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2

3

4 A Simple Partition

5 A partition can be more or less refined

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7

8 Partition A partition is the drawing of a (typically complex) fiat boundary over a certain domain

9 Artist’s Grid

10 Transparency A partition is transparent It leaves the world exactly as it is

11 Extension of Partitions via enlargement of domain (via gluing of partitions) via refinement via Cartesian product

12 Artist’s Grid

13 Label/Address System A partition typically comes with labels and an address system

14 Cerebral Cortex

15 Mouse Chromosome Five

16 Montana

17 A partition can comprehend the whole of reality

18 Universe

19 Universe

20 It can do this in different ways

21 Periodic Table

22 Perspectivalism Different partitions may represent cuts through the same reality which are skew to each other

23 Universe/Periodic Table

24 Fiat Fiat objects determined by partitions

25 Kansas

26 France

27 Bona Fide Bona fide objects

28 California Land Cover

29 Lake Tahoe Land Cover Form / Matter

30 Fiat vs bona fide The fiat boundaries which constitute a partition may or may not correspond to bona fide boundaries on the side of the objects in the domain of the partition

31 Fiat vs bona fide but since each partition is transparent (veridical) its fiat boundaries will correspond at least to fiat boundaries on the side of the objects in its domain

32 Partitions vs. 0bjects Partitions are artefacts of our cognition (of our theorizing, classifying, mapping activity)

33 Alberti’s Grid c.1450

34 Sets, groupings, mereological fusions, tesselations belong not to the realm of objects but to the realm of partitions

35 we have all been looking in the wrong direction

36 Dürer Reverse

37 Intentionality

38 Intentionality

39 Lakoff’s Big Error the road to idealism

40 Lakoff’s Big Error

41 Objects and cells objects are located in cells as guests are located in hotel rooms: L A (x, z) x  A :=  z (L A (x, z) object x is recognized by partition A

42 Defining  Sets are (at best) special cases of partitions

43 Set as List Partition A set is a list partition (it is, roughly, a partition minus labels and address system) The elements exist within the set without order or location —they can be permuted at will and the set remains identical

44 Against models transparent partitions vs. models and sets

45 Set Intentionality

46 D Lewis on Sets Set theory rests on one central relation: the relation between element and singleton. Sets are mereological fusions of their singletons (Lewis, Parts of Classes, 1991) But the relation between an element and its singleton is, as Lewis notes, “enveloped in mystery”

47 Mystery Lewis:... since all classes are fusions of singletons, and nothing over and above the singletons they’re made of, our utter ignorance about the nature of the singletons amounts to utter ignorance about the nature of classes generally.

48 L(x, z) An object can be located in a cell within a partition in any number of ways: – object x exemplifies kind K – object x possesses property P – object x falls under concept C – object x is in location L

49 –object x is a member of population P L(x, z) – object x has an observable attribute v in range R (of soil fertility, foliage density, exposure to sunlight, etc.) – object x is in ecological niche N L(x, z)

50 Cells form a partial order z  A z' := cell z is a sub-cell of the cell in partition A (compare: dog as sub-cell of mammal) not equivalent to  x(x  z  x  z' )

51 Empty Set Partition theory has no counterpart of the empty set Periodic Table

52 Union fails 1 We do not have z 1, z 2  A  z 1  z 2  A Consider: z 1 = Germany z 2 = France A = partition of states

53 Union fails 2 We do not have x 1, x 2  A  x 1  x 2  A Consider: x 1 = my cat Plato x 2 = your dog Aristotle A = the partition of the mammals

54 Better than Sets even in spite of all of these problems partitions are