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Single Idea 10217
[filed under theme 18. Thought / E. Abstraction / 3. Abstracta by Ignoring
]
Full Idea
One way to apprehend a particular structure is through a process of pattern recognition, or abstraction. One observes systems in a structure, and focuses attention on the relations among the objects - ignoring features irrelevant to their relations.
Gist of Idea
We can apprehend structures by focusing on or ignoring features of patterns
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 3.1)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.74
A Reaction
A lovely statement of the classic Aristotelian abstractionist approach of focusing-and-ignoring. But this is made in 1997, long after Frege and Geach ridiculed it. It just won't go away - not if you want a full and unified account of what is going on.
The
23 ideas
with the same theme
[creating abstractions by ignoring some features]:
9792
|
Mathematicians study quantity and continuity, and remove the perceptible features of things
[Aristotle]
|
9077
|
Mathematicians suppose inseparable aspects to be separable, and study them in isolation
[Aristotle]
|
9788
|
Mathematicians study what is conceptually separable, and doesn't lead to error
[Aristotle]
|
9094
|
Mathematical objects abstract both from perceived matter, and from particular substance
[Aquinas]
|
10505
|
We can just think of an apple's colour, because the apple is not part of the colour's nature
[Aquinas]
|
10504
|
Abstracting either treats something as separate, or thinks of it separately
[Aquinas]
|
10507
|
Numbers and shapes are abstracted by ignoring their sensible qualities
[Aquinas]
|
13186
|
Universals are just abstractions by concealing some of the circumstances
[Leibniz]
|
9189
|
Dedekind said numbers were abstracted from systems of objects, leaving only their position
[Dedekind, by Dummett]
|
9827
|
We derive the natural numbers, by neglecting everything of a system except distinctness and order
[Dedekind]
|
9890
|
The modern account of real numbers detaches a ratio from its geometrical origins
[Frege]
|
10803
|
Frege himself abstracts away from tone and color
[Yablo on Frege]
|
9988
|
If we abstract 'from' two cats, the units are not black or white, or cats
[Tait on Frege]
|
9579
|
Disregarding properties of two cats still leaves different objects, but what is now the difference?
[Frege]
|
9587
|
How do you find the right level of inattention; you eliminate too many or too few characteristics
[Frege]
|
8770
|
'Abstractionism' is acquiring a concept by picking out one experience amongst a group
[Geach]
|
8904
|
The Way of Abstraction says an incomplete description of a concrete entity is the complete abstraction
[Lewis]
|
9567
|
Maths deals with quantities of physical significance, ignoring irrelevant features
[Geroch]
|
8512
|
Abstractions come before the mind by concentrating on a part of what is presented
[Campbell,K]
|
10217
|
We can apprehend structures by focusing on or ignoring features of patterns
[Shapiro]
|
9554
|
We can focus on relations between objects (like baseballers), ignoring their other features
[Shapiro]
|
9626
|
A structure is an abstraction, focussing on relationships, and ignoring other features
[Shapiro]
|
17316
|
We can abstract to a dependent entity by blocking out features of its bearer
[Koslicki]
|