**Mathematical
Structures for Rational Discourse**

*Desmond
Fearnley-Sander*

dfs@hilbert.maths.utas.edu.au

Department of Mathematics

**University of Tasmania**

Australia

### Abstract

This paper reports the implementation of a system for the creation of *rational automata*: automata that communicate with one another, display curiosity, learn and are creative. The author calls the implemented automata *narrow minds*. The formal mathematical structure of a rational automaton is defined, a sketch is given of how they work and how they interact, and a sample dialogue between narrow minds is presented. Equation processing plays a central role. It is argued that basic everyday thinking and basic mathematical thinking, though very different in some ways, can be implemented using the same mechanisms.

The narrow minds system takes a step toward
the productive social interaction
of rational and, in particular,
mathematically capable autonomous
computer entities. Its implementation
rests upon two fundamental computational
paradigms: equational programming
and object-oriented programming,
the one for implementation of
simple intelligence, the other
for implementation of simple social
interaction.The system user can
control the level of intelligence
of the narrow minds that are created,
how they interact in dialogue,
how they speak, and, to some extent,
how they think. Parallelism is
intrinsic to the system, as it
is to human communities.

Some extensions are outlined, and
the relationship is considered
of the work reported here to automatic
theorem-proving.

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