1 True Grid Barry Smith

2 Leon Battista Alberti ( ) author of Della pittura ( ) the first scientific manual of painting and simultaneously a contribution to the ontology of visual representation

3 Leon Battista Alberti ( ) Alberti’s grid

4 Leon Battista Alberti ( ) 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

5 Dürer Underweysung der Messung (1525) the problem of measuring the surfaces of reality

6 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

7 ‘ true ’ or correct perspective = what is captured on a plane intersecting the visual pyramid

8 ‘ true ’ or correct perspective = what is captured by a transparent grid

9 Practical problem of perspective solved by Brunelleschi in 1425 with a painting of the Baptistery of St. John in Florence

10 Baptistery

11 Brunelleschi’s Peepshow

12 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

13 How did Alberti solve the theoretical problem of linear perspective ?

14 And why did mankind have to wait 1700 after Euclid’s Geometry and Optics for this solution?

15 The answer belongs to the history of cartography

16 Ptolemy’s Geographia (c. 140 A.D.) uses a regular mathematical grid system to map the entire known world

17 Ptolemaic World Maps

18 Ptolemy’s Regional World Divisions

19 Example of a Pre-Ptolemaic Map

20 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

21 The Rediscovery of Ptolemy’s Geographia Greek text arrived in Florence from Constantinople in 1400

22 Florence by 1424 a center of cartographic and geographic study commentaries on Florentine versions of the Geographia influenced Columbus

23 Uccello: Gridded Challice c. 1450

24 Ptolemy’s grid system not just mathematical regularity also transparency... the grid helps us to see the world aright

25 Grids of Reality (Mercator 1569)

26 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.

27 Giotto

28 Ideal City (Grid)

29 School of Athens

30 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

31 the result of this selection is perfectly objective compare what happens on the stage in the theater selection does not imply distortion

32 Degen’s Law If a well-formed diagram is transparent to reality, then so are all its well-formed parts From:

33 we can validly infer:

34 Mereological fallacies Inferring that a part is the whole Concluding, given a true representation, that truth implies completeness

35 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

36 Selection implies distortion only if the mistake is made of assuming that the selected part is identical with the whole

37 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)

38 How to Tell the Truth with Maps There are maps of different scales There are transparent grids of different granularities

39 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

40 Some nets are regular

41 Some nets are irregular

42 Some nets are many-sorted

43 … containing labeled and non-labeled cells formed by: linear and non-linear icons icons representing spatial regions

44 Most maps contain two grids of cells projecting simultaneously onto the same underlying reality

45 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?

46 Problem How many cells does this map contain?: Wyoming

47 Is the Western half of Wyoming represented on this map? Wyoming

48 Is the central square mile of Wyoming represented on this map? Wyoming

49 Is the capital of Wyoming represented on this map? Wyoming

50 Is the Texas panhandle represented on this map?

51 Is Hot Springs County represented on this map? Wyoming

52 Are the molecules of Wyoming represented on this map? Wyoming

53 Optical Projection

54 Cartographic Projection

55 Projection is involved wherever there is intentionality

56 intentionality = the directedness towards objects of a mental act

57 selection, foregrounding, labeling, classification The theory of transparent grids can help us to understand how intentional directedness works

58 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

59 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

60 Vagueness comes to awareness through ontological zooming (from coarse- grained to fine-grained partitions)

61 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

62 Foreground/Background granular partitions are involved wherever there is a division of reality into foreground and background

63 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’

64 Counting many-rayed intentionality counting involves plus granularity

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

66 Intentional directedness … is effected via partitions we reach out to the objects themselves because our partitions are transparent

67 A granular partition is like an open window we use partitions because they help us to see the world aright

68 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...)

69 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“

70 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

71 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

72 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

73 Fit happens

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

75 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

76 The railway tracks on the Circle Line are not in fact yellow:

77 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

78 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)

79 Almost all of our partitions are transparent intentional directedness succeeds... our job is to understand it

80 THE END