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

192 michael detlefsenanother basic question—namely, how the notion of topic ought rightly to beconceived.One facet of this question concerns how to judge what it is that a problem isabout. Suppose, for instance, that a problem is formulated as a problem concerningthe real roots of a certain type of polynomial equation. Suppose, in addition,that these roots can only be found by methods that make ineliminable use ofcomplex numbers.²⁵ What should we then say our problem is about? Since allthe roots we’re interested in are reals, this suggests that the problem is about realnumbers. That is, it is about those things to which the terms appearing in it refer.On the other hand, we might rather take our problem to be about thosethings and concepts we seemingly must bring into our reasoning in order tosolve it.²⁶The fourth paper mentioned above (i.e. Woo 1971) gives a proof of atheorem from measure theory (the Lebesgue Decomposition Theorem) andclaims that this proof is purer than previous proofs in that it makes no appealto measure theory beyond the definitions needed to state the theorem. Hereagain, though, it’s not entirely clear what the motive is. Specifically, it’s notclear whether purity is being assumed to provide an epistemically superiorjustification for the theorem proved, or a more effective instrument forfurthering certain larger epistemic (including, perhaps, pedagogic) projects.²⁷In all these cases we see a concern for purity of a broadly topical type. Themotives are not always the same, though they seem generally to be directedtowards epistemic concerns—concerns, particularly, for the quality of solutionsoffered to problems and the efficacy of such solutions as a foundation for thefurther development of our mathematical knowledge.7.4 ConclusionThe ideal of purity that seems to figure most significantly in modernmathematical practice is what I’ve referred to here as topical purity. This²⁵ I’m obviously thinking of the casus irreducibilis or irreducible case of the general cubic here.²⁶ There are, of course, questions to be asked concerning the meaning of ‘must’ and ‘solve’. Thatthis is so, though, doesn’t make the question any less relevant, only more intricate.²⁷ As regards pedagogic projects, the idea seems roughly to be that proofs ought to be conductedwith as little overhead as possible. Aristotelian purity would seem to constitute a minimum here. Surelyit would make no sense to try to prove a theorem if not enough defining had been done to make thecontent of the theorem intelligible. On the other hand, once such a point has been reached, a proofbuilt upon exactly that learning would represent a remarkable pedagogical efficiency. For exampleslinking purity with such motives see, Mullin (1964), Luh (1965), Spitznagel (1970), and Jenkins (1982).