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

the euclidean diagram (1995) 107never ambiguity as to the appearance of the diagram formed under properdiscipline from given data, adequacy of case distinction still requires controlover complete families of diagrams.But how are participants to see whether a given selection of variantdiagrams (after a construction complicates the diagram, or even at the outset ofargument) is exhaustive of the possibilities which require separate argument? Ifit is not exhaustive, how to locate further alternatives? Traditional practice lacksprocedure either to certify completeness of case distinctions, or to generatevariants. Except in special situations (negation of equalities) there are nodiscursive arguments to this effect; and while one would expect to look tothe diagram for suggestions in locating variant appearances, there is no hint ofclear-cut diagram-based procedures to do so.Traditional geometrical practice largely lacks artifact support that wouldallow participants to survey diagram appearances once and for all, or evento recognize when they had done so. First, the possibilities for free choices,whether at the beginning or along the way in building up a diagram, areunsurveyable. Second, (even without ambiguity as to the appearance ofdiagrams properly constructed from given data) the relationship between thosefree metric choices and ultimate diagram appearance is unsurveyable. Diagramscome one at a time; having made one set of choices, we may hope to tell theconsequences; but we thereby usually gain little grasp of the range of possiblealternative outcomes from other choices.These limitations of artifact support in traditional geometry may be broughtout by contrast with a quite limited but historically significant exception: locusdescriptions of curves. A good case is the well known result (extending III.21and converse) that the locus of points O that make a fixed given angle POQwith two given points P and Q is a circle through P and Q. Inaweaksense, this gives a survey of all possible ways, given P, Q, and the angle, oflocating O to satisfy the condition. It allows us to read off a little about (say)how triangles OPQ of this type would interact and form regions with furtherdiagram elements. We can tell what the possibilities are for O to lie withrespect to other regions in the same diagram, by seeing which such regionscontain points of the diagram; we can find all such triangles OPQ with sideparallel to a given line by intersecting the circle with the lines through Por Q parallel to the given one... But we still can’t read off all such trianglesOPQ; we can draw them one at a time, but if we draw more than one ortwo, the diagram becomes too cluttered to be of any use (we lose appearancecontrol).The relationship between case distinctions and discrepancies in capacityfor uniform proceeding is, however, quite general. One may also come to