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The Structure of Wholeness

Michael Rahnfeld
Universitšt Kiel

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ABSTRACT: Using a part-whole-calculus the vague concept of wholeness is rendered precisely as the structure of an atomic boolean lattice. The so-defined prototypical structure of wholeness has the status of a category, since every element of our experience may be considered as an intended application of it. This will be illustrated using examples from different ontological spheres. The hypothetical and therefore fallible character of the structure is shown in its inadequacy in grasping quantum logical facts. This demands a differentiation of wholeness. The defined structure may be seen as circular in two respects: On the one hand it is the precondition for the understanding of its own syntactic and semantic basics, on the other hand there exists a mutual defineability between its atoms, which leads us to the thesis that wholeness cannot be defined in a non-circular manner.

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In this paper I attempt both to explicate the popular, but vague notion of wholeness and to point out its meaning for ontology. To begin with, I’ll give a brief survey of the essentials: In accord with an elementary intuition of ‘wholeness’ I introduce an implicit axiomatic definition of its structure, which proves to be a familiar Boolean-lattice. This internal view of the concept of wholeness is followed by a more philosophical external view, which looks at the structure in its context. It will be shown that the structure corresponds to the criteria of an ontological category, namely consistence, adequacy, content and coherence, so that we are justified in speaking of the ‘category of wholeness’. This feature leads to some interesting results: As a consequence of the adequacy of a category the structure turns out to be a model on its own. The self-application leads on the level of the axioms to the boolean lattice of all substructures and on the level of the terms of axioms to semantical boolean lattices, which may seen as basic units for the whole language. Thus the understanding of the structure of ‘wholeness’ takes for granted that there is a pre-understanding of the very same. Furthermore, there is another kind of circular understanding on the level of the atoms of the structure, because there exists a mutual defineability between the atoms, which cannot be eliminated without leaving the wholeness. But even if we try to leave it, we enter another wholeness, so that circularity is inevitable in the end.

A. Intuition

First of all, let me describe the leading intuition of ‘wholeness’. Indeed, the concept of wholeness has a broad spectrum of different meanings, so that every attempt of defining an intuitive core of its meanings requires some preliminary decisions. My first decision is to use an algebraical approach for the sake of clarity, and my second decision is not to bring to light an eternal idea of wholeness, but just some minimal criteria, open to further specifications: The criteria are the following: A wholeness consists of parts and operations on the parts, called ‘inner operations’. On the one hand there is a largest part — the whole — which cannot be enlarged by any operations, and on the other hand there is a null-part, which is part of all parts; this part is not only required for systematic reasons, but has often a distinct meaning. A wholeness shows a complete connection, that means there are no isolated parts, which cannot be reached by the inner operations. A wholeness cannot be left from within, that is there are no inner operations, which lead to parts, which are not parts of the whole. Moreover, we assume that there are atomical parts, called atoms, so that every part is a complex of the latter. We take for granted that all combinations of atoms are possible for the generation of complex parts and that the identity of parts depends just on the atoms but not of their order. Of course, this is a forced and somewhat artificial criterion for identity, however, it offers us the opportunity for a classification of wholenesses according to their inner restrictions on combinations. For example, in this respect a jigsaw-puzzle is a very restricted wholeness, while the mixed colour white is not restricted and therefore a perfect wholeness in the defined sense. As this is not the place for constructing a whole family of wholenesses of different degrees of scope, I confine myself for reasons of simplicity and suitability to the defined prototype. Please note, that I do not take a nominalistic point of view, even though I speak of ‘parts’ and ‘wholes’; on the contrary I postulate that all conceivable ontological spheres exist and with them those wholenesses, which can be found therein. Hence colours and concepts may be regarded as wholenesses just as physical things or events, the only condition is that all parts of one and the same wholeness belong to the same ontological sphere, which means that ontologically mixed entities — for example sets, including elements which are not sets — are to be rejected. The following discourse is based on these intuitive preconditions.

B. Definitions

Now I give a precise definition of the structure of wholeness in accord with the intuition just sketched. The structure of wholeness is introduced by an implicit axiomatic predicate, that is — roughly speaking — a predicate of second order that consists of a set of linked axioms with non-interpreted primitive terms. It is, so to speak, the pure form of all those propositions that characterise the structure of the entity to be defined. Interpreting the variables of the primitives — so that all axioms become true simultaneously — offers a so-called ‘model’. One and the same implicit axiomatic predicate may produce different models, if there are different suitable interpretations. For our purposes it is important to stress that models for the predicate of wholeness can be found across all ontological spheres.

Using an implicit axiomatic predicate the concept of wholeness is rendered precisely as the structure of a complete atomic Boolean lattice. Some remarks on it.

