True Grid

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True Grid

• Barry Smith
• http//ontology.buffalo.edu/smith

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Leon Battista Alberti (1404-1472)
author of Della pittura (1435-36) the first
scientific manual of painting
and simultaneously a contribution to the
ontology of visual representation
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Leon Battista Alberti (1404-1472)
Albertis grid
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Leon Battista Alberti (1404-1472)
The goal of the artist is to produce a picture
that will represent the visible world as if the
observer of the picture were looking through a
window
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Dürer
Underweysung der Messung (1525)
the problem of measuring the surfaces of reality
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Panofsky

• one can properly speak of a perspectival
  intuition of space only where
• a whole picture is as it were transformed into a
  window through which we should then believe
  ourselves to be looking into the space

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true or correct perspective
what is captured on a plane intersecting the
visual pyramid
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true or correct perspective
what is captured by a transparent grid
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Practical problem of perspective

• solved by Brunelleschi in 1425
• with a painting of the Baptistery of St. John in
  Florence

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Baptistery
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Brunelleschis Peepshow
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Theoretical problem of perspective
solved by Alberti in Book 1of Della pittura The
solution, captured in the diagram of the
reticolato, belongs to projective geometry
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How did Alberti solve the theoretical problem of
linear perspective ?
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And why did mankind have to wait 1700 after
Euclids Geometry and Optics for this solution?
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The answer belongs to the history of cartography
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Ptolemys Geographia (c. 140 A.D.)

• uses a regular mathematical grid system to map
  the entire known world

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Ptolemaic World Maps
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Ptolemys Regional World Divisions
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Example of a Pre-Ptolemaic Map
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Ptolemys grid system

• transformed the relationship between astronomy
  and sublunar physics
• ... this made the world below for the first time
  susceptible to uniform mathematical treatment

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The Rediscovery of Ptolemys Geographia

• Greek text arrived in Florence from
  Constantinople in 1400

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Florence by 1424 a center of cartographic and
geographic study

• commentaries on Florentine versions of the
  Geographia influenced Columbus

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Uccello Gridded Challice c. 1450
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Ptolemys grid system

• not just mathematical regularity
• also transparency
• ... the grid helps us to see the world aright

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Grids of Reality (Mercator 1569)
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Alberti extended Ptolemys method to pictures

• Alberti the veil affords the greatest
  assistance in executing your pictures,
• since you can see any object that is round and
  in relief, represented on the flat surface of the
  veil.

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Giotto
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Ideal City
(Grid)
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School of Athens
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Albertis Ontology of Painting

• 1. The grid of the reticolato and the grid of
  the objective reality beyond are linked together
  by a projective relation
• 2. The grid effects a selection, from the
  totality of surfaces in objective reality, of
  those parts which will be foregrounded in the
  painting

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the result of this selection is perfectly
objective

• compare what happens on the stage in the theater
• selection does not imply distortion

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Degens Law

• If a well-formed diagram is transparent to
  reality, then so are all its well-formed parts

From
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we can validly infer
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Mereological fallacies

• Inferring that a part is the whole
• Concluding, given a true representation, that
  truth implies completeness

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Algebra

• Algebraic ontologists are correct the world
  contains processes
• they err only when they add and nothing else
• Field ontologists are correct the world
  contains fields
• they err only when they add and nothing else

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Selection implies distortion

• only if the mistake is made of assuming that the
  selected part is identical with the whole

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The world contains fields

• Evidence this assumption supports successful
  predictions
• ? The world contains only fields and nothing
  else
• This conclusion rests on a mereological fallacy
  (and also on a mistaken understanding of the role
  of granularity)

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How to Tell the Truth with Maps

• There are maps of different scales
• There are transparent grids of different
  granularities

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How to Tell the Truth with Maps

• Albertis reticolato casts its transparent net
  over the array of planes out there in objective
  reality in such a way as to cast into relief a
  visual scene.
• A good map casts its transparent net over
  reality in such a way as to cast into relief a
  certain portion of the surface of the earth

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Some nets are regular
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Some nets are irregular
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Some nets are many-sorted
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containing labeled and non-labeledcells formed
by

• linear and non-linear icons
• icons representing spatial regions

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Most maps contain two grids of cells

• projecting simultaneously onto the same
  underlying reality

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The analogy between maps and pictures

• has nothing to do with perspective
• but rather with the highly general concept of a
  transparent grid and with an associated highly
  general notion of projection
• But how are we to understand this notion of
  projection?

