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| wholes, systems, levels |
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Location: http://www.cs.mun.ca/~ulf/gloss/wholes.html.
By Ulf Schünemann since 2002.
Please mail any comments.
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| navigation bar: Wholes and Parthood | |||
| wholes & system composition levels | part-whole relations
different kinds of parthood and compositeness | aggregates (concrete) =/= sets (conceptual) | ontological status of wholes vs. parts
holism, atomism (allied with reductionism), systemism |
«[I]t is possible to define in a precise manner the words
"whole," "part," and "sum" so that "The whole is unequal to the sum of its parts"
is not only not logically absurd, but it is in fact logically true.
There is, therefore, no a priori reason for dismissing such statements
as inevitable nonsense, and the real issue is to determine,
when such an assertion is made, in what sense if any the crucial words
in it are being used in the given contexts» [WSOU 139]. ...
«[W]hen a given system has a special type of organization or structure,
a useful definition of "addition," if such can be given,
must take into account that mode of organization ...
Finally ... though a system has a distinctive structure,
it is not in principle impossible to specify that structure
in terms of relations between its elementary constituents,
and moreover in such a manner that the structure can be correctly
characterized as a "sum" whose "parts" are themselves specified
in terms of those elements and relations» [WSOU 140].
«[S]ome writer apparently understand by "sum" [in the context of melodies]
the unordered class of individual tones ...
Accordingly, if the word "sum" is used in this sense in contexts
in which the word "whole" refers to a pattern or configuration
formed by elements standing to each other in certain relations,
it is perfectly true, though trivial, to say that
the whole is more than the sum of its parts.
As has been noted, however, this fact does not preclude the possibility
of analyzing such wholes into a set of elements related
to one another in definite ways; nor does it exclude the possibility
of assigning a different sense to "sum" so that a melody might
then be construed as a sum of appropriately selected prats.
It is evident that at least a partial analysis of a melody
is effected when it is represented in the customary musical notation»
[WSOU 143]
«Consider Köhler's example of what one might regard as the very
antithesis of a `true' whole: three stones, selected at random,
lying in different continents. Although each of the stones is,
to be sure, part of the group of three, this group is not,
on the usual understanding of the term, a whole.
[footnote: The term aggregate has been employed to characterize
such failure to qualify under the intuitive notion of a `true' whole.
Actually, the term `aggreagte' has as many senses as there are criteria composing
the intuitive notion of a whole: for failure to meet certain of these criteria
yields a corresponding type of aggregate.]
Let us summarise the intuitive requirements or conditions of adequacy
which underlie talk of wholes. To begin with, the things we are
considering as wholes, must, of course, consist (in some sense) of
parts. In addition, it would seem that the following conditions
must be met:
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In object-oriented modeling, Henderson-Sellers makes similar demands
on the use of any kind of aggregation relationship:
«The whole ... exists because of its parts: without the parts the whole has no meaning. The whole provides at least one emergent capability not found in the parts, and is therefore more than the sum of its parts» [OML/RR, underlining added]. «[F]or the aggregation relationship to be valid, an aggregate (aka., composite) must have at least one emergent propery (a property of the aggregate that cannot be deduced from the evaluation of the properties of the individual components) [Kilov&Ross: IM; 1994]. In addition, at least one property of the composite/aggregate should be dependent upon the value of one or more properties of its components/parts. Furthermore, aggregations are generally considered to be antisymmetric and transitive [Rum+: OOMD]. Also some operations to the aggregate are propagated to the parts [K&R: IM]. An important use of aggregation is in support of abstraction, leading to layered diagrams at different granularities [JMEdwards, BHenderson-Sellers: A coherent notation for OO software engineering; TOOLS 5; Prentice-Hall 1991].» [OPEN/CC, underlining added]
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Aggregate << System. «Whether or not two things form a system, they can be assumed to associate (or add physically) to form a third thing. Thus thing a and b, no matter how distant and indifferent, my be assumed to form thing c = a °+ b. In other words, the set of things is closed under the operation °+ of association, physical addition, or juxtapos » [MB4 14].
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Consider a gravitional system like the solar system:
«However
what makes such a system into more than
just a naturally integrated whole is the relation of
complex interdependence among the determinables
[of different parts!]. ... We may thus schematically define a dependence system, following Rescher and Oppenheim [-> see below], as a collection of objects which form a family under a relation to which a class of determinables apply, such that each member of the family has some determinable from the class which is functionally dependent upon some or all of the determinables of some or all of the remaining members. Grelling and Oppenheim, who were the first to offer a definition of such systems, first called them `determinational systems' (Wirkungssystem), later preferring Koffka's term `functional whole'.» [Parts 345] |
«An aggregate or assemblage is a collection of items not held together by
bonds, and therefore lacks integrity or unity. Aggregates can be either
conceptual or concrete (material). A conceptual aggregate is a set.
