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

eyond unification 161||x|| = √ x 2 1 +···+x2 nB n (x, r) ={y ∈ R n |||y − x|| < r}(open ball)The Euclidean topology on R n is the topology for which open balls form abasis of open subsets. Complements of open sets are closed. Polynomials arecontinuous with respect to the Euclidean topology.6.3.2 A simple theorem and a variety of proofs (and systematizations)Now we are in a position to state a simple theorem and compare differentpossible approaches to its proof or to ways of systematizing its instances.Theorem. A polynomial f (x 1 , ... , x n ) assumes a maximum value on any boundedclosed semi-algebraic set S ⊂ R n .(I) Within RCF the result cannot be proved as stated for we cannot quantifyover polynomials and semi-algebraic sets. The best we could do is to proveits instances. For instance, RCF proves that x 3 has a maximum on [0, 2].We could, in principle, arrive at this proof by running the Tarski–Seidenbergdecision procedure and getting 1 as output. From there it would be a mechanicaltask to provide the explicit proof in RCF. And every single instance of thetheorem could be obtained in this way. In other words, if we let P ⊂ RCFbe the set containing all the instances of the theorem in RCF, then wecould thus arrive at a systematization of the whole set P on the basis of theTarski–Seidenberg decision procedure.(II) The following proof strategy draws on transcendental methods. Thisapproach is based not just on the Tarski–Seidenberg decision procedure butrather on one of its consequences, namely that RCF is a complete theory. Thisgives rise to the following transfer principle. Ifasentenceϕ in the languageof RCF can be shown to hold in one particular real closed field, say, the realnumbers R, then it must be true for any real closed field because of thecompleteness of RCF.Now, the theorem can of course be established for R relying on, amongother things, the Bolzano–Weierstrass theorem (every bounded sequence hasa convergent subsequence) and the least upper bound principle.³ These basicproperties of R don’t hold in general for real closed fields. For instance,they both fail for R alg . However, once the theorem has been proved forR, by whatever means, we can conclude by appeal to the transfer principlethat all its instances, i.e. all sentences in P, hold for real closed fields ingeneral.³ Cf. Courant, 1971, vol.Ip.60f, vol. II p. 86.