1. True Grid • Barry Smith • http://ontology.buffalo.edu/smith

2. 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

3. Leon Battista Alberti (1404-1472) Alberti’s grid

4. 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

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. Theoreticalproblem of perspective solved by Alberti in Book 1of 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 influencedColumbus

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-labeledcells 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. Optical Projection

47. Cartographic Projection

48. Projection is involved wherever there is intentionality

49. intentionality = the directedness towards objects of a mental act

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