Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

180 michael detlefsen76a37–40, 77a26–35, 78a10–13). Perhaps most fundamental, though, wasAristotle’s view of the preconditions required for the extension of knowledgethrough argumentation.All instruction given or received by way of argument proceeds from pre-existentknowledge. ... The mathematical sciences and all other speculative disciplines areacquired in this way ...The pre-existent knowledge required is of two kinds. In some cases admission ofthe fact must be assumed, in others comprehension of the meaning of the termused, and sometimes both assumptions are essential.Aristotle (anc1), 71a1–4; 12–14For Aristotle, then, the development of mathematical knowledge was inlarge part development by reasoning and such development depended on priorknowledge of its topic or subject. Specifically, it demanded knowledge of the‘what’ of its topic.Conservation of topic was thus built into Aristotle’s conception of howknowledge develops through inference and was itself an important aspect ofpurity. There was in addition, however, another basis for purity, namely, theneed for a necessary connection between the subject and predicate of a theorem.We ... possess unqualified scientific knowledge of a thing ... when we ... knowthe cause (aitia) on which the fact depends as the cause of the fact and of noother, and, further, that the fact could not be other than it is.Aristotle (anc1), 71b9–12To secure such a connection, mathematical demonstrations had ultimatelyto be based on knowledge of their subjects’ essences.The ‘why’ is referred ultimately ... in mathematics ... to the ‘what’, to the definition[horos] of straight line or commensurable or the like ...Aristotle (anc4), 198a16–18²In Aristotle’s view, then, purity increased epistemic quality. A pure proofprovided knowledge that the predicate of its conclusion (the minor term of theproof ) held of its subject (the major term) solely because of what the subjectin itself was. It showed the very whatness (i.e. the essence) of the subject of atheorem to be the ‘cause’ of its having the property expressed by its predicate.² See also Aristotle (anc2), 90a31–33, where it is stated that:... to know what a thing is [ti estin] is the same as to know why it is [dia ti estin]...and this isequally true of things in so far as they are said without qualification to be as opposed to beingpossessed of some attribute, and in so far as they are said to be possessed of some attribute suchas equal to right angles, or greater or less.