Presentation on theme: "Adjoint Typing: Is it Whitehead’s Category of the Ultimate?"— Presentation transcript:
1Adjoint Typing: Is it Whitehead’s Category of the Ultimate? Michael Heather & Nick RossiterNorthumbria University, UK
2Process 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 ofHis cosmology is developed in terms of a Categoreal SchemeDescribed 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”
3Structure of Categories These three special categories are composed ofeight categories of existence,twenty seven categories of explanationnine ‘categoreal obligations’
4At first sight there seems to be a hierarchical typing relationship among these categories which mightlook like this:Numbers are count of categories of each type
5What 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).
6The Categoreal Scheme Is the Structure of PR Comprising 4 categories UltimateExistenceExplanationCategoreal ObligationsThe whole of PR rests on this categoreal scheme
7Speculative NatureWhitehead 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 timehe was giving the Gifford lectures andfor the remaining 20 years of his lifethere was no formal presentation available for PRas he and Russell were able to provide for PM.
8Perhaps no longer a speculative philosophy New formal techniques are availableSo 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]
9CT Foreshadowed by ANWCategory Theory is a theory foreshadowed in Whitehead’s Category of the ultimatequite comprehensively in the sense of his preface to PR at p. vi:Motivation for a complete cosmologyto 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]
10Problems with Hierarchy From the formal theory of categoriesCan understand the need for interdependence between categoriesNot achieved in a hierarchySo Whitehead presumably dismissed the use of hierarchies.The relationship is more complex than the hierarchyin the same way Russell used the phrase ‘ramified type-theory’ rather than ‘hierarchical type-theory’ although both words contain the sense of a tree.
11Structure is not simple hierarchy In fact there is a lateral type hierarchy as well as in the vertical. Forexistence explanation obligationA 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.
12The previous diagram might be drawn more fully using the concept of adjointness as below
13Properties 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 diagramThere 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.
14Adjointness in more detail For F:L R and G:R LF,G are FunctorsL, R are CategoriesF┤G that is F is left adjoint to GIf we can defineUnit of adjunction η: L GFLCounit of adjunction ε: FGR RCommuting diagrams involving η, ε, F, GWhere L,R are objects in L,R respectivelyF is a free functor (creativity) with change ηG is an underlying functor (applies rules) with change εSpecial caseNo change in η or ε then equivalence relationship between F and G
15Adjointness -- Motivation Adjointness between functors provides a formal basis for relationships which for applied ctescapes the clutches of Gödel’s undecidability to provide a metaphysical approach to higher-order logic.enables relationships to be specified that are ‘less than’ equivalencebut which are common in real worlde.g. language translationis natural with respect to composition
16Cartesian Closed Categories Cartesian Closed Categories (CCC) are regarded as basic constructions. These have:initial and terminal objectsInitial provides entry point giving identity functorLimits and colimits exist as structure boundedall productsbasis of relationshipsexponentiationone pathCCC are regarded as minimal specification for reality
17View of CCC as adjunction CCC is an adjoint relationship between functors F and G:F ┤ GwhereFree functor F creates binary productsUnderlying functor G checks for exponentials (one path)
18Two 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.
192-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.
20Monads Monads represents the process or behaviour of a system through three cycles of an endofunctor (same source and target)e.g. GF is an endofunctorSimilar to information system transactionsThe 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
21Principles of Monads Represent behaviour Employ three cycles T T2 T3Defined as a 3-tuple:<T, η, μ>whereT 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)
22Principles of Comonads As for monads but tuple is:<S, ε, δ>whereS 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)
23Monad/Comonad Relationship Between monad <T, η, μ> and comonad <S, ε, δ>Functor F takes monad to comonadFunctor G takes comonad to monadThere is adjointness F ┤ G
24Categories are LCCCIf 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.
25CLXC RIn standard terms the functors are identifiable respectively with the existentialquantifier, the pullback functor and the universal quantifier.LXC R is the relationship of L with R in the context of CC includes L + R
26Category of UltimateWe can then write the Category of the Ultimate as:Existence ┤ Explanation ┤ Obligationwhere the reverse logic gate ┤distinguishes the left from the right adjoint
27Concluding 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].
28Defining ‘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’.
29Future WorkConsider taking Whitehead’s concepts and representing them in ct directlyUseful exercisePossible trap of categorificationApproach 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.