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

72 kenneth manders3.2 Geometric generalityA traditional quandary about geometrical demonstration, famously discussedby Locke, Berkeley, and Kant, is how one particular drawn diagram can justifya general claim.1. Some20th century commentaries on geometrical generality may missthe mark because our ‘strict-universal’ standards for universal quantification,by including singular (in the mathematical sense) instances, disagree withtraditional mathematical usage.A traditional geometrical demonstration only claims to establish its propositionfor non-trivial instances of its initial diagram, in which items such as anglesand triangles contain proper regions (respecting the co-exact component ofthese notions, explained above). The demonstration of I.1, for example, doesnot purport to construct an equilateral triangle on a ‘side’ consisting of a singlepoint. Individuating claims in this way is rational: as the example suggests,direct argument for (analogs of) claims in such limiting cases tends to be mucheasier than the demonstration at hand.Perhaps because later algebraic and calculus methods establish (most) limitingcases automatically, non-triviality understandings on primitive geometricnotions eventually vanished. From this later point of view, singular situationsbecame ‘exceptions’; traditional general claims and arguments have the force of‘admitting’ exceptions, i.e. not applying to certain singular instances.Strict-universal usage is surprisingly recent: ‘The subject treated ... illustrateswell one of the most striking tendencies of modern algebraic and analyticwork, namely, the tendency not to be satisfied with results that are merelytrue ‘‘in general’’, i.e. with more or less numerous exceptions, but to strivefor theorems which are always true.’ ‘The great importance of this tendencywill be apparent if we remember that when we apply a theorem, it is usuallyto a special case. If we merely know that it is true ‘‘in general’’, we mustfirst consider whether the special case ... is not one of the exceptional cases inwhich the theorem fails.’ (Bôcher (1901)) For the phenomenon of theorems‘with exception’, see also Sorensen (2005).2. Whether particular drawn diagrams suffice to justify general geometricalclaims (in the sense just recovered) depends on how demonstrations usediagrams. Particularity is not an incurable infectious agent.Beth (1956) suggested, and Hintikka developed, the idea that the setting-outof the diagram in ancient demonstration be understood by analogy to thesetting-aside of letters in universal generalization (UG) rules in modern logic,say as opening a scope for UG in a natural deduction system. But does analogy