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Single Idea 9890

[filed under theme 18. Thought / E. Abstraction / 3. Abstracta by Ignoring ]

Full Idea

From geometry we retain the interpretation of a real number as a ratio of quantities or measurement-number; but in more recent times we detach it from geometrical quantities, and from all particular types of quantity.

Gist of Idea

The modern account of real numbers detaches a ratio from its geometrical origins

Source

Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §159), quoted by Michael Dummett - Frege philosophy of mathematics

Book Ref

Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.262

A Reaction

Dummett glosses the 'recent' version as by Cantor and Dedekind in 1872. This use of 'detach' seems to me startlingly like the sort of psychological abstractionism which Frege was so desperate to avoid.

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]