A concrete or material aggregate, on the other hand, is a compound thing,
the components of which are not coupled, linked, connected, or bonded,
such as a field constituted by two superposed fields, a celestrial
constellation, and a random sample of a biological population.
Because the components of an aggregate do not interact
- or do not interact appreciably -
the behaviour of each is independent of the behaviour of the others. Consequently,
the history of the aggregate is the union of the histories of its members.
On the other hand
the components of a concrete system are linked [ie. interact],
whence the history of the whole differs from the union of the histories
of its parts. We shall take the last statement to [be?] an accurate version
of the fuzzy slogan of holistic metaphysics,
namely The whole is greater than the sum of its parts. ...
A system, then, is a complex object,
the component of which are interrelated rather than loose.
If the components are conceptual, so is the system; if they are concrete
or material, then they constitute a concrete or material system.
A theory is a conceptual system, a school is a concrete system of the
social kind. ...
[Whether conceptual or concrete,] a system may be said to have
a definite composition, a definite environment, and a definite structure.
The composition of a system is the set of its components; the environment,
the set of items with this it is connected; and the structure, the
relations among its components as well as among these and the enviroment.
For example, a theory is composed of propositions or statements; its
environment is the body of knowledge to which it belongs (e.g. algebra or
ecology); and its [internal] structure
is the entailment of logical consequence relation.
The merger of these three items is a propositional system,
ie. a system T composed of a set P of propositions,
embedded in a certain conceptual body B, and glued together by
the relation |-- of entaiment: in short, T = (P,B,|--).
And the composition of a school is the union of its staff and pupils;
the environment is the natural and social mileu,
and the structure consists of the relations of teaching and learning,
managed and being managed, and others.
The environment must be included in the description of a system
because the behavior of the latter depends critically on the nature
of its milieu. But of course in the case of the universe the environment
is empty, and so it is in the case of the important fiction known as the
free particle (or field)» [MB4 3f, underlining added].
Degrees of systemicity/integration:
[MB4 35, underlining added]
«Whatever is a system is also a whole but not conversely:
an aggregation of independent components is a whole
but not an integrated or unitary one. (Compare a living being with its ashes.)
Now systemicity or integration comes in degrees:
some systems are more tightly knit than others.
The degree of integration depends on the connections or links among a system's components
relative to the disintegrating actions of the environment.
If the inner couplings are "positive" (or "attractive") and strong,
the degree of integration is high; if the links are still positive but weak,
the degree of integration is low; and it the links are "negative" (or "repulsive"),
there is no systemicity or integration at all.
Finally, if some of the links are "positive" while others are "negative",
the degree of integration depends on which of them are overriding.
For example, a stable atomic nucleus is held together by nuclear forces
that overcome the electrical repulsions; and a stable humand community is
held together by participation in enterprises of common interest,
the value of which is greater than that of rivalry or competition
- until of course the latter gets the upper hand.
In the case of physical, chemical, and perhaps also biological systems,
what measures their degree of integration is their binding energy or,
what comes to the same, their dissociation energy.
This is the minimal energy required to dissociate the system into its components.
It is zero for an aggregate. But such a measure is not universal:
it fails to apply to systems where information links
play at least as important an integrating role as forces proper -
as is the case with social systems. In short,
there is no universal measure of the degree of integration or cohesion of a system.»
System: Integration (structural) << Coordination (functional)
[MB4 38] «Finally, another concept relevant to that of systemicity
is the notion of coordination, which must be distinguished from that of integration.
If integration fails the system undergoes structural breakdown.
On the other hand coordination concerns the relation among
either components or functions resulting in functional maintainance.
If coordination fails the system undergoes functional breakdown.
There can be integration without coordation but not conversely.
... »
Coordination << Control
[MB4 39] «Coordintation does not exclude inhibition. Quite on the contrary:
when coordination is a result of control it includes feedback,
which, when negative, is a kind of inhibition. ....
But of course there can be coordination without the intervention of a control system.
For example the corpus callosum bridges the two brain hemispheres
in vertebrates and thus renders their coordination possible
but is not a control system iteelf.»
Configurational Whole (= whole with some coordination?).
