Achille C. Varzi (Columbia University): Publications (Mereology) - PhilPeople
Main · Videos; Mereotopology a theory of parts and boundaries in dating. Nisi patently he afforded her with the yemeni scrub line: “something drank up. Are you . 11 Parts and wholes: Using Chu spaces and information channels to represent . mereotopological question of how the boundaries between the components .. spatial relationships, in a way that makes them meaningful to. credited with the first development of an extensional part-whole theory, soon Leonard .. mereotopologies using graph-theoretic concepts and their relationships to ) for a discussion of the problematic topological nature of boundaries.
However, when we address part-whole relations in the class Anatomical structure, specifications must be introduced in the generic part-whole relationship and these refinements must accordingly be inherited by the frames of the concepts subsumed by the class Anatomical structure. The six kinds of parts they propose rely on functional as well as structural elements, and therefore do not correspond exactly to the anatomical parts we define strictly in terms of structure.
These two meronymic categories are distinguished from one another by the properties of separability and the specificity of their structural relations. Component objects can be separated from one another without altering the identity of the whole or the part like separating the handle from the main part of the cup, or wheels from the carand they exhibit specific patterns of spatial or structural relationships to one another, which cannot be rearranged; whereas stuff objects like porcelain in a cup or the steel in a wheel lack both these attributes.
Other authors substitute countability for separability. The task of matching AT classes of Anatomical structure with component or stuff object is far from straightforward. The representation of the partonomy of anatomical structures, at all levels of the AT, must distinguish parts established by gene expression from those defined by arbitrary criteria, and also designate parts that are shared or unshared by higher level structures.
Moreover, all such parts can be viewed in different contexts and these views must be accommodated in a generalizable resource such as the FMA. It is these modeling challenges that have motivated us to modify and extend the meronymic categories of Winston et al. In order to avoid confusion in the classification we have implemented, we redefined and renamed these categories in terms of the different partitions current in anatomical and clinical discourse, and further specified these categories by attributes.
Before describing partitions and attributed part relations, however, we summarize the rules we established for assuring the consistent and comprehensive representation of all parts of an anatomical structure in any meronymic relationship or partition. Rules for Part-Whole Relationships For any general or specific meronymic relationship to be valid for anatomical structures, it has to conform to five rules: Partition Consistency, which specifies that any given decomposition of an anatomical structure should be constrained to a single defined context; 4.
Transitivity, specified by Winston et al. In other words, the specific partition will not be valid unless it can account for the whole. Component and stuff objects can be regarded as two of such possible partitions, although as already intimated, they require elaboration and translation into an anatomical context.
Partitions We define a partition as the decomposition of the entire body or any anatomical structure in a given context, meaning a particular viewpoint. Right side and left side of the heart are functional or clinical partitions, whereas subdividing the heart into cardiac chambers or into its walls and cavities are two overlapping anatomical partitions.
Yet another kind of partition relates the heart itself to the cardiovascular system. We rank ordered the different contexts according to which anatomical structures can be decomposed into primary and secondary partitions. We begin with primary partitions of anatomical structures, relate them to component and stuff objects, and then cite examples of secondary partitions that overlap with the primary ones.
Primary Partitions Except for Cell, Cell part, Biological macromolecule and Acellular anatomical structure, entities in all other subclasses of Anatomical structure are constituted, in the ultimate analysis, by cells and body substances which fill anatomical spaces enclosed within and among anatomical structures. However, body substances e. Thus the second of Winston et al. A further conflict arises in that, in accord with Winston et al. Consistent with the rule of transitivity, all parts constituted by cells should be considered as component objects, which is a generalization too broad to be useful.
For example, both the lung parenchyma and a lobe of the lung are partly constituted by cells, yet parenchyma and lobe are different kinds of parts of the lung. Moreover, both the fundus of the stomach and smooth muscle in its wall are partly made up of cells, yet they are not only distinct from one another, but fundus is a different kind of part than a lobe, and the same is true for smooth muscle and parenchyma.
Since the component and stuff object distinction seems to conflict with the ontogenetically self-elaborated structural organization of the body, we replaced them in the FMA with constitutional part and regional part Figure 1. We define these parts before illustrating them with examples. If we choose the open option then these do not belong to the desk; if the closed option, then they do. This choice has no physical meaning in bodily experience — an infinitesimally thin layer of points has no physical thickness to experience unlike a sheet of paper or even the thinnest possible sheet of gold leaf.
In topology, however, the difference between including and excluding this physically intangible layer is important in the mathematical theory. There is more at stake here than the topological boundary having no thickness. The sensation of contact between my hand and the desk suggests we should not find points which are between my hand and the desk yet belong to neither. As physical experience it is clear that if my hand touches the desk then there is no intervening space. If both hand and desk include the points on their boundaries, that is are closed sets, then this denial of intervening space forces the surfaces to share points.
This suggests that the desk and my hand are actually fused together, whereas we know experientially that they can be moved apart without any rupture to their physical boundaries. Alternatively, if they are open sets, then their boundary points belong to neither.
Further options exist, such as one hand being modelled as an open set and the other closed, or both being neither open nor closed. These considerations do not demonstrate that modelling the body as subsets of the conventional topological model of everyday three-dimensional space is impossible. What they do show is that such a topological model will be problematic in encouraging distinctions that do not exist in the body itself.
The topological model provides an elaborate and sophisticated language allowing us to talk about points and to distinguish open and closed sets and to have sets that are neither open nor closed.
To use this language, via some computational representation, to calculate with the body means a considerable overhead in translating back from the topological language to the body as experienced.
The possibility of having a computational model that does not encourage a plethora of phenomenologically meaningless distinctions has been a significant motivation in developing ways of representing the commonsense world in which such distinctions are not available. One of these ways is the computational representation not of point-set topology but of mereotopology. Thus we move next to the examination of the Region-Connection Calculus as a computational representation of mereotopology. Certain properties of connection were noted but there was no attempt by Whitehead to provide a set of axioms, such as we have for topological spaces for example.
The Region-Connection Calculus RCCas this formulation became known, is a logical system which posits the existence of entities, called regions, which have between them a relation called connection. One feature of the RCC that is common to other forms of qualitative spatial and temporal representation is that it leads to a categorisation of ways that the entities it deals with can relate to each other.
In the case of the RCC a number of different categorisations are possible. The simplest is just whether two regions are connected or not. A more useful set is the eightfold categorisation known as RCC These are illustrated in Figure 1. Eight possible relations between regions in RCC, one shaded vertically and one horizontally. What does it mean to have a computational representation of spatial relationships such as these?
If there are no lengths or distances, no shapes or angles associated with regions, no numerical values, of what does computation consist?
Representing Complexity in Part-Whole Relationships within the Foundational Model of Anatomy
The data in such a computation are statements, for example that two specific objects have a certain spatial relationship to each other, or that one object is not spatially connected to any other objects. One aspect of computation is then the derivation of further statements from these. In a very simple and purely mereological example, from B being a part of A and from C being a part of B, we can derive the statement that C is a part of A.
More generally if we know which one of the RCC-8 relations holds between B and A and which one between C and B then we can compute what the possibilities are for the relationship between C and A.
Mereotopology as a lense Mereotopology was partly motivated by the discrepancy between bodily spatial experience and point-based mathematical representations of space. We have seen that mereotopology has produced a computational counterpart in the Region-Connection Calculus.
While logic allows computation in the RCC, it is by no means clear that basing a theory of space on regions automatically means that it will support human spatial thinking more effectively than Cartesian co-ordinate geometry. Indeed the value of computational mereotopology in this respect has been called into question. From a psychological perspective, the cognitive plausibility of certain aspects of qualitative spatial relations has been examined 19and the question of whether the spatial ontologies arising from region-based accounts of space are actually simpler than the conventional mathematical approaches was considered by Pratt and Lemon Having introduced computational mereotopology and traced its origin in spaces of bodily experience, it remains unclear how it might actually be used in a concrete scenario of bodily computation.
To examine one possibility we turn to a use of mereotopology where it functions as a conceptual lens to view and relate ways of thinking about movements of the body in dance. Perhaps surprisingly mereotopology does not figure in this account, but as we have already stressed, a mereotopological model of a spatial situation is not a topological space.
Mereotopology is not a kind of topology in the usual mathematical point-set sense. The understanding that results from this theorisation comes through the use of mereotopological terminology and concepts rather than through the use of formal logic However, we shall see that as the relationship of logic to language is more fundamental than that of logic to number, the representation of statements as logical propositions allows mereotopological description in language to have a computational representation.
Could a practical computational mereotopology be applied in this setting?How Far is "Too Far" Physically When Dating? [Christian Boundaries]
Just concentrating on the spatial, as opposed to spatio-temporal aspects, neck and shoulder are spatial regions but where do they end?
Whilst tracing a route down my neck to my shoulder can I detect when my finger leaves the neck region or enters the shoulder region? Are bodies equipped with neat demarcations, like the boundaries between joints of meat in diagrammatic renderings of cattle, pigs, etc in cookery books? It is even less clear that an arbitrary gesture, a spatio-temporal rather than a purely spatial region, could be identified and made an object of computation. Of course systems already exist that recognise a specific range of gestures that have been learned in advance, but the extraction of meaningful movements as spatio-temporal regions from an unconstrained movement of a body is quite another matter.
