Managing Informal Mathematical Knowledge: Techniques from Informal Logic

Much work in MKM depends on the application of formal logic to mathematics. However, much mathematical knowledge is informal. Luckily, formal logic only represents one tradition in logic, specifically the modeling of inference in terms of logical form. Many inferences cannot be captured in this manner. The study of such inferences is still within the domain of logic, and is sometimes called informal logic. This paper explores some of the benefits informal logic may have for the management of informal mathematical knowledge. 1 Informal Mathematical Knowledge What sort of mathematical knowledge does mathematical knowledge management manage? A distinction between knowledge that and knowledge how is frequently deployed in epistemology. In mathematics this corresponds to the distinction between knowing mathematical propositions and knowing how to conduct mathematical proofs, that is being acquainted with mathematical practice. Each of these two sorts of knowledge may be related to a problem for MKM. In the first case, the problem may be expressed as ‘How can a computer represent the truths of mathematics?’. In recent years this problem has been tackled with increasing success. In the second case, the problem may be expressed as ‘How can a computer represent the proofs of mathematics?’. If this question is understood as ‘How can a computer perform the proofs of mathematics?’, then the progress in automated theorem proving provides a ready answer. However, this would be to misunderstand the original question, which did not ask how mathematics could be done by a machine, but how it is and has been done by mathematicians. The traditions of formalization and automated theorem proving upon which much work in MKM has been based are heavily indebted to the methods of formal logic. However, formal logic is a poor guide to mathematical practice, as mathematicians seldom use it to write proofs. Although most mathematical proofs may in principle be formalized, the process is often arduous and can dramatically reduce intelligibility. For this reason such formalization is rarely