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1 Was heisst mathematische Existenz ? Barry Smith Sinn und Bedeutung in den Grundlagen der Mathematik.

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Präsentation zum Thema: "1 Was heisst mathematische Existenz ? Barry Smith Sinn und Bedeutung in den Grundlagen der Mathematik."—  Präsentation transkript:

1 1 Was heisst mathematische Existenz ? Barry Smith Sinn und Bedeutung in den Grundlagen der Mathematik

2 2 Becker über mathematische Existenz Hilberts rein formal-mathematische Gegenständlichkeiten sind keine ausweisbaren Phänomene, sondern transphänomenale Gesetzheiten; sie können auch nicht zu Phänomenen werden (S )

3 3 Das Mathematische ist eine sinnvoll doppeldeutiger Ausdruck. Es bezeichnet einerseits die, das Leben im Vollzug mathematischer Erwägungen und andererseits den Gegenstand dieser Erwägungen selbst. (S. 759) Becker über mathematische Existenz

4 4 Unsere Betrachtungen haben den Vorrang der ersten noetischen Bedeutung von gezeigt... Becker über mathematische Existenz

5 5 ist als Phänomen allenfalls der mathematische Gedanke, eigentlicher aber das mathematische D e n k e n als lebendiger Vollzug selbst, – nicht aber sein etwaiger transzendenter Gegenstand. Becker über mathematische Existenz

6 6 ist als Phänomen allenfalls der mathematische Gedanke, eigentlicher aber das mathematische D e n k e n als lebendiger Vollzug selbst, – nicht aber sein etwaiger transzendenter Gegenstand. Becker über mathematische Existenz

7 7 B e z u g s p h ä n o m e n V o l l z u g d i e s e s B e z u g s Die genauere phänomenologische Analyse erwies, dass das Mathematische primär ein B e z u g s p h ä n o m e n ist. Als solches hat es seinen ontischen Schwerpunkt im V o l l z u g d i e s e s B e z u g s, in der konkreten Weise daseienden Lebens, in der dieser Bezug allein gelebt werden kann. (S. 760) Becker über mathematische Existenz

8 Counting

9 9 Husserl, Philosophie der Arithmetik When we count we perform an act of colligation or grouping This generates a partition of the objects we are intending to count

10 10 Counting Eine einfache Aufteilung

11 11 A simple partition

12 12 A simple partition Eine einfache, endliche Aufteilung mit zwölf Zellen

13 13 A simple partition

14 14 A partition can be the extension of another partition

15 15 A simple partition

16 16 A partition can be more or less refined

17 17

18 18 A refined partition

19 19 Partition A partition is the drawing of a (typically complex) boundary over a certain domain

20 20 GrGr

21 21 A partition is transparent It leaves the world exactly as it is

22 22 Artists Grid

23 23 Label/Address System A partition typically comes with labels and/or an address system

24 24 Montana

25 25 The Counties of England: An Irregular Partition

26 26 Cerebral Cortex

27 27 Mouse Chromosome Five

28 28 Some partitions are trivial

29 29 The DER-DIE-DAS partition DER (masculine) moon lake atom DIE (feminine) sea sun earth DAS (neuter) girl fire dangerous thing

30 30 A partition can comprehend the whole of reality

31 31 It can do this in different ways

32 32 Die Spinoza Aufteilung

33 33 Universe

34 34 Periodic Table

35 35 A Angst vor Relativismus? All partitions are equal but some are more equal than others

36 36 Perspectivalism Different partitions may represent cuts through the same reality which are skew to each other

37 37 Universe/Periodic Table

38 38 California Land Cover Reciprocal partitions

39 39 A partition can sometimes create the very objects it partitions fiat objects = objects created by partitions

40 40 Kansas

41 41 Flevoland, NL

42 42 = objects which exist independently of our partitions (objects with bona fide boundaries) bona fide objects

43 43 globe

44 44 Examples

45 45 Grids of Reality (Mercator 1569)

46 46 a partition is transparent it leaves everything in reality exactly as it is

47 47 Albertis Grid

48 48 a partition is transparent = its fiat boundaries correspond at least to fiat boundaries on the side of the objects in its domain if we are lucky they correspond to bona fide boundaries (JOINTS OF REALITY)

49 49 Partitions are artefacts of our cognition = of our referring, perceiving, classifying, sorting, listing, naming, counting, mapping activity

50 50... rookbishoppawnknight... JohnPaulGeorgeRingo... updowncharmstrange... Other partitions

51 51 Partitions always have a certain granularity

52 52... your partition does not recognize parts beneath a certain size... this is why it is compatible with a range of possible views as to the ultimate constituents of the objects included in its foreground domain

53 53 Partitions always have a certain granularity When I see an apple my partition does not recognize the molecules in the apple Tax authorities do not tax the separate molecules in our bodies

54 54 Granularity the partition does not recognize the molecules in the coffee

55 55 It is the coarse-grainedness of our partitions which allows us to ignore questions as to the lower-level constituents of the objects foregrounded by our uses of singular terms. This in its turn is what allows such objects to be specified vaguely Our attentions are focused on those matters which lie above whatever is the pertinent granularity threshold. Granularity the source of vagueness

56 56 John

57 massively increased... normal increased chronic......

58 58... rookbishoppawnknight... JohnPaulGeorgeRingo... updowncharmstrange...

