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

eyond unification 163cover also the sentences in RCF. Moreover, the elementary (sub)theory is byno means rendered peripheral. Indeed, the very notion of semi-algebraic set,which is fundamental to semi-algebraic geometry, is characterized in termsof first-order formulas. It is a salient fact, an immediate consequence of theTarski–Seidenberg theorem, that the semi-algebraic sets coincide with the setswhich are definable by formulas in the language of RCF. In other words, anyconstruction out of semi-algebraic sets that can be expressed in terms of anelementary sentence yields again a semi-algebraic set. This fact is of majorimportance in the light of its ‘consequences which are hard to obtain if oneuses the definition of a semialgebraic set directly’ (Andradas et al., 1996, p.9).Even G. W. Brumfiel, who decidedly rejects the Tarski–Seidenberg transferprinciple as a proof method, agrees that this application of the Tarski–Seidenbergtheorem is ‘a very efficient tool’ (Brumfiel, 1979, p.165) and he uses it himselfoccasionally in later parts of his book.⁶It is clear that elementary formulas demand special attention of anybodywho relies on the Tarski–Seidenberg transfer principle as an essential toolin the study of semi-algebraic geometry (cf. Bochnak et al., 1998; Andradaset al., 1996). Yet, they are not just ‘interesting’ for the purely methodologicalreason that the transfer principle is applicable only to elementary formulas.Even Brumfiel, who is not in any way restricted by such methodologicalconsiderations, is well aware of the fact that many of the theorems he proves(by his preferred methods) can be expressed by elementary sentences.⁷ In short,the theory RCF is anything but trifling.Let us now return to the task of ranking different systematizations ofRCF and the assessment of how Kitcher’s model fares in this respect. Ourstarting point, the set to be systematized, is the theory RCF. Tobeinlinewith Kitcher’s terminology let us call this set K. As was pointed out above,systematizations of a (consistent and deductively closed) set of sentences S mayhave to go beyond S and draw on premises from S ∗ , a consistent superset ofS. That’s what necessitated a slight modification of Kitcher’s original model⁶ Concerning the use of the Tarski–Seidenberg theorem to show that a set defined in terms ofsemi-algebraic sets by an elementary sentence is semi-algebraic Brumfiel emphasizes the following.‘[T]his type of application actually provides a proof that the asserted set is semi-algebraic, simultaneouslyfor all real closed fields, in fact an elementary proof. The reason is, any single elementary sentence isjust a special case of the theorem’ (Brumfiel, 1979, p.165). In contrast, proofs based on the transferprinciple lack precisely this kind of uniformity. However, as pointed out above already Brumfiel is not anadvocate of developing real algebraic geometry exclusively by elementary methods let alone by relyingon the Tarski–Seidenberg decision procedure.⁷ ‘We admit that many of our proofs are long and could be replaced by the single phrase‘‘Tarski–Seidenberg and true for real numbers’’. However, we feel the effort is worthwhile’ (Brumfiel,1979, p.166).