Collective Discovery Events: Web-Based Mathematical Problem-Solving with Codelets - Computational Creativity Research: Towards Creative Machines - page 370

or Web-based (group) brainstorming [ 2 ], by which a group tries to find a proof for
a posed mathematical problem by culling a list of spontaneously generated ideas
contributed by its members. This feature makes it possible to take advantage of
the complementarity of mathematical background and capabilities of the different
agents involved in a Web-based proof-event. In a recent study of the third Mini-
Polymath project [ 19 ], Alison Pease and Ursula Martin [ 15 ] have explored the kinds
of mathematical reasoning behind the online collaboration and classified them into
four kinds of comment that focus on mathematical concepts, examples, conjectures
and proof strategies and relate them, wherever relevant, to corresponding theories of
mathematical practices elaborated by Pólya [ 16 ] and Lakatos [ 11 ].
Subsequent to the brainstorming phase or the collection of ideas, parts of proof
or incomplete proofs (as practiced in the Polymath project) is the phase of their
categorization and integration. This task is allocated to a member of the group of
agents or is undertaken by a supervising agent. Thus the roles of the supervising agent
and the integrator are crucial. The supervising agent is themaster of the code of ethics,
and monitors the whole process of the mathematical problem-solving activity. The
integrator acts as proof checker or (global, above individual provers) validator and
integrates the different parts of the proof into a unified whole.
In Web-based proof-events the agents involved suggest possible contributions to
the proof, which can potentially be integrated into the final proof, or disregarded.
Thus,itisa modular contribution to proving that resembles the open-source coop-
erative activity or crowdsourcing (in case the proof-event is initiated by a single
agent). These features of Web-based proof-events expand the capacity of the agents
involved (i.e. the effectiveness of the agent system), ensuring deeper insight into the
problem. The final outcome of a Web-based proof-event cannot be credited to its
initiator or the supervising agent; it is a kind of collective intellectual property, in
which the individual contribution of every agent is clearly identifiable in the history
of the proof, represented as a sequence of proof-events.
18.5 A Codelets-Based Model for Web-Based Proof-Events
Web-basedmathematical problem-solving is a process based primarily on the prover-
interpreter interaction over the Web. During this process, an initial interpreter inserts
into a pool of unresolved issues a list of issues that, if resolved, amount to the solu-
tion of the initial problem. For example, if the problem to be solved is the proof
of a theorem, then the list of unresolved issues that are inserted into the pool can
be the highest-level pieces of the proof, as envisioned by the initial interpreter. The
pool is immediately available to the Web-based community of participants, who
are informed of its existence by mediating system software and may start acting as
codelets when they select pieces that they deem solvable. (Henceforth, for simplic-
ity, the participants in the Web-based problem-solving event will be referred to as
“codelets”.) When codelets form by selecting an unresolved issue they do so either
because they feel capable of solving it by their own means, or because they see how