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Was heisst “mathematische Existenz” ?
Barry Smithhttp://ontology.buffalo.edu
Sinn und Bedeutung in den Grundlagen
der Mathematik
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Becker über mathematische Existenz
Hilberts rein formal-mathematische Gegenständlichkeiten sind keine ausweisbaren Phänomene,
sondern transphänomenale Gesetzheiten;
sie können auch nicht zu Phänomenen werden (S. 755-6)
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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
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Unsere Betrachtungen haben den Vorrang der ersten ‚noetischen‘ Bedeutung von gezeigt ...
Becker über mathematische Existenz
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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
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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
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Die genauere phänomenologische Analyse erwies, dass das Mathematische primär ein B e z u g s p h ä n o m e nB 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 V o l l z u g d i e s e s B e z u g sB 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
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1 2 3 4
Counting
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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
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Counting
Eine einfache Aufteilung
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A simple partition
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A simple partition
Eine einfache, endliche Aufteilung mit zwölf Zellen
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A simple partition
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A partition can be the extension of another partition
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A simple partition
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A partition can be more or less refined
A partition can be more or less refined
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A refined partition
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Partition
A partition is the drawing of a (typically complex) boundary over a certain domain
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GrGr
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A partition is transparent
It leaves the world exactly as it is
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Artist’s Grid
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Label/Address System
A partition typically comes with labels and/or an address system
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Montana
Montana
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The Counties of England: An Irregular Partition
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Cerebral Cortex
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Mouse Chromosome Five
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Some partitions are trivial
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The DER-DIE-DAS partition
DER
(masculine)
moon
lake
atom
DIE
(feminine)
sea
sun
earth
DAS
(neuter)
girl
firedangerous thing
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A partition can comprehend the whole of reality
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It can do this in different ways
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Die Spinoza Aufteilung
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Universe
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Periodic Table
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AAngst vor Relativismus?
All partitions are equal but some are more equal than others
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Perspectivalism
Perspectivalism
Different partitions may represent cuts through the same reality which are skew to each other
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Universe/Periodic Table
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California Land Cover
Reciprocal partitions
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A partition can sometimes create the very objects it partitions
fiat objects = objects created by partitions
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Kansas
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Flevoland, NL
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= objects which exist independently of our partitions
(objects with bona fide boundaries)
bona fide objects
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globe
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Examples
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Grids of Reality (Mercator 1569)
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a partition is transparent
it leaves everything in reality exactly as it is
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Alberti’s Grid
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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)
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Partitions are artefacts of our cognition
= of our referring, perceiving, classifying, sorting, listing, naming, counting, mapping activity
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... rook bishop pawn knight ...
John Paul George Ringo
... up down charm strange ...
Other partitions
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Partitions always have a certain granularity
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... 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
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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
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Granularity
the partition does not recognize the molecules in the coffee
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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
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John
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... -20-10 -10 0 0 10 10 20 ...
massivelyincreased... normal increased chronic ...
... ...
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... rook bishop pawn knight ...
John Paul George Ringo
... up down charm strange ...
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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
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The theory of partitions
is a theory of foregrounding,
of setting into relief
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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
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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
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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
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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
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Foreground/Background
our use of partitions involvesalso a moment of delineation
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Mont Blanc from Lake Annecy
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Mont Blanc from Chatel
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Mont Blanc (Tricot)
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Intentionalityinvolves:
transparency
setting into relief
granularity
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Intentionality
the correct view
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corrected
content, meaningrepresentations
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Intentionality can be Many-Rayed
‘people’
‘my three sons’
‘Benelux’
‘die Deutschen’
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Counting
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Frege
referent
expression
sense
the correct view Fregeanized
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Idealism
propositions, senses, meanings
noemata, contents ...
the incorrect viewpretends that meanings can be in the target position
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Idealism
propositions, senses, meanings
noemata, ...
the road to philosophical pseudo-problems
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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 ?
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Intentional directedness
… is effected via partitions
we reach out to objects because partitions are transparent
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Beliefs, desires etc. are not ‘propositional attitudes’
rather they are object attitudes
= attitudes mediated by partitions(marked by granularity, delineation and transparency)
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And what of das Mathematische?
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we have all been looking in the wrong direction
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Dürer Reverse
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The mistaken view
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Intentionality
the correct view
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Language can generate partitions
„Blanche is shaking hands with Mary“
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Maps can generate partitions
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A map, too, is a Bezugsphänomen
self
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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
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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
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We should conceive our mathematical tools as we conceive our maps:
= in their projective relation to the world
(in their application to reality)
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The correct view
mathematical structures belong here
„das Mathematische ist primär einB 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 “
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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
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The correct view
objects
net of mathematicalstructures
self
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1 2 3 4
Counting
Frege: numbers belong
to the realm of concepts
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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
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Partitions are always partial
(This is something we can learn from Frege)
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David Lewis on Sets
Set theory rests on one central relation: the relation between element and singleton.
(Lewis, Parts of Classes, 1991)
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Cantor’s Hell
... the relation between an element and its singleton is “enveloped in mystery” (Lewis, Parts of Classes)
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Demolition
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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
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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
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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
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Artist’s Grid
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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”
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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
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Dummett
the realm of sense is a very special region of reality; its denizens are, so to speak, things of a very special sort.
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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
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etwaige mathematische Objekte ‚können auch nicht zu Phänomenen werden‘
Becker über mathematische Existenz
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das Mathematische ist primär ein
B e z u g s p h ä n o m e nB 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 V o l l z u g d i e s e s B e z u g sB 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
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THE END
THE END
