Sketchy Polytopes

Part, Site, Analysis

Notes on Casati and Varzi’s book, “Parts and places: the structures of spatial representation” (MIT Press).

Introduction

  • Leibniz’s contention against Newton: spatial entities - objects or events - are fundamentally prior to space.
  • Questions:
    • How relevant is the decomposition of objects into parts?
    • How do mereological notions interact with truly spatial concepts such as contact, containment, and relative distance?
    • How do the above relate to geometry, topology, and morphology? One could add dynamics.
    • Why are spatial boundaries more important for material objects than events?
    • What distinguishes parthood from constitution?
  • “Our contention is that the shape of the theory of space depends dramatically on the answers one gives.”
  • Elimination of subjectivity and perpectival facts: concern focused on “detched” concepts such as ‘part of’, ‘contained in’, or connected with’.
  • Mereology is ontologically neutral: unconcerned whether entities under investigation abstract or concrete.
  • Mereology as a starting point for theories of greater complexity: topology, morphology, kinematics…
    • “…the distinction between mereology and topology depends on the ontological fauna tha one is willing to countenance.”
  • “Our positions is that boundaries are ontologically on a part with (albeit parasitic upon) extended parts. But unlike extended parts, spatial boundaries have a peculiar relation to space (just as temporal boundaries have a peculiar relation to time): they are located in space, yet do not take up any space.”
  • Potential entities: “entities that do not quite count except as parts of wholes (e.g. the left half of a cigar).
  • Maps: spatial objects that represent other spatial objects, and hence have semantics.

Spatial Entities

  • “Relations amongs part are not necessarily parthood relations”: hence the motivation to use mereotopology instead of only mereology to describe spatial entities.
  • Solid wholes (table) and scattered objects (broken glass): “mereology is about parthood, hence about a relational property. By contrast wholeness is a monadic property…. the latter notion cannot be explained in terms of the former.”
  • There is no way, mereologically, to ‘rule out “bad” wholes consisting of scattered or ill-assorted entities.’ Alternatively, we lose the ability to distinguish between scattered from integral entities.
    • A. N. Whitehead’s ontology in 1919-20 was meant to admit only wholes (events, in Whitehead’s original presentation) made up on parts joined or connected together.
    • x is connected with y if and only if there exists some z that overlaps with both x and y, and that has no part that overlaps neither x not y.”
    • Whitehead’s problem: “x and y are not connected unless the overlapping piece z is itself assumed to be self-connected,” which makes the original argument circular.

2.2 The Mereological Option

  • “Connectedness is a topological relation… it cannot be defined in mereological terms.”
  • Different senses of individual integrity:
    • Aristotle: Continuity, rigidity, uniformity, and qualitative similarity.
    • Husserl: “a range of contents which are all covered by a single foundation with the help of further contents… every content is foundationally associated with every content.”
  • Some concepts of wholes:
    • causally unitary: operations on certain parts have systematic effects on other parts (e.g. pendulum wave).
    • functionally unitary: a system closed under certain functional relationships (e.g. electric circuit - telecom grid is a great example of a scattered whole).
    • teleologically unitary: sharing of a common goal (e.g. a sports team).
    • unitary by way of dependence: …
  • The whole considered here is topologically connected.
  • Whitehead’s 1929 redefinition of mereology in terms of topology: x is part of y if and only if everything connected to x is also connected to y. [3]
    • The notion of “connection” here implies more than overlaps. What’s the connections between two halves of a brick?
    • If, like Whitehead, we limit ourselves to considering only spatiotemporal regions instead of objects/entities, then we can say that regions x and y are connected if they have at least one point in common. A special kind of overlap i.e. a boundary.
  • What if we don’t limit ourselves to regions? “Either we insist that all spatial entities can be mapped onto their regions…, or we maintain that a topological apparatus defined with respect to regions can be applied holus bolus to ordinary things and events alike.”
    • The idea that entities have a 1:1 correspondence with their regions (what Locke used as a basis for the criterion of identity) may apply to material objects but not events. E.g. Caeser’s death (an event) took place in the same region where Caeser’s body (a physical object) was located, yet they are two distinct entities.

2.3 The Hole Trouble

  • To circumvent the issue of the ontological status of holes, “use holes as well as events and ordinary objects as dummies for spatial entities that are not themselves regions of space”
  • Reasons for not reducing spatial entities to the regions that they occupy?
    • the relationship between the body and space is trivialized
    • morphological features are ignored (the question of holes as entities cannot even be raised)
  • What kind of spatial entities are holes?
    • Substantivalist reduction: special regions of space
      • Harder to explain the individuality of a hole changing in time and/or space
    • Adjectivalist reduction: property of the objects that host them
      • Blind to topologically insignificant holes, such as superficial hollows, grooves, and indentations
      • Needs a lot of adjectives to characterize different configurations
        • “The cheese has at least one hole in it” vs “The cheese is at least singly-perforated”
  • “The expressive power of the predicate ‘connection’ is safe. But this doesn’t save us from explicit reference to immaterial entities.”

2.4 The Compositional Approach

  • Two lessons so far:
    • “Mereology alone is too weak; topology alone is too strong.”
      • Topology is clearly also two weak to consider morphological features other than holes
    • Entities should be considered w.r.t their mutual spatial relationships and their relationships to space
      • Again, only because otherwise morpholopgy is ignored
  • Recognition-by-Components (RBC) theory:
    • Aims to support both visual object representation and object recognition
    • Offers a spatial syntax where every object is composed of “geonic” (geometrical ion) elements
    • Given a vector of spatial attributes per geon, three geons can potentially describe 1.4 billion distinct object shapes

References

  1. Kostic D. 2020 General theory of topological explanations and explanatory asymmetry. Phil. Trans. R. Soc. B 375: 20190321. link
  2. D. Koehler. The city as an element of architecture. link
  3. B. Clark. A calculus of individuals based on “connection”. Notre Dame J. Formal Logic 22(3): 204-218 (July 1981). link
  4. Jones, N. (2021). Mereological Composition in Analytic and Buddhist Perspective. Journal of the American Philosophical Association, 1-22. doi:10.1017/apa.2020.41. link