Was heisst “mathematische Existenz” ?

Präsentation zum Thema: "Was heisst “mathematische Existenz” ?"—  Präsentation transkript:

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

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 Becker über mathematische Existenz
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)

4 Becker über mathematische Existenz
Unsere Betrachtungen haben den Vorrang der ersten ‚noetischen‘ Bedeutung von  gezeigt ...

5 Becker über mathematische Existenz
 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.

6 Becker über mathematische Existenz
 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.

7 Becker über mathematische Existenz
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)

8 Counting 1 2 3 4

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 Eine einfache Aufteilung
Counting Eine einfache Aufteilung

11 A simple partition

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

13 A simple partition

14 A partition can be the extension of another partition

15 A simple partition

16 A partition can be more or less refined

17

18 A refined partition

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

20 From: http://www. cinemedia
from P. Le Dubreuil, La Perspective Pratique (Paris 1642) GrGr

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

22 From: http://www. cinemedia
from P. Le Dubreuil, La Perspective Pratique (Paris 1642) Artist’s Grid

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

24 Montana Montana

25 The Counties of England: An Irregular Partition

26 Cerebral Cortex

27 Mouse Chromosome Five

28 Some partitions are trivial

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

30 A partition can comprehend the whole of reality

31 It can do this in different ways

32 Die Spinoza Aufteilung

33 Universe

34 Periodic Table

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

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

37 Universe/Periodic Table

38 Reciprocal partitions
California Land Cover Reciprocal partitions

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

40 Kansas

41 Flevoland, NL

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

43 globe

44 Examples

45 Grids of Reality (Mercator 1569)

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

47 Alberti’s Grid

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

50 ... ... rook bishop pawn knight John Paul George Ringo up down charm
Other partitions ... rook bishop pawn knight John Paul George Ringo ... up down charm strange

51 Partitions always have a certain granularity

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 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 Granularity the partition does not recognize
the molecules in the coffee

55 Granularity the source of vagueness
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.

56 John

57         massively increased normal increased chronic

58 ... ... rook bishop pawn knight John Paul George Ringo up down charm
strange

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

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

61 Setting into Relief 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

62 Setting into Relief 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

63 Setting into Relief 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

64 Setting into Relief 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

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

66 Mont Blanc from Lake Annecy

67 Mont Blanc from Chatel

68 Mont Blanc (Tricot)

69 Intentionality involves: transparency setting into relief granularity

70 Intentionality the correct view

71 corrected content, meaning representations

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

73 Counting

74 the correct view Fregeanized
expression referent sense the correct view Fregeanized

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

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

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

79

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

81 And what of das Mathematische?

82 we have all been looking in the wrong direction

83 Dürer Reverse

84 The mistaken view

85 Intentionality the correct view

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

87 Maps can generate partitions

88 A map, too, is a Bezugsphänomen
self

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

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

92 mathematical structures belong here
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 “

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 The correct view objects net of mathematical structures self

95 Counting 1 2 3 4 Frege: numbers belong to the realm of concepts

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

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

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

99 Cantor’s Hell ... the relation between an element and its singleton is “enveloped in mystery” (Lewis, Parts of Classes)

100 Demolition

101 The mystery arises because sets are made to belong to the realm of objects where they do not belong proper understanding, here, of Cantor’s 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 Cantor’s 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 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 From: http://www. cinemedia
from P. Le Dubreuil, La Perspective Pratique (Paris 1642) Artist’s Grid

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

110 Becker über mathematische Existenz
das Mathematische ist primär ein B e z u g s p h ä n o m e n. 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)

111 THE END THE END