1. 1 True Grid, or: The Windowing of Attention in Pictures Barry Smith

2. 2 Leon Battista Alberti (1404-1472) architect moral philosopher cryptographer painter mathematician Papal adviser and Doctor of Canon Law land surveyor

3. 3 Alberti (Medal)

4. 4 Della pittura 1435–36

5. 5 The goal of the artist: to produce a picture that will represent the visible world as if the observer of the picture were looking through a window

6. 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. 7 Alberti’s Grid c.1450

8. 8 Machines for seeing for measuring the visible surfaces of external reality ‘reticolato’ ‘grill’ (graticola) ‘veil’ (velo)

9. 9 Dürer’s treatise on measurement Underweysung der Messung (1527)

10. 10 Dürer

11. 11 Dürer

12. 12 Dürer

13. 13 Artist’s Grid transparent grid

14. 14 Practical problem of perspective solved by Brunelleschi in 1425 with painting of Baptistery of St. John in Florence

15. 15

16. 16 Peepshow

17. 17

18. 18 Theoretical problem of perspective solved by Alberti in Book 1 of Della pittura

19. 19

20. 20 Alberti influence on Dürer Piero della Francesca Leonardo da Vinci transformed painting in realist direction, freed European art from bad geometry

21. 21 Ideal City

22. 22 The Flagellation

23. 23 School of Athens

24. 24 Giotto

25. 25 Giotto

26. 26 School of Athens

27. 27 How did Alberti solve the theoretical problem of linear perspective ?

28. 28 Rediscovery of Ptolemy’s Geographia (c. 140 A.D.) Greek text arrived in Florence from Constantinople in 1400 Ptolemy used regular mathematical grid system to map the entire known world

29. 29 Florence by 1424 a center of cartographic and geographic study commentaries on Florentine versions of the Geographia influenced Columbus

30. 30 Ptolemaic World Map 12th-13th Century

31. 31 Ptolemaic World Map, 13th Century

32. 32 Ptolemaic World Map, J. Scotus 1505

33. 33 Ptolemy’s grid system not just mathematical regularity also transparency... the grid helps us to see the world aright

34. 34 Ptolemy’s grid system transformed relationship between astronomy vs. sublunar physics for the first time makes the world below susceptible to uniform mathematical treatment

35. 35 Ptolemy‘s Regional World Divisions

36. 36 Hecataeus 6th Century B.C:

37. 37 Ptolemy‘s Regional World Divisions

38. 38 Ptolemy’s grid system began to be used as basis for territorial boundary-demarcation during the wars of 1420 a longitudinal line established as the boundary between Milan and Florence first occasion when an imaginary mathematical line was recognized as a political-territorial limit.

39. 39 Kansas

40. 40 Periodic Table

41. 41 Ptolemy’s ‘third cartographic method’: how to make a picture based on a projection from a single point representing the eye of an individual human beholder.

42. 42 Alberti extended Ptolemy’s method to pictures Ptolemy applied his perspective construction only in the construction of maps and in stage design 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.

43. 43 Land Surveying Descriptio urbis Romae a ‘picture’ of Rome. Piero della Francesca gives an account of a perspective construction – based on plan and elevation drawings connected by lines from a point of sight and cut by a perpendicular – which is essentially Alberti’s surveying method moved indoors to the drawing board.

44. 44 The grid is transparent... it helps us to see the world aright

45. 45 Uccello: Gridded Challice c. 1450

46. 46 Alberti’s Ontology of Painting Two kinds or levels of matter linked together by projective geometry

47. 47 Alberti’s Ontology of Painting 1. the three-dimensional matter of the observable world (macrocosm) composed of surfaces in three-dimensional reality 2. the two-dimensional matter of the painting (microcosm, simulacrum) composed of marks on a flat plane

48. 48 Two kinds of matter Compare Gibson’s 1980 “essay on the perception of surfaces versus the perception of markings on a surface.”

49. 49 Two kinds of matter the two-dimensional matter of the painting exists in the form of an istoria constructed out of points, lines and planes (marks) grouped together to form limbs, bodies and groups of bodies in a way that is analogous to the logical structure of words, phrases, sentences and paragraphs in a story

50. 50 The artist’s job is to project the objective array of planes into the microcosm of the painting in such a way as to achieve a maximally beneficial (moral) effect

51. 51 Rules for manipulating the elements of an istoria dignità varietà modestia verisimilitudo together with geometry, these four principles constitute the basis of a rational art

52. 52 Leonardo: Non mi legga chi non e matematico. (‘Let no one read me who is not a mathematician.’)

