Mont Blanc

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A Simple Partition
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A partition can be more or less refined

• A partition can be more or less refined

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(No Transcript)
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Partition

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

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Artists Grid
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Transparency

• A partition is transparent
• It leaves the world exactly as it is

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Extension of Partitions

• via enlargement of domain
• (via gluing of partitions)
• via refinement
• via Cartesian product

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Artists Grid
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Label/Address System

• A partition typically comes with labels and an
  address system

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Cerebral Cortex
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Mouse Chromosome Five
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Montana
Montana
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A partition can comprehend the whole of reality
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Universe
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Universe
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It can do this in different ways
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Periodic Table
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Perspectivalism

• Perspectivalism

Different partitions may represent cuts through
the same reality which are skew to each other
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Universe/Periodic Table
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Fiat

• Fiat objects determined by partitions

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Kansas
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France
France
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Bona Fide

• Bona fide objects

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California Land Cover
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Lake Tahoe Land Cover
Form / Matter
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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

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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

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Partitions vs. 0bjects

• Partitions are artefacts of our cognition
• (of our theorizing, classifying, mapping
  activity)

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Albertis Gridc.1450
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Sets, groupings, mereological fusions,
tesselations belong not to the realm of objects
but to the realm of partitions
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we have all been looking in the wrong direction
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Dürer Reverse
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Intentionality
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Intentionality
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Lakoffs Big Error
the road to idealism
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Lakoffs Big Error
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Objects and cells

• objects are located in cells as guests are
  located in hotel rooms

• LA(x, z)

object x is recognized by partition A
x ? A ?z (LA(x, z)
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Defining ?
Sets are (at best) special cases of partitions

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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
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Against models

• transparent partitions
• vs.
• models and sets

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Set Intentionality
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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

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Mystery

• Lewis
• ... since all classes are fusions of singletons,
  and nothing over and above the singletons theyre
  made of, our utter ignorance about the nature of
  the singletons amounts to utter ignorance about
  the nature of classes generally.

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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 falls under concept C

• object x possesses property P

• object x is in location L

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L(x, z)
L(x, z)

• object x is a member of population P

• object x is in ecological niche N

• object x has an observable attribute v in range
  R (of soil fertility, foliage density, exposure
  to sunlight, etc.)

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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' )

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Empty Set

• Partition theory has no counterpart of the empty
  set

Periodic Table
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Union fails 1

• We do not have
• z1, z2 ? A ? z1 ? z2 ? A
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• Consider z1 Germany
• z2 France
• A partition of states
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Union fails 2

• We do not have
• x1, x2? A ? x1 ? x2 ? A
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• Consider
• x1 my cat Plato
• x2 your dog Aristotle
• A the partition of the mammals
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Better than Sets
even in spite of all of these problems
partitions are