We take as primitive terms:

  1. a set of all ontological spheres, therein
  2. a special ontological sphere of individuals,e.g. a sphere of colours or of concepts, and therein
  3. a set of atoms, then
  4. a sum-functions, a product-function and a complement-function and finally
  5. two individuals in the special sphere, namely the whole and the null.

Within the special sphere of individuals the following axioms are essential, please note, that they are intended as empirical statements, but not as definitions:

  1. The sum of all atoms constitutes the whole.
  2. The product of all atoms constitutes the null.
  3. For every individual exists a set of atoms, so that the individual is the sum of them.
  4. Every subset of atoms forms an individual.
  5. The sum of two individuals x and y equals a third individual z if and only if the union of the sets of atoms of x and y equals the set of atoms of z.
  6. The product of x and y equals z if and only if the intersection of the sets of atoms of x and y equals the set of atoms of z,
  7. The complement of x equals y if and only if the complementary set of atoms of x equals the set of atoms of y.
  8. The set of individuals is the base for the laws of a Boolean lattice.
  9. Two options concerning the absoluteness of the atoms and the whole are given.

More explicitly:


Indeed, this feature reminds one of the trivial structure of a Boolean powerset algebra. But, formal triviality does not exclude automatically philosophical productivity.

C. Category

In the next step I intend to qualify the structure of wholeness as a prototypical ontological category. This requires the structure to comply with the criteria of an ontological category. The criteria I use do not fully coincide with the traditional view on categories, I will point out the variations.

1. The first criterion of an ontological category is obvious: The structure must be free of any contradictions, that is tantamount to the fact that it has at least one model. Certainly the structure of wholeness meets this criterion, and I will give some models later.

2. It is not difficult to find models for the structure of wholeness in all ontological spheres, e.g. the conceptual sphere, the attributive sphere or the physical sphere. That means that the structure corresponds to the second criterion, the criterion of adequacy. It says that there should be a multitude of paradigmatic models across all spheres of experience. Please note, that this an important difference to the traditional concept of a category like causality or substance and accidence, which do not meet this criterion, for the former has no models in the conceptual sphere and the latter has no models in the attributive sphere.

3. In addition to adequacy the structure must show a distinct content — otherwise the poorest tautologies or the emptiest notions would be the most perfect categories. For this reason I start with a strong and far-reaching structure, rather than with a weaker one like a semi-group, which is part of the structure of a boolean lattice and therefore much more harmless. In traditional philosophy the question concerning the content of categories has been of no importance and therefore philosophy — e.g. contrary to physics — could not contribute to the empirical investigation of the world.

4. The last criterion covers the coherence of categories. Isolated categories make no sense, only their sound and mutual dependency may reflect appropriately the interconnections of reality. As I deal with the structure of wholeness separately from others in this context, I will just give a little example that shows how the structure of wholeness is to be combined with another formal structure, a so-called ‘tree’, which can be seen as a category as well.The combination of the two is a category again. First we specify the structure of wholeness to a cell wholeness containing a core and a body, then we specify the structure of a tree to a division tree with exactly two successors for each knot , finally we combine the two by replacing the knots by replacing the knots by cell wholenesses: as a result we get a cell-division-tree. The example given belongs to the physical sphere, but similar examples may be found for the other spheres.

In conclusion, we may ascertain that the structure of wholeness fulfils the requirements of an ontological category in the defined sense. Let us therefore replace the term ‘structure of wholeness’ with the more substantial and precise term ‘category of wholeness’. So, henceforth I will use the term ‘category of wholeness’.

D. Models

I will now look at some models of the category of wholeness. To emphasise its adequacy I select three simple and paradigmatic models out of three different ontological spheres. These are: a sun-earth-moon system from the physical sphere, the basic-colours from the attributive sphere and the classification of quadrangles from the conceptual sphere. In every case the structure of the boolean lattice is obvious.


Suppose that the whole is a sun-earth-moon system from the physical sphere, then the atoms are the sun, the earth and the moon, the null part is a fictitious nothing which is demanded by systematic reasons. Let 7 be the binding by gravity, 6 the overlapping of parts and ‘ (complementation) the remainder of the system. Another but more complicated physical model —  which I just mention here incidentally — is the Newton phase space for many particles.


The second model comes from the attributive sphere, the sphere of sensory phenomena. It is a matter of the basic colours: white, magenta, cyan, yellow, blue, red and green. The last three are the primitive-colours and hence also the atoms. The whole is white and the null part is the non-colour black. 7 is interpreted as additive mixture of colours, 6 as subtractive mixture and ‘ as complementary-colour. In this example the whole and the atoms are absolute.


The conceptual sphere is represented by the classification of quadrangles. The notion of a square is identified with the whole. Let the notion of a quadrangle be the null part. The semantic atoms are the notions ‘quadrangle with diagonals of the same length’, ‘quadrangle with a perpendicular diagonal’, ‘oblique rhombus’ and ‘trapezoid’. Let 7 be the intersection, 6 the join and ‘ the complement on the notions.