53. 53 global selection the visible scene made of finite planes or surfaces the totality of planes in the macrocosm exists objectively it changes from moment to moment with changes in the ambient light

54. 54 „fiat lux“

55. 55 „fiat lux“

56. 56 „fiat lux“

57. 57 „fiat lux“

58. 58 „fiat lux“

59. 59 „fiat lux“

60. 60 „fiat lux“

61. 61 „fiat lux“

62. 62 „fiat lux“

63. 63 „fiat lux“

64. 64 „fiat lux“

65. 65 „fiat lux“

66. 66 „fiat lux“

67. 67 „fiat lux“ light effects global selection

68. 68 The subject effects local selection

69. 69 but even the result of local selection is still perfectly objective compare what happens on the stage in the theater

70. 70 Alberti’s mathematics of projection mathematicians examine the form of things separated from their matter. Those who wish the object to be seen ‘will use a more sensate wisdom.’ version of Euclid’s geometry in terms of concrete visible ‘signs’ or ‘marks.’

71. 71 Points and lines points and lines among painters are not as among mathematicians, [who think that] in a line there fall infinite points

72. 72 Tolerance geometry... a point is a sign [signum] which one might say is not divisible into parts. I call a sign anything which exists on a surface so that it is visible to the eye. … Points joined together continuously in a row constitute a line. So for us a line will be a sign whose length can be divided into parts If many lines are joined closely together like threads in a cloth, they will create a surface.

73. 73 Rays of marvelous subtlety qualities of color, shape and size of planes in the objective array are ‘measured with sight.’ rays that serve sight carry the form of the thing seen to the sense ‘by a certain marvelous subtlety’ they penetrate the air and ‘all thin and clear objects’

74. 74 Optical Projection

75. 75 Rays of marvelous subtlety... until they strike against something dense and opaque, where they strike with a point and adhere to the mark they make.

76. 76 Intromission vs. extromission Among the ancients there was no little dispute whether these rays come from the eye or the plane. This dispute is very difficult and is quite useless for us. It will not be considered. We can imagine those rays to be like the finest hairs of the head, or like a bundle, tightly bound within the eye where the sense of sight has its seat.

77. 77 Intromission vs. extromission The rays, gathered together within the eye, are like a stalk; the eye is like a bud which extends its shoots rapidly and in a straight line on the plane opposite. intromissionist vs. extromissionist views of visual perception

78. 78 Extromissionists: Pythagorians, Euclid, Stoics, Ptolemy, Galen Euclid’s geometry relates not to rays of light in the physical sense but to extromissionist ‘visual rays’ Galen: the eye’s crystalline lens is a transmitter of visual force

79. 79 Atomist argument for extromissionism The effluxes of, say, a camel or a mountain could not very well pass through the tiny pupil of the eye How could every point on so large a visual surface be transmitted simultaneously to the eye, with its finite compass, via atoms of light?

80. 80 The intromissionist answer Alhazen: refraction and the curvature of the lens of the eye work to filter out excess information in the light, every point on the surface of an object can convey its form to the seat of vision within the eye – in an exact one-for-one, place- for-place proportionate way.

81. 81 Lux gratiae Robert Grosseteste, Roger Bacon and John Pecham: the new optical theories of the transmission of light provide a model of how God spreads the light of grace to his subjects in the world. Grosseteste: light stands in the same relation to the natural world as abstract space stands to geometry.

82. 82 Physics and physiology, too, are now thoroughly intromissionist

83. 83 Yet extromissionism lives on, in cognitive science through the arrow of intentionality

84. 84 Intentionality

85. 85 Intentionality

86. 86 corrected content, meaning representations

87. 87 concepts belong here

88. 88 Foreground/Background ‘ many-rayed intentionality ’ foreground/background

89. 89 Extromission lives When an archer considers in which direction to point his bow, he traverses in his mind or in reality a tight bundle of marvelously subtle rays extending out to the plane of his target.

90. 90 Against conventionalism, relativism, Nelson Goodman, Marcel Duchamp, French philosophy and other bad things

91. 91 Panofsky Perspective as Symbolic Form 1927 perspective is not a true theory of the way light is projected by three-dimensional surfaces onto a two-dimensional plane rather: a system of conventions bound to a certain time and culture.

