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3. As already mentioned, the work of some codelets may turn out to be superfluous
or even useless. The outputs of such codelets are not ultimately integrated into
the final structure of the formal mathematical proof. Nevertheless, they cannot
be considered totally irrelevant, because they might have revealed unexpected
relationships with other mathematical concepts or statements or elucidate the
independence of some assumption of the mathematical statement to be proved or
uncover the need of a weaker or refined assumption.
4. Particular codelets, or their derivative contributions, may vary in location and
weight in a process of generation of a Web-based proof-event. A codelet may
turn out to be prerequisite, refinement, simple correction or even counterexample
for the contribution of another codelet. Therefore, they are arranged neither in
parallel, nor in sequential order. They have a complex, graph-like structure that
follows the eventual formal structure of the provisional mathematical proof.
5. Administrators do not know in advance the final outcomes of Web-based proof
events, so they can't provide deterministic guidance. They are trusted by the com-
munity of the contributors in view of their reputation in the academic world. At
the final stage ofWeb-based proof events administrators can potentially intervene,
evaluate, correct, filter and integrate all kinds of contributions.
18.6 Collective Codelet Generation: An Example Using
the Pythagorean Theorem
Consider Pythagoras' Theorem:
a
2
c
2
whenever
a
and
b
are the lengths
of the legs of a right triangle and
c
is the length of the hypotenuse. While not an
example of a research problem in contemporary mathematics, we will use this widely
known property of right triangles to illustrate our approach. Here, for the purposes of
illustration, we imagine an unrealistic scenario in which there exists a mathematical
community that has all the skills and training that mathematicians usually have, but
which lacks the knowledge of the truth of the Pythagorean Theorem. This is necessary
for our discussion because we want to avoid discussing some other, sophisticated
problem, which is understandable only by the real mathematical community of our
times. Thus, suppose that the problem of finding proofs of the Pythagorean Theorem
is publicly posted, and the members of this hypothetical mathematical community
may participate in the quest for a proof (or proofs!). In what follows, we will draw
from material from the webpage of Bogomolny [
1
], a collection of over one hundred
different proofs of the theorem from various sources.
Initially there should be a single task:
PT
(proving the Pythagorean Theorem),
initiated by a single agent:
A
. This marks the beginning of a proof-event that is going
to evolve in time, as long as other codelets (acting as provers) would attempt to
present a proof of
PT
. We shall use the notation
b
2
+
=
agent
,
task
for codelets that enter
the system pool. Thus,
A
,
PT
will be the initially single codelet in the pool. This
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