Title: True Grid
1True Grid
- Barry Smith
- http//buffalo.ontology.edu
2Leon 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
3Leon Battista Alberti (1404-1472)
Albertis grid
4Leon 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
5Dürer
Underweysung der Messung (1525)
the problem of measuring the surfaces of reality
6true or correct perspective
what is captured on a plane intersecting the
visual pyramid
7true or correct perspective
what is captured by a transparent grid
8Practical problem of perspective
- solved by Brunelleschi in 1425
- with a painting of the Baptistery of St. John in
Florence
9Baptistery
10Brunelleschis Peepshow
11Theoretical problem of perspective
solved by Alberti in Book 1of Della pittura The
solution, captured in the diagram of the
reticolato, belongs to projective geometry
12How did Alberti solve the theoretical problem of
linear perspective ?
13Why did mankind have to wait 1700 after Euclids
Geometry and Optics for this solution?
14The answer belongs to the history of cartography
15Ptolemys Geographia (c. 140 A.D.)
- uses a regular mathematical grid system to map
the entire known world
16Example of a Pre-Ptolemaic Map
17Ptolemaic World Maps
18Ptolemys Regional World Divisions
19Ptolemys grid system
- transformed the relationship between astronomy
and sublunar physics - ... this made the world below for the first time
susceptible to uniform mathematical treatment
20The Rediscovery of Ptolemys Geographia
- Greek text arrived in Florence from
Constantinople in 1400
21Uccello Gridded Challice c. 1450
22Florence by 1424 a center of cartographic and
geographic study
- commentaries on Florentine versions of the
Geographia influenced Columbus
23Ptolemys grid system
- not just mathematical regularity
- also transparency
- ... the grid helps us to see the world aright
24Alberti 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.
25Giotto
26Ideal City
(Grid)
27School of Athens
28Albertis 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
29the result of this selection is perfectly
objective
- selection does not imply distortion
30Mereological fallacies
- Inferring that a part is the whole
- Concluding, given a true representation, that
truth implies completeness
31Algebra
- 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
32The 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)
33How to Tell the Truth with Maps
- There are maps of different scales
- There are transparent grids of different
granularities
34How 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
35Some nets are regular
36Some nets are irregular
37Some nets are many-sorted
38 containing labeled and non-labeledcells formed
by
- linear and non-linear icons
- including icons representing spatial regions
-
39Most maps contain two grids of cells
- projecting simultaneously onto the same
underlying reality
40The 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)
41Problem
- How many cells does this map contain?
42Is the Western half of Wyoming represented on
this map?
43Is the capital of Wyoming represented on this map?
44Is Hot Springs County represented on this map?
45Are the molecules of Wyoming represented on this
map?
46Is the Texas panhandle represented on this map?
47Cartographic Projection
48Optical Projection
49intentionality the directedness towards objects
of a mental act
50selection, foregrounding, labeling, classification
51Intentional directedness
- we can reach out to objects because our
partitions are transparent
52and 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
53This 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
54CountingItIt
it explains how counting is possible
counting involves
many-rayed intentionality
plus granularity
55Intentional directedness
- is effected via partitions
- we are able to reach out to the objects
themselves because our partitions are transparent
56A granular partition is like an open window
57Some 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 ...)
58Against 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
59The 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
60Fit happens
61Fit happens
- There are structures out there in the world
accessible at different levels of granularity - (There are maps of different scales)
62Every 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
63The railway tracks on the Circle Line are not in
fact yellow
64There 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
65There 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)
66Almost all of our partitions
- are transparent
- intentional directedness succeeds
- ... our job is to understand it
67THE END