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

82 kenneth mandersand know to articulate are broadly but constrainedly ‘inferential’. It is plainthat we are missing a lot. Present philosophical resources for understandingthe understanding seem insufficient to explain why intercourse with diagramsremains essential to geometrical thought. While their role is much less clear-cutin 20th-century geometrical theories, one look at the diagrams Hilbert entersnext to his axioms of order should convince us of this—what are they doingthere, and throughout the rest of his Foundations of Geometry?4.1 Euclidean diagrams: artifacts of controlor semantics?At its most basic, a mathematical practice is a structure for cooperative effortin control of self and life. In geometry, this takes many forms, starting withthe acceptance of postulates, and the unqualified assent to stipulations—andas it appears, for now, to conclusions—required of participants. Successes ofcontrol may be seen in the way we can expect the world to behave according tothe geometer’s conclusions; the way one geometer centuries later can pick upwhere another left off; the way geometers can afford not to accept contradiction.When the process fails to meet the expectations of control to which the practicegives rise, I speak of disarray, or occasionally, impotence. Such occurrences aredisruptive of mathematical practices; they tend to reduce the benefits toparticipants and to deter participation. At best, they motivate adjusting artifactuse, modifying the practice to give similar benefits with less risk of disarray.The notion that the intellectual might be seen as some kind of game hasrecently been at the center of contention. Can’t games just be made up?Kids make up games all the time; grown-ups get elected and paid to do it. Ifintellectual practices are games, what about the image of rigorous hard-workingScience giving us a solid grip on Reality?Theoretical geometry, which after all can be practiced in many forms andfor many purposes, is hardly a context in which to address many such concerns,and facing any head-on would anyhow detract from our investigation. Theimpression, however, that making up a game involves excessive freedom,reflects a lack of grasp how hard it is to get a good game going, how hardit is to satisfy the varied and stringent constraints on an effective intellectualpractice, especially one that must handle a fairly subtle and informative theory(such as Euclidean geometry) with minimal risk of disarray. Along the way,we shall become acquainted with such constraints and the forces shaping thepractice to which they give rise.