«In an illustrative device of A Meyer, magnetic needles of equal strength
are inserted in pieces of cork, and floated, with all like poles upwards,
in a basin of water. A strong magnet of unlike pole is placed in a fixed
position overhead. The floating magnets arrange or rearrange themselves
in a symmetric pattern of one or more concentric circles ...
In this type of configuration there exist various dependendecies;
for example, the distance of one cork from the nearest adjacent cork
depends upon the magnetic strength and the number of other magnets»
[Gestalt 95].
«In the general characterisation of the concept of depdendence here
exemplified we will have to refer to a set of object which, by virtue of
standing in certain specified relations, are said to form a configurational
whole, or briefly, a configuration. ...
[T]hose relations jointly may always be viewed as constituting one more
complex relation, R. An ordered set of objects,
p1, p2, ... pn,
which stand in the relation R to each other, i.e. for which
R(p1, p2, ... pn) holds,
will be said to form a configuration of kind R.
...
Dependence ... will consist in a (more or less complex) relationship
phi--in many cases a functional relationship--between the value
of the depdendet attribute for p1, and the values of
certain attributes for the parts of the configuration»
[Gestalt 95].
«A quantitative attribute f of the part p1 in a configuration of kind R consisting of the n objects p1, p2, ... pn is phi-depdendent upon the class G of quantitative attributes of these objects pi, if phi is a relationship such that in every configuration of kind R the f-value of the first member of this configuration is related by phi to the values which the quantitative attributes in G assume for the parts p1, p2, ... pn.«Electrostatics is the source of the classical example of this concept, employed by Köhler. The charge-density at any point of a given charged, well-insulated electric conductor is phi-dependent upon the charge-densities at the remaining points, relative to a complex condition specifiable in electrostatics. Biological homeostasis also provides an illustration. For example, ... the content of individual ions in one volume of the fuild of an animal is very nearly the same as the ion-content of other, equal volumes» [Gestalt 97].
Thus to assert a dependence relation of this type is to assert a lawlike connection» [Gestalt 95f].
Dependence system (= each part a bit coordinated).
«A configuration is a phi-dependence system
relative to a set G of attributes if each part of the configuration
has some G-attribute which is phi-dependent upon (some or all of)
the G-attributes of (some or all of) the remaining parts»
[Gestalt 98].
«The concept of dependence system furnishes a logical reconstruction
of the informal conception of an `organic' or `functional' or `integrated'
whole. For this is based upon the intuitive requirement that the parts of
a whole must stand in some special and characteristic relation of
interdependence with one another, in virtue of their status as parts of the
whole.
«What we have termed a `dependence system' provides a precisely defined
counterpart of what has been described as systems `the behavior of which is
not determined by that of their individual elements, but where the
part-processes are themselves determined by the intrinsic nature of the whole'.
Any dependence system furnishes an example of a whole whose parts have
features which are such that knowledge of them cannot, relative to the
available theoretical information, be acquired by studying their parts in
isolation, but requires information regarding other parts. In consequence,
the concept of a dependence system also provides a natural housing for the
organismic biologists' claim that `analysis of living processes into the
behaviors of distinguishable parts of organisms entails a radical distortion
of our understanding of such processes'. The study of the logic of such
dependence systems would, thus, be an integral part of a General System
Theory in the sense of von Bertalanffy» (1951)
[Gestalt 99].
| Definition: (Un)Derivable Attribute |
|---|
|
Let Q be an attribute, in the most general sense, of whole w.
The definition is relative to some decomposition D of w into objects p which are its Pt-parts, i.e., which are in a (part) relation Pt with w. Q is a derivable/underivable attribute of w relative to a set G of attributes and some "theory" T (and relative to D) if `Q(w)' is/is-not deducible by means of T from the G-characterisation of the D-parts of w.
«(We construe `A is deducible from B my means of C' to exclude the cases
An attribute is logically (un)derivable if it is (un)derivable relative to an empty theory T. |
| Don't confuse derived with shared vs. unshared |
|---|
|
- Q is an unshared attribute [TYPE] of w (relative to D)
if Q is possessed by no D part: «If a pile (w) of round stones (D) has the attribute (Q) of being conical, this shape is an instance of a D-unshared attribute.» An unshared attribute is essentially the same as Broad's `collective property'. - Q is a shared attribute [TYPE] of w (relative to D) if Q is possessed by all of the D parts: «The attribute of having weight (Q) posses by a pile (w) of stones (D) is a D-shared attribute of the pile.» Being a homeomerous whole [parts are of same type as whole] implies having shared attributes (?) |
Underivable = Emergent = Holistic = Gestaltqualität.