59 59 An object can be located in a cell within a partition in any number of ways: – object x exemplifies kind K – object x possesses property P – object x falls under concept C – object x is in spatial location L – object x is in measurement-band B contrast the meagre resources of set theory

60 60 The theory of partitions is a theory of foregrounding, of setting into relief

61 61 You use the name Mont Blanc to refer to a certain mountain You see Mont Blanc from a distance In either case your attentions serve to foreground a certain portion of reality Setting into Relief

62 62 You use the name Mont Blanc to refer to a certain mountain You see Mont Blanc from a distance In either case your attentions serve to foreground a certain portion of reality Setting into Relief

63 63 You use the name Mont Blanc to refer to a certain mountain You see Mont Blanc from a distance In either case your attentions serve to foreground a certain portion of reality Setting into Relief

64 64 You use the name Mont Blanc to refer to a certain mountain You see Mont Blanc from a distance In either case your attentions serve to foreground a certain portion of reality Setting into Relief

65 65 Foreground/Background our use of partitions involves also a moment of delineation

66 66 Mont Blanc from Lake Annecy

67 67 Mont Blanc from Chatel

68 68 Mont Blanc (Tricot)

69 69 Intentionality involves: transparency setting into relief granularity

70 70 Intentionality the correct view

71 71 corrected content, meaning representations

72 72 Intentionality can be Many-Rayed people my three sons Benelux die Deutschen

73 73 Counting

74 74 Frege referent expression sense the correct view Fregeanized

75 75 Idealism propositions, senses, meanings noemata, contents... the incorrect view pretends that meanings can be in the target position

76 76 Idealism propositions, senses, meanings noemata,... the road to philosophical pseudo-problems

77 77 Examples of Pseudo-Problems What is the ontological status of meanings? What are the identity criteria for meanings? How can we ever transcend the realm of meanings / contents / ideas / sensations / noemata and reach out to the realm of objects in themselves ?

78 78 Intentional directedness … is effected via partitions we reach out to objects because partitions are transparent

79 79

80 80 Beliefs, desires etc. are not propositional attitudes rather they are object attitudes = attitudes mediated by partitions (marked by granularity, delineation and transparency)

81 81 And what of das Mathematische?

82 82 we have all been looking in the wrong direction

83 83 Dürer Reverse

84 84 The mistaken view

85 85 Intentionality the correct view

86 86 Language can generate partitions Blanche is shaking hands with Mary

87 87 Maps can generate partitions

88 88 A map, too, is a Bezugsphänomen self

89 89 Mathematics can generate partitions Mathematics is as much a part of the natural history of mankind as maps, or language, or Albertis reticolato … Anthropologismus … is a problem only if you commit the genetic fallacy

90 90 Das Mathematische belongs not to the realm of objects but to the realm of partitions (the realm of senses) partitions are mathematical tools for talking about reality

91 91 We should conceive our mathematical tools as we conceive our maps: = in their projective relation to the world (in their application to reality)

92 92 The correct view mathematical structures belong here das Mathematische ist primär ein B e z u g s p h ä n o m e n B e z u g s p h ä n o m e n

93 93 The problem of the applicability of mathematics to reality is a pseudo- problem mathematics arises in the nexus of veridical intentional directedness effected via partitions mathematics is part of the scientific net

94 94 The correct view objects net of mathematical structures self

95 Counting Frege: numbers belong to the realm of concepts

96 96 The mystery of set theory arises from supposing that sets are objects This is the root, also, of Freges problem in the Grundgesetze This is the root of the catastrophic high- rise projects of post-Cantorian set theory

97 97 Partitions are always partial (This is something we can learn from Frege)

98 98 David Lewis on Sets Set theory rests on one central relation: the relation between element and singleton. (Lewis, Parts of Classes, 1991)

99 99 Cantors Hell... the relation between an element and its singleton is enveloped in mystery (Lewis, Parts of Classes)

100 100 Demolition

101 101 The mystery arises because sets are made to belong to the realm of objects where they do not belong proper understanding, here, of Cantors continuum problem, which arises because we try to insert the set-theoretic grid of cells into the realm of objects, where it does not belong

102 102 Cantors Hell or we confuse the fiat boundaries generated by our partitions (e.g. of the real numbers) with the bona fide boundaries possessed by objects themselves

103 103 Does this imply Kantianism? We cannot know what objects are like (e.g. mathematically), because our partitions always get in the way? No: PARTITIONS ARE TRANSPARENT They are designed, like spectacles, to reveal the world as it is

104 104 Artists Grid

105 105 Christian Thiel: The Fregean allowance of a participation of ontology in the doctrine of sense and reference is a completely unacceptable contamination to be sign, sense or reference is only a role, which certain entities take on when they enter into semantic contexts

106 106 senses, too, can play the role of referents therefore it might seem: if the mathematical belongs to the realm of sense, then it, too, is a matter of objects, of referents but this is mistaken

107 107 Dummett the realm of sense is a very special region of reality; its denizens are, so to speak, things of a very special sort.

108 108 David Hume (roughly): We cannot turn our eyeballs in our sockets Whenever I enter most intimately into what I call myself, I always stumble on objects … I can never catch my self at any time without objects, I can never observe senses or contents or noemata without their being directed towards objects

109 109 etwaige mathematische Objekte können auch nicht zu Phänomenen werden Becker über mathematische Existenz

110 110 das Mathematische ist primär ein B e z u g s p h ä n o m e n B e z u g s p h ä n o m e n. V o l l z u g d i e s e s B e z u g s Als solches hat es seinen ontischen Schwerpunkt im V o l l z u g d i e s e s B e z u g s, in der konkreten Weise daseienden Lebens, in der dieser Bezug allein gelebt werden kann. (S. 760) Becker über mathematische Existenz

111 111 THE END


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