Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for

Three topics are discussed that relate to the teaching of linear algebra using computers (here the term computers includes calculators). The rst topic is the variation in notation and terminology both between books and computer systems, and between dierent computer systems. The second topic is the importance of numerical linear algebra, and how it becomes more dicult to avoid numerical aspects of the subject once computers are used in teaching. The nal topic is the Turing factoring of a matrix. This is a factoring approach to row reduction, and if it is taught in courses, students can transfer what they know to computers with a minimum of diculty. 1