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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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Partition

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

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Artist’s 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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Artist’s 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

PerspectivalismDifferent 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

25Kansas

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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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Alberti’s Grid

c.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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Lakoff’s Big Error

the road to idealism

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Lakoff’s Big Error

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Objects and cells

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

LA(x, z)

x A := z (LA(x, z)

object x is recognized by partition A

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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 withoutorder 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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MysteryLewis:

... 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.

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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 possesses property P

– object x falls under concept C

– object x is in location L

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– 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)

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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

 

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 Consider:

x1 = my cat Plato

x2 = your dog AristotleA = the partition of the mammals

 

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Better than Sets

even in spite of all of these problems

partitions are