Was heisst “mathematische Existenz” ? Sinn und Bedeutung in den Grundlagen der Mathematik   Barry Smith http://ontology.buffalo.edu

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)

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)

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

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.

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.

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)

Counting 1 2 3 4

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

Eine einfache Aufteilung Counting Eine einfache Aufteilung

A simple partition  

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

A simple partition  

A partition can be the extension of another partition

A simple partition  

A partition can be more or less refined

A refined partition  

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

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

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

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

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

Montana Montana

The Counties of England: An Irregular Partition

Cerebral Cortex http://rprcsgi.rprc.washington.edu/~atlas

Mouse Chromosome Five

Some partitions are trivial

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

A partition can comprehend the whole of reality

It can do this in different ways

Die Spinoza Aufteilung

Universe

Periodic Table

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

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

Universe/Periodic Table

Reciprocal partitions California Land Cover http://www.tahoecons.ca.gov/library/rip_data/rd_grnd_samp.html Reciprocal partitions

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

Kansas

Flevoland, NL

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

globe

Examples

Grids of Reality (Mercator 1569)

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

Alberti’s Grid

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)

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

... ... 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

Partitions always have a certain granularity

... 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

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

Granularity the partition does not recognize the molecules in the coffee

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.

John  

...     ... ... -20-10 -10  0 0  10 10  20 ... massively increased ... normal increased chronic ...

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

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

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

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

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

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

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

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

Mont Blanc from Lake Annecy

Mont Blanc from Chatel

Mont Blanc (Tricot)

Intentionality involves: transparency setting into relief granularity

Intentionality the correct view

corrected content, meaning representations

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

Counting

the correct view Fregeanized expression referent sense the correct view Fregeanized

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

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

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 ?

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

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

And what of das Mathematische?

we have all been looking in the wrong direction

Dürer Reverse

The mistaken view

Intentionality the correct view

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

Maps can generate partitions

A map, too, is a Bezugsphänomen self

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

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

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

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 “

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

The correct view objects net of mathematical structures self

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

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

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

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

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

Demolition

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

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

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

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

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”

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

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

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

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

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)

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