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

174 johannes hafner and paolo mancosuvocabulary which remains is idling. The presence of that nonlogical vocabularyimposes no constraints on the expressions we can substitute for the dummysymbols. (Kitcher, 1981, p.528)Now, is the pattern TSP in the same ball park as the other spuriousunifications we discussed? Does it, for instance, parallel in all the relevantrespects the pattern of the religious fanatics? Apparently TSP can in fact beruled out on the basis of the new requirement by a move which is analogousto the ones previously considered. If we change the filling instructiontoReplace ‘ψ’ bys, ifF(s) = 1andreplace‘ψ’ by¬s, ifF(s) = 0.Replace ‘ψ’ by any sentence (in the language of ordered fields).we obtain a pattern which, apparently, allows the derivation of any old sentence(in the given language). The problem with such a quick dismissal of TSP asachieving merely a spurious unification is that this argument pattern imposesmore constraints on ‘ψ’ over and above what is expressed by the fillinginstruction concerning this dummy letter. Hence the simple modificationof the filling instruction above is not sufficient to license the derivation ofany old sentence. Even if the filling instruction concerning ‘ψ’ is relaxedcompletely, the part of the classification of the argument pattern is still in placewhich specifies that sentence (2), i.e. ψ, follows from (1) bymetatheory.And thisrequirement is certainly not fulfilled just by any arbitrary sentence ψ. Bydefinition, the classification identifies the logical structure that instantiationsof a pattern must exhibit, which, to recall, is one of the two constraints thatdetermine the stringency of a pattern. Hence, in short, TSP is too stringent tobe ruled out by the new requirement.¹⁸ A valid complaint one might haveabout TSP, however, is that it is too compressed, its classification somewhatopaque, leaving in a perhaps misleading way certain elucidations to the fillinginstructions. This can be remedied to some extent by bringing out moreclearly the structure and functioning of the argument pattern in the followingreformulation TSP ′ .(1) F is some fixed specification of a decision algorithm for RCF.(2) F(ϕ) = x¹⁸ If—on a mistaken interpretation of the workings of filling instructions—one allowed thepossibility that the modified filling instruction overrides the classification in any given conflict betweenthem, such that indeed any sentence ψ could be derived, then this would yield in effect an incoherentargument pattern or, rather, one wouldn’t be left with an argument pattern any more (in the originalsense of the word). If, on the other hand, the (modified) filling instruction does not interfere with theclassification—as it should be—then the modification has no effect on the functioning of the pattern.In particular, this modification cannot render the pattern spurious.