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

162 johannes hafner and paolo mancosu(III) Another way of establishing the theorem relies on purely algebraicmeans exploiting the fact that if A ⊂ R n is a closed and bounded semi-algebraicset and g : A → R p a continuous semi-algebraic mapping, then g(A) is a closedand bounded semi-algebraic set.⁴Since a polynomial f (x 1 , ... , x n ) is a continuous semi-algebraic mappingand we assume a closed and bounded semi-algebraic set S ⊂ R n to be given,it follows that f (S) is a closed and bounded semi-algebraic set. But sincef (S) ⊂ R, it is a finite union of points and of closed and bounded intervals.And so f (S) has a maximum.6.4 Assessment of Kitcher’s modelThe previous section exemplified different approaches to systematizing knowledgeabout real closed fields. More precisely, in the following we will beconcerned with systematizations of the theory RCF, which,torecall,istheconsistent and deductively closed, in fact complete, set of elementary sentencestrue in any real closed field (or, equivalently, following from the axiomsof RCF).Before entering into the discussion of how Kitcher’s account would rankdifferent systematizations of RCF according to their unifying power let’s pausefor a moment to address a worry one might have, at the very outset, concerningthe choice of RCF as the body of statements to be systematized. To be sure,mathematicians working in semi-algebraic geometry are as a rule not onlyinterested in (systematizing) elementary sentences but also sentences that gobeyond RCF by, for instance, quantifying over functions and sets, or sentencescoming from a wider framework in which RCF is embedded like real analysisor category theory (we’ll return to this below). The question might thus beraised how faithful to mathematical practice our focus on RCF indeed is.⁵However, this worry can easily be dispelled.First of all, the choice of some non-elementary context for the studyof real closed fields—be it a more expressive language or certain moreencompassing mathematical theories, etc.—doesn’t make the elementary partof semi-algebraic geometry disappear or irrelevant. Whatever the context,RCF still forms a (precisely definable) subset of K, the totality of acceptedsentences in the respective context. Hence any systematization of K has to⁴ Cf. Brumfiel, 1979, p.207; or Bochnak et al., 1998, p.40f.⁵ This point has been urged particularly by Jeremy Avigad.