Collective Discovery Events: Web-Based Mathematical Problem-Solving with Codelets - Computational Creativity Research: Towards Creative Machines - page 368

by modification of definitions or explication of implicit concepts in order to achieve
higher standards of rigor in the purported proof.
The prover and the interpreter, involved in a sequence of proof-events, may act
in different geographical locations, be surrounded by different environments at dif-
ferent times and belong to different mathematical cultures. However, they are in
communication by sharing a common interpersonal space , so that communication
becomes possible. The environments in which the agents are located provide the
means of communication between the agents, i.e. written texts, conditions for verbal
or non-verbal communication, as well as for communication through practices. The
“text” communicated by a prover can be encoded in any (already available, or ad hoc
invented) semiotic code (usually in a blend of the natural language and some formal
language) and is articulated in a specific style that serves communication functions.
Styles can be personal for provers or for the schools they belong to or for a whole
mathematical tradition; it may be also mimicry of the style of a renowned authority.
The style is a meta-code that determines the selection of a particular code and the
combination of blending principles to produce the “text” by the prover [ 18 ].
Different agents can ascribe different meanings to the same “text”, communicated
by a prover. Themeaning that the prover may ascribe to the “text” (the intended mean-
ing ) is generally different from the perceived meaning that the interpreter may ascribe
to the same “text”. Moreover, the prover and the interpreter may follow different kinds
of logic in their reasoning. Understanding is achieved when the meaning perceived by
the interpreter corresponds to the intended meaning of the prover, that is, whenever
a mapping called semiotic morphism (or “translation”) can be established between
the semiotic space of the prover into the semiotic space of the interpreter, as defined
in [ 4 , 5 ].
The interpreter's understanding of a “text” is a prerequisite to its interpretation; yet
this does not guarantee the validity of the proof. Interpretation means the determina-
tion of the meaning of the signs in which a transmitted proof or what is thought of to
be a proof is encoded. Interpretation is an active process, during which the interpreter
may amend the initial proof by adding new concepts (definitions) or fill possible gaps
in the proof by elaborating new parts of the proof, etc. In some sense, interpretation
is a reconstruction of meaning or conscious reproduction of the information content
conveyed by the “text”. During this process, the interpreter may also choose a new,
different code to express more adequately the meaning of the prover's “text”. How-
ever, during this transformation the original “text” loses its stylistic peculiarities. The
style of the interpreter may be completely different from that of the prover (although
it may incorporate some elements of the style of the prover). In general, the item
produced by a prover may lead to different communication outcomes:
For a mathematician engaged in proving, the most satisfactory outcome is that all participants
agree that 'a proof has been given.' Other possible outcomes are that most are more or less
convinced, but want to see some further details; or they may agree that the result is probably
true, but that there are significant gaps in the proof event; or they may agree the result is
false; and of course, some participants may be lost or confused (Goguen [ 6 ]).
A sequence of proof-events is considered as terminated (finite) when the agents
(interpreters) involved in it conclude that they have understood the proof and agree