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

eyond unification 159elementary proofs of sentences formulated in the language of that theory—atleast in principle, since, as Brumfiel remarks, ‘[i]t certainly might be verytedious, if not physically impossible, to work out this elementary proof’(Brumfiel, 1979, p.166).Another proof method consists in using a so-called transfer principle whichallows to infer the truth of a sentence for all real closed fields from its beingtrue in one real closed field, like the real numbers. Despite the fact that thetransfer principle is a very useful proof method, Brumfiel does not make anyuse of it, and he is very clear about this.In this book we absolutely and unequivocally refuse to give proofs of this secondtype. Every result is proved uniformly for all real closed ground fields. Ourphilosophical objection to transcendental proofs is that they may logically provea result but they do not explain it, except for the special case of real numbers.(Brumfiel, 1979, p.166)Brumfiel prefers a third proof method which aims at giving non-transcendentalproofs of purely algebraic results. This does not mean that he restrictshimself to just elementary methods; he does use stronger tools, but it is crucialthat they apply uniformly to all real closed fields. It is also clear from thecontext that Brumfiel does not consider proofs obtained from applying thedecision procedure for RCF as explanatory.² And in fact they are almostnever carried out in practice because their length makes them unwieldy andunilluminating.To illustrate Brumfiel’s point and subsequently confront Kitcher’s theory ofexplanation with it, we would like to give a short exposition of a theorem,which is also mentioned by Brumfiel himself (p. 207), and various (sketchesof) proof types corresponding to the classification above.6.3.1 Some concepts from semi-algebraic geometryLet’s start with the definition of ‘real closed field’: a field which admits aunique ordering, such that every positive element has a square root and every² The (elementary) proofs one can construct on the basis of the decision procedure prove theirconclusion uniformly for all real closed fields because the decision procedure is carried out independentlyof the ground field. Despite their uniformity, however, Brumfiel does not take this kind of proofas the optimal model which should guide mathematical research—on the contrary. First, as alreadymentioned these proofs are not always feasible; due to their size it may in certain cases not even bephysically possible to construct them. Second, Brumfiel as a rule prefers different uniform (in generalnon-elementary) proof techniques not only for studying the rich non-elementary theory of real closedfields but also for dealing with elementary sentences (i.e. first-order sentences in the language of orderedfields), even in cases where one could, in principle, find an elementary proof, i.e. ‘even if a statementturns out to be equivalent to an elementary statement, it may be unnatural to dwell on this fact, andeven worse to be forced to depend upon it’ (Brumfiel, 1979, p.166).