Adjoint Typing: Is= it Whitehead’s Category of the Ultimate?

Michael Heather &a= mp; Nick Rossiter

CEIS, Northumbria University, Newcastle, NE1 8ST, UK.

michael.heather@cantab.net; = nick.rossiter@unn.a.c.uk<= /o:p>

http://computing.unn.ac.uk= /staff/cgnr1/

̵= 6;An Essay in Cosmology’ is the subtitle given by Alfred North Whitehead t= o his celebrated Gifford lectures on PROC= ESS & REALITY (PR) at Edinb= urgh in the session of 1927-28. His cosmology is developed in terms of a Categor= eal Scheme which he describes as his speculative philosophy. The foundation of = his whole scheme of cosmology is the Category= of the Ultimate. A category in process terms is a typing and this fundamen= tal category of his “expresses the general principle presupposed in the t= hree more special categories”. These three special categories are composed= of eight categories of existence, twenty seven categories of explanation and nine &#= 8216;categorial obligations’. At first sight there seems to be a hierarchical typing = relationship among these categories which might look like this:

Howev= er, Whitehead does not provide such a diagram (nor i= ndeed any diagram in PR). Nor does he state that there is a hierarchical relationship. Whitehead does not even explain what he means by the term ‘category’. It seems it is defined by the Category of the Ultim= ate itself and therefore is self-referencing.&= nbsp; If he has no difficult= y with a category being a member of itself, then a cate= gory is not to be identified with a set, the concept earlier promoted in his oth= er magnum opus (co-authored with Bert= rand Russell) the PRINCIPIA MATHEMATICA = (PM).

In fa= ct there is a lateral type hierarchy as well as in the vertical. For

existence à explanation à obligation

progresses in logic from propositional logic through predicate logic to modal logic. Note= too that for the third level, Whitehead moves from categories as a noun to the adjective categoreal to qualify obligations. This reveals the horizontal dependence at the semantic level and explains why the hierarchical structur= e we have drawn above would have been inadequate for Whitehead. For a classical tree structure in = set theory requires horizontal independence between atomic entities. These conc= epts are neither atomic nor independent.

White= head seems very conscious of the speculative nature of his philosophy at this st= age. The whole of Part I of PR is he= aded ‘a Speculative Scheme’. It is speculative perhaps because at the time he was giving the Gifford lectures and for the remaining 20 years of h= is life there was no formal presentation available for PR as he and Russell were able to provide for PM. This speculat= ive state of affairs no longer holds. There is now a formal theory of categorie= s [Heather & Rossiter, Process Category Theory, Salzburg International Whitehead Conference, University of Salzburg, 2006] only just beginning at the time of Whitehead’s death in 1947 but now maturing. It is a theory foreshadow= ed in Whitehead’s Category of the ultimate and quite comprehensive in the sense of his preface to PR at p= . vi:

Also, i= t must be one of the motives of a complete cosmology, to construct a system of ide= as which bring the aesthetic, moral and religious interest into relation with those concepts of the world which have their origin in natural science. [Wh= itehead PR Part I]. <= /p>

We ca= n now understand from this formal theory of categories the point about interdependence and w= hy Whitehead might not want to use the concept of a hierarchy. It is a more complex relationship than the hierarchy in the same way as Russell used the phrase ‘ramified type-theory’ rather than ‘hierarchical type-theory’ although both words contain the sense of a tree. These d= ifferent categories are not independently defined from one another and the relations= hips go both ways. The above diagram might be drawn more fully using the concept= of adjointness as follows:

This structure is a partial order but that still may not be general enough for t= he Category of the Ultimate. For partial orders are equiva= lent to quotients of a preorder which cannot be fully drawn because of their non-reductionist status. However, some idea of the notion may be gle= aned from the second diagram. There is no obvious starting or finishing point, t= hat is top or bottom, as the Category of the Ultimate may be either, depending = on the viewpoint.

Adjointness provides a formal basis for relationships escaping the clutches of Göd= el’s undecidability to provide a metaphysical approach to higher-order logic. Cartesian Closed Categories advance= beyond the Category of Sets to represent relationships between systems as adjoint functors. Two formal constructions for adjunctions both involve composition: the first that of distinct functors giving 2-cells, the second that of endofunctors giving monads.

The former [Rossiter & Heather, Data Structures in Natural Computing: Datab= ases as Weak or Strong Anticipatory Systems, AIP Conference Proceedings 718, 392= -405 (2004)] represents composition across a number of levels. For example we ma= y compose data in turn with metadata and metameta data so that the adjoint relationsh= ip is represented across four levels of category. That is three levels of mapp= ing, from data values to data abstractions like aggregation and inheritance.

The latter [Heather, Rossiter, & Sisiaridis, The Semantics of Jitter in Anticipating Time Itself within Nano-Technology, AIP Conference Proceedings (2008)] represents the process or behaviour of a system; for example in transactions where an endofunctor in three cycles constitutes a monad and comonad [Mac Lane, S, Categories for the Working Mathematician, 2nd ed, Springer (1998) pp.137-142].

These two constructions are complementary: the first handles principally data structures, their values and methods; the latter the behaviour of the data objects. It is highly significant that three levels are involved in each construction: in limit constructions in category theory three levels are of= ten used. The monadic structure has a particular robustness with respect to G&o= uml;del's theorems. Monadic higher-order functions are complete and decidable unlike dyadic higher-order ones.

If Whitehead’s categories represent the real world they are cartesian closed with products, limits and colimits. T= hey are also locally cartesian closed with the following relationship holding between categories L and R in the context of the th= ree functor categories Existence, Explanation and Obligation. (In standard terms these are identifiable respectively with the existential qualifier, the pullback functor and the universal quantifier.)

We can then write the Category of the Ultimate as:

Existence ┤ Explanation ┤ Obligation

where the reverse logic gate ┤distinguishes the left fro= m the right adjoint.

White= head concludes the section of his Preface quoted above with:

The doc= trine of necessity in universality means that there is an essence to the universe which forbids relationships beyond itself, as a violation of its rationalit= y. Speculative philosophy seeks that essence [Whitehead PR, Part I, Chapter I Speculative Philosophy, Section I p.4, The Speculative Scheme].

Whitehead is in effect defining God, the ultimate limit which for Cartesian Closed Ca= tegories constitutes the boundaries of a relationship or process bringing ‘the aesthetic, moral and religious interest into relation with those concepts of the world which have their origin in natural science’.