ClDo _ IIIIIIIoaIIIics bu _ die 'EI JDDi, sij'_ . . . . -. . . _. je _ . . . . . lIbupalaJllllllllll __ D'y_poa~: wbae it beIoap. . . . die . . ". . . ,. . _ DOD to dlecluly __ __ . ~ 1110 _ is dioapaI; -. . e _ may be EricT. BeD IbIetodo--'_iL O. 1feaoriIide Mathematics is a tool for dloogIrt. A bighly necessary tool in a world where both feedback and noolineari ties abound. Similarly, all kinds of parts of IIIIIIhcmatiI:s serve as tools for odIcr parts and for ocher sci eoccs. Applying a simple rewriting rule to the quote on the right above one finds suc:h stalements as: 'One ser vice topology has rcncIerM mathematical physics . . . '; 'One service logic has rendered computer science . . '; 'One service category theory has rmdcn:d mathematics . . . '. All arguably true. And all statements obrainable this way form part of the raison d'etm of this series. This series, Mathmlatics tDIII Its Applications, saaned in 1977. Now that over one hundred volumcs have appeared it seems opportune to reexamine its scope. AI. the time I wrote "Growing spccialization and divenification have brought a host of monographs and textbooks on incJeasingly specialized topics. However, the 'tree' of knowledge of JJJatbcmatics and reIatcd ficIds docs not grow only by putting forth new bnDdIcs. It also happens, quite often in fact, that brancbes which were thought to be comp1etcly disparate am suddenly seen to be rdatcd.

Inhalt
I Classical Methods.- 1 Approximation Theorems For Valuations On Fields.- 2 Valuations On Commutative Rings.- 3 Ordered Groups And Homomorphisms.- 4 Approximation Theorems For Multi-Structures.- II Categorical Methods.- 5 Categorical Logic.- 6 Approximation Theorems In Categories.- Index of Notation.