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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)

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

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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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Brunelleschi’s 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 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

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The Flagellation

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

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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selection, foregrounding,

labeling, classification

The theory of transparent grids can help us to

understand how intentional directedness works

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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 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 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

many-rayed intentionalitycounting involves

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 transparentintentional directedness succeeds

... our job is to understand it

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THE END

THE END