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

164 johannes hafner and paolo mancosuand the different systematizations connected with (I), (II), and(III) are casesin point.Systematization (I) uses some metatheoretical machinery in addition to K(= RCF ). The decision algorithm for sentences in K is not itself a theorem ofthat theory but a statement in its metatheory. Let KI∗ be K together with somemetatheory of K. We leave it open to some extent how much metatheoryK ∗ Ihas to contain—certainly at least the decision algorithm but perhaps more,like induction, to actually establish the main properties of the algorithm. In (I)we indicated how the set P, i.e. the set of all instances in K of the theorem,could be construed as the conclusion set of a certain systematization based onthe Tarski–Seidenberg decision algorithm. This approach can be generalizedto apply to all of K, since the decision algorithm works for all elementarysentences. Somewhat streamlined we thus have the following systematizationof K relative to KI∗ which is given by a simple argument pattern:(1) F(ϕ) = x(2) ψHere ‘ϕ’, ‘x’, and ‘ψ’ are dummy letters, whereas F denotes (some fixedspecification of) a decision algorithm for K.Filling instructions:Replace ‘ϕ’ byasentences in the language of RCF.Replace ‘x’ bythevalue,0or1,ofF at s.Replace ‘ψ’ bys, ifF(s) = 1andreplace‘ψ’ by¬s, ifF(s) = 0Classification:Thesentence(1) isapremise.Thesentence(2) follows from (1) by metatheory, i.e. by using facts aboutthe functioning of F.The proof of the theorem along the lines of (II) employs transcendentalmethods, like the use of completeness and compactness, which hold for Rbut not for real closed fields in general. Beyond that the transfer principleis used to argue that the instances of the theorem hold in every real closedfield. Hence in this case the superset KII ∗ includes some part of the theory ofthe classical real numbers and a formulation of the Tarski–Seidenberg transferprinciple. Because this requires the use of model-theoretic concepts and methodsin addition to syntactical ones in order to talk about truth in certain realclosed fields, notably the real numbers, the metatheoretic part of KII ∗ is moreencompassing than the one of KI ∗ . Again, however, we don’t need to specify