Euclid's Elements and the Axiomatic Method

algebra, analytic geometry, real number theory, and mathematical logic. It is not possible to describe here the evolution of the idea of structure in the modern era. An important step in it is the realisation that Euclidean geometry has an interpretation in the universe of the real numbers-a realisation made possible by the existence of analytic geometry and making possible the unification of two apparently diverse subjects, the study of space and the study of number. One effect of this unification was the clear conception of space as a class of points and of geometry as the study of the relations existing between them. Given the interpretability of the class of points as the class of real numbers (pairs or 1 The distinction between logically correct reasoning and reasoning based on logic shows the inconclusiveness of Szab6's (I964, PP. 42-8) argument that the use of indirect proof is a sign of the non-intuitive character of Greek mathematics. One cannot have an image of what is not the case geometrically, but one can use an image or diagram to show that something is geometrically impossible, as in Socrates' argument in the Meno or in the proof of I, 6. This content downloaded from 207.46.13.137 on Fri, 22 Apr 2016 05:53:28 UTC All use subject to http://about.jstor.org/terms Euclid's Elements and the Axiomatic Method 299 triples of real numbers) and of the geometric relations as relations between real numbers, it is a short step to the view that geometry is nothing but the study of any class of objects and relations on those objects isomorphic to the class of points and the original geometric relations. In other words, it is a short step to the view that geometry is the study of abstract structure. Logic made possible the perfection of this view by providing a theory of structure-preserving inference and a set of formal rules enabling one to derive from a set of axioms all and only the sentences true under every interpretation making the axioms true (in every structure in which the axioms are true). It is important to realise, however, that these formal rules depend on the notion of structure, interpretation, or model for their justification. I do not believe that the Greeks possessed the notion of mathematical structure in this sense. The descriptions of mathematics which have survived from antiquity never employ notions like that of structure. The mathematical practice of Euclid and other mathematicians of antiquity suggests strongly the definition of geometry as the science of magnitudes (megethe). The geometrical magnitudes which Euclid studies are not 'structural objects', but, as I have argued, the intuitively perceived spatial objects which are characterised in his definitions. There is no indication that he considered these objects (e.g. points) as constituting a system or structure. Had he so conceived them, he might equally have had the idea of an isomorphic system; and had he had this idea, it is hard to see why the Elements should contain so many 'logical gaps'. For it is precisely the development of this idea which made it possible for the moderns to discover these gaps. To attribute an understanding of abstract structure to Euclid or his contemporaries is to obscure, if not to obliterate completely, the revolutionary character of nineteenth-century mathematics. The absence of an understanding of mathematical structure among the ancients makes it misleading and probably impossible to call Euclid's argumentation logical or formal. For it is in terms of structure that the idea of logical or formal argumentation is given a precise sense. Perhaps the major obstacle to an acceptance of the interpretation of Euclid's arguments as thought experiments is the belief that such arguments cannot be conclusive proofs. In particular, one might ask how consideration of a single object can establish a general assertion about all objects of a given kind. Part of the difficulty is due, I think, to failure to distinguish two ways of interpreting general statements like 'All isosceles triangles have their base angles equal'. Under one interpretation the statement refers to (talks about, presupposes) a definite totality-that is, the class of all isosceles triangles-and it says something about each one of This content downloaded from 207.46.13.137 on Fri, 22 Apr 2016 05:53:28 UTC All use subject to http://about.jstor.org/terms