ABSTRACT
A distinctive feature of mathematics, that feature in virtue of which it stands as a paradigmatically rational discipline, is that assertions are not accepted without proof. To establish, for example, that the internal angles of a triangle add up to two right angles, it is neither sufficient nor appropriate to go round measuring the internal angles of triangular figures. By proof is meant a deductively valid, rationally compelling argument which shows why this must be so, given what it is to be a triangle. But arguments always have premisses so that if there are to be any proofs there must also be starting points, premisses which are agreed to be necessarily true, self-evident, neither capable of, nor standing in need of, further justification. The conception of mathematics as a discipline in which proofs are required must therefore also be a conception of a discipline in which a systematic and hierarchical order is imposed on its various branches. Some propositions appear as first principles, accepted without proof, and others are ordered on the basis of how directly they can be proved from these first principles. Basic theorems, once proved, are then used to prove further results, and so on. Thus there is a sense in which, so long as mathematicians demand and provide proofs, they must necessarily organize their discipline along lines approximating to the pattern to be found in Euclid's Elements.