We may pursue this line of reasoning further, thus leaving this primitive level. First we can study structures with elements of a different logical type, e.g.structures of two place relations, including the all-relation as the whole and the null-relation as the null part. Furthermore: As is well-known there are some operations on the structure of a boolean lattice that produce new lattice-structures of the same type, especially this is done by forming sub-structures, direct products and homomorphisms. These operations enable us to rise from lower to higher wholenesses and moreover, to combine all spheres. This way ontological work is enriched considerably. The following instance may give a slight impression of the operations: Starting from the attributive sphere, we select two tiny sub-lattices, one of the basic-colours ‘white’ and ‘black’, and one of three visual-field-dots, then we form the product and come to the lattice of the white and black colour-spots. Now we construe a homomorphism to the conceptual sphere by a structure-preserving map from the lattice of colour-spots into the lattice of truth-values. At the end we define the concept of a white spot as the set of those colour-spots that are mapped on the truth-value ‘true’ Obviously all entities mentioned here are wholenesses. In a similar way all colour-spots and their concepts may be defined. I may add that this approach of forming wholenesses allows a precise specification of important traditional distinctions of parts, in particular ‘pieces’ and ‘moments’ in the sense of Husserl’s ‘Logical investigations’. It is evident that the sub-lattices of a lattice correspond to the ‘pieces’ of a wholeness and the factors of the lattice-product correspond to the ‘moments’ of a wholeness.

The given examples as diagrams:

HASSE-DIAGRAMs are to be read:











E. Exceptional Cases

It is a truism that there is no real certainty that our categories correspond to reality in every respect. Even if a situation is so clear, that there seem to be no alternative categories to its description, the used categories may prove inapplicable. Indeed, there are a lot of examples which show that not everything is a model of a wholeness, this is often the case when the atoms are not really independent from each other, even though they appear to be so. Everybody who is familiar with baking cakes knows that the axiom of associativity does not hold for the parts of dough, for we do not get the same result, if we first mix water and flour and then add fat, or if we first mix flour and fat and then add water. Or, with regard to the law of commutativty: it is not the same thng, if you ask your girlfriend first, whether she wants to go with you to the movies and then, whether she wants to marry you, or if you change the order of questions. These simple examples show that the category of wholeness is too narrow to reflect all possibilities within an atomic whole. So it makes sense to specify the prototypical structure for special cases. The most famous specification of the category of wholeness is found in quantum-logic. It is not necessary to go into details to see its non-Boolean-structure. Let’s look at the simplest case, namely the lattice of the two spin-states of an electron, which constitutes a so-called modular lattice. Just a few remarks about this: Quantum-theory — at least a certain interpretation of it — includes two assumptions, from which the invalidity of distribution follows. The first assumption says that the spin of an electron is quantized in two states, it may be ‘up’ or it may it may be ‘down’. The second assumption says that the spin can be measured exactly only in one direction of space. Either it is measured exactly in the X-direction or in the Y-direction; to do both is impossible. Therefore a proposition like ‘the spin in Y-direction is up and the spin in X-direction is down ’is necessarily wrong, because only of one its components can be verified. On the other hand, the proposition ‘the spin in Y-direction is up or the spin in Y-direction is down’ is true, because one of the alternatives always holds. Assuming, we have found out that the spin is up in X-direction, then the proposition ‘the spin is up in X-direction and (the spin is up in Y-direction or the spin is down in Y-direction)’ is obviously true. According to the law of distributivity the proposition follows: ‘(the spin in X-direction is up and the spin in Y-direction is up) or (the spin in X-direction is up and the spin in Y-direction is down)’. But this proposition is wrong, as each of its alternatives is wrong. Consequently the implication of the two sentences is wrong and the law of distributivity is invalid in this instance. The solution to the problem is to bring the structure into accord with experience, and that means to change the boolean structure into a poorer one, which is called ‘modular lattice’. There are grounds for the assumption that there is a whole family of wholenesses with different lattice structures.

F. Self-Application

As has already been mentioned the criterion of adequacy demands that a category must be qualified to reflect wide domains of our experience. One cannot deny that the category of wholeness itself is an essential element of our conceptual experience. Hence it does not surprise that the category of wholeness reflects itself. It does contradict any sense of adequacy if you have to exclude the categories from the experience interpreted by them, otherwise categories can be interpreted by no means; thus self-application is often unavoidable. For the sake of simplicity let’s focus only on the boolean laws of the category of wholeness.

