Several have asked whether categorical algebra can clarify the relation between syntax and semantics, two terms often paired in theoretical linguistics. The hope is that, as in previous work, focusing on mathematical experience will give information that can be investigated further and which is more richly detailed than abstract speculation about cognition in general has been able to provide. Whatever serious discussion we can have here will be limited to mathematics in particular, although the mathematics may suggest generalizations to other fields. Semantics has been previously analyzed as the contravariant 2-functor which relates the abstract general and the concrete general aspects of each general concept which is admissible in a given doctrine. In some doctrines semantics has an adjoint, called structure, a paradigmatic example of which is the structure that the system of cohomology operations naturally has. This adjoint does not have much directly to do with syntax, except insofar as the structure of something, being an abstract general, is in need of presentations to permit many kinds of calculation and reasoning about it. The word SYNTAX has the same Greek root has TACTICS . That suggests thinking of syntax as the tactics for manipulating symbols , specifically the symbols involved in presentations of abstract generals. More generally we might also include the word problems associated to individual algebraic categories relative to underlying set functors, but that case of 1-categories is distinct from the 2-categorical case I am emphasizing here. Presentations themselves are often objectified for mathematical study, and in all known cases that involves a choice of a further category with an adjoint pair connecting the latter with the category of objects to be presented. For example, in the doctrine where abstract generals are identified with single-sorted algebraic theories a standard choice is the category of sequences of sets, often called "signatures", with the functor assigning to each theory its sequence of ( )-ary operations; the left adjoint is then determined and hence a monad T on the chosen category. That monad has a crucial further role, in the construction of the third category Pres(T) of presentations wherein the signatures also play a further role as AXIOMS : namely this "syntactical" category has as objects pairs G,R of "signatures" equipped with a pair of maps from R to T(G) . The presentation functor applies the left adjoint, then takes the coequalizer in the first category; this presentation functor might also be considered part of syntax. Even if one considered (as was done for many years) that semantics was really the functor going all the way from syntax to the concrete generals, the fact that it has a preferred factorization would not remain concealed forever from mathematicians: the discovery of groups revealed a rich content beyond permutation symbols, and calculations with characteristic polynomials of linear transformations are clarified by, as well as present, the "abstract" rings whose representations are involved (There is a SLNM by Lambek about Linear Semantics). In these terms the choice of home for the notion "theory" can be clarified . There are apparently FOUR reasonable possibilities : 1) The presentations of abstract generals. This syntactical emphasis was the one most used for many decades by logicians. 2) The abstract generals themselves. This choice, exemplified by the term "algebraic theory", emphasizes that theories should be those objects which play the pivotal role of forming the one category which is functorially linked to both syntax and semantics. (Note that there is usually no way of getting from concrete generals to syntax). 3) The concrete generals themselves. The word "theory" has only occasionally been used in this sense, but note that is philosophically analogous to the use of "homology theory" to signify an objective functor. 4) But when a group theorist refers to "group theory" he does not usually mean either 1) or 2), but more like Presentations of his Concrete General ! For 3) has also an algebraic structure, but in a 2- dimensional sense; specifically, in the doctrine of algebraic theories,etc., the concrete generals all have the natural 2-structure presented by filtered colimits, reflexive coequalizers, and all small limits. Thus we could start with any small family of groups and apply any composite of those operations, apply also any other such composite functor, and compare the results homomorphically. 2-syntax as "theory"? _________________________________________________________________ MSN Photos is the easiest way to share and print your photos: http://photos.msn.com/support/worldwide.aspx 15-Jan-2002 08:46:03 -0400,1594;000000000000-00000000