Of all the possible regions of space time that could be identified as regions, why should certain ones be counted as important? Mereotopology is unlikely to help make such decisions and this does not appear the best route to lead to mereotopological computation in the context of dance.
An alternative possibility lies in the way that computational mereotopology uses logical statements and not numerical values. Such data, could be augmented by a logical encoding of the properties of the connection relation and the properties of the purely spatial arrangement of the body, in terms of what is joined to what.
Assembling such data for a number of virtual choreographic objects opens the prospect of computation over the space of these virtual objects.
For example, what similarities are there within this space? What fragments of the sequences of gestures in one object appear in which other objects?
Such a representation could play a role analogous to the representations of movements in the languages of Laban 24 or Benesch but in a significantly more qualitative manner. Differentiated experience revealed in movement Computation can be a means of categorizing the world, of measuring associating numbers to things ; of distinguishing one thing from another — whether individuals are the same or not, whether something counts as large or small.
Computation can also be a tool to understand the world, not in a scientific determination, but by virtue of a lens through which the body can be viewed and from where an understanding of embodied space as well as the distinctions marked by one body in relation to another can be conveyed. Could this theoretical characterisation work in practice?
Could a mereotopological computation determine whether a body is black, female, gay, handicapped, obese? What does it mean to compute such a thing? How might the distinctive category of a specific body manifest itself through an aspect of the body open to computational representation?
And could mereotopology have a role in constructing this representation? Iris Young reported 25 a number of distinctions between feminine and masculine modes of bodily movement and further interpreted what these differences reveal about underlying structures in the society some women inhabit.
- Article information
- Annotated SUMO History
- Download options
Young starts from a study by Straus 26 of differences between boys and girls in ways of throwing a ball. Her analysis expands from this specific task to consider other bodily movements directed to specific ends. Thus focus upon ways in which the feminine body frequently or typically conducts itself in such comportment or movement may be particularly revelatory of the structures of feminine existence.
Modes of being-in-the-world are manifested through manners of bodily movement that are apparent to others, and possibly — in different ways — apparent to the individual herself. The analysis Young provides depends on qualitative distinctions in movement. Young shows how traces of inner experience — ways of being-in-the-world — can be read from the outward movement of particular bodies in everyday tasks.
Similar kinds of inner experience would be expected to leave similar kinds of traces. Underlying categories interact with transient inner worlds such as happy, sad, tired, anxious, hopeful, and so on. We can imagine these interactions as complex interweavings of bodily expressions that might be read as messages by others. Deconstructing such messages, or signals, in a quantitative way — a sophisticated kind of Fourier analysis splitting the movements into combinations of atomic constituents like neat sine waves — seems a hopelessly unrealistic idea.
But perhaps mereotopology could lead to a computational representation of these signals in a qualitative way?
Achille C. Varzi (Columbia University): Publications - PhilPeople
If so what would the regions be, and how might they be extracted from the world and represented as data? In that case I suggested that discourse might provide the raw material that computation could manipulate. This possibility arises because discourse and analysis of choreographic movement necessarily refers explicitly to spatial configurations, even if the lens is not the mereotopological one used by Portanova In the case of experience as revealed by the body, however, the correlation of spatial configurations and changing gestures with inner worlds is not often explicitly formulated as discourse by the individual.
Analyses, such as that by Young might be used as a source of discourse for these sub-conscious gestures but these are likely to be much less detailed than in the case of the deliberate and explicit gestures of choreography.
Seeking a discourse yielding terms representable via a logical representation is likewise not an obvious possibility at the level of individual bodies. The role of computational mereotopology here, however, may lie not in the direct application of region-based models of space, but instead in the use of computational representations of spatio-temporal phenomena that use the relational approach to deal with the qualitative.
Computational mereotopology based on spatial regions is only one aspect of the more general qualitative approach to both space and time which we examine next. For example, given two regions they fall into exactly one of the eight relations described earlier.
Although we need have no numerical information about the location of the regions, we can still represent the qualitative spatial relationship computationally through the ability of the computer to encode logic just as readily as quantitative data. The use of qualitative relations as a computational representation of the world is by no means restricted to the representation of space conceived as regions alone.
As a first example consider not regions of space but intervals of time. In what ways can such intervals be related qualitatively? Some possibilities are readily seen: In terms of bodily movement an event, such as raising the left hand from touching the left thigh to being above the head, can have one of these temporal relationships to the event of raising the right hand in a similar way.
A set of thirteen possible relationships between time intervals in this fashion has been determined by Allen This is similar to the eightfold classification in the Region-Connection Calculus in that the collection of entities, spatial regions on the one hand and temporal intervals on the other, has exactly one of the relations between every pair of the entities. It is also similar that the entities have boundaries and we may speak of connection between temporal intervals in exactly the same way as between spatial regions.
