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

eyond unification 169Now, can there be an embedding of B II into B III ? Without further informationon the argument patterns in the two bases we are in no position todevise any specific translation or embedding. Yet, that certainly doesn’t ruleout the existence of such an embedding. Above, in (II) and (III) we sketchedtwo different types of proof of the statement that some given polynomialassumes a maximum on a given closed and bounded semi-algebraic set. Therespective argument patterns of which these proofs are instantiations mightbe seen as correlated by the kind of translation we are looking for. And ifthis kind of correlation could be extended to supply for each pattern in B II acorresponding one in B III (such that—at least—conclusion sets are preserved),then this would indeed yield an embedding of B II into B III . However, itis crucial to notice that while at this abstract level an embedding might inprinciple be possible along the indicated lines, an embedding of B II into aproper subset of B III is definitely excluded. The reason is this. Assume thatB II embeds into a proper subset S ⊂ B III .Let S be the set of all instantiationsof the argument patterns in S. ( S is a subset of E III (K).) SinceB II is the basis for E II (K) and the embedding preserves conclusion sets, wehave C(E III (K)) = K = C( S ).HenceallofK is systematized by S and,moreover, S is generated by a proper subset, S, ofB III . This contradicts thefact that E III (K) is the best systematization of K (relative to KIII ∗ ), i.e. thesystematization whose basis ranks best with respect to the criterion of paucityof patterns.¹⁶ There must be redundancies in B III if the same conclusion setcan be gotten from a proper subset of it. In other words, B III must be inflatedand hence could not have the greatest unifying power among all bases and,in turn, E III (K) would not be the set of arguments which best unifies K(relative to KIII ∗ ).Turning now to the case B III ⊂ B II the reasoning is very similar. In principle,no methodological constraint precludes the possibility that B III can be embeddedinto B II (or even, in this case, that B III is a subset of B II ). As Brumfielhimself indicates one can indeed find a certain correspondence between hisown, algebraic, approach and the approach that uses transcendental methods.‘We admit that many of our proofs are long and could be replaced by bythe single phrase ‘‘Tarski–Seidenberg [transfer principle] and true for the realnumbers’’ ’ (Brumfiel, 1979, p.166). This might be taken, in our context, asa hint towards the possibility of constructing a correlation between patterns¹⁶ In case S is not only a generating set for S but also the basis for it, then the assumption thatS is a proper subset of B III leads directly to a contradiction by applying Kitcher’s corollary (in itsoriginal formulation). Because in this case we have two systematizations, E III (K), S ,ofK such thatC(E III (K)) = C( S ) and the basis for S is a proper subset of the basis for E III (K). HencewehaveE III (K) ̸= E III (K).