Making Conjectures

Abstract

Conjectures are statements about various concepts in a theory which are hypothesised to be true. If the statement is proved to be true, it is a theorem; if it is shown to be false, it becomes a non-theorem; if the truth of the statement is undecided, it remains an open conjecture. Making and proving conjectures automatically in mathematics has been a long term goal of Artificial Intelligence, dating back to Simon and Newell’s 1958 prediction, [Simon & Newell 58], that within ten years a computer would discover and prove an important mathematical theorem. There has been considerable work in automated theorem proving, but much less research into the problem of discovering conjectures automatically. As in many sciences, mathematical conjectures often arise from empirical observations of data. In mathematics, patterns found in the examples of concepts can result in a conjecture that the pattern is not just true of the small sample in the data, but is true of all the examples possible for the concepts. We discuss four ways to identify such patterns in the examples of mathematical concepts.