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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]