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Single Idea 10203
[filed under theme 15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
]
Full Idea
The epistemological account of mathematical structures depends on the size and complexity of the structure, but small, finite structures are apprehended through abstraction via simple pattern recognition.
Gist of Idea
We apprehend small, finite mathematical structures by abstraction from patterns
Source
Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.11
A Reaction
Yes! This I take to be the reason why John Stuart Mill was not a fool in his discussion of the pebbles. Successive abstractions (and fictions) will then get you to more complex structures.
The
26 ideas
with the same theme
[singling out properties and parts of reality for special attention]:
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9092
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Abstracting A from B generates truth, as long as the connection is not denied
[Aquinas]
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9101
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Abstractive cognition knows universals abstracted from many singulars
[William of Ockham]
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9102
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If an animal approached from a distance, we might abstract 'animal' from one instance
[William of Ockham]
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10499
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We know by abstraction because we only understand composite things a part at a time
[Arnauld,A/Nicole,P]
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6711
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The mind creates abstract ideas by considering qualities separated from their objects
[Berkeley]
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10581
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I can only combine particulars in imagination; I can't create 'abstract' ideas
[Berkeley]
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9079
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We can focus our minds on what is common to a whole class, neglecting other aspects
[Mill]
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20363
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Leaves are unequal, but we form the concept 'leaf' by discarding their individual differences
[Nietzsche]
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9575
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Husserl identifies a positive mental act of unification, and a negative mental act for differences
[Husserl, by Frege]
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9032
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Before we can abstract from an instance of violet, we must first recognise it
[Price,HH]
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9035
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If judgement of a characteristic is possible, that part of abstraction must be complete
[Price,HH]
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9034
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There may be degrees of abstraction which allow recognition by signs, without full concepts
[Price,HH]
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9036
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There is pre-verbal sign-based abstraction, as when ice actually looks cold
[Price,HH]
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9037
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Intelligent behaviour, even in animals, has something abstract about it
[Price,HH]
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8775
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A big flea is a small animal, so 'big' and 'small' cannot be acquired by abstraction
[Geach]
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8776
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We cannot learn relations by abstraction, because their converse must be learned too
[Geach]
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15493
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Memory requires abstraction, as reminders of what cannot be fully remembered
[Martin,CB]
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15450
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Maybe abstraction is just mereological subtraction
[Lewis]
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18525
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Mental abstraction does not make what is abstracted mind-dependent
[Heil]
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10203
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We apprehend small, finite mathematical structures by abstraction from patterns
[Shapiro]
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8966
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Properties are facets of objects, only discussable separately by an act of abstraction
[Lowe]
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16561
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We can abstract by taking an exemplary case and ignoring the detail
[Machamer/Darden/Craver]
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10998
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The mind abstracts ways things might be, which are nonetheless real
[Read]
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4460
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Abstractions are formed by the mind when it concentrates on some, but not all, the features of a thing
[Moreland]
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8954
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Geometrical circles cannot identify a circular paint patch, presumably because they lack something
[Szabó]
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19736
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Neural networks can extract the car-ness of a car, or the chair-ness of a chair
[New Sci.]
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