.
• Theoretical background work on linguistics and mathematics.
• Several versions of Arxana (an Emacs-based hypertext system) have been developed, which will eventually integrate with the above.

In order to progress to stage 4: (a) Round-tripping and exporting from LaTeX documents and Emacs to/from PlanetMath; (b) finishing some full interactive examples of network programming / network math inside Emacs.

Joe says: To give some idea of what we're talking about...

Here's an overview of a possible computational agents approach.

My aim is to formalize what I've learned about groups of human co-learners in a system of computational agents that can collaborate to solve mathematical problems. There is some precedent for this sort of work within the knowledge-rich artificial intelligence tradition (e.g. Marvin Minsky's Society of Mind), although not typically with a mathematics focus.

In my thesis, I observed that paragogy has structural similarities to Imre Lakatos's description of mathematical argumentation, as embodied in the dialogs in his Proofs and Refutations. There are also parallels to Martin Nowak's work on the evolution of cooperation in a game theoretic setting. Previous research on Lakatos-style computational agents was carried out Alison Pease, a philosopher of mathematics whom I interviewed during the requirements-gathering phase of my thesis. However, this prior work does not fall within the knowledge-rich AI tradition, but instead focuses on theory construction from axioms.

My plan connects the new theory of paragogy with my earlier writing on hypertext and knowledge representation for mathematics (Corneli & Krowne (2005), Corneli & Puzio (2013), Corneli (2003, unpublished), Corneli (2004, unpublished)). This work provides both technical and theoretical foundations, but full articulation of the project will take considerable time and effort. The results could transform mathematical practice as well as the way we think about learning.

See also

• http://arxana.net
• Hyperreal Dictionary of Mathematics page on AsteroidMeta

[holtzermann17]

See MathHubInfo/Legacy-planetary#416