That is, two intervals are connected when they either overlap, sharing a common sub-interval, or when they touch at a boundary, or in other words, meet at an endpoint.
Drag to reposition
However, a key feature of the Region-Connection Calculus is the way that the relation of connection alone can be used to define other relations. The fact that two intervals touch at their boundary but do not overlap does not tell us which interval is earlier than the other.
Twelve of the thirteen Allen Relations between temporal intervals. The Region-Connection Calculus leads to but is not the same as a qualitative language of eight relations. The significance of this from a computational point of view lies in the mechanism this provides for dealing with qualitative knowledge about the world rather than in the connections with the mereotopological view of space itself. Systems of qualitative relations are commonplace in knowledge representation.
The many examples include cardinal directions, but one further case is needed here. Consider relations between trajectories in space as explored by Kurata and Egenhofer Here the entities being related are lines in two-dimensional space — imagine traces of bodily movement, the path taken by a person across a public space for example.
Just as with the thirteen Allen relations and the eight RCC relations we can consider how one trajectory relates qualitatively to another. In Figure 3, the first diagram shows two arrows starting at the same point, crossing over once and ending in different places. The lengths of the arrows, their directions, and exactly where they cross are all irrelevant, although any specific instance will have particular values for all these things.
Similarly, in the last example in Figure 3, the key qualitative feature is that the solid arrow meets first the body of the other arrow, and then the end of it and then the starting point of the other arrow before finally reaching its own end. Any two arrows having an arrangement that fits this description stand in the same qualitative relationship to each other.
Example relations between arrows in the head-body-tail calculus of Kurata and Egenhofer. It is this wider field that may well offer the most immediate route to forms of computing with the body that are capable of including representations of the phenomenological experience revealed in bodily movement as considered by Young.
In the final section I examine one way in which QSR has found practical application in artificial intelligence. This involves the re-introduction of co-ordinate geometry, not in opposition to mereotopology but in collaboration with the more general QSR. This will lead finally to a concluding vision of how phenomenological experience could be introduced into a computational representation.
Co-Ordination with Geometry QSR with a combination of space and time has been used in computing activities from video observations This work aims to tell from purely visual evidence what activity is being observed. This might be unloading an aircraft or a simpler domestic scenario such as making a hot drink. In this work a sequence of frames is given which is then processed by computer vision techniques to obtain objects that are tracked through time.
Objects considered in the specific setting include, for example, planes, air-bridges, and vehicles involved in activities at an airport. In an individual video frame each object is reduced to its bounding box, the smallest rectangle drawn in the space of the video frame that encloses the object. The interaction between these rectangles in each frame is assigned one of three qualitative spatial relations: These interactions are computed for each pair of objects in each frame and the temporal relationships between these spatial relationships are also computed.
This computational representation of objects moving in space uses two QSR calculi: However, the spatial relationships are not between the actual objects themselves but between rectangles in the co-ordinate space of each video frame which are tied to the objects.
The mechanism by which spatial and temporal relationships is combined is too sophisticated to review in detail here. However a bodily example should help to clarify the idea. Imagining the bounding boxes in a video we would see the box corresponding to the face contains within it the box corresponding to the fist and this relationship would persist for a number of frames. For a sequence of frames within the ones showing the fist in front of the face, we can also see that the boxes corresponding to the head and the right hand remain separate.
The computational representation describes for each pair of objects in a spatial relationship the temporal relationship between the intervals in which the spatial relationship between those two objects was maintained.
Philosophically in opposition to point-based geometry, and in particular to the computational representation of co-ordinate geometry, the practical application of computational mereotopology has been found not as an alternative to this geometry but as a component of a lens through which the geometry is viewed.
This is the lens of abstraction, of taking a less-detailed view; it replaces the quantitative data of where exactly in the frame-space each bounding box is located with the qualitative description of how each pair of bounding boxes is spatially related. At the temporal level, the lens replaces quantitative information about the precise frames for which spatial relationships are maintained which qualitative statements about relationships between the intervals in which this happens.
In this setting the practical role of QSR does not lie in an independent way of computing with space and time. Instead QSR provides a way of abstracting from the geometric information; a way of avoiding the problem of not being able to see the wood for the trees. Having seen how QSR can be co-ordinated with geometry, we return to the question of the possibility of a computational representation of some aspects of phenomenological bodily experience.
Relational computation and the body In the extraction of knowledge about activities from purely visual evidence just described, two particular sequences of events in the world are recognised as instances of the same abstract activity when the lens provided by QSR for viewing the detailed geometric representation detects no difference in the two sequences.
Although the application of these techniques has been, so far, in the practical domain of knowledge extraction from video in artificial intelligence, it is possible to envisage how they might play a role in a type of computational representation of phenomenological experience. Consider the description of Steinbock: I carve out a world through this habitual body ….