Mereotopologies: Fiat and Bona Fide Boundaries

1. Mereotopologies: Fiat and Bona Fide Boundaries

2. A substance has a complete physical boundary • The latter is a special sort of part of a substance • … a boundary part • something like a maximally thin extremal slice

3. boundary substance interior

4. A substance takes up space. • A substance occupies a place or topoid (which enjoys an analogous completeness or rounded-offness) • A substance enjoys a place at a time

5. A substance has spatial parts • … perhaps also holes

6. Each substance is such as to have divisible bulk: • it can in principle be divided into separate spatially extended substances

7. By virtue of theirdivisible bulk • substances compete for space: • (unlike shadows and holes) • no two substances can occupy the same spatial region at the same time.

8. Substances vs. Collectives • Collectives = unified aggregates: families, jazz bands, empires • Collectives are real constituents of reality (contra sets) • but still they are not additional constituents, over and above the substances which are their parts.

9. Collectives inherit some, but not all, of the ontological marks of substances • They can admit contrary moments at different times.

10. Collectives, • like substances, • may gain and lose parts or members • may undergo other sorts of changes through time.

11. Qualities and processes, too, may form collectives • a musical chord is a collective of individual tones • football matches, wars, plagues are collectives of actions involving human beings

12. Collectives/heaps • are the duals of undetached parts • Both involve fiat boundaries

13. Substances, Undetached Parts and Heaps • Substances are unities. • They enjoy a natural completeness • in contrast to their undetached parts (arms, legs) • and to heaps or aggregates • … these are topological distinctions

14. substance undetached part collective of substances

15. special sorts of undetached parts • ulcers • tumors • lesions • …

16. physical (bona fide) boundary fiat boundary Fiat boundaries

17. Holes, too, involve fiat boundaries

18. hole A hole in the ground • Solid physical boundaries at the floor and walls but with a lid that is not made of matter:

19. Holes involve two kinds of boundaries • bona fide boundaries which exist independently of our demarcating acts • fiat boundaries which exist only because we put them there

20. Examples • of bona fide boundaries: • an animal’s skin, the surface of the planet • of fiat boundaries: • the boundaries of postal districts and census tracts

21. Mountain • bona fide upper boundaries • with a fiat base:

22. where does the mountain start ? ... a mountain is not a substance

23. nose ...and it’s not an accident, either

24. Examples • of bona fide boundaries: • an animal’s skin, the surface of the planet • of fiat boundaries: • the boundaries of postal districts and census tracts

25. Mountain • bona fide upper boundaries • with a fiat base:

26. Architects Plan for a House • fiat upper boundaries • with a bona fide base:

27. where does the mountain start ? ... a mountain is not a substance

28. nose ...and it’s not a process, either

29. One-place qualities and processes • depend on one substance • (as a headache depends upon a head)

30. kiss John Mary • Relational qualities and processes stand in relations of one-sided dependence to a plurality of substances simultaneously

31. Examples of relational qualities and processes • kisses, thumps, conversations, • dances, legal systems • Such real relational entities • join their carriers together into collectives of greater or lesser duration

32. Mereology • ‘Entity’ = absolutely general ontological term of art • embracing at least: all substances, qualities, processes, and all the wholes and parts thereof, including boundaries

33. Primitive notion of part • ‘x is part of y’ in symbols: ‘x ≤ y’

34. We define overlap as the sharing of common parts: • O(x, y) := z(z ≤ x  z ≤ y)

35. Axioms for basic mereology • AM1 x ≤ x • AM2 x ≤ y  y ≤ x  x = y • AM3 x ≤ y  y ≤ z  x ≤ z • Parthood is a reflexive, antisymmetric, and transitive relation, a partial ordering.

36. Extensionality • AM4 z(z ≤ x  O(z, y))  x ≤ y • If every part of x overlaps with y • then x is part of y • cf. status and bronze

37. Sum • AM5 x(x)  • y(z(O(y,z) x(x  O(x,z)))) • For every satisfied property or condition  there exists an entity, the sum of all the -ers

38. Definition of Sum • x(x) := yz(O(y,z) x(x  O(x,z))) • The sum of all the -ers is that entity which overlaps with z if and only if there is some -er which overlaps with z

39. Examples of sums • electricity, Christianity, your body’s metabolism • the Beatles, the population of Erie County, the species cat

40. Other Boolean Relations • x  y := z(z ≤ x  z ≤ y) binary sum • x  y := z(z ≤ x  z ≤ y) product

41. Other Boolean Relations • x – y := z (z ≤ x  O(z, y)) difference • –x := z (O(z, x)) complement

42. What is a Substance? • Bundle theories: a substance is a whole made up of tropes as parts. • What holds the tropes together? • ... problem of unity

43. Topology • How can we transform a sheet of rubber in ways which do not involve cutting or tearing?

44. Topology • We can invert it, stretch or compress it, move it, bend it, twist it. Certain properties will be invariant under such transformations – • ‘topological spatial properties’

45. Topology • Such properties will fail to be invariant under transformations which involve cutting or tearing or gluing together of parts or the drilling of holes

46. Examples of topological spatial properties • The property of being a (single, connected) body • The property of possessing holes (tunnels, internal cavities) • The property of being a heap • The property of being an undetached part of a body

47. Examples of topological spatial properties • It is a topological spatial property of a pack of playing cards that it consists of this or that number of separate cards • It is a topological spatial property of my arm that it is connected to my body.

48. Topological Properties • Analogous topological properties are manifested also in the temporal realm: • they are those properties of temporal structures which are invariant under transformations of • slowing down, speeding up, temporal translocation …

49. Topological Properties

50. Topology and Boundaries • Open set: (0, 1) • Closed set: [0, 1] • Open object: • Closed object: