Adjoint Typing: Is it Whitehead’s Category of the Ultimate? Michael Heather & Nick Rossiter Northumbria University, UK

Process and Reality An Essay in Cosmology subtitle given by Alfred North Whitehead to his celebrated Gifford lectures: PROCESS & REALITY (PR) at Edinburgh in the session of 1927-28. His cosmology is developed in terms of a Categoreal Scheme Described 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 fundamental category of his “expresses the general principle presupposed in the three more special categories”

Structure of Categories These three special categories are composed of eight categories of existence, twenty seven categories of explanation nine ‘categoreal obligations’

At first sight there seems to be a hierarchical typing relationship among these categories which might look like this: Numbers are count of categories of each type

What Whitehead does not say However, Whitehead does not provide such a diagram (nor indeed 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 Ultimate itself and therefore is self-referencing. If he has no difficulty with a category being a member of itself, then a category is not to be identified with a set, the concept earlier promoted in his other magnum opus (co-authored with Bertrand Russell) the PRINCIPIA MATHEMATICA (PM).

The Categoreal Scheme Is the Structure of PR Comprising 4 categories Ultimate Existence Explanation Categoreal Obligations The whole of PR rests on this categoreal scheme

Speculative Nature Whitehead seems very conscious of the speculative nature of his philosophy at this stage. The whole of Part I of PR is headed ‘a Speculative Scheme’. It is speculative perhaps because at the time he was giving the Gifford lectures and for the remaining 20 years of his life there was no formal presentation available for PR as he and Russell were able to provide for PM.

Perhaps no longer a speculative philosophy New formal techniques are available So this speculative state of affairs may no longer hold. There is now a formal theory of categories only just beginning at the time of Whitehead’s death in 1947 but now maturing [Heather & Rossiter, Process Category Theory, Salzburg International Whitehead Conference, University of Salzburg, 2006]

CT Foreshadowed by ANW Category Theory is a theory foreshadowed in Whitehead’s Category of the ultimate quite comprehensively in the sense of his preface to PR at p. vi: Motivation for a complete cosmology to construct a system of ideas which bring the aesthetic, moral and religious interest into relation with those concepts of the world which have their origin in natural science. [Whitehead PR Part I]

Problems with Hierarchy From the formal theory of categories Can understand the need for interdependence between categories Not achieved in a hierarchy So Whitehead presumably dismissed the use of hierarchies. The relationship is more complex than the hierarchy in the same way Russell used the phrase ‘ramified type-theory’ rather than ‘hierarchical type-theory’ although both words contain the sense of a tree.

Structure is not simple hierarchy In fact there is a lateral type hierarchy as well as in the vertical. For existence  explanation  obligation A progress in logic from propositional through predicate to modal. 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 structure we have drawn above would have been inadequate for Whitehead. For a classical tree structure in set theory requires horizontal independence between atomic entities. Such concepts are neither atomic nor independent. Requirement: these different categories are not independently defined from one another and the relationships go both ways.

The previous diagram might be drawn more fully using the concept of adjointness as below

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

Adjointness in more detail For F:L R and G:R  L F,G are Functors L, R are Categories F┤G that is F is left adjoint to G If we can define Unit of adjunction η: L  GFL Counit of adjunction ε: FGR  R Commuting diagrams involving η, ε, F, G Where L,R are objects in L,R respectively F is a free functor (creativity) with change η G is an underlying functor (applies rules) with change ε Special case No change in η or ε then equivalence relationship between F and G

Adjointness -- Motivation Adjointness between functors provides a formal basis for relationships which for applied ct escapes the clutches of Gödel’s undecidability to provide a metaphysical approach to higher-order logic. enables relationships to be specified that are ‘less than’ equivalence but which are common in real world e.g. language translation is natural with respect to composition

Cartesian Closed Categories Cartesian Closed Categories (CCC) are regarded as basic constructions. These have: initial and terminal objects Initial provides entry point giving identity functor Limits and colimits exist as structure bounded all products basis of relationships exponentiation one path CCC are regarded as minimal specification for reality

View of CCC as adjunction CCC is an adjoint relationship between functors F and G: F ┤ G where Free functor F creates binary products Underlying functor G checks for exponentials (one path)

Two Process Techniques Two formal constructions for adjunctions both involve composition: the first that of distinct functors giving 2-cells, the second that of endofunctors giving monads/comonads.

2-cells 2-cells represents composition across a number of levels. For example we may compose data in turn with metadata and metameta data so that the adjoint relationship is represented across four levels of category. That is three levels of mapping, from data values to data abstractions such as aggregation and inheritance.

Monads Monads represents the process or behaviour of a system through three cycles of an endofunctor (same source and target) e.g. GF is an endofunctor Similar to information system transactions The monadic structure has a particular robustness with respect to Gödel's theorems. Monadic higher-order functions are complete and decidable unlike dyadic higher-order ones. Also dual comonad with endofunctor FG

Principles of Monads Represent behaviour Employ three cycles T  T2  T3 Defined as a 3-tuple: <T, η, μ> where T is GF (endofunctor) η is unit of adjunction (defines change in source category on one cycle, measures creativity) μ is multiplication (looks back T2  T, measures creativity)

Principles of Comonads As for monads but tuple is: <S, ε, δ> where S is FG (endofunctor) ε is counit of adjunction (defines change in target category on one cycle) δ is comultiplication (looks forward S  S2, anticipation, in conjunction with monad)

Monad/Comonad Relationship Between monad <T, η, μ> and comonad <S, ε, δ> Functor F takes monad to comonad Functor G takes comonad to monad There is adjointness F ┤ G

Categories are LCCC If Whitehead’s categories represent the real world they are Cartesian Closed (CCC) with products, limits and colimits. They are also Locally Cartesian Closed (LCCC) with the following relationship holding between categories L and R in the context of the three functor categories Existence, Explanation and Obligation.

C LXC R In standard terms the functors are identifiable respectively with the existential quantifier, the pullback functor and the universal quantifier. LXC R is the relationship of L with R in the context of C C includes L + R

Category of Ultimate We can then write the Category of the Ultimate as: Existence ┤ Explanation ┤ Obligation where the reverse logic gate ┤distinguishes the left from the right adjoint

Concluding Remark by Whitehead Whitehead concludes the section of his Preface quoted above with: The doctrine of necessity in universality means that there is an essence to the universe which forbids relationships beyond itself, as a violation of its rationality. Speculative philosophy seeks that essence [Whitehead PR, Part I, Chapter I Speculative Philosophy, Section I p.4, The Speculative Scheme].

Defining ‘God’ Whitehead is in effect defining ‘God’, the ultimate limit which for Cartesian Closed Categories 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’.

Future Work Consider taking Whitehead’s concepts and representing them in ct directly Useful exercise Possible trap of categorification Approach here has been to show the potential of ct for representing processes and relationships, including aspects such as adjoint relations, creativity, anticipation, looking back, identity and limits.