True Grid
Barry Smith
http://ontology.buffalo.edu/smith
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Leon Battista Alberti (1404-1472)
author of Della pittura
(1435-36)
Alberti (Medal)
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)
Alberti’s 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
5
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
6
‘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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Brunelleschi’s Peepshow
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Theoretical problem of perspective
solved by Alberti in Book 1
of 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 Euclid’s Geometry and
Optics for this solution?
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The answer belongs to the history of
cartography
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Ptolemy’s 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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Ptolemy’s Regional World
Divisions
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Example of a Pre-Ptolemaic Map
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Ptolemy’s 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 Ptolemy’s
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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Ptolemy’s 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 Ptolemy’s 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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Alberti’s 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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Degen’s Law
If a well-formed diagram is transparent to
reality, then so are all its well-formed parts
The Flagellation
From:
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we can validly infer:
The Flagellation
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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
Alberti’s 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-labeled
cells 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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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 frog’s skin
when we think about Mary, we do not think
about the molecules in Mary’s nose
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Vagueness comes to awareness
through ontological zooming (from coarsegrained 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 Mary’s 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 ‘God’s 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
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