 Boethius begins with an argument against universals as an object of enquiry. Everything that really exists is one in number, but nothing that is common to many at the same time can be one in number. But universals are common to many at the same time. And so universals do not exist in reality, but in thought alone. Thoughts, Boethius continues, are of two sorts: those which derive from their object in the way it is (call them ‘corresponding thoughts’) and those which do not. If the thoughts that are universals were corresponding thoughts, then universals would also exist in reality. Since they do not, universals are non-corresponding thoughts, and non-corresponding thoughts are empty. Enquiry into universals (and therefore into the five predicables studied in the Isagoge) should therefore be abandoned. Boethius’s way of tackling this objection is to challenge just the very final stage. Non-corresponding thoughts, he argues, are not empty if they are abstractions. Consider a mathematical object such as a line or a point, which the mathematician contemplates by abstracting from the material body of which it is part. No such thing exists in reality as an immaterial line or point, and yet the mathematician’s thought is not empty or misleading. The case is similar if we disregard the accidental features of some particular thing (John Marenbon, for instance) and are left just with his nature of man. This line of reply, as Alain de Libera (1999, 159–280) has shown, goes back to Alexander of Aphrodisias or his followers. Boethius, however, goes on to give it his own particular twist, by suggesting that the universals produced by abstraction are not merely the constructions of the mind, but do grasp reality as it is. Although this line fits oddly with the argument from which Boethius set out, he may already be anticipating the Principle of Modes of Cognition, which he proposes in the Consolation (see  Section 6  below).