«D-G-T-underivable attributes are
closely linked ... to what is usually discussed under the heading
of emergence and therefore the case in which the theoretical
part of T is a micro-structure theory is especially relevant.
... Indeed, our definition ... is adapted from the definition of
`emergence' given by Hempel and Oppenheim» (1948) [Gestalt 93].
«[O]ur discussion has connections with the holistic position in the
philosophy of biology ... Our concept of an underived attribute
of a whole could, in this connection, be taken as a precise explication
of the informal concept of a holistic property.»
«Special interest attaches to the concept of attributes of a whole
underivable from information regarding its parts even by using
the accepted laws and theories of the sciences.
In his pioneering study, `Über Gestaltqualitäten',
Christian von Ehrenfels first directed the attention of psychologists
to instances of perceptual properties of objects which are characteristic
attributes of wholes in the sense considered in this section--the
so-called Gestaltqualitäten. It is to this end that the first
condition for wholes given by von Ehrenfels is directed--the so-called
First Ehrenfelscriterion, i.e. the condition that:
`The whole is more than the sum of its parts.'» [Gestalt 94]
Emerging properies may be sortals, i.e., give raise to new identity criteria. Different identity criteria are another way of defining the ontological levels (strata) of (concrete) particulars [OntPrinc]:
| level | example | individuation | persistence | |
| extensional | atomic | a minimal grain of matter | minimal spatial or temporal dimensions | spatio-temporal continuity |
| static | a configuration of atoms, a situation | mereological sum of atoms | unchanged non-temporal properties | |
| mereological | an amount of matter (possibly disconnected) | mereological sum of entities | same parts | |
| physical | topological | a piece of matter | self-connection | similar topology |
| morphological | a cubic block | proximity | similar spatial shape / temporal pattern | |
| based on interaction with external world | functional | an artifact | purpose | persistence of functionality |
| biological | a human body | presence of life | persistence of life | |
| intentional | a person or a robot | intentional behavior | persistence of (capability to) intentional behavior | |
| social | a company | inter-agent connections (rules, conventions) | persistence of inter-agent connections |
Complexes.
«Now, in scientific discussion, we often do not know (or care) about the
individual parts which occupy the various positions.
An example from chemistry: when discussing the molecule H-O-H,
it suffices that some O atom should occupy the place corresponding to the
center space of the representation; it makes no difference in our discussion
which particular oxygen atom this is. These considerations lead us to the
concept of a complex. We will say that whenever any structured
whole is considered with a view to the types of its parts, rather than its
specific parts themselves, it is viewed as a complex»
[Gestalt 100].
«A complex is characterised by the following three features:
Isomorphism. «Two complexes (X1,G1,f1) and (X2,G2,f2) are isomorphic if
Complexical Feature.
Typical groups of isomorphisms preserve certain properties.
«In the case of our musical example, the isomorphic complexes--transposed
compositions--have a significant common structural characteristic, they
have the same melody» [Gestalt 103].
«The musical composition (complex) has an attribute (is a melody)
which is unchanged or invariant under transpositions belonging
to a subgroup of the group of all abstractly possible transpositions
(the transpositions in the sense of music).» [Gestalt 104].
«Given some subgroup Z of the group of all transpositions of
a complex, we designate as a complexical feature (relative to Z)
of this complex any attribute which it shares with all complexes differing
from it only by Z-transpositions, i.e. any attribute invariant
under these transpositions» [Gestalt 104].
«In the applications of this schematism of complex and
isomorphism, it may happen that a complexical feature reflects
an attribute of a complex which is underivable in the sense of our
earlier discussion ... and this, consequently, is of special interest.
...
It should be remarked that the possession of such attributes,
which are invariant under transpositions, is another condition for
Gestalten, the so-called Second Ehrenfels-criterion ...»
[Gestalt 104].
«An important application of these concepts is to cases in which
something is constant under varying processes or modifications.
For if the object in question can be represented as a complex in our sense,
and its modifications as transpositions, the constancy will amount to a
complexical feature. Significant instances in the applications are furnished
by physical systems that attain a state which is indepdendent of time, a
stationary state or state of equilibrium: a state left invariant by the
processes of nature. ...
An object is considered a good Gestalt in the degree to which it exhibits
invariance or stability ...
... Thus in the case of the Gestalten of visual perception, a premium is put
on symmetry, simplicity of linear structure, and unifomity of component
structures, since all of these lead to invariances under certain groups
of geometric transpositions.» [Gestalt 105f]