True Grid

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True Grid

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

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Leon Battista Alberti (1404-1472)
author of Della pittura (1435-36) the first
scientific manual of painting
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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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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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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Example of a Pre-Ptolemaic Map
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Ptolemaic World Maps
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Ptolemys Regional World Divisions
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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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Uccello Gridded Challice c. 1450
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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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Ptolemys grid system

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

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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

• selection does not imply distortion

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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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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
• including 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 the associated highly
  general notions of selection and foregrounding
  ( granularity and 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 capital of Wyoming 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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Is the Texas panhandle represented on this map?
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Cartographic Projection
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Optical Projection
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intentionality the directedness towards objects
of a mental act
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selection, foregrounding, labeling, classification
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Intentional directedness

• we can reach out to objects because our
  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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This granularity of our partitions

• explains also (how we are able to cope with) the
  phenomenon of vagueness
• 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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CountingItIt
it explains how counting is possible
counting involves
many-rayed intentionality
plus granularity
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Intentional directedness

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

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A granular partition is like an open window
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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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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 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, globally (though they may
  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 no 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