We have to distinguish between two kinds of self-application: In a macroscopic view, the atoms of the structure of the boolean lattice are its axioms, provided that they are independent of each other. The parts of the structure are all possible subsets of the set of the axioms. The operations on these sets are the join, the intersection and the negate. Roughly speaking: This way you get the boolean-lattice of all sub-structures of the boolean lattice. Some sub-structures are more interesting than others. For example, the familiar structure of a simple lattice is composed of the independent axioms of commutativitiy, associativity and absorption. The whole is the lattice and the null some tautology which has no additional content. Then, the lattice may be seen as null for further specifications which lead to a boolean lattice, which is the whole.

The microscopic view focuses on the terminal conceptual components of axioms. It is usual to distinguish between semantical and syntactical components. Let’s first attend to the syntactical components. If we take logical grammar there are only two syntactical basic-types sufficient to produce all syntactically correct expressions. These are (1) the type of sentences S and (2) the type of proper names N. In principal every expression can be seen as a function of the combinations of these two types. For example the simple combination S(N) (S applied to N) is read: that type, added to a proper name, makes a sentence. This type is a one-place predicate. A further example: S(S(N)) (S applied to (S applied to N)). This combination is read: that type, added to a one-place predicate, makes a sentence, and that is a quantifier. We won’t set out to define the details of logical grammar, but to prove that the terminal constituents, ’proper name’ and ‘sentence’, correspond to the category of wholeness. This follows from the lattice-diagram, which says that the generic term ‘basic-type’ is completely exhausted by the two disjointed atomic notions ‘proper name’ and ‘sentence’. Another simple example in regard to semantics: The generic term ‘quantify’ — essantial to the understanding of quantifiers — falls into ‘particular’ and ‘universal’. As the semantic distinction is complete, the product of the partial notions is the null-notion, and the sum is the whole, the generic notion etc. Admittedly, the given explanations and examples are simple and few, but they are sufficient to indicate a result which is very important for ontology. Understanding the syntactical and semantical basics of the category of wholeness presupposes that we are always in a position to make a meaningful use of the very same.


G. Circular Definition of Atoms

Now I turn to the question, how can the whole and its parts be determined within the structure. Obviously, the parts can be broken down to the atoms by a hierarchical chain-definition. The atoms are an exception. As they are all on the lowest level, mutual and circular definition is unavoidable. Looking back at the last example, the circle becomes apparent directly:

universal := quantity & not particular

particular := quantity & not universal

The microsopic view shows that the whole syntax and semantics of a language is based on wholenesses of such a kind. The truth-values — true and false — or negation and affirmation are other examples. Hence there seems to exist no understanding of a language, without circular knowledge. You can raise the objection that the circularity may be eliminated by a change of the kind of definition. As a substitute you may suppose an ostentative definition which works with direct indications, examples and counter-examples for defining the atoms. But I can assure you that this step leads us into a very broad field of further severe difficulties and gives no solution in the end. Just one hint: If I want to define e.g. the color ‘red’ by pointing at it, I leave the wholeness of colours and enter the wholeness of actions. But if I want to define the atomic action ‘to indicate or to point at something’ I just have to use this kind of action and I must take for granted that its meaning is well-known. So, the circle appears again.

H. Wittgenstein

One of the most famous opponents of circular definitions is Wittgenstein in his ‘Tractatus’. From the possibility of the ‘crosswise-definition’ of logical primitive terms Wittgenstein draws the conclusion that terms which are defined in a circular manner cannot really be primitives and have to be eliminated in an ideal language. Wittgenstein says so very clearly in sentence 5.42 of his ‘Tractatus’. To use the last example of quantification, I refer Wittgenstein’s proposal to replace the universal quantifier with a ‘long conjunction’ of sentences (P & Q & R...) and the particular quantifier with a ‘long disjunction’ of sentences (P v Q v R...). Even if you admit that the quantifiers run in a finite domain, the proposal is misleading. Let me give a simple example: If I want to say e.g. that all my neighbours a, b and c are rich, this can be done by the sentence ‘a is rich’, ‘b is rich’ and ‘c is rich’. But that makes sense only because I know that a, b and c are all my neighbours and to express this, I need another sentence including a universal quanifier, namely: ‘and no other entity is my neighbour’. As quantifiers are not allowed, I have to replace ‘no other entity is my neighbour’ with another long sentence of the former kind: ‘d is not my neighbour’, ‘e is not my neighbour’, ‘f is not my neighbour’ and so on, until all entities are checked. In view of the fact that the domain of entities is potentially infinite, it cannot be done. Therefore Wittgenstein’s attempt to get rid of the circular definition of quantifiers this way fails. So, to be brief, I think that it is not possible to escape from semantical circularity at all.

Let me conclude this paper by saying that circular definitions within the bounds of wholenesses are not crimes but virtues, and necessary for understanding the very same. A wholeness leaves us just the choice: to ignore it, or to jump right into it.

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