92. 92 Perspective space is foreign to human experience quantitative, homogeneous, isotropic and boundless (rational, knowable) space vs. qualitative, heterogeneous, anisotropic bounded space of real (authentic) life

93. 93 Arguments for conventionalism: 1. many non-Westernized people have difficulty reading perspectival drawings. 2. the theory of simple geometric perspective took so long to develop

94. 94 3. ‘true’ perspective is not really true: assumes that vision is monocular abstracts from all movement on the part of the observer abstracts from size constancy scaling treats horizontal and vertical lines (railway tracks and telegraph poles) in the same way

95. 95 Windowless monads post Duchamp: visual arts are freed from connection to everyday life (and to beauty and harmony) recontextualized in the museum

96. 96 The Domain of Arnheim

97. 97 The Fair Captive

98. 98 After Duchamp 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

99. 99 Pipe

100. 100 The Production of Space The fact is that around 1910 a certain space was shattered. (Henri Lefebvre)

101. 101 Ways of Worldmaking Renaissance men were living in a different world from the world of their predecessors.... they were ‘producing’ or ‘shattering’ a certain space.

102. 102 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 approximation. best explanation for this: the mathematical forms captured in the geometry of perspective are out there in the world

103. 103 The realist response the strange fascination which perspective had for the Renaissance mind ‘was the fascination of truth.’ (Pirenne 1952)

104. 104 Reasons for Realism the objective, geometrical relationship between an object and its image on the picture plane obtains independently of whether there is an eye at the vanishing point (cf. laser-guided missiles) the laws of perspective hold independently of the existence of subjects, observers, artists or cultures

105. 105 Reasons for Realism the laws of perspective govern the way light, space and the surfaces of objects are related together, laws having nothing to do with neurology or psychology. the picture drawn in perspective aims not at representing anything like the retinal image or any pattern of nervous stimulation it aims to send to the eye the same distribution of light as that which the object itself would send.

106. 106 Gibsonian definition of a picture a surface treated so that it yields light to a particular station point, usually on a normal to the picture surface, which could have come from a scene in the real world.

107. 107 Reasons for Realism the fact that there are pictures embodying various types of distortions and symbolic elements does not imply that all pictures are lacking in transparency the fact that there are rays which deviate from their proper path does not imply that there is no central ray which truly hits its target

108. 108 Reasons for Realism the fact that there are maps which deviate from the reality which they are purporting to depict does not imply that there are no correct maps which truly hit their targets

109. 109 How to Tell the Truth with Maps A good map casts a transparent net over the surface of the earth 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..... the windowing of attention in pictures

110. 110 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

111. 111 Alberti’s Grid c.1450

112. 112 Ontology of Alberti’s Grid 1.eye (or point of projection), 2.projective rays, 3.the artist’s grid itself, 4.the constituent cells of the artist’s grid, 5.the totality of objective visible surfaces, 6.the target grid: a selection from this totality effected by projection through the artist’s grid, 7.the constituent cells within the target grid

113. 113 Optical Projection

114. 114 Cartographic Projection

115. 115 Semantic Projection, or: The Picture Theory of Pictures

116. 116 Semantic Projection Blanche is shaking hands with Mary

117. 117 Wittgenstein’s picture theory of meaning I call the sign with which we express a thought a propositional sign. – And a proposition is a propositional sign in its projective relation to the world. (3.12) true proposition = picture or map of a state of affairs in reality (atomic) proposition = complex of simple signs (names) which stand in a projection relation to corresponding objects in the world.

118. 118 Wittgenstein’s picture theory of meaning truth = isomorphisms The proposition is a complex of names arranged in a certain order The Albertian grill is a complex of cells arranged in a certain order

119. 119 Semantic Projection Already every simple list is such as to constitute a true grid in the sense here intended The membership of the category ‘cat,’ like that of ‘Mama,’ is a natural unit in nature, to which the concept cat does something like pointing, and continues to point despite large changes in the properties the thinker represents the unit as having.... The difficulty is to cash in the metaphor of ‘pointing’ in this context. (Millikan)

120. 120 Grids fail either because the projective relation is not well-defined (vagueness, ambiguity, presupposition failure) or because the cells of the grid do not stand to each other in relations isomorphic to the relations between the corresponding target objects.

121. 121 Directions of Fit Three sorts of tiling: 1. the imputed grid depends exclusively on the grid of the map (a map-to-world direction of fit), 2. the grid on the map reflects a pre-existing grid in reality (a world-to-map direction of fit) 3. map grid and target grid are mutually dependent upon each other.

122. 122 Directions of Fit 2a. the grid of the map reflects bona fide boundaries on the side of the target objects, 2b. the grid of the map reflects pre-existing fiat boundaries on the side of the target objects, 2c. the grid of the map reflects some combination of bona fide and fiat boundaries.

123. 123 Globe

124. 124 Globe

125. 125 True grids can have different resolutions true maps of the very same reality can be of different scales

126. 126 Therefore: No ‘God’s eye perspective’ No ‘view from nowhere’  every single one of the myriad perspectives we enjoy embodies a false view of reality This inference from partiality to falsehood is valid only in a world without windows.

127. 127 FIT HAPPENS

128. 128 THE END