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

172 johannes hafner and paolo mancosuhow stringency should be weighted against the other criteria, size of conclusionset and size of the basis, it is impossible to incorporate the stringency criterioninto the determination of the best systematization of a given K. AtonepointKitcher even suggests that the notion of stringency could be clarified throughthe notions of explanation and unification (cf. Kitcher, 1981, p.519), thuscourting circularity. Due to all of these difficulties the stringency criterioncould not be employed in our discussion so far.The only context in which Kitcher actually puts to use the conceptof stringency and formulates a workable requirement in terms of it is hisdiscussion of ‘spurious unifications’ which threaten to trivialize his account.The problem is posed by trivial yet maximally encompassing systematizationsof our beliefs on the basis of single patterns like ‘from α&B infer α’ or,even more simple, ‘from α infer α’, where ‘α’ istobereplacedbyanysentence we accept. In order to exclude such patterns Kitcher invokes thecriterion of stringency. As he points out, the mentioned systematizations areindeed highly successful according to the criteria of generating a large numberof beliefs on the basis of a small number of patterns but they fail badlyin terms of stringency. That is, ‘both of the above argument patterns arevery lax in allowing any vocabulary whatever to appear in the place of α’(Kitcher, 1981, p.527). Hence they are (intuitively) non-stringent and shouldbe excluded.Now, on the face of it, it might seem that a very similar case could be madeout against our systematization E I (K). To recall, E I (K) unifies the set K byusing the following single argument pattern, ‘TSP’ for short, which relies onthe Tarski–Seidenberg decision algorithm.(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.