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

128 kenneth mandersgeometrical claims. This, rather than unqualified assent, is to be shared acrossspace, time, and participants. Probing is an integral part of that responsibility,its element of criticism.The normative structure of this dissent, with its insistence on uniformlyacceptable resolution of differences, explains how the practice is compatiblewith an agentless conception of mathematics even though dissent implies morethan one epistemic role. For, given that the practice was reasonably successfulover the long term in providing artifact support for uniform responses, thedistinct roles we recognized do not give rise to sustained distinct-commitmentstances of the sort which would force us to admit multiple epistemic agents inour long-term perspective. We were able to construe I.7, eachofitscasesinfact, as a response to probing the outcome of certain compass constructionsin extremes of sensitivity; where a uniqueness question arises gratifyinglysimilar to Protagoras’ challenge on the intersection of circle and tangent. Wemight notice, however, that such appearance control probing tends to beurgent primarily in extremes, in atypical diagrams. Our discussion of III.2confirmed this: sensitivity problems in telling whether the interior of a chordlies inside the circle arise only in extreme situations. In particular, neithernatural target of probing from which III.2 might arise, in the arguments forIII.1 and III.13, involves points near the end of a chord. Thus, only extremelyshort and hence atypical chords raise an appearance control concern in thosearguments.We must therefore cast around for what drives the practice to developexplicit proofs for the many propositions that conclude some explicit co-exactcondition that could be read off from all but quite atypical diagrams. A practicemight conceivably hold disarray at bay by simply not considering such atypicaldiagrams.²² Or rather than not considering such diagrams, one might hope toassess them in light of ‘neighboring’ diagrams—ones related by continuousvariation of the diagram which encounters no recognizable change of diagramappearance—which are clear; leaving the assimilations to be probed. Forexample, one could presume that very short chords of circles give the sameappearance as longer ones, just as one presumes that diagrams too large to drawgive the same appearance as smaller ones; all the while accepting the obligationto probe whether some boundary is crossed on the way to some extreme case,at which the appearance would change; and to probe for appearance-changingdegeneracy of extreme cases themselves.²² An analogous attempt was made by the Italian School in algebraic geometry at the turn of thecentury; which made tremendous progress by limiting its attention mainly to ‘generic cases’; and whilethis is now discredited, a similar strategy underlies the focus on ‘stable’ singularities in current singularitytheory (Arnold (1992) Ch.8.).