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

eyond unification 165the necessary metatheory precisely but can leave its concrete delimitationsomewhat open.The proof in (III) belongs to a systematization of K relative to a widerframework, KIII ∗ , of real algebraic geometry. Brumfiel stresses that K∗ III is notconfined to elementary methods. ‘In fact, we use Dedekind cuts, total orders,and signed places⁸ repeatedly. The point is, in the form we use these conceptsthey apply uniformly to all real closed fields. One advantage to developing suchtechniques is precisely that one is not tied down to ‘‘elementary sentences’’ ’(Brumfiel, 1979, p.166).Now we can ask what the best systematization of K is. If we can find asystematization that makes use of only one argument pattern to generate allof K, then any other systematization which uses a greater number of patternsis inferior. It turns out, then, that the best systematization of K is the oneprovided by the Tarski–Seidenberg decision procedure for RCF, i.e. the oneexemplified by (I). The uniformity of the procedure shows that we havein principle a single argument pattern which we can use to generate all ofK. Comparing this situation with those in which we try to prove arbitraryelementary sentences of K by means of proofs such as those given in (II) and(III) makes it obvious that we can’t get by with just one pattern of argument.Consider proofs of type (III). The specific argument used to determine thevalidity of the proposition asserting that a polynomial assumes a maximumon any arbitrary closed and bounded semi-algebraic set would be useless forderiving, say, the algebraic form of Brouwer’s Fixed Point Theorem.⁹Similarly in the case of proofs of type (II). For in this case we need todetermine the truth of ϕ under consideration in R, or some other real closedfield. And only after this task has been accomplished can we appeal to thecompleteness of RCF. But the first task, the determination of the truth valueof ϕ in some real closed field, is not a uniform process; rather we need, aspointed out with respect to case (III), different argument patterns for differentclasses of ϕ. In short, whereas with respect to the specific problem concerningmaxima of polynomials all three systematizations exhibit a certain uniformity inthe treatment of sentences that are instances of this problem, only the strategythat relies on the Tarski–Seidenberg decision procedure can be extended to⁸ Cf. Brumfiel, 1979, p.144f: Let be a totally ordered field. We adjoin symbols ∞ and −∞ to and extend the operations of addition, subtraction, multiplication, and division between ∞, −∞ andelements a ∈ (e.g. ∞+∞=∞, a = 0, etc. Some expressions, like 0 ·∞, remain undefined). If∞K is a field by a signed place, withvaluesin, we mean a function f : K → ∪{∞, −∞}, such thatf (x + y) = f (x) + f (y), f (xy) = f (x)f (y), andf (1) = 1, whenever the terms are defined.⁹ Let S be a closed, bounded, convex, semi-algebraic set in R n , for some real closed field R. Thenevery continuous rational function mapping S to itself has a fixed point in S.