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Problem

• How many cells does this map contain?

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Is the Western half of Wyoming represented on
this map?

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Is the central square mile of Wyoming represented
on this map?
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Is the capital of Wyoming represented on this map?
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Is the Texas panhandle represented on this map?
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Is Hot Springs County represented on this map?
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Are the molecules of Wyoming represented on this
map?
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Optical Projection
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Cartographic Projection
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Projection is involved wherever there is
intentionality
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intentionality the directedness towards objects
of a mental act
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The theory of transparent grids can help us to
understand how intentional directedness works
selection, foregrounding, labeling, classification
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Intentional directedness

• is effected in every case via something like
  an Albertian grid a cognitive artifact which we
  shall call a granular partition
• we can reach out to objects because partitions
  are transparent

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and such partitions

• are always granular
• when we perceive a frog we do not perceive the
  molecules in the frogs skin
• when we think about Mary, we do not think about
  the molecules in Marys nose

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Vagueness comes to awareness

• through ontological zooming (from coarse-grained
  to fine-grained partitions)

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This granularity of our partitions

• explains also (how we are able to cope with) the
  phenomenon of vagueness
• when we think about Mary, we do not think about
  the molecules in Marys nose
• when we think about Mount Everest, we do not
  think about where, precisely, the mountain begins
  or ends in its foothills

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Foreground/Background
granular partitions are involved wherever there
is a division of reality into foreground and
background
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That granular partitions have multiple cells
corresponds to the fact that intentionality can
be many-rayed

• people
• my three sons
• Benelux
• the Germans
• COSIT participants

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Counting
counting involves
many-rayed intentionality
plus granularity
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Granular partitions

• are involved in simple acts of naming,
  classifying, seeing, recognizing, mapping
• All (veridical) databases and information
  systems involve granular partitions

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Intentional directedness

• is effected via partitions
• we reach out to the objects themselves because
  our partitions are transparent

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A granular partition is like an open window
we use partitions because they help us to see the
world aright
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Some would deny the veridicality of intentionality

• partitions, concepts, contents are not
  transparent, they say ...
• we can never see objects as they really are, they
  say ...
• because we must always use those human artifacts
  called partitions (concepts, ideas, words,
  metaphors, image schemata ...)

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Against the veridicality of intentionality

• and whenever we grasp an object by means of a
  concept we somehow change the object,
• hence we can never know how the object really is
  in itself
• call this Midas-touch epistemology

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After Duchamp

• there is no place for talk of correct
  perspectival representation, with its implication
  to the effect that there is some single detached
  master point of view
• no method of painting can be true or
  correct for there is no single notion of
  reality against which its results could be matched

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The realist response

• even granting the simplifying assumptions of
  geometrical optics, perspective paintings
  correspond to the way we see the world around us
  with a very high degree of accuracy.
• The best explanation for this is the
  mathematical forms captured in the geometry of
  perspective are out there in the world

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The realist response

• even granting the simplifying assumptions
  involved when we use a grid of cells of a certain
  granularity, our intentional reference gives us
  access to the world around us with a very high
  degree of accuracy.
• The best explanation for this is our granular
  partitions are transparent to the structures out
  there in the world

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Fit happens
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Fit happens

• There are structures out there in the world
  accessible at different levels of granularity
• (There are maps of different scales)

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Every one of the standard map projection systems
is correct

• the point is merely to use them properly
• maps do not lie (but they may be old, or embody
  local errors)
• intelligence of the projective technique vs.
  stupidity of the interpreter

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The railway tracks on the Circle Line are not in
fact yellow
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There is no Gods eye perspective no view
from nowhere

• No super-partition encapsulating the entirety of
  human knowledge
• But this does not mean that every one of the
  myriad perspectives we enjoy embodies a false
  view of reality
• Rather, it means that we must take distinct
  (granular) perspectives together

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There is super-partition encapsulating the
entirety of human knowledge
Yet the claims of the scientific method to yield
knowledge of reality still stand the mistake
would be to claim that we can know reality only
through science (or through Haskell-programming,
or whatnot)
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Almost all of our partitions

• are transparent
• intentional directedness succeeds
• ... our job is to understand it

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THE END

• THE END