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Single Idea 9145
[filed under theme 18. Thought / E. Abstraction / 2. Abstracta by Selection
]
Full Idea
We call 'cardinal number' the general concept which, by means of our active faculty of thought, arises when we make abstraction from an aggregate of its various elements, and of their order. From this double abstraction the number is an image in our mind.
Gist of Idea
We form the image of a cardinal number by a double abstraction, from the elements and from their order
Source
George Cantor (Beitrage [1915], §1), quoted by Kit Fine - Cantorian Abstraction: Recon. and Defence Intro
Book Ref
-: 'Journal of Philosophy' [-], p.599
A Reaction
[compressed] This is the great Cantor, creator of set theory, endorsing the traditional abstractionism which Frege and his followers so despise. Fine offers a defence of it. The Frege view is platonist, because it refuses to connect numbers to the world.
The
24 ideas
with the same theme
[mental acts which create abstract concepts]:
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9789
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You can't abstract natural properties to make Forms - objects and attributes are defined together
[Aristotle]
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9070
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We learn primitives and universals by induction from perceptions
[Aristotle]
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10506
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Mathematics can be abstracted from sensible matter, and from individual intelligible matter
[Aquinas]
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9104
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A universal is the result of abstraction, which is only a kind of mental picturing
[William of Ockham]
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23640
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Only mature minds can distinguish the qualities of a body
[Reid]
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23653
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If you can't distinguish the features of a complex object, your notion of it would be a muddle
[Reid]
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19577
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Everything is a chaotic unity, then we abstract, then we reunify the world into a free alliance
[Novalis]
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9145
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We form the image of a cardinal number by a double abstraction, from the elements and from their order
[Cantor]
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13454
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Cantor says (vaguely) that we abstract numbers from equal sized sets
[Hart,WD on Cantor]
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9976
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Frege accepts abstraction to the concept of all sets equipollent to a given one
[Tait on Frege]
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9361
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We have to separate the mathematical from physical phenomena by abstraction
[Lewis,CI]
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10644
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A 'felt familiarity' with universals is more primitive than abstraction
[Price,HH]
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10646
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Our understanding of 'dog' or 'house' arises from a repeated experience of concomitances
[Price,HH]
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9031
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The basic concepts of conceptual cognition are acquired by direct abstraction from instances
[Price,HH]
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10557
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Abstract objects are captured by second-order modal logic, plus 'encoding' formulas
[Zalta]
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12657
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Abstractionism claims that instances provide criteria for what is shared
[Fodor]
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9149
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To obtain the number 2 by abstraction, we only want to abstract the distinctness of a pair of objects
[Fine,K]
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9150
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We should define abstraction in general, with number abstraction taken as a special case
[Fine,K]
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9981
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Abstraction is 'logical' if the sense and truth of the abstraction depend on the concrete
[Tait]
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9982
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Cantor and Dedekind use abstraction to fix grammar and objects, not to carry out proofs
[Tait]
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10141
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Many different kinds of mathematical objects can be regarded as forms of abstraction
[Fine,K]
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10229
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Simple types can be apprehended through their tokens, via abstraction
[Shapiro]
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9919
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The old debate classified representations as abstract, not entities
[Burgess/Rosen]
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8917
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The Way of Abstraction used to say an abstraction is an idea that was formed by abstracting
[Rosen]
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