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

88 kenneth mandersto remedy this by strengthening the discursive text. Felix Klein comments: ‘Thesignificance of these axioms of betweenness must not be underestimated ... ifwe wish to develop geometry as a really logical science, which, after the axiomsare selected, no longer needs to have recourse to intuition and to figures for thededuction of its conclusions... Euclid, who did not have these axioms, alwayshad to consider different cases with the aid of figures. Since he placed so littleimportance on correct geometric drawing, there is real danger that a pupil ofEuclid may, because of a falsely drawn figure, come to a false conclusion.’⁴Twentieth-century philosophers have tended to follow mathematicians intheir assessments of traditional geometry. Ayer, for example: ‘diagrams ...provide us with a particular application of the geometry, and so assist us toperceive the more general truth that the axioms of the geometry involvecertain consequences.... most of us need the help of an example...[T]he appealto intuition ... is also a source of danger to the geometer. He is tempted tomake assumptions which are accidentally true of the particular figure he istaking as an illustration.... It has, indeed, been shown that Euclid himself wasguilty of this, and consequently that the presence of the figure is essential tosome of his proofs.’⁵Thus, the first task of an account of geometrical reasoning in which diagramsare treated as genuinely inferentially engaged ‘textual’ elements is to augmentthe well-established logical construal of the discursive text by a reconstructionof the standards for reading and producing diagrams.⁶Extant ancient texts give little explicit discussion of these standards, onwhich we anyhow cannot expect full unanimity over the centuries of Greekmathematical practice. Some light is shed on them by occasional recordsof dispute on geometrical propriety, and by records of proposed cases orobjections.⁷The key to reconstructing standards for producing and reading diagramsis the realization that diagram and text contribute differently, so as to makeup for each other’s weaknesses. Some things may be read off directly from⁴ Klein (1939, p.201); quoted from Mueller (1981, p.5).⁵ (Ayer, 1936, p.83). (My emphasis. I thank Ali Behboud for pointing out this passage to me.)Philosophers’ concern about diagram use in geometry go back at least as far as Plato’s Cratylus:‘... geometric diagrams, which often have a slight and invisible flaw in the first part of the process, andare consistently mistaken in the long deductions that follow’ ( Jowett transl., at 436D).⁶ Wilbur Knorr has pointed out to me that the standards for diagrams in solid geometry, and in itsapplications in mathematical astronomy, must be quite different indeed from those in force in planegeometry. Only plane geometry is considered here.⁷ The most informative remaining source is Proclus’ Commentary on the First Book of Euclid. Fordiscussion, see especially Heath (Euclid,vol.1) and Mueller (1981). There may be some bias in focussingon the foundationally oriented commentary tradition represented by Proclus; see the comments onmathematical impact of the Alexandrian foundational school, in Knorr (1980, esp. pp. 174–177).